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+{"text":"\\section{Introduction}\n\nIn the centre of most galaxies lies a supermassive black hole (SMBH) with a mass exceeding $10^5M_{\\odot}$ \\citep{reines14}. It is believed that SMBH binaries in galactic nuclei form by successive coalescences following the merger of their respective host halos, and by accretion of gas present in the central regions of galaxies. The mechanisms assumed to form a SMBH binary after the merger of two galaxies are i) binary shrinking due to stellar or gas dynamical processes; followed by ii) coalescence due to low frequency gravitational wave emission ($\\leq 10^{-3}$ Hz) \\citep{mayeretal07, fillouxetal}. If the interacting galaxies are rich in gas, an accretion disc will form from gas rapidly falling towards the black holes \\citep{armitage02}. \n\nThe interaction between the SMBH binary and the surrounding material may be observable. Gas plays a key role in the binary merger process. It also powers the black hole luminosity via accretion. SMBH mergers are thus promising for multi-messenger astronomy since they produce an electromagnetic counterpart to a gravitational wave burst.\n\n \\citet{armitage02} showed, using 1D simulations, that rapid accretion of the inner disc due to the tidal effect of the merging secondary causes an accretion rate exceeding the Eddington limit prior to the merger of the SMBH binary. By contrast, the 2D hydrodynamical simulations of \\citet{baruteal12} suggested that such a `snowplough mechanism' is unlikely to work, because the binary shrinking time driven by gravitational waves is shorter than the viscous time-scale, meaning that fluid elements in the inner disc get funneled to the outer disc across the secondary gap via horseshoe orbits. However, 3D simulations carried out by \\citet{ceriolietal} demonstrated the presence of a squeezing mechanism caused by the compression of the inner disc gas as the secondary companion spirals towards the more massive black hole. The resulting accretion luminosity in the final stages of the merger exceeded the Eddington rate.\n\nRecent detections of gravitational waves have provided tests of General Relativity and direct measurements of compact objects parameters like mass and spin. These have been demonstrated in detections of gravitational radiation from stellar mass black holes binaries (events GW150914, GW151226, GW170104, GW170814 and GW170608) and by the first direct evidence of a link between a neutron star merger and short $\\gamma$-ray bursts (GW170817) detected by LIGO and Virgo observatories \\citep{abbott01, abbott02, abbott03}. However, the detection of gravitational waves from SMBH binaries will be possible only from future space detectors, such as LISA \\citep{amaro}. \n\nMost previous simulations of a SMBH binary coalescence in a gaseous environment have modeled aligned disc-binary systems \\citep{ceriolietal}. In the present work, we generalize the analysis to the case of misaligned disc-binary systems. Inclined binaries are expected when the black holes are spinning at an inclined angle with respect to the orbital plane. We model the disc-binary interaction with inclined discs at 1, 5, 10, 20, 30 and 180 degrees, using the 3D Smoothed Particle Hydrodynamics (SPH) code {\\sc phantom} \\citep{lodatoeprice10, priceefede10, price12, priceetal18}.\n\nThe paper is organized as follows: in Section~\\ref{section2} we introduce the interaction process in misaligned disc-binary systems. Section~\\ref{section3} describes the numerical method and initial conditions. Section~\\ref{section4} shows our results varying the disc inclination angle and the corresponding accretion rates. We discuss and conclude in Section~\\ref{section5}.\n\n\\section{ACCRETION DYNAMICS IN MISALIGNED DISC-BINARY SYSTEMS}\n\\label{section2}\n\nPrevious numerical simulations and analytic work has shown that the tidal torques of the binary secondary component can strongly influence the gas disc \\citep{linpapa79, goldtre79, lubowetal15}. The dynamics of a binary embedded in a gas disc depend on the mass ratio of the binary and the gas mass. Binaries with nearly equal mass black holes have tidal torques carrying angular momentum between the gas and the binary orbit. However, for binaries with small mass ratios, the secondary tidal force can only be felt when the gas is close enough to the secondary object. Therefore, angular momentum transfer mechanism between the secondary and the gas in the accretion disc can be determined by studying the close encounters through impulsive approximation. These encounters occur on a timescale much smaller than the typical orbital time \\citep{linpapa79, linpapa7902}.\n\nThe angular momentum exchange with the accretion disc can make the secondary component migrate. However, there are different types of migration depending on the relative mass of the companion. In particular, in this work we consider a low mass secondary SMBH embedded in a circumprimary disc. The low mass companion falls towards the supermassive primary. When the binary reaches separations smaller than $10^{-3}$pc the dominant process for the loss of energy and angular momentum is gravitational wave emission \\citep{peters63}. The energy dissipation results in a negative torque shrinking the binary down to merger. If we assume a circular orbit, the decay rate of the binary separation due to gravitational wave emission is given by \\citep{peters64}\n\\begin{equation}\n\t\\left(\\dfrac{da}{dt}\\right)_{\\rm{gw}} \\simeq -\\dfrac{64}{5} \\dfrac{G^3 M_p^3 q_1}{c^5 a^3},\n\t\\label{eq:decayrategw}\n\\end{equation}\nwhere $M_p$ is the primary SMBH mass, $a$ is the binary separation and the $q_1$ parameter is defined as $q_1$=$q$(1+$q$), where $q$ is the binary mass ratio and the final merger time is\n\\begin{equation}\n    \\tau_{\\rm{gw}} \\approx \\dfrac{5}{256} \\dfrac{c^5 a_0^4}{G^3 M_p^3 q_1}\\cdot\n\t\\label{eq:mergertime}\n\\end{equation}\n\nWhen the disc and binary black hole orbits are aligned, gravitational radiation emission causes the binary separation to decrease, producing an increase in luminosity that may be detectable as an electromagnetic precursor. Previous investigations for this orbital configuration yielded accretion rates of the order of $10^2$ times the Eddington limit \\citep{armitage02, ceriolietal}.\n\n\\subsection{Exploring misaligned disc-binary systems}\n\n\nFor a SMBH binary in a circular orbit, the interaction between the misaligned disc and the binary is analogous to Lense-Thirring precession acting on an accretion disc around a single, spinning black hole \\citep{lt1918, lodatoepringle06,nealonetal15}. Misaligned disc-binary interaction involves the additional effect of inclination decay due to viscous dissipation effects. For a primary black hole surrounded by an inclined disc, the gas flow is dominated by viscous torques. In this case, tilted disc-binary systems are not expected to produce an electromagnetic counterpart to the binary merger.\n\nBinary interaction in a gaseous environment depends on the disc properties. The decoupling radius refers to the radius at which the gravitational torque becomes comparable to the viscous torque. When the initial distance between the two black holes is much larger than the decoupling radius, the binary can be surrounded by a circumbinary disc. By contrast, when the evolution is dominated by gravitational wave emission (i.e, when the black hole initial separation is small compared to the decoupling radius), then an individual disc surrounds only the more massive black hole. For a circumbinary disc, the angular momentum vector of the disc does not always coincide with that of the binary. The initial disc orientation is set by the angular momentum distribution of the gas rather than by the angular momentum of the binary \\citep{hayasaki13}. \n\n\\citet{lubowetal15} probed the effects of binary-disc inclination on Lindblad resonant tidal torques acting on a circumbinary disc. The authors studied the 2:1 inner Lindblad resonance (for m = 2) that dominates the tidal truncation of coplanar discs  by a prograde binary. For that resonance, they found a rapid decrease of the torque with inclination angle --- by a factor of about 2 for 30 degrees, by a factor of about 20 for 90 degrees and to zero for 180 degrees (counter-rotating case). \n\nViscous torques can dominate Lindblad resonant torques (for m = 2) if the binary is counter-rotating. They can also dominate for smaller inclination angles, if the disc is sufficiently viscous. In this case, the gas in the disc can be captured by the secondary, flowing either into a circumbinary disc or escaping. In a inclined circumsecondary disc, the weakened tides allow mass transfer from the secondary component to the central one. Inclined discs are expected to be larger than coplanar discs due to the decrease in the resonant tidal torques \\citep{lubowetal00, lubowetal15}. \n\nIn the present work, it is assumed that the most important effect for the misalignment between the disc and the orbital plane, is that the direction of the orbital angular momentum vector does not match the spin of the primary \\citep{bardeen}. Therefore, since by the Bardeen-Petterson effect the disc is aligned with the primary spin axis, this results in a misalignment with the orbital plane of the companion.\n\n\\section{NUMERICAL SIMULATIONS}\n\\label{section3}\n\nWe model the evolution of a misaligned disc-binary system using the {\\sc phantom} smoothed particle hydrodynamics (SPH) code. {\\sc phantom} solves the hydrodynamics equations in Lagrangian form using SPH \\citep{price07, lodatoeprice10, priceefede10, price12, priceetal18}. The disc viscosity $\\nu$ is modeled according to the \\citet{ss73} $\\alpha$-prescription, given by\n\\begin{equation}\n    \\nu \\simeq \\alpha_{\\rm SS} c_{\\rm s} H,\n\t\\label{eq:sspresciption}\n\\end{equation}\nwhere $\\alpha_{\\rm SS}$ is a dimensionless scale parameter, $c_{\\rm s}$ is the sound speed of the gas in the disc and $H$ = $c_{\\rm s}\/\\Omega$ is the disc thickness, where $\\Omega$ is the Keplerian velocity. In order to model the \\citet{ss73} viscosity we use the SPH artificial viscosity formalism set by the relation\n\\begin{equation}\n    \\alpha_{\\rm SS} \\simeq \\dfrac{1}{10} \\alpha^{\\rm AV} \\dfrac{\\langle h \\rangle}{H},\n\t\\label{eq:artviscosity}\n\\end{equation}\nwhere $\\alpha^{\\rm AV}$ is the artificial viscosity coefficient and $\\langle h \\rangle$ is the azimuthally averaged smoothing length. SPH employs this artificial viscosity term primarily to resolve shocks, but we use this to represent a source of viscous diffusion following \\citet{lodatoeprice10}. \n\nThe gas in the disc follows a locally isothermal equation of state with the sound speed described by the power law\n\\begin{equation}\n    c_{\\rm s} \\simeq c_{{\\rm s},0} R^{-\\beta},\n\t\\label{eq:sppowerlaw}\n\\end{equation}\nwhere $R$ is the radial distance from the centre of mass of the binary (in cylindrical coordinates) and $c_{{\\rm s},0}$ is a normalization that determine the disc thickness. The disc surface density profile is given by\n\\begin{equation}\n    \\Sigma \\simeq \\Sigma_0 R^{-\\gamma},\n\t\\label{eq:discsfprofile}\n\\end{equation}\nwhere $\\Sigma_0$ is chosen in order to set the disc mass.\n\nIn equations~(\\ref{eq:sppowerlaw}) and~(\\ref{eq:discsfprofile}), we chose the parameters $\\beta$ = 3\/2 and $\\gamma$ = 3\/4 in order to uniformly resolve the disc, given that the smoothing length $h \\propto \\rho^{-\\frac13}$ (where $\\rho$ is the density) is proportional to the thickness $H \\propto R^{\\frac34}$, so the ratio $\\langle h \\rangle$\/$H$ in equation~(\\ref{eq:artviscosity}) is constant with respect to radius.\n\n\\subsection{Initial conditions}\n\\label{initialcond}\n\n We set up a binary system of SMBHs with unequal masses on a circular orbit. The binary mass ratio is $q$ = $M_{s}$\/$M_{p}$ = $10^{-3}$, where $M_p$ = $10^8 M_{\\odot}$ is the mass of the primary and $M_s$ = $10^5 M_{\\odot}$ is the mass of the secondary. Based on 3D SPH simulations by \\citet{ceriolietal}, we chose $a_0$ = $4.75 GM_p$\/$c^2$ (in code units) for the binary initial separation, corresponding to $a_0$ $\\simeq$ $2.28 \\cdot 10^{-5}$ pc. Moreover, since we were interested in estimating the mass accretion rate of the primary and secondary black holes, we set their accretion radii to $R_{ar,p}$ = $2 GM_p$\/$c^2$ and $R_{ar,s}$ = $0.2 GM_p$\/$c^2$. We assumed an initial binary orbital separation smaller than the decoupling radius ($a_{dec} \\simeq 36.4 GM_p$\/$c^2$) implying a circumprimary disc only. We adopt the inner disc radius $R_{\\rm in}$ = $R_{{\\rm ar},p}$ = $2 GM_p$\/$c^2$, equal to the Schwarzschild radius and the outer radius $R_{\\rm out}$ = $4.1 GM_p$\/$c^2$, according to the initial separation of the black holes. Assuming the values for initial separation and accretion radii, the binary coalescence time by gravitational decay is $\\tau_{\\rm gw} \\simeq$ 9476 $GM_p$\/$c^3$ (in code units), or approximately 54 days.\n\nWe focus on inclined disc models to investigate the effects of the tidal torques by the secondary component embedded in a geometrically thin gas disc. We assume an initial disc mass of $M_{\\rm disc}$ $\\approx$ $1 M_{\\odot}$, flat and misaligned by a specified inclination angle. The disc is truncated at an outer radius related to the intensity of the tidal torques. We neglect the outer disc, i.e. we assume it is frozen behind the companion such that the disc inner edge is located at the decoupling radius (the disc may then viscously move inwards).\n\nWe assume a disc aspect ratio $H$\/$R$ = 0.01 and Shakura-Sunyaev viscosity parameter $\\alpha_{\\rm SS}$ = 0.01. We consider discs inclined by 1, 5, 10, 20, 30 and 180 degrees. Moreover, we also consider a control simulation of the disc without the secondary black hole. We performed seven simulations with $5 \\cdot 10^5$ SPH particles for all angles mentioned above, including the simulation without no companion. The choice for the number of particles is for computational efficiency, but does not affect the quality of the numerical results as we will see in the next section. We adopted the following code units: for the length, the gravitational radius ($R_g$=$G M_p$\/$c^2$); for the mass, the primary black hole mass ($M_p$) and for the time, $t$=$GM_p$\/$c^3$. \n\n\\subsection{Physical validity of the disc model in the context of AGN accretion}\n\nThe accretion rates exceeding the Eddington rate presented in this article show that the primary SMBH disc is radiatively efficient. Currently, the mechanism powering Active Galactic Nuclei (AGN) is believed to come from gas accretion onto a SMBH. The standard thin accretion disc models are normally associated with radiatively efficient AGNs. A significant fraction of the energy generated by viscous dissipation in the flow is radiated locally and the advection of energy is negligible, as a consequence the disc is geometrically thin. So that AGN discs are in general quite thin, ranging H\/R $\\sim 0.001-0.01$ and the bolometric luminosities correspond to about 10 per cent of the rest-mass energy of the accreting matter and approximately 40 per cent for rapidly spinning black holes \\citep{ss73}.\n\n\\subsection{Disc aspect ratio}\n\nMotivated in the investigation of electromagnetic precursors just prior to binary merger, we chose the disc aspect ratio in the inner disc based on the following reasons. \\cite{tl15} estimated the fossil disc mass just prior to a SMBH merger and they found that for a $10^8 M_{\\odot}$ primary black hole the inner disc mass at decoupling is of the order of $1 M_{\\odot}$. This result shows that the rapid accretion of the whole circumprimary disc would produce peak luminosities of order 1--20 times Eddington luminosity. Merging binaries are expected to have thinner accretion discs and to provide an electromagnetic signature from the squeezing of the inner disc.\n\n\\cite{ceriolietal} performed simulations with thin discs for the aligned disc-binary orbital planes and they found an increase in luminosity exceeding the Eddington limit. In addition, the authors confirmed that if gas is present between the SMBHs it is squeezed and quickly accreted by the primary black hole during the orbital decay. On the other hand, if the disc is thick, gas can more easily flow outwards through the gap.\n\nBy contrast, \\cite{baruteal12} found that the snowplough effect does not occur for thicker discs. Those authors showed that a small fraction (about 20 per cent) of the disc mass is accreted by the primary SMBH. This quantity does not cause any rise in the luminosity prior to binary merger and it does not excite the formation of peaks in the surface density of the disc.\n\n\\begin{figure*}\n\t\\includegraphics[width=\\columnwidth]{primary} \\includegraphics[width=\\columnwidth]{sec5}\n    \\caption{SMBH accretion rates in code units with discs at different inclination angles for simulations with $5 \\cdot 10^5$ particles. Left panel: primary accretion rates as a function of time, show super-Eddington peaks in $\\dot{M}$ for discs inclined by 1, 5 and 10 degrees. Right panel: secondary accretion rates as a function of time show squeezing phenomenon only in discs inclined by 1 and 5 degrees only. In both figures the black horizontal line shows the Eddington limit.} \n    \\label{fig:allanglesaccrate}\n\\end{figure*}\n\nLong before the merger of black holes embedded in thicker discs, the binary hollows out any gas present and shrinks slowly compared to the viscous timescale of a circumbinary accretion disc. So that a small fraction of the energy liberated during the merger can go into heating the gas, producing an electromagnetic afterglow \\citep{miphinney}.\n\n\\section{RESULTS}\n\\label{section4}\n\n We investigate the accretion rates induced as the secondary orbit shrinks towards the primary. We compare our results with previous investigations for aligned disc-binary orbital planes.\n\nAfter binary decoupling from the gas disc, the tidal torques dominate over viscous torques. At this moment, gravitational wave emission is the dominant process driving the binary to merger, with a possible electromagnetic signature in the last stages of coalescence. However, for misaligned disc-binary orbital planes, that path does not necessarily occur. Some discs are expected to be inclined, for instance when the black holes are spinning at an inclined angle with respect to angular momentum vector of the disc. \n\n\n\\subsection{Varying disc inclination angles}\n\nFigure~\\ref{fig:allanglesaccrate} shows primary and secondary accretion rates as a function of time (in code units) for all simulated disc inclination angles. For discs with small inclination angles (1, 5 and 10 degrees) we found an increase in luminosity exceeding the Eddington rate and very close to the aligned disc case \\citep{ceriolietal}. Table~\\ref{tab:accratefinal} presents results for the angles less than 10 degrees with two times ($t_1$ and $t_2$) corresponding to the two peaks marked by episodes of strong accretion on the primary (near the binary merger). The times of the first and second spikes differ between simulations because the binary evolves embedded in a gas disc with different tilts. The double peak in the primary accretion rate occurs as the tidal torques dominate viscous torques, because the viscous time-scale is shorter than the gravitational decay time-scale. The tidal torques provide the disc angular momentum loss that drives the gas accretion onto the primary component.\n\n\\begin{figure} \n\t\\includegraphics[width=\\columnwidth]{inc180_limitEdd}\n    \\caption{Primary (black line) and secondary (red line) accretion rates with circumprimary disc inclined at 180 degrees. The black horizontal line shows the Eddington limit.}\n    \\label{fig:allanglesprimary}\n\\end{figure}\n\nThe disc inclined at 10 degrees shows a less pronounced first spike in the primary accretion rate ($dM_p$\/$dt$ $\\simeq 10.6 M_{\\odot}$\/$\\rm{yr}$) compared to discs with smaller angles $d{M}_p$\/$dt$ $\\simeq 23.1 M_{\\odot}$\/$\\rm{yr}$ and  $d{M}_p$\/$dt$ $\\simeq 24.7 M_{\\odot}$\/$\\rm{yr}$ (1 and 5 degrees, respectively). In this case, tidal torques decline due to higher inclination (10 degrees) and the primary accretion rate reaches about $d{M}_p$\/$dt$ $\\approx 2.35 \\dot{M}_{\\rm Edd}$. The inclined disc with 1 degree yielded $d{M}_p$\/$dt$ $\\approx 5.49 \\dot{M}_{\\rm Edd}$ and with 5 degrees it obtained $d{M}_p$\/$dt$ $\\approx 5.13 \\dot{M}_{\\rm Edd}$. Moreover, when the disc is inclined at 10 degrees, the evolution between first and second spikes occurs more slowly. On the other hand, the second peak on the primary accretion rate at 10 degrees shows a narrower shape, reaching in $t_2$ a accretion rate of $d{M}_p$\/$dt$ $\\approx 6.58 \\dot{M}_{\\rm Edd}$, exceeding all other accretion rates (see Table~\\ref{tab:accratefinal}).\n\nWe also obtain two spikes at the initial times of binary evolution in the secondary black hole accretion rate, but these spikes occurred only in discs inclined at 1 and 5 degrees. The spikes are marked by the rapid decay of the secondary towards the primary, squeezing the inner disc gas. Our choice of a low-mass secondary ($10^5 M_{\\odot}$) resulted in its interaction with the inner disc implying in an exchange of angular momentum between the secondary component and the gas.\n\n\\begin{table*}\n\t\\centering\n\t\\label{tab:accratefinal}\n\t\\begin{tabular}{lccccccr}\n\t\t\\hline\n\t\t$Inclination$ & $t_1$ & $t_2$ & ($d{M}_p$\/$dt$)\/($M_{\\odot}$\/yr) [$t_{1}$] & ($d{M}_p$\/$dt$)\/($M_{\\odot}$\/yr) [$t_2$] & ($d{M}_p$\/$dt$)\/$M_{Edd}$ [$t_1$] & ($d{M}_p$\/$dt$)\/$M_{Edd}$ [$t_2$]\\\\\n\t\t\\hline\n\t\n\t\t$\\theta$=$1^o$ & 8560 & 8902 & 23.11 & 19.23 & 5.13 & 4.27\\\\\n\t\t$\\theta$=$5^o$ & 8569 & 8930 & 24.68 & 25.80 & 5.49 & 5.73\\\\\n        $\\theta$=$10^o$ & 8513 & 8975 & 10.58 & 29.61 & 2.35 & 6.58\\\\\n        \\hline\n        \\end{tabular}\n        \\caption{Primary black hole accretion rates at final stages of the binary merger. The first column shows inclination angle of the disc, the second and the third columns show the two different times $t_1$ and $t_2$, that correspond to the first and second spikes in the accretion rate, respectively, the fourth and the fifth columns show the accretion rates in units of $M_{\\odot}$\/$\\rm{yr}$ for each spike, the sixth and seventh columns show primary accretion rates in units of the Eddington rate ($d{M}_{Edd}$\/$dt$)=$4.5 M_{\\odot}$\/$\\rm{yr}$ for both times.}\n\\end{table*}\n\nFigure~\\ref{fig:allanglesaccrate} also shows the accretion rates with discs tilted at 20 and 30 degrees. However, discs with inclinations between these two angles showed a much less pronounced rise in the mass accretion rate, because the snowplough effect is important for misalignment up to 10 degrees, much larger than $H$\/$R$. In particular, for 20 \ndegrees of inclination we find only the first spike, with a primary accretion rate of order $1.1\\dot{M}_{\\rm Edd}$. Previous work by \\citet{lubowetal15} predicted such results based on analytic calculations of tidal torques acting on misaligned discs. By contrast, the primary accretion rate without the secondary presented in Figure~\\ref{fig:allanglesaccrate} shows no peak in the accretion rate due to the absence of the companion black hole and of the snowplough mechanism.\n\nFigure~\\ref{fig:allanglesprimary} reports the accretion rates in a circumprimary disc initially inclined at 180 degrees. No peaks are observed in the primary and secondary accretion rates. For a counter-rotating disc the tidal torques decline to zero, because viscous torques dominate over binary gravitational torques. These results are also in accordance with the analytic work by \\citet{lubowetal15}.\n\nFigure \\ref{fig:residual} shows the residual between the accretion rates of the primary with and without the companion. This figure clearly demonstrate what is the amount of the accretion rate due the presence of the secondary SMBH and of the snowplough effect. When comparing the accretion rate of the primary with and without the secondary, it can be seen that they are equal until $t \\sim 5000$ (in code units). After this time, the snowplough effect for small inclination angles becomes important and the accretion rate increases with respect to the accretion rate without the companion.\n\nFigure~\\ref{fig:allanglessnapshots} shows snapshots of column density in the $xy$ plane. In this plot each row presents the misaligned disc-binary evolution with discs inclined at 1, 5, 10, 20 and 30 degrees, from top to bottom. The second and fourth columns show the times corresponding to the first and second spikes in the primary accretion rate. In particular, the disc inclined at 20 degrees shows only a more pronounced first peak at time $t$=9025, reaching a primary accretion rate of the order of the Eddington rate. Discs with inclination angles at 1, 5 and 10 degrees produce an increase in luminosity exceeding the Eddington rate. The snapshots shown in Figure~\\ref{fig:allanglessnapshots}, with misaligned discs with $\\theta >10$ degrees show more extended discs compared those inclined with angles smaller than 5 degrees.\n\nFigure \\ref{fig:allangle180snapshots} shows the evolution for a counter-rotating disc ($\\theta$=180 degrees) showing a gap where the gas moves without falling towards the central black hole. In this case, the final merger take place in vacuum \\citep{miphinney}.  \n\n\\subsection{Effect of thicker discs}\n\nThe disc thickness can affect the amount of gas ejected outside the orbit of the secondary black hole. In order to investigate if the thicker discs just prior to the binary merger show electromagnetic evidences, we performed simulations increasing the disc aspect ratio. In these simulations, we used inclination angles with 1 and 5 degrees, H\/R=0.05, $5 \\cdot 10^5$ SPH particles and the same initial conditions shown in subsection \\ref{initialcond}.\n\nFigure \\ref{fig:thickerdisc} shows the accretion rates of the primary and secondary black holes for a thicker disc (H\/R=0.05), with different inclination angles, normalized by initial disc mass. In both figures, we do not see any effects of the forced compression, independent of the disc inclination. All the mass is swallowed before it could be squeezed by the secondary black hole. These results are in agreement with previous simulations performed by \\cite{ceriolietal} for the same value of disc aspect ratio.\n\n\\begin{figure} \n\t\\includegraphics[width=\\columnwidth]{residual_final2}\n    \\caption{Residual between the accretion rates of the primary SMBH with and without the secondary.}\n    \\label{fig:residual}\n\\end{figure}\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{evolutioninc5}\n    \\caption{Snapshots of column density (logarithmic scale). Each row indicates the evolution of a misaligned binary-disc system at 1, 5, 10, 20 and 30 degrees, respectively. The supermassive black holes are represented by black filled circles with sizes  corresponding to the primary ($R_{ar,p}$=2 $GM_p$\/$c^2$) and secondary ($R_{ar,s}$=0.2 $GM_p$\/$c^2$) accretion radius. Time (in code units) is shown in the top right corner. The length unit is the gravitational radius $R_g$=$G M_p$\/$ c^2$, where $M_p$=$10^8 M_{\\odot}$ is the primary black hole mass.}\n    \\label{fig:allanglessnapshots}\n\\end{figure*}\n\n\\subsection{Numerical resolution}\n\nUnder the SPH formalism the spatial resolution of a simulation increases in denser regions due to the Lagrangian formulation. The choice of initial conditions is largely regulated by the available computational resources. Here, all our simulations used $5 \\cdot 10^5$ SPH particles. Using higher resolution does not change the appearance of the peaks in accretion rates, but it shows improved values for the rates. Previous hydrodynamical simulations performed by \\citet{ceriolietal}, for the case of an aligned disc, examined different numbers of SPH particles and resolution effects. The authors emphasized that the two spikes in primary accretion rate are present at different resolutions ($5 \\cdot 10^5$, $1 \\cdot 10^6$ and $2 \\cdot 10^6$ SPH particles), indicating this is not a numerical artefact. However, the simulation with $2 \\cdot 10^6$ particles had the most significant enhancement in the primary accretion rate, with 4.5 times better the accretion spike compared to a resolution similar to ours. But, as seen in Figures \\ref{fig:allanglesaccrate} and \\ref{fig:allanglesprimary}, the low resolution did not affect our overall conclusions.\n\n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{evolutioninc180}\n    \\caption{Snapshots of column density during the evolution of a circumprimary disc inclined at 180 degrees. The code unit for length assumed is the gravitational radius $R_g$=$G M_p$\/$c^2$.}\n    \\label{fig:allangle180snapshots}\n\\end{figure*}\n\n\\section{CONCLUSIONS}\n\\label{section5}\n\nWe performed SPH simulations for misaligned disc-binary systems, varying the disc inclination angle, in order to investigate a possible electromagnetic precursor during the final inspirals of a binary system of SMBHs. We concentrated on a model with a gas disc surrounding the primary black hole, while the secondary black hole spirals towards the primary. We assumed a binary mass ratio $q$=$10^{-3}$, a thin accretion disc with aspect ratio $H$\/$R$=0.01 and an inner disc mass of the order of $\\approx 1 M_{\\odot}$. We chose the disc aspect ratio in the inner disc based on the previous work of \\citet{tl15}. Those authors estimated the fossil disc mass just prior to a SMBH merger and they found that for a $10^8 M_{\\odot}$ primary black hole the inner disc mass at decoupling is of the order of $1 M_{\\odot}$. Thus merging binaries are expected to have thinner discs.\n\nWe found that inclined discs with inclination angles of 1, 5 and 10 degrees lead to the mass accretion rates exceeding the Eddington rate between the times $t \\simeq 8500 - 8975$, e.g., near binary merger time $\\tau_{gw} \\simeq 9476$. By contrast, inclined circumbinary discs between 20 and 30 degrees showed a much less pronounced rise in the accretion rates (see Figure \\ref{fig:allanglesaccrate}). On the other hand, discs with inclination at 180 degrees showed no increase in the accretion rates (see Figure \\ref{fig:allanglesprimary}). The inner mass of the disc is rapidly accreted just before of the merger, leading to an abrupt increase in the mass accretion rate above the Eddington limit. However, for inclination larger than 10 degrees the accretion rate fell abruptly with primary accretion rates smaller than the Eddington rate. These results show that the snowplough effect is important for misalignment up to 10 degrees, much larger than $H$\/$R$.\n\nMoreover, secondary accretion rates for inclined discs at 1 and 5 degrees presented double peaks confirming the squeezing phenomenon at initial times of binary evolution. The companion black hole quickly migrates towards the central black hole sweeping up the gas in the inner disc. Angles larger than 5 degrees did not present any pronounced rise in the secondary accretion rates during the binary evolution (see right panel from Figure \\ref{fig:allanglesaccrate}) and the misaligned discs show somewhat larger than the those with inclinations smaller than 5 degrees (see Figure \\ref{fig:allanglessnapshots}). \n\nOur results agree with previous analytic works by \\citet{lubowetal15}, who obtained a decrease in resonant tidal torques with increasing inclination angle; we also compared the results of the electromagnetic precursors studied by \\citet{ceriolietal} and found an increase in luminosity for small inclination angles (< 10 degrees), exceeding the Eddington rate. Furthermore, from the simulations we were able to identify the presence of the squeezing mechanism (for 1 and 5 degrees) during the secondary migration, such as found by \\citet{ceriolietal} for aligned disc-binary systems. It should be noted, however, that we performed low resolution simulations ($N_{\\rm part}$ = $5 \\cdot 10^5$), while \\citet{ceriolietal} considered different numerical resolutions. In particular, for a high resolution simulation ($N_{\\rm part}$ = $2 \\cdot 10^6$) they obtained an improvement in the accretion rates of the order of about 10 times the Eddington rate when compared with our results.\n\nWe also performed simulations with counter-rotating discs, which we found to be in concordance with the results of \\citet{lubowetal15}. Those authors found that the tidal torque is zero, because viscous torques dominate over the resonances. In this case, since the disc is inclined at 180 degrees, there is an angular momentum cancellation leading to direct gas accretion on a dynamical time-scale, since in this case the gas only needs to move into the created gap and not directly on to the primary black hole. The final black hole merger thus occurs in vacuum \\citep{miphinney}, as shown in Figure \\ref{fig:allangle180snapshots}.\n\nWe ran simulations with a thicker disc (H\/R = 0.05) motivated in the investigation of electromagnetic signatures just prior to binary merger. We found no effect of the forced compression, independent of the disc inclination (see Figure \\ref{fig:thickerdisc}). These results are in agreement with previous work of \\citet{baruteal12}. Those authors found that the snowplough effect does not occur for thicker discs. In order that a small fraction of the energy liberated during the merger can go into heating the gas, producing an electromagnetic afterglow \\citep{miphinney}.\n\nAlthough we have investigated tidal transport of a small mass ratio $q$=$10^{-3}$, with the secondary component migrating via gravitational waves emission towards the primary, in low resolution simulations, our results are in agreement with earlier studies. A wider investigation at higher resolution may increase the predicted mass accretion rates for small inclination angle values. \n\nWe also concluded that our results can be applied to electromagnetic counterparts from stellar mass black hole mergers. In that case, if even a fraction of gas remains on the disc, the accretion rates produced when the gas disc is pushed into the primary black hole by the decaying secondary during the merger, lead to accretion peaks comparable or exceeding to Eddington rate. The electromagnetic signature can occur about 8-10 days before the binary final merger. \n\n\\begin{figure*}\n\t\\includegraphics[width=\\columnwidth]{hr005inc1} \\includegraphics[width=\\columnwidth]{hr005inc53}\n    \\caption{SMBH accretion rates in code units for a thicker disc with disc aspect ratio H\/R=0.05. Left panel: primary and secondary accretion rates as a function of time for a disc inclined at 1 degree. Right panel: primary and secondary accretion rates for a disc inclined at 5 degrees. The black horizontal line shows the Eddington limit.}\n    \\label{fig:thickerdisc}\n\\end{figure*}\n\nA scenario where mergers of stellar mass black hole binaries driven by gravitational radiation producing an electromagnetic signature had already been suggested by \\citet{mink17}. Those authors argued that scenario happens if a low mass accretion disc survives until coalescence. Moreover, they proposed that the disc responds to some processes of the final evolutionary phases (such as sudden mass loss and gravitational wave recoil) within hours of the merger. The electromagnetic signal will possibly arise in medium energy X-rays, perhaps extending to the infrared and last at least a few hours. In addition, \\citet{martinetal18} investigated the evolution of a disc around a merging stellar mass black hole binary in the two extreme limits of an accretion disc and a decretion disc. They concluded that dynamical readjustment of the disc after the black holes merger is probably to release significant energy in electromagnetic form, corroborating with previous results by \\citet{mink17}.\n\nIn conclusion, the 3D SPH simulations performed in this work predict the existence of electromagnetic precursors from the primary and from the secondary in a binary system of SMBHs when the angle of misalignment of the circumprimary accretion disc is small (less than 10 degrees), though we have only investigated a limited parameter space. This work suggests a link between electromagnetic signals and the gravitational radiation potentially detected by ground-based and future space detectors.\n\t\n\n\\section*{Acknowledgements}\n\nFACP acknowledges financial support from CAPES (88881.133179\/2016-01). \nMESA would like to thank the Brazilian agency FAPESP for financial support (grant 13\/26258-4). DJP acknowledges financial support from the Australian Research Council (FT130100034). All figures in this paper were produced using SPLASH \\citep{price07}. \n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhether a universal quantum computer is sufficiently powerful to be able\nto perform quantum field-theoretical computations efficiently has been a\nlong-standing and important open question.\nEfficient quantum algorithms for simulating quantum many-body systems have been\ndeveloped theoretically \\cite{Lloyd_science,Abrams_Lloyd,Zalka} \nand implemented experimentally \\cite{Lanyon:2011,Mueller:2011,Barreiro:2011}, \nbut quantum field theory\npresents additional technical challenges, such as the formally\ninfinite number of degrees of freedom per unit volume. In earlier work\n\\cite{phi4, longversion}, we presented and analyzed a quantum\nalgorithm for simulating a bosonic quantum field theory called\n$\\phi^4$ theory. That algorithm runs in a time that is polynomial in\nthe number of particles, their energy, and the desired precision, and\napplies at both weak and strong coupling. Hence, it offers exponential\nspeedup over existing classical methods at high precision or strong\ncoupling. In this paper, we extend our work to fermionic quantum field\ntheories, exemplified by the massive Gross-Neveu model, a theory in\ntwo spacetime dimensions with quartic interactions. Although our\nanalysis is specific to this theory, our algorithm can be adapted to\nother massive fermionic quantum field theories with only minor\nmodification while retaining polynomial complexity.\n\nOur quantum algorithm generates scattering events: it takes (as the\ninput) the momenta of the incoming particles and, sampling from the\nprobability distribution of possible outcomes, returns (as the\noutput) the momenta of the outgoing particles produced by the\nphysical scattering process. Physical quantities of interest, such as\nscattering cross sections, can thus be approximated by repeated\nruns of the simulation, together with statistical data analysis\nsimilar to that used for particle-accelerator experiments.\n\nThe features of fermionic field theories not present in bosonic theories pose\nnew technical problems, the solutions to which require different techniques.\nPerhaps the most obvious difference is the anticommutation, rather than\ncommutation, of fermionic fields.\nThis forces a change in the representation of the state by qubits: we\nuse an encoding method for fermionic mode occupation numbers introduced by\nBravyi and Kitaev \\cite{Bravyi_Kitaev}. In \\cite{longversion}, it was shown \nthat simulation of Hamiltonian time evolution via Suzuki-Trotter formulae\nhas efficiency advantages when applied to spatially local Hamiltonians.\nFermionic anticommutation makes it more difficult to gain efficiency\nby exploiting spatial locality. Nevertheless, we obtain a construction\nthat gives quasi-linear asymptotic scaling in time and the number of\nlattice sites, as in the bosonic case.\n\nIn contrast with bosonic field theories,\ndiscretization of fermionic field theories leads to the well-known\n``fermion doubling'' problem, in which spurious fermion species not in the\ncontinuum theory appear in the discretized theory.\nOne solution used in lattice gauge theory is to add to the action\nthe so-called Wilson term, a second-derivative operator that vanishes\nin the naive continuum limit. The Wilson term can also\nbe accommodated in our quantum algorithm; in particular, we show how\nit can be turned on during the preparation of the ground state.\n\nIn general, state preparation is a demanding task.\nThe algorithm in \\cite{phi4,longversion} uses a three-step procedure.\nFirst, the free vacuum is prepared. For the free scalar theory, this is a\nmultivariate Gaussian wavefunction.\nNext, wavepackets are excited within the free theory. In order that\nonly single-particle states are created, an ancillary qubit is used,\ntogether with a particular Hamiltonian that acts on the enlarged space.\nFinally, the interaction is turned on via a generalization of adiabatic \nstate preparation that can be applied to superpositions of eigenstates.\nThis procedure intersperses backwards time evolutions governed by \ntime-independent Hamiltonians into the turn-on to undo the different \ndynamical phases, which otherwise would cause undesirable propagation \nand broadening of wavepackets.\n\nThe state-preparation method analyzed here differs from that of\n\\cite{phi4,longversion} in two main ways.\nPreparation of the free vacuum requires modification because the vacuum\nof the free fermionic theory is different from that of the free\nbosonic theory. For this purpose, we incorporate a separate adiabatic\nturn-on step. Furthermore, sources are used to create particle\nexcitations after the coupling constant is adiabatically turned on,\nrather than before. (This difference is not required by the\nfermionic nature of the theory.) This method has the advantage that it\nworks when bound states are possible, in which case the adiabatic\nwavepacket preparation of \\cite{phi4, longversion} might fail. Another\nconsequence is that the procedure no longer requires the interleaving\nof backwards time evolutions to undo dynamical phases. \nOn the other hand, a disadvantage is that the \npreparation of each particle has a significant probability of producing \nno particle. In the case of two-particle scattering, one can perform \nadditional repetitions of the simulation, and recognize and discard\nsimulations in which fewer than two particles have been created. However,\nthe procedure is not well suited to processes involving\nmore than two incoming particles.\n\nWe analyze two different measurement procedures to be used as the last step\nof the simulation. The first method is to return adiabatically to the free\ntheory and then measure the number operators of the momentum\nmodes. For unbound states, this procedure yields complete information\nabout particle momenta, but is not well-suited to detecting bound\nstates or resolving spatial information. The second procedure is to\nmeasure charge within local regions of space. These measurements can\ndetect charged bound states, although they are blind to neutral\nones. Which of these measurement schemes is preferable depends on the\ndesired application.\n\nThere is a substantial body of work on analog quantum simulation\nof quantum systems, including lattice field theories. (See \\cite{Wiese}\nfor a recent review.) In such work, proposals are made for the\nengineering of experimental systems so that they mimic systems of\ninterest, that is, so that the  Hamiltonians of the laboratory systems\napproximate Hamiltonians of interest. The proposed quantum simulators\ncan be thought of as specialized quantum computers. In contrast, we\naddress digital quantum algorithms, namely, algorithms to be run on a\nuniversal, fault-tolerant, digital quantum computer. Our work thus\nprobes the fundamental asymptotic computational complexity of quantum\nfield theories. \n\nThere is also an extensive literature on the study of quantum field \ntheories on classical computers via lattice field theory. \n(See Ch.~17 of \\cite{Beringer:1900zz}\nfor a review of its results and status.)\nHowever, classical lattice algorithms rely on analytic continuation\nto imaginary time, $t \\to -i\\tau$. Thus, they are useful for \ncomputing static quantities such as mass ratios, but are unsuitable\nfor calculating dynamical quantities such as scattering cross\nsections. In contrast, our quantum algorithm simulates the dynamics of\nquantum field theories, a problem that is expected to be \n$\\mathsf{BQP}$-complete and thus impossible to solve by \npolynomial-time classical algorithms.\nAlthough our algorithm draws upon some concepts from lattice field \ntheory, new techniques are needed, particularly for state preparation and\nmeasurement. \n\n\nThe work presented in this paper is another step towards the goal of \nobtaining an efficient quantum algorithm for simulating the Standard Model\nof particle physics. Such an algorithm would establish that, except for\nquantum-gravity effects, the standard quantum circuit model suffices to\ncapture completely the computational power of our universe.\n\nThe rest of this paper is organized as follows. Section 2 introduces\nthe massive Gross-Neveu model, gives an overview of our quantum algorithm\nfor computing the theory's scattering amplitudes, and analyzes\nthe algorithm's complexity. Section 3 describes in detail the\nefficient simulation of the Hamiltonian time evolution in the quantum\ncircuit model. Section 4 presents our procedures for state preparation\nand measurement. Finally, Section 5 addresses some field-theoretical\naspects, namely, the effects of a non-zero lattice spacing and the\nrenormalization of mass, which are crucial elements in our complexity\nanalysis.\n\n\n\\section{Quantum Algorithm}\n\nIn this section we describe the massive Gross-Neveu model\n(\\sect{sec:MGN}), outline the steps in our algorithm for simulating\nparticle scattering processes within this model (\\sect{sec:alg}), and\ngive an overview of the algorithm's complexity\n(\\sect{sec:complexity}). The run time is polynomial in the inverse of\nthe desired precision and in the momenta of the incoming particles.\nThe detailed analysis of the steps of the algorithm that contribute\nto the overall complexity stated in \\sect{sec:complexity} is given in\nlater sections.\n\n\\subsection{The Massive Gross-Neveu Model}\n\\label{sec:MGN}\n\nThe theory we consider is a generalization of the Gross-Neveu model to\ninclude an explicit mass term in the Lagrangian. The (original) \nGross-Neveu model \\cite{Gross:1974jv} is a quantum field\ntheory in two spacetime dimensions consisting of\n$N$ fermion species with quartic interactions. It has a rich\nphenomenology. Like quantum chromodynamics (QCD), the theory governing\nthe strong interactions, it has the remarkable property of asymptotic\nfreedom, whereby the interaction becomes weaker at higher\nenergies. The theory has a discrete chiral symmetry, \n$\\psi \\rightarrow \\gamma^5 \\psi$, where \n\\begin{equation}\n\\gamma^5 = \\left[ \\begin{array}{cc} 1 & 0 \\\\ 0 & -1\n  \\end{array} \\right].\n\\end{equation}\nThis symmetry is spontaneously broken by the\nnon-perturbative vacuum. (The related theory known as the chiral\nGross-Neveu model has a continuous chiral symmetry, $\\psi \\rightarrow\ne^{i\\theta\\gamma^5}\\psi$.) Correspondingly, mass is generated\ndynamically, and the theory admits a topological soliton, the\nCallan-Coleman-Gross-Zee (CCGZ) kink. Non-topological\nsolitons also exist \\cite{Dashen:1975xh}.\n\nThese interesting characteristics have attracted intense study\nand led to applications not only in particle physics but also in\ncondensed-matter physics, including studies of\nferromagnetic superconductors \\cite{Machida:1984zz},\nconducting polymers, and systems of strongly correlated electrons\n\\cite{Lin:1998zz}.\n\nThe Gross-Neveu model, together with the chiral Gross-Neveu model,\nwas originally solved in the limit $N \\to \\infty$ \\cite{Gross:1974jv}.\nVia inverse scattering methods \\cite{Neveu:1977cr}, and later through\na generalized Bethe Ansatz \\cite{Andrei:1979sq}, integrability was\ndemonstrated for general values of $N$, a feature related to the existence of\ninfinitely many conserved currents \\cite{Brezin:1979am}.\nThe model's $S$-matrix is factorizable \\cite{Zamolodchikov:1978xm,\nKarowski:1980kq}: the $n$-body $S$-matrix is expressible as the product of\ntwo-body $S$-matrices.\n\nIn contrast, the massive Gross-Neveu model, in which there is an explicit\nbare mass, is thought not to be integrable for arbitrary values of $N$.\nThis theory still exhibits asymptotic freedom, but it does not admit\nsolitons: for any non-zero mass, the CCGZ kink becomes infinitely massive\nand disappears \\cite{Feinberg:1996kr}.\nThe asymptotic freedom and non-zero bare mass make a rigorous perturbative\nconstruction of the theory satisfying the Osterwalder-Schrader axioms\npossible \\cite{Feldman:1985ar,Feldman:1986ax}.\n\n\\smallskip\n\nThe massive $N$-component Gross-Neveu model is given by the following\nLagrangian in two spacetime dimensions:\n\\begin{equation} \\label{eq:MGN}\n{\\cal L} = \\sum_{j=1}^N \\bar{\\psi}_j (i\\gamma^\\mu \\partial_\\mu - m) \\psi_j \n+ \\frac{g^2}{2} \\bigg( \\sum_{j=1}^N \\bar{\\psi}_j\\psi_j \\bigg)^2 \\,,\n\\end{equation}\nwhere each field $\\psi_j(x)$ has two components,\n$\\gamma^\\mu$ is a two-dimensional representation of the Dirac algebra,\nand $\\bar{\\psi} = \\psi^\\dag \\gamma^0$.\\footnote{\nThe Dirac matrices satisfy \n$\\{\\gamma^\\mu,\\,\\gamma^\\nu\\} \\equiv \\gamma^\\mu\\gamma^\\nu \n+ \\gamma^\\nu\\gamma^\\mu = 2 g^{\\mu\\nu} \\mathds{1}$, and $\\psi_j(x)$ is a spinor,\nthat is, its Lorentz transformation is such that \\eq{eq:MGN} is\nLorentz-invariant. We use the metric $g^{\\mu,\\nu} = \\mathrm{diag}(+1,-1)$.} \nWe use the Majorana representation, namely,\n\\begin{equation}\n\\label{imp}\n\\gamma^0 = \\left[ \\begin{array}{cc} 0 & -i \\\\\ni & 0 \\end{array} \\right] \\,,\\quad \\quad \\gamma^1 =\n-\\left[ \\begin{array}{cc} 0 & i \\\\ i & 0 \\end{array} \\right]\n\\,.\n\\end{equation}\nThe components of the field operator associated with the particle species\n$j \\in \\{1,2,\\ldots,N\\}$ will be denoted by $\\psi_{j,\\alpha}$,\n$\\alpha \\in \\{0,1\\}$.\nIn units where $\\hbar = c = 1$, any quantity has units of some power of \nmass, referred to as the mass dimension.\nWe shall use bold-face to represent spatial vectors, such as $\\mathbf{p}$ \nand $\\mathbf{x}$, to distinguish them from spacetime vectors $x^\\mu =\n(t,\\mathbf{x})$ and $p^\\mu = (E,\\mathbf{p})$. Note, however, that we are\nconsidering 1+1 dimensions; thus, spatial vectors have only one\ncomponent. \n\nThe dimensionless parameter $g$ determines the strength of the interaction. \nWhen $g=0$, the $\\psi_j$ are free fields obeying the Dirac equation,\n$(i\\gamma^\\mu \\partial_\\mu - m_0) \\psi_j(x) = 0$. \nThen one can write\n\\begin{equation}\n\\psi_j (x) = \\int\\frac{d\\mathbf{p}}{2\\pi}\\frac{1}{\\sqrt{2 E_{\\mathbf{p}}}} \n\\left( a_j(\\mathbf{p}) u(\\mathbf{p}) e^{-ip\\cdot x}\n + b_j^\\dag(\\mathbf{p}) v(\\mathbf{p}) e^{ip\\cdot x} \\right)\\,, \n   \\label{eq:psi}\n\\end{equation}\nwhere \n\\begin{equation}\nE_{\\mathbf{p}} = \\sqrt{\\mathbf{p}^2 + m_0^2} \\,,\n\\end{equation}\n$ a_j(\\mathbf{p}),\\, b_j^\\dag(\\mathbf{p})$ are creation and annihilation operators,\nand $u,v$ satisfy\n\\begin{eqnarray}\n(m_0 \\gamma^0 + \\mathbf{p} \\gamma^0 \\gamma^1) u(\\mathbf{p}) & = &\n  E_{\\mathbf{p}} u(\\mathbf{p}) \\,, \\label{ident1}\\\\\n(m_0 \\gamma^0 - \\mathbf{p} \\gamma^0 \\gamma^1 ) v(\\mathbf{p}) & = & -\n  E_{\\mathbf{p}} v(\\mathbf{p}) \\,, \\label{ident2}\\\\\nu^\\dag(\\mathbf{p}) u(\\mathbf{p}) = v^\\dag(\\mathbf{p}) v(\\mathbf{p}) \n& = & 2 E_{\\mathbf{p}} \\,, \\label{ident3}\\\\\nu(\\mathbf{p})^\\dag v(-\\mathbf{p}) & = & 0 \\,, \\label{ident4}\\\\\n\\bar{u}(\\mathbf{p}) u(\\mathbf{p}) = - \\bar{v}(\\mathbf{p})\nv(\\mathbf{p}) & = & 2 m_0 \\,, \\label{ident5}\\\\\n\\bar{u}(\\mathbf{p}) v(\\mathbf{p}) = \\bar{v}(\\mathbf{p})\nu(\\mathbf{p}) & = & 0 \\,. \\label{ident6}\n\\end{eqnarray}\nIn the Majorana representation \\eq{imp}, one \nhas the following concrete solution:\n\\begin{equation}\n\\label{concrete}\nu(\\mathbf{p}) = \\left[ \\begin{array}{c} \\sqrt{E_{\\mathbf{p}} -\n      \\mathbf{p}} \\\\\ni \\sqrt{E_{\\mathbf{p}} + \\mathbf{p}} \\end{array} \\right] \\, \\,, \\quad\nv(\\mathbf{p}) = \\left[ \\begin{array}{c} \\sqrt{E_{\\mathbf{p}} -\n      \\mathbf{p}} \\\\\n-i \\sqrt{ E_{\\mathbf{p}} + \\mathbf{p}} \\end{array} \\right] \\,.\n\\end{equation}\n\n\n\n\\subsection{Description of Algorithm}\n\\label{sec:alg}\n\nTo represent the field using qubits, we first discretize the quantum\nfield theory, putting it on a spatial lattice. (Discretization errors are\nanalyzed in \\sect{EFT}.) Having done that, our algorithm consists of six\nmain steps, which we analyze in subsequent sections.\n\n\\begin{enumerate}\n\\item Prepare the ground state of the Hamiltonian with\n  both the interaction term ($g_0^2$) and the\n  nearest-neighbor lattice-site interactions turned off. This can be\n  done efficiently because the ground state is a tensor product of the\n \n  ground states of the individual lattice sites.\n\\item Simulate, via Suzuki-Trotter formulae, the adiabatic turn-on of\n  the nearest-neighbor lattice-site interactions, thereby obtaining\n  the ground state of the non-interacting theory.\n\\item Adiabatically turn on the interaction term, while\n  adjusting the parameter $m_0$ to compensate for the renormalization\n  of the physical mass.\n\\item Excite particle wavepackets, by introducing a\n  source term in the Hamiltonian. The source term is chosen to be\n  sinusoidally varying in time and space so as to select the desired\n  mass and momentum of particle excitations by resonance.\n\\item Evolve in time, via Suzuki-Trotter formulae,\n  according to the full massive Gross-Neveu Hamiltonian. It is during this\n  time evolution that scattering may occur.\n\\item Either use phase estimation to measure local charge observables,\n  or adiabatically return to the free theory and then use phase\n  estimation to measure number operators of momentum modes. (The\n  choice between these forms of measurement depends on the application.)\n\\end{enumerate}\n\n\n\\subsection{Complexity}\n\\label{sec:complexity}\n\nIn this section we bound the asymptotic scaling of the number of gates\nneeded to simulate scattering processes as a function of the momentum\n$p$ of the incoming particles and the precision $\\epsilon$ to which\nthe final results are desired. The effect of discretization, via a \nlattice of spacing $a$, is captured by (infinitely many) terms in the\neffective Hamiltonian that are not present in the continuum massive \nGross-Neveu theory (\\sect{EFT}). Truncation of these terms, \nwhich make contributions of $O(a)$ to scattering cross sections,\ntherefore constitutes an error. Thus, \nto ensure any cross section $\\sigma'$ in the discretized quantum field\ntheory matches the continuum value $\\sigma$ to within\n\\begin{equation}\n\\label{criterion}\n(1-\\epsilon) \\sigma \\leq \\sigma' \\leq (1+\\epsilon) \\sigma,\n\\end{equation}\none must choose the scaling $a \\sim \\epsilon$ in the high-precision limit, \nthat is, the limit $\\epsilon \\to 0$. Similarly, in the large-momentum limit, \none must choose the scaling $a \\sim p^{-1}$ in order to ensure that the \nwavelength of each particle is large compared with the lattice spacing.\n\nIt suffices to use an adiabatic\nprocess of duration\n\\begin{equation}\nT = O \\left( \\frac{L^2}{a^4 m^3 \\epsilon} \\right)\n\\end{equation}\n(where $L$ is the length of the spatial dimension and $m$ is the physical\nmass)\nto prepare a state within a distance $\\epsilon$ of the free vacuum \n(\\sect{freeprep}). \nUsing Suzuki-Trotter decompositions of the form described in \\sect{trotter},\nwe can simulate this adiabatic time evolution using a number of quantum gates\nscaling as\n\\begin{eqnarray}\nG_{\\mathrm{prep}} & = & O\\left( \\left( \\frac{TL}{a^2} \\right)^{1+o(1)}\n\\epsilon^{-o(1)} \\right) \\\\\n& = & O \\left( \\left( \\frac{L^3}{a^6 m^3 \\epsilon} \\right)^{1+o(1)}\n\\right) \\,. \\label{gprep}\n\\end{eqnarray}\n\nThe next state-preparation step is to simulate adiabatic turn-on of\nthe coupling, thereby obtaining the interacting vacuum.\nThis can be achieved in a time (\\sect{turnon})\n\\begin{equation}\nT_{\\mathrm{turn-on}} = O \\left( \\frac{L^2}{a^4 m^3 \\epsilon }\n\\right).\n\\end{equation}\nApplying Suzuki-Trotter formulae, one obtains a gate count of\n\\begin{equation}\nG_{\\mathrm{turn-on}} = O \\left( \\left( \\frac{L^3}{a^6 m^3 \\epsilon}\n\\right)^{1+o(1)} \\right). \\label{gturnon}\n\\end{equation}\n\nThe final state-preparation step is to excite particle wavepackets. We\ndo this by applying a time-dependent perturbation $\\lambda W(t)$ for\ntime $\\tau$. It is necessary to choose $\\tau$ large enough and\n$\\lambda$ small enough to suppress the production of particle\npairs. The choice of small $\\lambda$ means that there will be a substantial\nprobability that no particle is produced. Let $p_1$ denote the probability\nthat exactly one particle is produced. In a typical simulation one\nwishes to produce an initial state of two spatially separated incoming\nparticles. The probability that both of these are produced is\n$p_1^2$. The simulations in which one or both initial particles has failed \nto be created can be detected at the final measurement stage of the\nsimulation and discarded. This comes at the cost of a factor of\n$1\/p_1^2$ more repetitions of the simulation. \nThe probability $p_1$ is independent of momentum and scales with \nprecision as $p_1 \\sim \\epsilon$ (\\sect{exciting}). \nAlso, in \\sect{exciting} one finds that the total\nnumber of quantum gates needed for the excitation step is\n\\begin{equation}\nG_{\\mathrm{excite}} = \\left\\{ \\begin{array}{ll}\n\\epsilon^{-4-o(1)} \\,, & \\textrm{as} \\,\\,\\, \\epsilon \\to 0 \\,, \\\\\np^{3+o(1)} \\,, & \\textrm{as} \\,\\,\\, p \\to \\infty \\,.\n\\end{array} \\right.\n\\end{equation}\n\nIn both the high-momentum and high-precision limits, the dominant costs in\nthe algorithm are the two adiabatic state preparation steps, whose\ncomplexity is given in \\eq{gprep} and \\eq{gturnon}.\nIn the high-precision limit, to compute physical quantities such as\nscattering cross sections to within a factor of $(1+\\epsilon)$, one\nmust choose $a$ to scale as $\\epsilon$ (\\sect{EFT}). Also, in this limit,\nthe complexity contains a further factor of $1\/\\epsilon$ owing to\npostselection of simulations in which both wavepacket excitations have been\nsuccessful (\\sect{exciting}). \nSubstituting\n$a \\sim \\epsilon$ into \\eq{gprep} and including this extra factor of\n$1\/\\epsilon$ yield a total complexity of\n$O(\\epsilon^{-8-o(1)})$. \nIn the  high-momentum limit, $a$ must\nscale as $1\/p$ to ensure that the particle wavelength is long compared\nto the lattice spacing, and $L$ must scale as $p$ to accommodate the\nexcitation step (\\sect{exciting}). \nIn summary, we obtain\n\\begin{equation}\nG_{\\mathrm{total}} = \\left\\{ \\begin{array}{ll}\nO(\\epsilon^{-8-o(1)}) \\,, & \\textrm{as} \\,\\,\\, \\epsilon \\to 0 \\,, \\\\\nO(p^{9+o(1)}) \\,, & \\textrm{as} \\,\\,\\,  p \\to \\infty \\,. \\end{array} \\right.\n\\end{equation}\nNote that these are only upper bounds on the complexity, and it may be\npossible to improve them by using more detailed analysis, such as more\nspecialized adiabatic theorems.\n\n\n\\section{Qubits and Quantum Gates}\n\nWe divide the problem of simulating Hamiltonian time evolutions\nin the massive Gross-Neveu model into three subproblems. \nThe first subproblem is to represent the state of the field with qubits. \nWe do this by choosing a complete set of commuting observables and encoding \ntheir eigenvalues with strings of bits (\\sect{rep}). The second subproblem\nis to simulate local fermionic gates on the degrees of freedom defined by\nthe commuting observables. Achieving this in an efficient manner is\nnon-trivial because of the fermionic statistics. For this purpose, we employ\na technique due to Bravyi and Kitaev \\cite{Bravyi_Kitaev}, which\nimplements fermionic statistics with only logarithmic overhead in the\nnumber of lattice sites (\\sect{BK}). The third subproblem is to decompose the\ntime evolution governed by the massive Gross-Neveu Hamiltonian into a product\nof local fermionic gates. We do this using high-order\nSuzuki-Trotter formulae \\cite{Suzuki90} with optimizations tailored to\nthe fermionic statistics and the spatially local nature of the Hamiltonian \n(\\sect{trotter}). The local unitary transformations act on at most\n$2^{2N}$-dimensional Hilbert spaces and can therefore be efficiently\ndecomposed into elementary gates for any constant number of particle\nspecies, $N$, via the Solovay-Kitaev algorithm \\cite{Kitaev97, Dawson_Nielsen}.\n\n\n\\subsection{Representation by Qubits}\n\\label{rep}\n\nFirst, we put the massive Gross-Neveu model on a spatial lattice\n\\begin{equation}\n\\Omega = a \\mathbb{Z}_{\\hat{L}}  \\,.\n\\end{equation}\nFor simplicity, we impose periodic\nboundary conditions, so that $\\Omega$ can be considered a circle of\ncircumference $L = a \\hat{L}$. \nThe Hamiltonian is\n\\begin{equation}\n\\label{h}\nH = H_0 + H_g + H_W \\,,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nH_0 & = & \\sum_{\\mathbf{x} \\in \\Omega} a \\sum_{j=1}^N \\bar{\\psi}_j(\\mathbf{x}) \n\\left[ -i \\gamma^1 \\frac{\\psi_j(\\mathbf{x} + a)\n- \\psi_j(\\mathbf{x}-a)}{2a} + m_0 \\psi_j(\\mathbf{x}) \\right] \\label{h0} \\,, \\\\\nH_g & = & -\\frac{g_0^2}{2} \\sum_{\\mathbf{x} \\in \\Omega} a \\bigg( \\sum_{j=1}^N\n\\bar{\\psi}_j (\\mathbf{x}) \\psi_j(\\mathbf{x}) \\bigg)^2 \\label{hg} \\,, \\\\\nH_W & = & \\sum_{\\mathbf{x} \\in \\Omega} a \\sum_{j=1}^N \\left[ - \\frac{r}{2a} \n\\bar{\\psi}_j(\\mathbf{x}) \\left( \\psi_j(\\mathbf{x}+a) - 2 \\psi_j(\\mathbf{x}) + \\psi_j(\\mathbf{x}-a) \\right) \n\\right] \\,. \n\\label{hw} \n\\end{eqnarray}\nHere, $H_g$ is the interaction term, and $H_W$ is the Wilson term, used to\nprevent fermion doubling \\cite{Wilson74}. Correspondingly, $0 < r\n\\leq 1$ is called the Wilson parameter. $H$ is spatially local in the sense \nthat it consists only of single-site and nearest-neighbor terms on the\nlattice.\n\nLet $\\Gamma$ denote the momentum-space lattice corresponding to\n$\\Omega$, namely,\n\\begin{equation}\n\\Gamma = \\frac{2\\pi}{L} \\mathbb{Z}_{\\hat{L}} \\,.\n\\end{equation}\nWe can deduce the spectrum $H_0 + H_W$ using\n\\begin{eqnarray}\n\\label{psij}\n\\psi_j(\\mathbf{x}) & = & \\sum_{\\mathbf{p} \\in \\Gamma} \\frac{1}{L}\n\\frac{1}{\\sqrt{2 E_{\\mathbf{p}}}} \\left( a_j(\\mathbf{p}) u(\\mathbf{p}) e^{i \\mathbf{p} \\cdot \\mathbf{x}}\n  + b_j^\\dag(\\mathbf{p}) v(\\mathbf{p}) e^{-i \\mathbf{p} \\cdot \\mathbf{x}} \\right) \\,,\\\\\n\\label{psidagj}\n\\bar{\\psi}_j(\\mathbf{x}) & = & \\sum_{\\mathbf{p} \\in \\Gamma} \\frac{1}{L}\n\\frac{1}{\\sqrt{2 E_{\\mathbf{p}}}} \\left( a_j^\\dag(\\mathbf{p}) \\bar{u}(\\mathbf{p}) e^{-i \\mathbf{p}\n  \\cdot \\mathbf{x}} + b_j(\\mathbf{p}) \\bar{v}(\\mathbf{p}) e^{i \\mathbf{p} \\cdot \\mathbf{x}} \\right) \\,.\n\\end{eqnarray}\nThe inverse transformation is\n\\begin{eqnarray}\na_j(\\mathbf{p}) & = & \\frac{1}{\\sqrt{2 E_{\\mathbf{p}}}} u^\\dag(\\mathbf{p}) \\sum_{\\mathbf{x} \\in \\Omega}\na e^{-i \\mathbf{p} \\cdot \\mathbf{x}} \\psi_j(\\mathbf{x}) \\,, \\label{adef}\\\\\nb_j^\\dag(\\mathbf{p}) & = & \\frac{1}{\\sqrt{2 E_{\\mathbf{p}}}} v^\\dag(\\mathbf{p}) \\sum_{\\mathbf{x}\n  \\in \\Omega} a e^{i \\mathbf{p} \\cdot \\mathbf{x}} \\psi_j(\\mathbf{x}) \\label{bdef}\\,.\n\\end{eqnarray}\nSubstituting \\eq{psij} and \\eq{psidagj} into \\eq{h0} and \\eq{hw} and\nneglecting the vacuum energy, we obtain\n\\begin{equation}\nH_0 + H_W = \\sum_{j=1}^N \\sum_{\\mathbf{p} \\in \\Gamma} \\frac{1}{L} E^{(a)}_{\\mathbf{p}}(m_0)\n\\left( a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p}) + b_j^\\dag(\\mathbf{p}) b_j(\\mathbf{p}) \\right) \\,,\n\\end{equation}\nwhere\n\\begin{equation}\nE^{(a)}_{\\mathbf{p}}(m_0) = \\sqrt{\\left( m_0 + \\frac{2r}{a} \\sin^2 \\left(\n  \\frac{\\mathbf{p} a}{2} \\right) \\right)^2 + \\frac{1}{a^2} \\sin^2 (\\mathbf{p} a)} \\,.\n\\label{eq:Epa}\n\\end{equation}\nFrom the canonical fermionic anticommutation relations\n\\begin{eqnarray}\n\\label{anticanon1}\n\\{ \\psi_{j,\\alpha}(\\mathbf{x}), \\psi^\\dag_{k,\\beta}(\\mathbf{y})\\} & = & a^{-1}\n\\delta_{\\mathbf{x}, \\mathbf{y}} \\delta_{j,k} \\delta_{\\alpha, \\beta} \\mathds{1} \\,, \\\\\n\\label{anticanon2}\n\\{ \\psi^\\dag_{j,\\alpha}(\\mathbf{x}), \\psi^\\dag_{k,\\beta}(\\mathbf{y}) \\} & = &\n\\{\\psi_{j,\\alpha}(\\mathbf{x}), \\psi_{k,\\beta}(\\mathbf{y}) \\}  =  0 \\,,\n\\end{eqnarray}\nit follows that\n\\begin{eqnarray}\n\\label{anticanonp1}\n\\{ a_j(\\mathbf{p}), a_k^\\dag(\\mathbf{q}) \\} & = & L \\delta_{\\mathbf{p}, \\mathbf{q}} \\delta_{j,k}\n\\mathds{1} \\,, \\\\\n\\label{anticanonp2}\n\\{ b_j(\\mathbf{p}), b_k^\\dag(\\mathbf{q}) \\} & = & L \\delta_{\\mathbf{p}, \\mathbf{q}} \\delta_{j,k}\n\\mathds{1} \\,,\n\\end{eqnarray}\nwith all other anticommutators involving $a$ and $b$ operators equal to\nzero. We thus have the following interpretation: there are $N$\nindependent fermion species, created (with momentum $\\mathbf{p}$) by\n$a_1^\\dag(\\mathbf{p}),\\ldots,a_N^\\dag(\\mathbf{p})$ and annihilated by\n$a_1(\\mathbf{p}),\\ldots,a_N(\\mathbf{p})$. Similarly, for each species $j$, \n$b_j^\\dag(\\mathbf{p})$ and $b_j(\\mathbf{p})$ are the creation and annihilation\noperators for a corresponding antifermion. Thus, $H$ acts\non a Hilbert space of dimension $2^{2N\\hat{L}}$.\n\n\\medskip\n\nWe can specify a basis for the Hilbert space of field states by\nchoosing a complete set of commuting observables. The basis is then\nindexed by the set of eigenvalues of these observables. \nThe fermionic anticommutation relations $ \\{a,a^\\dag\\} = \\mathds{1} ,\\,\n\\{a,a\\} = 0$ imply that the algebra generated by $a$ and $a^\\dag$ has\nthe irreducible representation\n$\na \\to \\left[ \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right]\n\\,,\\,\na^\\dag \\to \\left[ \\begin{array}{cc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right]\n$,\nwhich is unique up to the choice of basis. Hence, the eigenvalues of\n$a^\\dag a$ are $0$ and $1$. The two basis vectors for the space on\nwhich $a$ and $a^\\dag$ act are interpreted as the presence or absence\nof a fermion.\n\nThus, by \\eq{anticanon1} and \\eq{anticanon2},\n\\begin{equation}\nS_x = \\{a \\psi^\\dag_{j,\\alpha}(\\mathbf{x})\n\\psi_{j,\\alpha}(\\mathbf{x})|j=1,\\ldots,N; \\ \\alpha=0,1; \\ \n\\mathbf{x} \\in \\Omega \\}\n\\end{equation}\nis a set of $2N\\hat{L}$ commuting observables, each of which has\neigenvalues zero and one. Similarly, by \\eq{anticanonp1} and\n\\eq{anticanonp2}, \n\\begin{equation}\nS_p = \\{ L^{-1} a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p})| j=1,\\ldots,N;\n\\ \\mathbf{p} \\in \\Gamma \\} \\cup \\{ L^{-1} b_j^\\dag(\\mathbf{p})\nb_j(\\mathbf{p})| j=1,\\ldots,N; \\ \\mathbf{p} \\in \\Gamma \\}\n\\end{equation}\nis a set of $2N\\hat{L}$ commuting observables, each with eigenvalues\nzero and one. In the non-interacting theory, the eigenvalues of the\nelements of $S_p$ are interpreted as the fermionic occupation numbers \nof different momentum modes.\n\nThe Hamiltonian $H_0+H_W$ is called the free theory. The\neigenstates of the number operators in $S_p$ are eigenstates of\n$H_0+H_W$, and thus the particles do not interact. The rest mass\nof these non-interacting particles is $E_0^{(a)}(m_0) =\nm_0$. It is not known how to solve for the spectrum of\n$H_0+H_W+H_g$ analytically, but the eigenvalue spectrum of\n$H_0+H_W+H_g$ can still be characterized in terms of particles. The\nrest mass $m$ of the particles in $H_0+H_W+H_g$ is equal to the\neigenvalue gap between the ground state (also called the vacuum) and\nthe first excited state. In the interacting theory, it is no longer\ntrue that $m = m_0$. Rather, $m$ depends in a non-trivial way on $m_0$,\n$g_0$, and $a$; the mass is said to be renormalized. \nA quantitative analysis of this effect contributes to our\nanalysis of adiabatic state preparation and is given in\n\\sect{massren}.\n\nOne can represent the quantum state of the fermionic fields using\n$2N\\hat{L}$ qubits to store the eigenvalues of the elements of either\n$S_x$ or $S_p$. The ground state of\nthe free theory in the $S_p$ representation is thus \n$\\ket{000\\ldots}$. However, the ground state of the interacting theory\nis non-trivial in both\nrepresentations. We define our qubit basis in terms of the elements of\n$S_x$, because the Gross-Neveu Hamiltonian is local in this basis,\nwhich improves the scaling of the Suzuki-Trotter formulae used to\nimplement time evolution. \nHowever, we do not simply store the eigenvalues of the\nelements of $S_x$ directly as the values of the qubits. \nThis representation would be somewhat inefficient to act upon, because \ndirect implementation of the fermionic minus signs requires $O(\\hat{L})$ \ngates. Instead,\nwe apply the method of \\cite{Bravyi_Kitaev} to reduce this overhead to\n$O(\\log \\hat{L})$, as described next.\n\n\n\\subsection{Simulating Fermionic Gates}\n\\label{BK}\n\nThe implementation of fermionic gates using qubits can present a technical\nchallenge \\cite{Bravyi_Kitaev}. As an example, consider the unitary \ntransformation $U_{j,\\alpha}(\\mathbf{x}) = \\sqrt{a} \\big(\n  \\psi_{j,\\alpha}(\\mathbf{x}) + \\psi^\\dag_{j,\\alpha}(\\mathbf{x})\n\\big)$. This toggles the eigenvalue of $a\n\\psi_{j,\\alpha}(\\mathbf{x}) \\psi^\\dag_{j,\\alpha}(\\mathbf{x})$ between\nzero and one. Such a toggling can be implemented on qubits  with the\nNOT gate. However, to satisfy the fermionic anticommutation relations\n\\eq{anticanon1} and \\eq{anticanon2} the sign of the transition\namplitude between the zero and one state must depend on the occupation\nof other modes. A well-known way to satisfy\n\\eq{anticanon1} and \\eq{anticanon2} is to use a Jordan-Wigner\ntransformation, in which the modes are given an ordering and\n$U_{j,\\alpha}(\\mathbf{x})$ is represented by the operator $\\sigma_x\n\\otimes \\sigma_z \\otimes \\ldots \\otimes \\sigma_z$, where the\n$\\sigma_z$ operators apply to all preceding modes\\footnote{Note that\n  one can apply both the Jordan-Wigner and Bravyi-Kitaev methods for\n  implementing fermionic operators on quantum computers\n  in any number of spatial dimensions, using an arbitrary numbering of\n  modes.\n}\n\\cite{Jordan_Wigner}. Unfortunately, this method clearly has an $O(\\hat{L})$\noverhead. In \\cite{Bravyi_Kitaev}, Bravyi and Kitaev give a method \nwith only $O(\\log \\hat{L})$ overhead, which we briefly review\nhere. \n\nLet $n_i$ be the occupation number of the $i\\th$ fermionic mode according \nto some chosen numbering of the modes from 1 to $2N\\hat{L}$. To implement\nthe minus signs in $U_{j,\\alpha}(\\mathbf{x})$, one needs to know\n$\\sum_i n_i$, where the sum is over all preceding modes. Thus, a\nnatural encoding of fermionic mode occupation numbers is to store\nthe quantities $t_i = \\sum_{j=1}^i n_j$ instead of the quantities\n$n_i$. This encoding has the advantage that calculating the relevant signs has\nan $O(1)$ cost. However, it has the disadvantage that, if the\noccupation number of the $i\\th$ mode changes, then $i-1$ of the $t_i$ values\nmust be updated. Thus, updates have an $O(\\hat{L})$ cost. \nThe Bravyi-Kitaev encoding uses the following compromise, \nin which the calculation of the relevant signs and the\nupdate steps can both be performed in time $O(\\log \\hat{L})$.\n\nThe mode index $i \\in \\{1,\\ldots,2N\\hat{L}\\}$ can be represented by a\nbit string of length $l = \\lceil \\log_2 (2N\\hat{L}) \\rceil$. One can\ndefine the following partial order on these bit strings. Consider two\nbit strings $x = x_l x_{l-1} \\ldots x_1$ and $y = y_l y_{l-1} \\ldots\ny_1$. Then $x \\preceq y$ if, for some $r$, $x_j = y_j$ for $j > r$ and\n$y_{r-1} = y_{r-2} = \\ldots = y_1 = 1$. Now, let $k_j = \\sum_{s\n  \\preceq j} n_s$. Any total\noccupation number $t_i$ can be computed from the $k_j$ quantities in $O(\\log\n\\hat{L})$ time and changing the occcupation of any mode $n_j$\nrequires updating only $O(\\log \\hat{L})$ of the $k_j$ quantities \n\\cite{Bravyi_Kitaev}.\n\nIn fact, the Bravyi-Kitaev construction is\nrelevant only to the excitation of wavepackets\n(\\sect{exciting}). In all other parts of our algorithm, we simulate a\nHamiltonian in which every term is a product of an even number of fermionic\nfield operators, all acting on the same site or on\nnearest-neighbor sites in one dimension. In this case, traditional\nJordan-Wigner techniques incur only $O(1)$ overhead.\n\n\n\n\n\\subsection{Application of Suzuki-Trotter Formulae to Fermionic systems}\n\\label{trotter}\n\nIn this section, we describe how to construct efficient quantum\ncircuits that simulate time evolution induced by the Hamiltonian $H$\ndefined in \\eq{h}, \\eq{h0}, \\eq{hg}, and\n\\eq{hw}. We present the case in which $H$ is time-independent. By the results\nof \\cite{Suzuki93}, the same analysis applies to the simulation of the\ntime-dependent Hamiltonians that we use in adiabatic state\npreparation. (See also \\cite{Wiebe}.)\n\nUsing a $k^{\\mathrm{th}}$-order Suzuki-Trotter formula, one can implement\nHamiltonian time evolution of duration $t$ using a number of quantum\ngates that scales as $t^{1+\\frac{1}{2k}}$ \\cite{Suzuki90, Cleve_sim}. \nGenerally, applying a Suzuki-Trotter formula directly to a Hamiltonian of \nthe form\n\\begin{equation}\n\\label{manyterms}\nH = \\sum_{i=1}^m H_i\n\\end{equation}\nyields an algorithm with $O(m^{1+o(1)})$ timesteps, and\nhence $O(m^{2+o(1)})$ gates, if the $H_i$ are not mutually commuting. \nThus, \nit is often\nadvantageous to group terms in a Hamiltonian like \\eq{manyterms}\ninto as small a collection as possible of sets of mutually\ncommuting terms \\cite{Raeisi,longversion}. \n\nConsider the problem of simulating the Hamiltonian $H$ defined in\n\\eq{h}, \\eq{h0}, \\eq{hg}, and \\eq{hw}. By\n\\eq{anticanon1} and \\eq{anticanon2}, one\nsees that \n\\begin{equation}\n\\label{sitecom}\n[\\bar{\\psi}_j(\\mathbf{x}) \\psi_j(\\mathbf{x}), \\bar{\\psi}_k(\\mathbf{y})\n\\psi_k(\\mathbf{y})] = 0 \\,,\n\\end{equation}\nregardless of whether $j=k$ or $\\mathbf{x}=\\mathbf{y}$. Thus, we start by\ndecomposing $H$ as a sum of two parts, the single-site terms and the\nterms that couple nearest neighbors:\n\\begin{equation}\nH = H_{\\mathrm{ss}} + H_{\\mathrm{nn}} \\,,\n\\end{equation}\nwhere\n\\begin{equation}\nH_{\\mathrm{ss}} = \\sum_{\\mathbf{x} \\in \\Omega} a \\Bigg[ \\sum_{j=1}^N\n  \\left( m_0 \\bar{\\psi}_j(\\mathbf{x}) \\psi_j(\\mathbf{x}) + \\frac{r}{a}\n    \\bar{\\psi}_j(\\mathbf{x}) \\psi_j(\\mathbf{x}) \\right) +\n  \\frac{g_0^2}{2} \\bigg( \\sum_{j=1}^N \\bar{\\psi}_j(\\mathbf{x})\n    \\psi_j(\\mathbf{x}) \\bigg)^2 \\Bigg] \\,.\n\\end{equation}\nBy \\eq{sitecom}, $e^{-i H_{\\mathrm{ss}} \\delta t}$ decomposes into a\nproduct of local unitary transformations.\n\nAll terms in $H_{\\mathrm{nn}}$ are of the form\n\\begin{equation}\n\\label{form}\n\\psi_{j,\\alpha}^\\dag (\\mathbf{x}) \\psi_{j, \\beta}(\\mathbf{y}) +\n\\psi^\\dag_{j, \\beta}(\\mathbf{y}) \\psi_{j,\\alpha}(\\mathbf{x}) \\,,\n\\end{equation}\nfor $\\mathbf{x} = \\mathbf{y} \\pm a$. Terms with $\\alpha = \\beta$ and\nterms with $\\alpha \\neq \\beta$ are both present in $H_{\\mathrm{nn}}$.\n\nGiven an operator of the form \\eq{form}, let us refer to the\nsubset of $\\{1,\\ldots,N\\} \\times \\{0,1\\} \\times \\Omega$ on which it\nacts as its support. Because they consist of a product of an even number \nof fermionic operators, any two operators of the form \\eq{form} commute\nprovided they have disjoint support. Thus, we next decompose\n$H_{\\mathrm{nn}}$ as\n\\begin{equation}\n\\label{fourcolors}\nH_{\\mathrm{nn}} = H_1 + H_2 + H_3 + H_4 \\,,\n\\end{equation}\nwhere each of $H_1,\\ldots,H_4$ consists of a sum of terms with\nnon-intersecting support.\n\nIn $H_{\\mathrm{nn}}$ there is no coupling between different species,\nthat is, no products of $\\psi_j$ and $\\psi_k$ for $j \\neq\nk$. Thus, we can ignore the index $j$. We now construct a graph whose\nvertices correspond to the elements of $\\{0,1\\} \\times \\Omega$. We\ndraw an edge between two vertices if there exists a term in\n$H_{\\mathrm{nn}}$ with the corresponding support. One sees that this\ngraph is as shown in Fig.~\\ref{coloring}. The graph is\nedge-colorable with four colors, and therefore $H_{\\mathrm{nn}}$ is\ncorrespondingly decomposable as in \\eq{fourcolors} with each of\n$H_1,H_2,H_3,H_4$ consisting of a sum of commuting terms. (Because of\nthe periodic boundary conditions, this works only if $\\hat{L}$ is\neven, which we assume henceforth.)\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{coloring2.eps}\n\\caption{\\label{coloring} Vertices represent elements of $\\{0,1\\}\n  \\times \\Omega$ two vertices are connected by an edge if\n  $H_{\\mathrm{nn}}$ couples these sites. (Different species are never\n  coupled by $H_{\\mathrm{nn}}$, so the full graph with vertices\n  corresponding to elements of $\\{1,\\ldots,N\\} \\times \\{0,1\\} \\times\n  \\Omega$ would consist of $N$ disconnected copies of the graph\n  shown.) The edges can be colored with four colors such that each node\n  has no more than one incident edge of each color. \n  One can obtain the decomposition\n  $H_{\\mathrm{nn}} = H_1 + H_2 + H_3 + H_4$ \n  by choosing $H_1$ to be the sum of all interaction terms\n  along the edges labeled 1 (which are blue), $H_2$ to be the sum of\n  all the interaction terms along edges labeled 2 (which are red), and\n  so on.}\n\\end{center}\n\\end{figure}\n\nThe unitary time evolution induced by $H=H_{\\mathrm{ss}} +\nH_1 + H_2 + H_3 + H_4$ can be approximately decomposed via high-order\nSuzuki-Trotter formulae into a sequence of \n\\begin{equation}\n\\label{trottersteps}\nn_{\\mathrm{S-T}} = O\\big((t\/a)^{1+o(1)} \\hat{L}^{o(1)} \\epsilon^{-o(1)} \\big)\n\\end{equation}\ntime evolutions induced by individual members of $\\{H_{\\mathrm{ss}},\nH_1, H_2, H_3, H_4\\}$. The scaling with $t$ follows from\n\\cite{Suzuki90, Suzuki93}. The scaling with $\\hat{L}$ is a consequence\nof the spatial locality of $H$ (see \\S 4.3 of\n\\cite{longversion}), that is, the property that only \nnearest-neighbor sites are coupled. \nThe scaling with $a$ is a consequence of the fact\nthat the individual terms in the Hamiltonian each have norm at most of\norder $a^{-1}$. This affects the magnitude of the error term in the\nSuzuki-Trotter decomposition, which arises from commutators of these\nterms.\n\nEach member of $\\{H_{\\mathrm{ss}}, H_1, H_2, H_3, H_4\\}$ is a sum of\n$O(\\hat{L})$ commuting terms. The time evolution $e^{-i \\sum_i\n  M_i t}$ induced by \ncommuting terms $M_i$\ndecomposes as $e^{-i \\sum_i M_i t} = \\prod_i e^{-iM_i t}$. If each\n$H_i$ acts on only a constant number of qubits, then the individual\nfactors $e^{-iH_i t}$ in this product can each be simulated in\n$\\widetilde{O}(1)$ time, by the Solovay-Kitaev theorem \\cite{Kitaev97,\n  Dawson_Nielsen}. Thus, including a logarithmic overhead for fermionic\nstatistics, the cost of implementing $e^{-iJt}$ for any\n$J \\in \\{H_{\\mathrm{ss}}, H_1, H_2, H_3, H_4\\}$ is $\\widetilde{O}(\\hat{L})$.\nBy \\eq{trottersteps}, the total cost of time evolution is\n$O \\big( \\left( \\frac{tL}{a^2} \\right)^{1+o(1)} \\epsilon^{-o(1)}\n\\big)$ quantum gates.\n\n\n\\section{State Preparation and Measurement}\n\nWe divide the problem of state preparation into three steps, described\nin \\sect{freeprep}--\\sect{exciting}:\npreparing the free vacuum,\ntransforming the free vacuum into the interacting vacuum, and exciting\nwavepackets on the background of the interacting vacuum.\nTwo possible measurement procedures are described in \n\\sect{measurements} and \\sect{sec:charge}.\n\n\\subsection{Preparing the Free Vacuum}\n\\label{freeprep}\n\nAlthough the free Hamiltonian $H_0 + H_W$ is exactly solvable,\npreparing its ground state in the $S_x$ representation on a quantum computer \nis non-trivial. We do so using adiabatic state preparation, as\nfollows. Let\n\\begin{equation}\nH(s) = \\sum_{\\mathbf{x} \\in \\Omega} a \\sum_{j=1}^N \\bar{\\psi}_j(\\mathbf{x}) \\left[\n  -si\\gamma^1 \\frac{\\psi_j(\\mathbf{x} + a) - \\psi_j(\\mathbf{x} - a)}{2a} + m\n  \\psi_j(\\mathbf{x}) \\right] + s H_W.\n\\end{equation}\nThe energy gap of this Hamiltonian is equal to the parameter $m$ for all\n$s$. We set this equal to the physical mass of the particles whose\nscattering we ultimately wish to simulate.\n\n$H(0)$ is a sum of separate Hamiltonians acting on each lattice\nsite and each species of particle. Its ground state is therefore the\ntensor product of the ground states of the\nfour-dimensional Hilbert spaces associated with each pair \n$(\\mathbf{x},j) \\in \\Omega \\times \\{1,\\ldots,N\\}$. \n(Specifically, the ground state for a given site is\n$\\frac{1}{\\sqrt{2}} \\left( \\ket{01} + i\\ket{10} \\right)$, where\n$\\ket{b_0 b_1}$ with $b_0,b_1 \\in \\{0,1\\}$ denotes the state\nsatisfying $a \\psi^\\dag_{j,0}(\\mathbf{x}) \\psi_{j,0}(\\mathbf{x})\n\\ket{b_0 b_1} = b_0 \\ket{b_0 b_1}$ and\n$a \\psi^\\dag_{j,1}(\\mathbf{x}) \\psi_{j,1}(\\mathbf{x}) \\ket{b_0 b_1} =\nb_1 \\ket{b_0 b_1}$.) The cost of\nproducing this tensor product of $N \\hat{L}$ local states, \nincluding the cost of fermionic\nantisymmetrization via the encoding of \\cite{Bravyi_Kitaev},\nis $O(N \\hat{L} \\log (N \\hat{L}))$.\n\nAfter the ground state of $H_0$ has been prepared, the complexity of the\nremaining adiabatic state preparation is determined by the\nadiabatic theorem \\cite{Ruskai, Goldstone}.\n\n\\begin{theorem}\n\\label{adiabaticthm}\nLet $H(s)$ be a finite-dimensional twice differentiable Hamiltonian on\n$0 \\leq s \\leq 1$ with a non-degenerate ground state $\\ket{\\phi_0(s)}$\nseparated by an energy gap $\\gamma(s)$. Let $\\ket{\\psi(t)}$ be\nthe state obtained by Schr\\\"odinger time evolution according to the \nHamiltonian $H(t\/T)$ from the state $\\ket{\\phi_0(0)}$ at $t = 0$. \nThen, with an appropriate choice of phase for $\\ket{\\phi_0(t)}$, \nthe error $\\Delta \\equiv \\| \\Ket{\\psi(T)} - \\Ket{\\phi_0(1)} \\|$ satisfies\n\\begin{equation}\n\\label{adiabateq}\n\\Delta \\leq\n\\frac{1}{T} \\left[ \\frac{1}{\\gamma(0)^2} \\left\\| \\frac{\\mathrm{d} H}{\\mathrm{d} s} \\right\\|_{s=0}\n+ \\frac{1}{\\gamma(1)^2} \\left\\| \\frac{\\mathrm{d} H}{\\mathrm{d} s} \\right\\|_{s=1}\n+ \\int_0^1 \\mathrm{d} s \\left( \\frac{5}{\\gamma^3} \\left\\| \\frac{\\mathrm{d} H}{\\mathrm{d} s}\n\\right\\|^2 + \\frac{1}{\\gamma^2} \\left\\| \\frac{\\mathrm{d}^2 H}{\\mathrm{d} s^2}\n\\right\\| \\right) \\right]. \n\\end{equation}\n\\end{theorem}\n\nAnalyzing the adiabaticity of this process is relatively easy, because\n\\eq{psij} and \\eq{psidagj} diagonalize $H(s)$\n(and $\\frac{dH}{ds}$) for all $s$. One finds that the eigenvalue gap\nof $H(s)$ throughout the adiabatic path $0 \\leq s \\leq 1$ is always\nprecisely $m$. Furthermore,\n\\begin{equation}\n\\frac{dH}{ds} = \\sum_{j=1}^N \\sum_{\\mathbf{p} \\in \\Gamma} \\frac{1}{L}\nE^{(a)}_{\\mathbf{p}}(0) \\left( a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p}) + b_j^\\dag(\\mathbf{p}) b_j(\\mathbf{p})\n\\right).\n\\end{equation}\nThus,\n\\begin{equation}\n\\left\\| \\frac{dH}{ds} \\right\\| = 2N \\sum_{\\mathbf{p} \\in \\Gamma} E^{(a)}_{\\mathbf{p}}(0) \\,.\n\\end{equation}\nFor large $L$, $\\sum_{\\mathbf{p} \\in \\Gamma} \\frac{1}{L}$ becomes well\napproximated by the integral $\\int_0^{2 \\pi\/a} d \\mathbf{p}$. Thus, using\n\\eq{eq:Epa}, we obtain \n\\begin{eqnarray}\n\\left\\| \\frac{dH}{ds} \\right\\| & \\simeq &\n2NL \\int_0^{2 \\pi\/a} d \\mathbf{p} E^{(a)}_{\\mathbf{p}}(0) \\\\\n& = & 2NL \\int_0^{2 \\pi\/a} d \\mathbf{p} \\sqrt{\\frac{4r^2}{a^2} \\sin^4 \\left(\n  \\frac{\\mathbf{p} a}{2} \\right) + \\frac{1}{a^2} \\sin^2 (\\mathbf{p} a)} \\\\\n& = & \\frac{2NL}{a^2} \\eta(r) \\,, \\label{finalderiv}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\eta(r) = \\int_0^{2 \\pi} d \\hat{p} \\sqrt{4 r^2 \\sin^4 \\Big(\n  \\frac{\\hat{p}}{2} \\Big) + \\sin^2 \\left( \\hat{p} \\right)} \\,.\n\\end{equation}\nWe can therefore substitute $\\frac{d^2 H}{ds^2} = 0$, $\\left\\|\n  \\frac{dH}{ds} \\right\\| = O(L a^{-2})$ and $\\gamma = m$ into\n  \\eq{adiabateq}. Theorem \\ref{adiabaticthm} then shows that we can\nprepare a state with distance no more than $\\epsilon_{\\mathrm{prep}}$\nfrom the exact state using\n\\begin{equation}\nT = O \\left( \\frac{L^2}{a^4 m^3 \\epsilon_{\\mathrm{prep}}} \\right).\n\\end{equation}\nNote that the adiabatic theorem applied here, though convenient because \nof its generality, may not yield a tight upper bound on the run time.\n\n\\subsection{Preparing the Interacting Vacuum}\n\\label{turnon}\n\nGiven the ground state of the free theory, we can prepare the ground\nstate of the interacting theory by adiabatically varying the parameters\n$g_0^2$ and $m_0$ in the massive Gross-Neveu Hamiltonian, starting from\n$g_0^2 = 0$. For adiabaticity to be maintained, the physical mass must\nnot vanish at any point in the adiabatic path. By \\sect{massren}, \nthe physical mass varies with $g_0^2$ according to\n\\begin{equation}\n\\label{two-orders}\nm = m_0 - c_1 g_0^2 - c_2 g_0^4 + O(g_0^6) \\,,\n\\end{equation}\nwhere $c_1,c_2>0$ are given by\n\\begin{eqnarray}\nc_1 & = & \\frac{m}{2\\pi} \\log\\Big(\\frac{1}{ma}\\Big) + \\cdots \\,, \\\\\nc_2 & \\simeq & \\frac{m}{16\\pi^3}\\big(9.3 N - 0.07\\big)\\log^2(ma)  + \\cdots\\,. \n\\label{c2}\n\\end{eqnarray}\n(The coefficients in \\eq{c2} were determined numerically.)\nThe vanishing of the physical mass marks the location of a quantum phase \ntransition, which cannot be adiabatically crossed. \nEquation \\eq{two-orders} indicates that the phase diagram takes the\nschematic form as shown in Fig.~\\ref{paths}.\n\nAs in \\sect{freeprep}, we parametrize our adiabatic state\npreparation by a quantity $s$, which increases over time from $0$ to\n$1$. In this second adiabatic process, the Hamiltonian is the full\nmassive Gross-Neveu Hamiltonian with $s$-dependent parameters\n$g_0^2(s)$ and $m_0(s)$. We choose $g_0^2(0) = 0$ and $m_0(0)=m$ so\nthat the initial Hamiltonian of this adiabatic process matches the\nfinal Hamiltonian of the preceding adiabatic step. Thus, the ground\nstate at $s=0$ is the free vacuum prepared in the previous step of the\nalgorithm. To keep our analysis simple, we choose a linear adiabatic\npath, namely,\n\\begin{eqnarray}\ng_0^2(s) & = & s g_0^2 \\,, \\nonumber \\\\\nm_0(s) & = & m + s \\delta_m \\,. \\label{linearpath}\n\\end{eqnarray}\nWe choose $\\delta_m$ so that the physical mass at $s=1$ is\nequal to the physical mass at $s=0$. To second order in\n$g_0^2$,\n\\begin{equation}\n\\label{deltam}\n\\delta_m = c_1 g_0^2 + c_2 g_0^4 + \\cdots \\,,\n\\end{equation}\nas illustrated in Fig.~\\ref{paths}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{paths.eps}\n\\caption{\\label{paths} Our perturbative calculations of the physical mass\n  in the massive Gross-Neveu model indicate a phase diagram with the\n  qualitative features illustrated above. The phase above the dashed\n  curve is accessible adiabatically from the free theory but the phase below\n  is not. The arrow depicts our linear adiabatic path, described in\n  \\eq{deltam}. Our perturbative analysis shows that the first two\n  derivatives of the phase transition curve with respect to $g_0^2$\n  are both positive and diverge only as $\\mathrm{poly}(\\log(m_0 a))$\n  in the limit $a \\to 0$.\n}\n\\end{center}\n\\end{figure}\n\nBy \\eq{linearpath}, $\\frac{d^2 H}{ds^2} = 0$ and \n\\begin{equation}\n\\label{deriv}\n\\frac{dH}{ds} = \\sum_{\\mathbf{x} \\in \\Omega} a \\Bigg[\n  \\delta_m \\bar{\\psi}_j(\\mathbf{x}) \\psi_j(\\mathbf{x}) + \\frac{g_0^2}{2} \\bigg( \\sum_{j=1}^N\n\\bar{\\psi}_j (\\mathbf{x}) \\psi_j(\\mathbf{x}) \\bigg)^2 \\Bigg] \\,.\n\\end{equation}\nFurthermore, the minimal eigenvalue gaps occur at $s=0$ and $s=1$ and\nare equal to the final physical mass $m$. Thus, to apply Theorem\n\\ref{adiabaticthm} we need only bound $\\left\\| \\frac{d H}{ds}\n\\right\\|$. \n\nWe can deduce the spectrum of $\\frac{dH}{ds}$ by the following\ntransformation:\n\\begin{eqnarray}\na_j(\\mathbf{x}) & = & \\frac{1}{\\sqrt{2}} \\big( \\psi_{j,0}(\\mathbf{x}) - i\n  \\psi_{j,1}(\\mathbf{x}) \\big) \\,,\\\\\nb_j^\\dag(\\mathbf{x}) & = & \\frac{1}{\\sqrt{2}} \\big( \\psi_{j,0}(\\mathbf{x}) + i\n  \\psi_{j,1}(\\mathbf{x}) \\big) \\,.\n\\end{eqnarray}\nThis corresponds to\n\\begin{equation}\n\\label{localtrans}\n\\psi_j(\\mathbf{x}) = \\frac{1}{\\sqrt{2m_0}} \\left( a_j(\\mathbf{x}) u(0) + b_j^\\dag(\\mathbf{x})\nv(0) \\right) \\,,\n\\end{equation}\nwhere $u,v$ are defined in \\eq{concrete}. Using \\eq{anticanon1} and\n\\eq{anticanon2}, one can verify that\n\\begin{eqnarray}\n\\{ a_j(\\mathbf{x}), a_k^\\dag(\\mathbf{y}) \\} = \\{ b_j(\\mathbf{x}), b_k^\\dag(\\mathbf{y}) \\} \n& = & a^{-1} \\delta_{j,k} \\delta_{\\mathbf{x}, \\mathbf{y}} \\mathds{1} \\,,\\\\\n\\{ a_j(\\mathbf{x}), a_k(\\mathbf{y}) \\} = \\{ b_j(\\mathbf{x}), b_k(\\mathbf{y}) \\} & = & 0 \\,,\\\\\n\\{ a_j(\\mathbf{x}), b_k(\\mathbf{y}) \\} = \\{a_j^\\dag(\\mathbf{x}), b_k(\\mathbf{y}) \\} & = & 0 \\,.\n\\end{eqnarray}\nThus, $a_j(\\mathbf{x}),a_j^\\dag(\\mathbf{x}),b_j(\\mathbf{x}),b_j^\\dag(\\mathbf{x})$ are\ncreation and annihilation operators for $2N$ species of fermions\nlocalized on the spatial lattice. By \\eq{localtrans},\n\\begin{equation}\n\\bar{\\psi}_j(\\mathbf{x}) \\psi_j(\\mathbf{x}) = a_j^\\dag(\\mathbf{x}) a_j(\\mathbf{x}) - b_j(\\mathbf{x})\nb_j^\\dag(\\mathbf{x}) \\,, \n\\end{equation}\nfrom which we obtain\n\\begin{equation}\n\\Bigg\\| \\sum_{j=1}^N \\bar{\\psi_j}(\\mathbf{x}) \\psi_j(\\mathbf{x}) \\Bigg\\| = 2Na^{-1},\n\\end{equation}\nand hence\n\\begin{equation}\n\\left\\| \\frac{dH}{ds} \\right\\| = \\delta_m 2N\\hat{L} + \\frac{2\n  \\hat{L} g_0^2 N^2}{a} \\,. \n\\end{equation}\n\nBy the results of \\sect{massren}, we find that $\\delta_m =\nO(\\log^2(ma))$. Hence, recalling that $\\hat{L} = L\/a$, we obtain\n\\begin{equation}\n\\left\\| \\frac{dH}{ds} \\right\\| = O \\left( \\frac{L}{a^2} \\right).\n\\end{equation} \nTherefore, by Theorem \\ref{adiabaticthm} the diabatic error is at most\n\\begin{eqnarray}\n\\epsilon & = & O \\left( \\frac{1}{T_{\\mathrm{turn-on}}} \\frac{ \\left\\|\n      \\frac{dH}{ds} \\right\\|^2}{\\gamma^3} \\right) \\\\\n& = & O \\left( \\frac{L^2}{T_{\\mathrm{turn-on}} a^4 m^3} \\right).\n\\end{eqnarray}\nIt thus suffices to choose\n\\begin{equation}\nT_{\\mathrm{turn-on}} = O \\left( \\frac{L^2}{a^4 \\epsilon m^3} \\right).\n\\end{equation}\n\nIn the above procedure, we choose our adiabatic path so that the\ninitial and final physical masses equal some user-specified value\n$m$. To achieve this, one needs to tune the quantity\n$\\delta_m$ in accordance with \\eq{linearpath} and \\eq{deltam}. For\nsufficiently weak coupling, the proper choice of $\\delta_m$ can be\ndetermined by the perturbative calculations performed in\n\\sect{massren}. In the strongly coupled case, these perturbative\ncalculations no longer provide precise guidance as to a choice of\n$\\delta_m$. Instead, as previously discussed in \\cite{longversion},\nthe adiabatic path can be determined by the quantum\ncomputer. Specifically, one can measure the physical mass at a given\ncoupling strength $g_0$ by exciting a particle and measuring energy\nvia phase estimation. This measurement guides the choice of a suitable\nadiabatic path to a slightly larger coupling strength, at which\npoint the mass can be measured again. Iterating this process, one\ncan reach any coupling strength for which the corresponding vacuum is\nin the same quantum phase as the free vacuum.\n\n\\subsection{Exciting Wavepackets}\n\\label{exciting}\n\nAfter preparing the interacting vacuum, $\\ket{\\mathrm{vac}}$, we\nexcite wavepackets by simulating a source that varies sinusoidally in\nspace and time so as to induce excitations of some particular total energy \nand momentum by resonance. Given the physical rest mass $m$\nof the particles, we can choose this energy and momentum so that the\nonly corresponding state is a single-particle state. (For a given total\nmomentum, an unbound state of two particles will have greater energy\nthan the corresponding state of one particle. In the ultrarelativistic\nlimit, $p \\gg m$, this energy difference scales as $m^2\/p$.)\nIn the remainder of this section, we show that, using a source of\nspatial extent $l$ and duration $\\tau$, one can ensure that\nexcitations off resonance are suppressed as $\\sim \\exp \\left[\n - \\frac{1}{4}  \\left( l^2 (\\mathbf{p}-\\mathbf{p}_0)^2 + \\tau^2 (E-E_0)^2 \\right)\n\\right]$. Hence, by simulating a process of duration $\\tau \\sim p\/m^2$\nand spatial extent $l \\sim p\/m^2$, one can control the incoming momentum \nand ensure that the probability of obtaining more than one particle is small.\n\nThe creation of two incoming particles has only an\n$O(\\epsilon)$ success probability, which can be compensated for by\nrepeated attempts. (See the discussion following \\eq{w}.) \nThe total complexity of preparing two particles is\nthe cost of simulating the time evolution given in \\eq{bigR} a total\nof $1\/\\epsilon$ times. Thus, by the results of \\sect{trotter},\nthe complexity is $\\big( \\frac{\\tau l}{a^2 \\epsilon}\n\\big)^{1+o(1)}$. Thus, since $p \\sim a^{-1}$ for fixed\n$\\epsilon$ and $a \\sim \\epsilon$ for fixed $p$, the number of quantum\ngates $G_{\\mathrm{excite}}$ needed to excite the two initial particles\nis\n\\begin{equation}\nG_{\\mathrm{excite}} \\sim \\left\\{ \\begin{array}{ll}\n\\epsilon^{-3-o(1)}\\,, & \\textrm{as} \\,\\,\\, \\epsilon \\to 0 \\,, \\\\\np^{4+o(1)} \\,, & \\textrm{as} \\,\\,\\, p \\to \\infty \\,.\n\\end{array} \\right.\n\\end{equation}\nNote also that for the initial wavepackets to be well separated, $L$\nmust be larger than $2l$. Hence, in the high-momentum limit $L \\sim\np$, which affects the complexity of other steps of the algorithm.\n\n\\subsubsection*{Perturbative Expansion}\n\\label{dyson}\n\nThe resonant excitation can be analyzed with time-dependent perturbation\ntheory. Let\n\\begin{equation}\n\\label{bigR}\nR = T \\left\\{ \\exp \\left[-i \\int_0^\\tau \\mathrm{d} t \n\\left( H+ \\lambda W(t) \\right) \\right] \\right\\} \\,,\n\\end{equation}\nwhere $T\\{ \\cdot \\}$ denotes the time-ordered product, \n$H$ is given by \\eq{h},\n\\begin{equation}\n\\label{Wt}\nW(t) = \\int \\mathrm{d} \\mathbf{x} \\left( f(t,\\mathbf{x})\\psi_{i,\\alpha}(\\mathbf{x}) +\n  f^*(t,\\mathbf{x})\\psi_{i,\\alpha}^\\dag(\\mathbf{x}) \\right),\n\\end{equation}\n$i$ and $\\alpha$ are chosen\naccording to the desired type of particle, and $f(t,\\mathbf{x})$ is an \noscillatory function whose form we optimize in the next\nsubsection. The end product of the excitation process is $R\n\\ket{\\mathrm{vac}}$. One can expand this quantity using the Dyson\nseries, as follows:\n\\begin{equation}\n\\label{Dyson}\nR = \\mathds{1} - i \\lambda \\int_0^\\tau \\mathrm{d} t_1 W_I(t_1) + (-i \\lambda)^2\n\\int_0^\\tau \\mathrm{d} t_1 \\int_0^{t_1} \\mathrm{d} t_2 W_I(t_1) W_I(t_2) + \\cdots \\,,\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{WI}\nW_I(t) = e^{iHt} W(t) e^{-iHt}\n\\end{equation}\nand the $n\\th$-order term in $\\lambda$ is\n\\begin{equation}\n(-i \\lambda)^n \\int_0^\\tau \\mathrm{d} t_1 \\ldots \\int_0^{t_{n-1}} d t_n\nW_I(t_1) \\ldots W_I(t_n) \\,.\n\\end{equation}\nThe total contribution from orders $\\lambda^2$ and higher is bounded by\n\\begin{eqnarray}\n\\left\\| \\sum_{n=2}^\\infty (-i \\lambda)^n \\int_0^\\tau \\mathrm{d} t_1\n  \\ldots\n\\int_0^{t_{n-1}} \\mathrm{d} t_n W_I(t_1) \\ldots W_I(t_n) \\right\\|\n& \\leq & \\sum_{n=2}^\\infty \\frac{\\lambda^n \\tau^n}{n!} w^n \\\\\n& = & \\exp[\\lambda \\tau w] - 1 - \\lambda \\tau w \\,,\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{w}\nw = \\max_{0 \\leq t \\leq \\tau} \\left\\| W(t) \\right\\|.\n\\end{equation}\n\nFrom the above analysis, one sees that the Dyson series converges\nrapidly. The single-particle excitation amplitude is of order\n$\\lambda$, and the dominant error, other than non-excitation, is the\ntwo-particle excitation amplitude, which is of order\n$\\lambda^2$. Setting the two-particle excitation probability to\n$\\epsilon$, one obtains a single-particle excitation with probability\n$p_1 \\sim \\sqrt{\\epsilon}$, and non-excitation with probability on the order of\n$1-\\sqrt{\\epsilon}$. In a standard scattering simulation, one wishes\nto prepare as an initial state single-particle excitations at two\nspatially separated locations. The fraction of simulations in which\nthis is achieved (rather than one or both particles failing to be produced) \nis thus of order $p_1^2 \\sim \\epsilon$. One\ncan detect such instances and compensate by repeating the\nsimulation $O(1\/p_1^2)$ times and postselecting the instances in\nwhich both particles were produced.\n\nNext, we consider the first-order excitation amplitude in more\ndetail. Let $\\ket{E,\\mathbf{p}}$ be any state with total momentum $\\mathbf{p}$ and\nenergy $E$ above the vacuum energy, so that  $P \\ket{E,\\mathbf{p}} = \\mathbf{p}\n\\ket{E,\\mathbf{p}}$ and $H \\ket{E,\\mathbf{p}} = E \\ket{E,\\mathbf{p}}$, where $P$ is the total\nmomentum operator. (Here, we rely on the fact that $[H,P] = 0$.) Then,\nto first order in $\\lambda$, by \\eq{Dyson} and \\eq{WI},\n\\begin{eqnarray}\n\\bra{E,\\mathbf{p}} R \\ket{\\mathrm{vac}} \n& \\simeq & - i \\lambda \\int_0^\\tau \\mathrm{d} t \\bra{E,\\mathbf{p}} W_I(t)\n\\ket{\\mathrm{vac}} \\\\\n& = & - i \\lambda \\int_0^\\tau \\mathrm{d} t \\, e^{-iEt} \\bra{E,\\mathbf{p}} W(t)\n\\ket{\\mathrm{vac}} \\,.\n\\end{eqnarray}\nRecalling that the momentum operator is the generator of spatial\ntranslations, one has $\\psi_{i,\\alpha}(\\mathbf{x}) = e^{iP\\mathbf{x}} \\psi_{i,\\alpha}(0)\ne^{-iP\\mathbf{x}}$. Thus, to first order in $\\lambda$,\n\\begin{equation}\n\\bra{E,\\mathbf{p}} R \\ket{\\mathrm{vac}} \n\\simeq - i \\lambda \\int_0^\\tau \\mathrm{d} t \\int\n\\mathrm{d} \\mathbf{x} e^{-i(Et+\\mathbf{p}\\mathbf{x})} \\left[ f(t,\\mathbf{x}) \\bra{E,\\mathbf{p}} \\psi_{i,\\alpha}(0)\n  \\ket{\\mathrm{vac}} + f^*(t,\\mathbf{x}) \\bra{E,\\mathbf{p}} \\psi^\\dag_{i,\\alpha}(0)\n  \\ket{\\mathrm{vac}} \\right] \\,.\\\\\n\\end{equation}\n(Here we have used $P \\ket{\\mathrm{vac}} = 0$.) Defining $f(t,\\mathbf{x}) = 0$\nfor $t \\notin [0,\\tau]$, we can extend the time integration to\ninfinity and express $\\bra{E,\\mathbf{p}} R \\ket{\\mathrm{vac}}$ in terms of\n$\\tilde{f}$, the Fourier transform of $f$. For our choice of $f$,\ngiven in the next subsection, $\\tilde{f}$ is real, and therefore\n\\begin{equation}\n\\label{amp}\n\\bra{E,\\mathbf{p}} R \\ket{\\mathrm{vac}} =  - i \\lambda \n\\left[ \\tilde{f}(E,\\mathbf{p}) \\bra{E,\\mathbf{p}} \\psi_{i,\\alpha}(0) \\ket{\\mathrm{vac}}\n+ \\tilde{f}(-E,-\\mathbf{p}) \\bra{E,\\mathbf{p}} \\psi_{i,\\alpha}^\\dag(0)\n\\ket{\\mathrm{vac}} \\right] + O(\\lambda^2).\n\\end{equation}\n\n\\subsubsection*{Wavepacket Shaping}\n\\label{shaping}\n\nWe now show that a Gaussian wavepacket is a good choice for $f(t,\\mathbf{x})$.\nSpecifically, for chosen constants $\\alpha, \\beta > 0$, let\n\\begin{equation}\nf(t,\\mathbf{x}) = \\left\\{ \\begin{array}{cl}\n\\eta \\exp \\left[-(\\alpha t)^2 - (\\beta \\mathbf{x})^2 - i E_0 t + i \\mathbf{p}_0 \\mathbf{x} \\right] \\,, &\n-\\tau\/2 \\leq t \\leq \\tau\/2, -l\/2 \\leq \\mathbf{x} \\leq l\/2 \\,, \\\\\n0 \\,, & \\textrm{otherwise} \\,.\n\\end{array} \\right.\n\\end{equation}\n(For convenience, we have shifted the origin of the coordinate\nsystem.) Here $\\eta$ is a normalization factor\\footnote{It is \n  reasonable to choose $\\eta$ so that $\\int_0^{\\tau} dt W_I(t)\n  \\ket{\\mathrm{vac}}$ is a normalized state. In the ultrarelativistic\n  limit this implies that $\\eta \\sim \\left( \\alpha^2\n    \\beta^4 + \\alpha^4 \\beta^2 \\right)^{1\/4}$.} with mass dimension $3\/2$. \nWith this choice of $f$,\n\\begin{equation}\n\\label{1peak}\n\\tilde{f}(E,\\mathbf{p}) =  \\eta q_{\\beta,l}(\\mathbf{p}-\\mathbf{p}_0) q_{\\alpha,\\tau}(E-E_0) \\,,\n\\end{equation}\nwhere\n\\begin{equation}\nq_{\\rho,r}(d) = \\int_{-r\/2}^{r\/2} \\mathrm{d} \\mathbf{x} \\, e^{id\\mathbf{x}-(\\rho \\mathbf{x})^2}.\n\\end{equation}\nIn the limit $r \\to \\infty$, the function $q_{\\rho,r}(d)$ converges to a\nGaussian peak of width $\\sim 1\/\\rho$. Since $E$ must be positive, the\n$\\tilde{f}(-E,-\\mathbf{p}) \\bra{E,\\mathbf{p}} \\psi^\\dag_{i,\\alpha}(0)\n\\ket{\\mathrm{vac}}$ term in \\eq{amp} is exponentially small. Hence,\none obtains\n\\begin{equation}\n\\label{amp2}\n\\bra{E,\\mathbf{p}} R \\ket{\\mathrm{vac}} \\simeq  - i \\lambda \n\\tilde{f}(E,\\mathbf{p}) \\bra{E,\\mathbf{p}} \\psi_{i,\\alpha}(0) \\ket{\\mathrm{vac}}.\n\\end{equation}\nfor $E \\gg 1\/\\tau$ and $\\lambda \\ll 1$. By \\eq{amp2} and \\eq{psij}, one\nsees that $R \\ket{\\mathrm{vac}}$ is a antifermion wavepacket with\nmomentum centered around $\\mathbf{p}$. To create a fermion, one interchanges\n$\\psi$ and $\\psi^\\dag$ in \\eq{Wt}.\n\nUsing the asymptotics of error functions, we can furthermore bound the\ncontributions due to $r$ being finite. One finds that\n\\begin{equation}\n\\big|q_{\\rho,r}(d) - q_{\\rho,\\infty}(d)\\big|  \\leq  \\frac{2}{r \\rho^2}\ne^{-(\\rho r)^2\/4}.\n\\end{equation}\n\n\\subsection{Measuring Number Operators}\n\\label{measurements}\n\nRecall from \\sect{rep} that the free theory ($g_0^2 = 0$) \nis exactly solvable, with the number operators\n$L^{-1} a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p})$ counting fermions of species\n$j$ in momentum-mode $\\mathbf{p}$ and $L^{-1} b_j^\\dag(\\mathbf{p})\nb_j(\\mathbf{p})$ similarly counting antifermions. \nThus, as one possible set of measurements to perform on the final state of \nthe simulation, we propose, as in \\cite{longversion}, adiabatically returning \nto the free theory and then measuring number operators via the \nphase-estimation algorithm. \nWe analyze this measurement procedure in this section. \nAn alternative set of measurements that is more suitable when bound states\nare present is analyzed in \\sect{sec:charge}.\n\n\nThe adiabatic return to the free theory is performed in the presence\nof particle wavepackets, so the state being adiabatically\ntransformed is not an energy eigenstate. Different energy eigenstates\nin the superposition will acquire different dynamical phases\nduring the adiabatic process and thus, in physical terms, the simulated\nparticles will propagate. Such propagation is undesirable because we do \nnot want any scattering to occur while the interaction is being turned off. \n\nHence, we apply the same technique proposed in \\cite{longversion} to\nsuppress particle propagation: we interleave (simulated) backwards\ntime evolutions governed by time-independent Hamiltonians into the\nadiabatic process. By an analysis similar to that performed in\n\\cite{longversion}, one finds that, to ensure that a particle\npropagates no further than a distance $\\mathcal{D}$, it suffices to use\n\\begin{equation}\nJ = \\widetilde{O} \\left( \\frac{\\sqrt{\\tau}}{p \\mathcal{D}} \\right)\n\\end{equation}\nbackwards evolutions, where $\\tau$ is the duration of the original\nadiabatic process and $p$ is the momentum of the particle. Further,\none finds that the total probability of diabatically exciting one or\nmore particles is\\footnote{This result is based on the\n  adiabatic criterion of \\cite{Messiah} which\n  appears to be applicable \\cite{longversion} to our Hamiltonian \n  although it may not apply to all Hamiltonians.}\n\\begin{equation}\nP_{\\mathrm{diabatic}} = O\\left( \\frac{J^2 L p^2}{\\tau^2} \\right).\n\\end{equation}\nHence, setting $\\mathcal{D}$ to a constant\n$P_{\\mathrm{diabatic}}$ to  $\\epsilon$, one obtains\n\\begin{equation}\n\\tau = \\widetilde{O} \\left( \\frac{L}{\\epsilon} \\right). \n\\end{equation}\nA process of this duration can\nbe implemented with (\\sect{trotter})\n\\begin{equation}\nG_{\\mathrm{turn-off}} = O \\left( \\left(\n    \\frac{L^2}{a \\epsilon} \\right)^{1+o(1)} \\right)\n\\end{equation}\nquantum gates.\n\nThe phase-estimation algorithm \\cite{Kitaev95} enables one to\nmeasure in the eigenbasis of $L^{-1} a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p})$,\nprovided one can efficiently implement $e^{-i L^{-1} a_j^\\dag(\\mathbf{p})\n  a_j(\\mathbf{p}) t}$ for various $t$ using quantum circuits. By\n\\eq{adef} and \\eq{bdef}, one sees that the problems of simulating \n$e^{-i L^{-1} a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p}) t}$ and its\nantifermionic counterpart are largely similar to the problem of\nsimulating the time evolution $e^{-iHt}$, which was analyzed in detail\nin \\sect{trotter}. However, these number operators are spatially\nnonlocal, which means that the methods of \\sect{trotter} do not\nperform well as a function of $\\hat{L}$. Instead, it is more\nefficient to use recent techniques from \\cite{BCCKS}. \n\nIn \\cite{BCCKS}, a method is described for simulating sparse\nHamiltonians in which the matrix elements are given by an oracle. As\ndiscussed on pg. 2 of \\cite{BCCKS}, in the case where the sparse\nHamiltonian consists of a sum of $d$ terms each acting on $O(1)$ qubits,\nthe number of oracle queries and non-oracle-related quantum gates both\nscale as $O(d)$. A number operator for a momentum mode consists of\n$O(\\hat{L}^2)$ terms, acting between all pairs of spatial lattice sites.\nThus, if one ignored the fermionic statistics, the number of non-oracle-related\ngates needed to simulate the time-evolution induced by a number operator\nwould be $O(\\hat{L}^2 n) = O(\\hat{L}^3)$. The number of gates needed to\nimplement one oracle query to the sparse matrix defined by the number\noperator would be $O(n)$, and number of quantum gates needed to implement\nall of the oracle queries would be $O(\\hat{L}^3)$. Using the Bravyi-Kitaev\nencoding for fermionic statistics adds a logarithmic factor to the\ncomplexity. Measuring all $2N\\hat{L}$ of the number operators thus has total\ncomplexity $\\widetilde{O}(\\hat{L}^4) = \\widetilde{O}(L^4\/a^4)$.\n\n\\subsection{Measuring Local Charge}\n\\label{sec:charge}\n\nIn previous work \\cite{longversion}, we proposed measuring local\nenergy observables as an alternative to returning to the free theory\nand measuring number operators. This procedure has the advantage that\nit can detect bound states. It has the disadvantage that the local\nenergy observables have ultraviolet-divergent vacuum fluctuations \nthat represent a noise background above which particle excitations must be\ndiscerned.\nIn this paper, we instead measure simpler local observables, namely \ncharges, whose vacuum fluctuations are less difficult to\ncontrol. These observables can thus detect charged bound states,\nalthough they are blind to neutral ones.\n\nFrom the equation of motion of the massive Gross-Neveu model, one finds \nthat for each $j \\in \\{1,2,\\ldots,N\\}$ the\nquantity\n\\begin{equation}\nJ^\\mu_j(x) = \\bar{\\psi}_j(x) \\gamma^\\mu \\psi_j(x)\n\\end{equation}\nobeys\n\\begin{equation}\n\\partial_\\mu J_j^\\mu = 0.\n\\end{equation}\nHence,\n\\begin{equation}\n\\widetilde{Q}_j \\equiv \\sum_{\\mathbf{x}} J_j^0(\\mathbf{x}) = \\sum_{\\mathbf{x}}\n\\bar{\\psi}_j(\\mathbf{x}) \\gamma^0 \\psi_j(\\mathbf{x})\n\\end{equation}\nis a conserved charge. Note that, for any \n$b,c \\in \\mathbb{R}$, $Q_j = b \\widetilde{Q}_j + c$ is also\nconserved. We can calibrate the charge observable by demanding that\nthe vacuum have zero charge and that particle creation change the\ncharge by $\\pm 1$. One satisfies these criteria with the following\ndefinition:\n\\begin{equation}\nQ_j = \\sum_{\\mathbf{x} \\in \\Omega} a \\bar{\\psi}_j(\\mathbf{x}) \\gamma^0\n\\psi_j(\\mathbf{x}) - \\hat{L} \\mathds{1} \\,.\n\\end{equation}\nBy (\\ref{psij}), (\\ref{psidagj}), and (\\ref{anticanonp2}), one finds that\n\\begin{equation}\nQ_j = \\frac{1}{L} \\sum_{\\mathbf{p} \\in \\Gamma} \\left(\n  a_j^\\dag(\\mathbf{p}) a_j(\\mathbf{p}) - b_j^\\dag(\\mathbf{p})\n  b_j(\\mathbf{p})\\right).\n\\end{equation}\nFor any envelope function $f:\\Omega \\to [0,1]$, one can similarly\ndefine\n\\begin{equation}\nQ_j^{(f)} =  \\sum_{\\mathbf{x} \\in \\Omega} f(\\mathbf{x}) \\left( a\n  \\bar{\\psi}_j(\\mathbf{x}) \\gamma^0 \\psi_j(\\mathbf{x}) - \\mathds{1} \\right).\n\\end{equation}\nIf $f$ has support only in some region $R \\subset \\Omega$, then\n$Q_j^{(f)}$ can be thought of as an observable for the charge in that\nregion. \n\nThe most obvious choice of $f$ is a square function that is\nequal to one inside $R$ and zero elsewhere. However,\na better signal-to-noise ratio can be obtained by choosing $f$ to\ndecay from one to zero more smoothly at the boundary of $R$.\nSpecifically, calculations (in Appendix \\ref{fluctuations}) show that, \nwhen $f$ is chosen to be a\nGaussian of width $R$, the variance of the observable $Q_j^{(f)}$ in\nthe vacuum state is $O(1\/mR)$, independent of the lattice spacing $a$. \nHence the noise background above which particle excitations are to be\ndetected is nondivergent in $a$ and can be brought to an arbitrarily\nlow level at the cost of increasing the detector size. \nIn practice, one will use a truncated Gaussian, replacing the\nexponentially small tails with zero at distances greater than some constant\nmultiple of $R$. This modified $f$ then has support on a region of\nsize $O(R)$, but the corresponding operator is exponentially close to\nthe Gaussian case treated by our analysis.\n\n$Q_j^{(f)}$ has eigenvalues with $O(1)$ separations. Thus, measuring\n$Q_j^{(f)}$ by phase estimation entails simulating the unitary\ntransformation $\\exp\\big[i Q_j^{(f)} t\\big]$ for $t$ of order one. Because\n$Q_j^{(f)}$ is the sum of local terms, these unitary\ntransformations can be implemented by techniques similar to those\nin \\sect{trotter} with complexity $O(a^{-1-o(1)}\n\\epsilon^{-o(1)})$.\n\n\n\\section{Some Field-Theoretical Aspects}\n\nThis section describes some quantum field-theoretical calculations:\nanalysis of the effect of discretizing the spatial dimension of the \nmassive Gross-Neveu model, and the perturbative renormalization of \nthe mass in the discretized theory.\n\nIn our complexity analysis (\\sect{sec:complexity}), our criterion for\nchoosing the lattice spacing $a$ is that the scattering cross sections \nfor processes at a momentum scale $p$ in the discretized theory should \ndiffer from their continuum values by at most a factor of $(1+\\epsilon)$. \nThe results of \\sect{EFT} show that one can satisfy this criterion \nby choosing $a \\sim \\epsilon\/p$.\nThis choice then affects the overall scaling of the algorithm in the\nlarge-momentum and high-precision limits. As one would expect,\nhigher energies and greater precision require a smaller lattice spacing\nand thus a larger number of lattice sites (for fixed $L$).  \nConsequently, the number of quantum gates needed to simulate\ntime evolutions via Suzuki-Trotter formulae is larger.\n\nIn \\sect{massren}, we perturbatively calculate the relationship\nbetween the bare mass $m_0$, which is a parameter in the\nlattice Hamiltonian (see \\eq{h} and \\eq{h0}), and the physical mass\n$m$ of the particles in the theory. We need to know the behavior of\n$m$ in order to design and analyze the procedure for preparing the\ninteracting vacuum (\\sect{turnon}).  In particular, a suitable\nadiabatic path must maintain a non-zero mass, the magnitude of which\naffects the algorithmic complexity, as indicated by the adiabatic\ntheorem.\n\n\n\n\\subsection{Effects of Non-zero Lattice Spacing}\n\\label{EFT}\n\nThe effects of a non-zero lattice spacing can be analyzed via effective\nfield theory. The discretized Lagrangian can be thought of as the leading \ncontribution to an effective field theory, neglected terms of which \ncorrespond to discretization errors. Hence, the scaling of the error \nwith the lattice spacing is given by the scaling of the coefficients of \nthose terms. \n\nThe symmetries of the continuum theory restrict the possible operators \nin the effective field theory.\nConsider the discrete transformations parity (denoted $P$), \ntime reversal ($T$), and charge conjugation ($C$).\nParity changes the handedness of space and hence reverses the momentum.\nThus,\n\\begin{equation}\nP a(\\mathbf{p}) P = a(-\\mathbf{p}) \\,,\\qquad\nP b(\\mathbf{p}) P = -b(-\\mathbf{p}) \\,.\n   \\label{eq:P}\n\\end{equation}\nUsing \\eq{eq:psi} and \\eq{eq:P}, we then obtain\n\\begin{equation}\nP \\psi(t,\\mathbf{x}) P = \\gamma^0\\psi(t,-\\mathbf{x}) \\,,\\qquad\nP \\bar{\\psi}(t,\\mathbf{x}) P = \\bar{\\psi}(t,-\\mathbf{x})\\gamma^0 \\,.\n\\end{equation}\nLikewise, \n\\begin{equation}\nT a(\\mathbf{p}) T = a(-\\mathbf{p}) \\,,\\qquad\nT b(\\mathbf{p}) T = -b(-\\mathbf{p}) \\,.\n\\end{equation}\nIt turns out that time reversal needs to be an antilinear operator.\nThen\n\\begin{equation}\nT \\psi(t,\\mathbf{x}) T = \\gamma^1\\psi(-t,\\mathbf{x}) \\,,\\qquad\nT \\bar{\\psi}(t,\\mathbf{x}) T = -\\bar{\\psi}(-t,\\mathbf{x})\\gamma^1 \\,.\n\\end{equation}\nFinally, charge conjugation interchanges particles and antiparticles.\nThus,\n\\begin{equation}\nC a(\\mathbf{p}) C = b(\\mathbf{p}) \\,,\\qquad\nC b(\\mathbf{p}) C = a(\\mathbf{p}) \\,,\n\\end{equation}\nand\n\\begin{equation}\nC \\psi(t,\\mathbf{x}) C = \\psi^*(t,\\mathbf{x}) \\,,\\qquad\nC \\bar{\\psi}(t,\\mathbf{x}) C = \\psi^T(t,\\mathbf{x})\\gamma^0 \\,.\n\\end{equation}\nOne can verify that the Lagrangian (\\ref{eq:MGN}) is invariant under\neach of the transformations $P$, $T$ and $C$.\n\nNow consider the operator $\\psi^\\dagger \\mathbb{M} \\psi$, where\n$\\mathbb{M}$ is Hermitian. \nInvariance under $P$, $T$ and $C$ requires\n\\begin{eqnarray}\n\\mathbb{M} & = & \\gamma^0 \\mathbb{M} \\gamma^0 \\,, \\\\\n\\mathbb{M} & = & -\\gamma^1 \\mathbb{M}^* \\gamma^1 \\,, \\\\\n\\mathbb{M} & = & - \\mathbb{M}^T \\,. \n\\end{eqnarray}\nThese conditions imply that\n\\begin{equation}\n\\mathbb{M} = c \\gamma^0\\,, \\,\\, c \\in \\mathbb{R} \\,.\n\\end{equation}\nLikewise, for $i\\psi^\\dagger \\mathbb{M} \\partial_\\mu \\psi$,\nwhere $\\mathbb{M}$ is Hermitian, $P$, $T$ and $C$ invariance requires\n\\begin{eqnarray}\n\\mathbb{M} & = & (-1)^\\mu \\gamma^0 \\mathbb{M} \\gamma^0 \\,, \\\\\n\\mathbb{M} & = & -(-1)^\\mu \\gamma^1 \\mathbb{M}^* \\gamma^1 \\,, \\\\\n\\mathbb{M} & = & \\mathbb{M}^T \\,. \n\\end{eqnarray}\nThese conditions imply that, for $\\mu=0$,\n\\begin{equation}\n\\mathbb{M} = c \\mathds{1} = c (\\gamma^0)^2 \\,, \\,\\, c \\in \\mathbb{R} \\,,\n\\end{equation}\nwhile, for $\\mu=1$,\n\\begin{equation}\n\\mathbb{M} = c \\gamma^5 = -c\\gamma^0\\gamma^1 \\,, \\,\\, c \\in \\mathbb{R} \\,.\n\\end{equation}\nThus, the only $P$-, $T$- and $C$-invariant bilinears of Dirac fields are\n$\\bar{\\psi}\\psi$ and $i\\bar{\\psi}\\gamma^\\mu\\partial_\\mu \\psi$\n($\\mu=0$ or $1$).\n\nNow consider four-fermion operators, namely, products of two bilinears.\nThe set $\\{\\mathds{1},\\sigma^i\\}$ forms a complete basis, elements of which\nsatisfy the identity\n\\begin{eqnarray}\n\\delta_{\\alpha\\beta} \\delta_{\\gamma\\delta} \n& = & \\frac{1}{2}\\big(\\delta_{\\alpha\\delta} \\delta_{\\gamma\\beta} \n+ \\sum_{i=1}^{3} \\sigma^i_{\\alpha\\delta} \\sigma^i_{\\gamma\\beta}\\big)\n\\,.\n\\end{eqnarray}\nFor \n$\\gamma^0 = \\sigma^2$, $\\gamma^1 = -i\\sigma^1$, $\\gamma^5 = \\sigma^3$,\nthis is equivalent to\n\\begin{eqnarray}\n\\delta_{\\alpha\\beta} \\delta_{\\gamma\\delta} \n& = & \\frac{1}{2}(\\delta_{\\alpha\\delta} \\delta_{\\gamma\\beta} \n+ (\\gamma^\\mu)_{\\alpha\\delta} (\\gamma_\\mu)_{\\gamma\\beta}\n+ (\\gamma^5)_{\\alpha\\delta} (\\gamma^5)_{\\gamma\\beta})\n\\,.\n   \\label{eq:Fierz0}\n\\end{eqnarray}\nEquation~(\\ref{eq:Fierz0}) can be used to obtain Fierz identities.\nFor example,\n\\begin{eqnarray}\n\\bar{\\psi}_i\\psi_j \\bar{\\psi}_j\\psi_i \n& = & (\\bar{\\psi}_i)_\\alpha(\\psi_j)_\\beta \n(\\bar{\\psi}_j)_\\gamma(\\psi_i)_\\delta \\delta_{\\alpha\\beta}\\delta_{\\gamma\\delta} \n\\nonumber \\\\\n& = & -\\frac{1}{2} \\big(\\bar{\\psi}_i\\psi_i \\bar{\\psi}_j\\psi_j\n+ \\bar{\\psi}_i\\gamma^\\mu\\psi_i \\bar{\\psi}_j\\gamma_\\mu\\psi_j\n+ \\bar{\\psi}_i\\gamma^5\\psi_i \\bar{\\psi}_j\\gamma^5\\psi_j \\big)\\,,\n\\end{eqnarray}\nwhere the minus sign comes from fermion anticommutation.\nThus, any operator of the form \n$\\bar{\\psi}_i\\tilde\\Gamma_1\\psi_j \\bar{\\psi}_j\\tilde\\Gamma_2\\psi_i$ \ncan be rewritten as a sum of operators of the form \n$\\bar{\\psi}_i\\Gamma_1\\psi_i \\bar{\\psi}_j\\Gamma_2\\psi_j$,\nwith $\\Gamma_{1,2} \\in \\{\\mathds{1}, \\gamma^\\mu, \\gamma^5 \\}$.\n\nIf $\\Gamma_1 \\neq \\Gamma_2$, then \n$\\bar{\\psi}_i\\Gamma_1\\psi_i \\bar{\\psi}_j\\Gamma_2\\psi_j$ will violate\nat least one of the discrete symmetries.\nFurthermore, the $O(N)$ symmetry\\footnote{In fact, the massive Gross-Neveu \nmodel has an $O(2N)$ symmetry. \n} \nassociated with the $N$ fermion species restricts \nthe allowed form of operators to functions of \n$\\sum_{i=1}^N \\bar{\\psi}_i \\Gamma \\psi_i$.\nFor $i\\neq j$, $\\bar{\\psi}_i\\gamma^5\\psi_i \\bar{\\psi}_j\\gamma^5\\psi_j$\nis ruled out by invariance under $P$ (or $C$) of any single\nfield $\\psi_i$, and thus \n$\\big(\\sum_{i=1}^N \\bar{\\psi}_i \\gamma^5 \\psi_i\\big)^2$ is ruled out. \nLikewise, \n$\\bar{\\psi}_i\\gamma^\\mu\\psi_i \\bar{\\psi}_j\\gamma_\\mu\\psi_j$ ($i\\neq j$)\nand consequently\n$\\big(\\sum_{i=1}^N \\bar{\\psi}_i \\gamma^5 \\psi_i\\big)^2$ \nare ruled out.\n\nWe conclude that the only four-fermion operator (without derivatives) \nin the effective field theory is \n$(\\sum_{i=1}^N \\bar{\\psi}_i \\psi_i)^2$.\n\nEach extra derivative or factor of $\\bar{\\psi}\\Gamma\\psi$ in an operator \nwill increase its mass dimension by one; correspondingly, it will be \nsuppressed by an extra power of $a$.\nWe therefore conclude that no new unsuppressed operators are\ninduced in the effective field theory.\nThe spatial derivative in the continuum theory is approximated by a \ndifference operator, with an error of order $a$, and the Wilson term is\nalso formally of order $a$.\nSpatial discretization errors are hence of order $a$.\n\n\n\n\\subsection{Mass Renormalization}\n\\label{massren}\n\nIn this subsection, we calculate the renormalized (or physical) mass of the\ndiscretized theory, using second-order perturbation theory. \nA convenient way to obtain a suitable expression is to use a partially \nrenormalized form of perturbation theory (as was done in \\cite{longversion}), \nin which one uses the bare coupling but the renormalized mass. \n\nTo perform perturbative calculations, we need the Feynman rules\nfor the discretized theory.\nThe propagator is\n\\begin{equation}\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{fermiprop.eps} \n\\end{array} \n= \n\\frac{\\gamma^\\mu\\tilde{p}_\\mu+\\widetilde{m}(p)}{\\tilde{p}^2-\\widetilde{m}(p)^2} \n\\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\tilde{p}^\\mu = \\left( p^0,\\frac{1}{a}\\sin(a p^1) \\right) ,\\qquad \n\\widetilde{m}(p) = m + \\frac{2r}{a} \\sin^2\\left(\\frac{a p^1}{2}\\right)\n\\,.\n\\end{equation}\nFor convenience, we use the standard technique of introducing an auxiliary\nfield $\\sigma$ and rewrite the Lagrangian as\n\\begin{equation}\n{\\cal L} = {\\cal L}_0 + {\\cal L}_\\sigma \\,,\n\\end{equation}\nwhere ${\\cal L}_0$ is the discretized free Lagrangian and\n\\begin{equation} \n {\\cal L}_\\sigma = -\\frac{1}{2} \\sigma^2 - g \\sigma \\bar{\\psi}_j \\psi_j\n\\,.\n\\end{equation}\nThe corresponding Feynman rules are\n\\begin{equation} \\label{y}\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{scalarprop2.eps} \n\\end{array} \n = -i \\,,\\qquad\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{y.eps} \n\\end{array} \n = -ig \\,.\n\\end{equation}\n\nAt one-loop order,\n\\begin{eqnarray}\n-i M(p)\n& = & \n\\begin{array}{l} \\includegraphics[width=0.6in]{dia1b.eps} \n\\end{array} \n+\n\\begin{array}{l} \\includegraphics[width=0.6in]{countercircle.eps} \n\\end{array} \n\\,,\n  \\label{eq:diags}\n\\end{eqnarray}\nwhere the second diagram is the counterterm.\n\nThe first diagram gives\n\\begin{eqnarray}\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{dia1b.eps} \n\\end{array} \n& = & -g_0^2 \\int_{-\\infty}^{\\infty} \\frac{dk^0}{2\\pi} \n\\int_{-\\pi\/a}^{\\pi\/a} \\frac{dk^1}{2\\pi}\n\\frac{\\gamma^\\mu\\tilde{k}_\\mu+\\widetilde{m}(k)}{\\tilde{k}^2-\\widetilde{m}(k)^2} \n\\\\\n& = & \\frac{ig_0^2}{4\\pi a} \\int_{-\\pi}^{\\pi} dk^1\n\\frac{ma + 2r \\sin^2\\big(\\frac{k^1}{2}\\big)}{\n\\sqrt{ \\sin^2 k^1 + \\big( ma + 2r \\sin^2\\big(\\frac{k^1}{2}\\big) \\big)^2}\n}\n\\,.\n\\label{eq:m1}\n\\end{eqnarray}\nThe term in \\eq{eq:m1} proportional to $r$ scales as $1\/a$ and gives the \nmass correction to the doubler (spurious fermion). The term proportional to \n$m$ gives the following:\n\\begin{equation}\nm_0 = m - \\frac{g_0^2 m}{2\\pi} \\log(ma) + \\cdots \\,.\n\\end{equation}\n\nAt two-loop order, the 1PI amplitude has the additional contributions\n\\begin{eqnarray}\n\\begin{array}{l} \\includegraphics[width=0.6in]{rainbow3.eps}\n\\end{array}\n+\n\\begin{array}{l} \\includegraphics[width=0.6in]{countersun3.eps} \n\\end{array} \n+\n\\begin{array}{l} \\includegraphics[width=0.6in]{dia2b.eps}\n\\end{array}\n+\n\\begin{array}{l} \\includegraphics[width=0.6in]{dubub2.eps} \n\\end{array} \n\\,.\n\\nonumber\n  \\label{eq:diags2}\n\\end{eqnarray}\nThe renormalization condition satisfied at first order implies that\nthe first two diagrams cancel.\n\n\n\nThe last two diagrams give\n\\begin{equation}\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{dia2b.eps} \n\\end{array} \n = -\\frac{ig_0^4}{16\\pi^3}\\left(m I_1^{(a)} + \\frac{1}{a} I_1^{(b)}\\right)\n\\end{equation}\nand\n\\begin{equation}\n\\begin{array}{l}\n\\includegraphics[width=0.6in]{dubub2.eps} \n\\end{array} \n = \\,\\frac{ig_0^4 N}{16\\pi^3}\\left(m I_2^{(a)} + \\frac{1}{a} I_2^{(b)}\\right) \n\\,,\n\\end{equation}\nwhere $I_1^{(a)}$, $I_1^{(b)}$, $I_2^{(a)}$ and $I_2^{(b)}$ are given\nin Appendix \\ref{integrals}. Numerical evaluation of these integrals\nreveals the forms\n\\begin{eqnarray}\nI_{i}^{(b)} & = & c^{(b1)} - c^{(b2)} \\, m a + \\cdots \\,, \\\\\nI_{i}^{(a)} & = & c_{i}^{(a1)} \\log^2(ma) - c_{i}^{(a2)} \\log(ma)\n+ c_{i}^{(a3)} + \\cdots \\,, \n\\end{eqnarray}\nwith $c_i^{(j)} > 0$.\nWe thus obtain\n\\begin{equation}\nm = m^{(1)} - \\frac{g_0^4m^{(1)}}{16\\pi^3}\\big(N c_2^{(a1)}-c_1^{(a1)}\\big) \n\\log^2(m^{(1)}a)\n+ \\cdots \\,,\n\\end{equation}\nwhere $m^{(1)}$ denotes the physical mass at one-loop\norder. \n\n\\bigskip\n\\bigskip\n\n\\noindent \\textbf{Acknowledgments:}\nWe thank William George for help with numerical calculations. This\nwork was supported by NSF grant PHY-0803371, DOE grant\nDE-FG03-92-ER40701, and NSA\/ARO grant W911NF-09-1-0442. IQC and\nPerimeter Institute are supported in part by the Government of Canada\nthrough Industry Canada and by the Province of Ontario through the\nMinistry of Research and Innovation. The Institute for Quantum\nInformation and Matter (IQIM) is an NSF physics Frontiers Center with\nsupport from the Gordon and Betty Moore Foundation. S.J. and K.L. are\ngrateful for the hospitality of the IQIM (formerly IQI), Caltech,\nduring parts of this work. Portions of this work are a contribution of\nNIST, an agency of the US Government, and are not subject to US\ncopyright.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, there has been growing interest in the electronic and photonic systems displaying\na dispersionless flat band in the band structure \\cite{flach1,tang,im3}.\nFor the modes belonging to the flat band, the particle energy or wave frequency does not depend on the momentum or wave vector and the group velocity vanishes.\nThis has a strong effect on the behavior of quasiparticles and waves and can cause many interesting phenomena by greatly amplifying\nthe effects of various perturbations such as interactions and disorder \\cite{liu,der,goda,shukla,ley,luck}.\nExamples include superconductivity in magic-angle twisted bilayer graphene, flat-band ferromagnetism, and anomalous Landau levels \\cite{cao,lieb,mielke1,mielke2,tasaki1,pons,balents,im1}.\n\nMany models having one or more flat bands have been studied theoretically.\nTwo-dimensional (2D) lattices such as the Lieb, dice, and kagome lattices and one-dimensional (1D) lattices\nsuch as the stub, sawtooth, and diamond lattices are among the examples \\cite{dora,vic,flach,mizo}.\nThe low-energy physics of the aforementioned 2D lattices can be described by two Dirac cones intersected by a flat band\nand modeled by the pseudospin-1 Dirac equation in 2D \\cite{shen,urban,ocam,chan,kima}.\nThere also have been many recent attempts to realize flat-band systems experimentally \\cite{vicen,muk,zong,bab,slot,xie}.\n\nIn this paper, we show that there exists another interesting phenomenon which has avoided\nthe attention of researchers until now, though it should occur generically in all flat-band systems unless forbidden by symmetry.\nIn plasma physics, the phenomenon termed (somewhat ambiguously) as mode conversion has been known for a long time and has played\na crucial role in explaining a variety of processes including the heating of solar corona and fusion plasmas and\nthe sudden appearance or disappearance of specific wave modes in space plasmas \\cite{swan,mjo,hink,pp1,pp2,ehkim,ehkim2,yu1,yu2,jkps}. The simplest example is as follows.\nIn an inhomogeneous unmagnetized plasma where the plasma density $n$ varies smoothly along the $z$ direction,\nthe plasma frequency $\\omega_p$ ($=\\sqrt{4\\pi ne^2\/m}$), where $m$ and $e$ are the mass and charge of an electron, is also a function of $z$.\nLet us consider a situation where a $p$-polarized electromagnetic (EM) wave of frequency $\\omega$ is obliquely incident on this plasma and propagates within it.\nIf there exists a resonant region where $\\omega$ is matched to the local plasma frequency, then the local dielectric permittivity\nvanishes and the transverse wave excites\na longitudinal plasma oscillation there. Since the group velocity for the plasma oscillation mode is zero, the energy of the incident\nwave is continuously converted into that of the plasma oscillation mode and is accumulated at the resonant region. Ultimately this energy will be dissipated as heat\nand contribute to the heating of the plasma.\n\nWe point out that the plasma oscillation mode is an example of flat band. In an inhomogeneous plasma where this band crosses\nthe dispersive band describing EM waves, the energy can flow from the (fast) dispersive mode to the (slow) flat-band mode.\nWe will demonstrate that a precisely analogous phenomenon occurs in the systems described by the pseudospin-1 Dirac equation.\nFurthermore, we will\nshow that similar phenomena take place in many other systems with flat bands\nincluding pseudospin-2 Dirac systems, continuum models derived for 1D stub and sawtooth lattices, and a 2D model with a nearly flat band.\n\n\n\\section{Pseudospin-1 Dirac equation}\n\\label{sec_sp1}\n\n\\begin{figure}\n\\centering\\includegraphics[width=6cm]{fig1.eps}\n\\caption{Sketch of the configuration considered in Sec.~\\ref{sec_sp1}. A plane wave is incident at an angle $\\theta$ from the region\n$x>L$ where $U=0$ onto the nonuniform region in $0\\le x\\le L$ where $U=U(x)$ and then transmitted at an angle $\\theta_t$ to the uniform region $x<0$ where $U=U_t$.\nIf the wave is evanescent in the region $x<0$, then the transmittance $T$ vanishes and the angle $\\theta_t$ is undefined.}\n\\label{fig_c1}\n\\end{figure}\n\nThe effective Hamiltonian that describes massive pseudospin-1 Dirac particles moving in the 2D $xy$ plane\nin a 1D scalar potential $U=U(x)$ takes the form\n\\begin{eqnarray}\n{\\mathcal H}=v_F \\left(S_x p_x+S_y p_y\\right)+UI+M V,\n\\label{eq:ham1}\n\\end{eqnarray}\nwhere $v_F$ is the Fermi velocity and $M$ ($=m{v_F}^2$) is the mass energy.\nThe $x$ and $y$ components, $S_x$ and $S_y$, of the pseudospin-1 operator are represented by\n\\begin{eqnarray}\nS_x=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0& 1& 0\\\\ 1& 0& 1\\\\ 0& 1& 0\\end{pmatrix},~~\nS_y=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0& -i& 0\\\\ i& 0& -i\\\\ 0& i& 0\\end{pmatrix}\n\\end{eqnarray}\nand $I$ is the $3\\times 3$ unity matrix.\nThe $x$ and $y$ components of the momentum operator, $p_x$ and $p_y$, are\n\\begin{eqnarray}\np_x=\\frac{\\hbar}{i}\\frac{d}{dx},~~p_y=\\hbar k_y,\n\\end{eqnarray}\nwhere $k_y$ is the $y$ component of the wave vector.\nWe assume that the mass energy $M$ is a constant.\nThe mass term $MV$ describes the generation of the band gap between the conduction and valence bands and the position of the flat band.\nFor the matrix $V$, we choose\n\\begin{eqnarray}\nV=\\begin{pmatrix} 1& 0& 0\\\\ 0& -1& 0\\\\ 0& 0& 1\\end{pmatrix}.\n\\end{eqnarray}\nThen the flat band is located at the bottom of the conduction band if $M>0$\nand at the top of the valence band if $M<0$ \\cite{ocam}.\nThe size of the band gap is $2\\vert M\\vert$.\n\nThe time-independent Dirac equation in 2D for the three-component vector wave function $\\psi$ [$=\\left( \\psi_1, \\psi_2, \\psi_3\\right)^{\\rm T}$] is\n\\begin{eqnarray}\n{\\mathcal H}\\psi=E\\psi,\n\\end{eqnarray}\nwhere $E$ is the particle energy.\nWe can eliminate $\\psi_1$ and $\\psi_3$ using the equations\n\\begin{eqnarray}\n\\psi_1&=&-\\frac{i}{\\sqrt{2}}\\frac{\\hbar v_F}{E-M-U}\\left(\\frac{d}{dx}+k_y\\right)\\psi_2,\\nonumber\\\\\n\\psi_3&=&-\\frac{i}{\\sqrt{2}}\\frac{\\hbar v_F}{E-M-U}\\left(\\frac{d}{dx}-k_y\\right)\\psi_2,\n\\label{eq:ff}\n\\end{eqnarray}\nand obtain a single wave equation for $\\psi_2$ of the form\n\\begin{eqnarray}\n&&\\frac{d}{dx}\\left(\\frac{\\hbar v_F}{E-M-U}\\frac{d\\psi_2}{dx}\\right)\\nonumber\\\\&&~~~~+\\left[\\frac{E+M-U}{\\hbar v_F}\n-\\frac{\\hbar v_F{k_y}^2}{E-M-U}\\right]\\psi_2=0.\n\\label{eq:we1}\n\\end{eqnarray}\nWe assume that a plane wave described by $\\psi_2$ is incident obliquely from the region $x>L$ where $U=0$\nonto the nonuniform region in $0\\le x\\le L$ where $U=U(x)$ and then transmitted to the uniform region $x<0$ where $U=U_t$.\nThen the wave number $k$ and the {\\it negative} $x$ component of the wave vector, $p$, in the incident region\nand the constant of motion $k_y$ are given by\n\\begin{eqnarray}\nk=\\frac{\\sqrt{E^2-{M}^2}}{\\hbar v_F},~~ p=k\\cos\\theta, ~~k_y=k\\sin\\theta,\n\\end{eqnarray}\nwhere we assume that $E >M \\ge 0$ and $\\theta$ is the incident angle. A sketch of the configuration considered here is shown in Fig.~\\ref{fig_c1}.\n\nWe introduce the dimensionless parameters $\\epsilon$ and $\\mu$ defined by\n\\begin{eqnarray}\n\\epsilon=1-\\frac{U}{E-M},~~\\mu=1-\\frac{U}{E+M},\n\\end{eqnarray}\nwhich are equal to each other in the massless case.\nIn the incident region, we have $\\epsilon=\\mu=1$.\nIn terms of the parameters $\\epsilon$ and $\\mu$, the wave equation, Eq.~(\\ref{eq:we1}), can be written as\n\\begin{eqnarray}\n\\frac{d}{dx}\\left(\\frac{1}{\\epsilon}\\frac{d\\psi_2}{dx}\\right)+k^2\\left(\\mu-\\frac{\\sin^2\\theta}{\\epsilon}\\right)\\psi_2=0.\n\\label{eq:we0}\n\\end{eqnarray}\nWe notice that if we replace $\\psi_2$, $\\epsilon$, and $\\mu$ with the $z$ component of the magnetic field $H_z$, the dielectric permittivity,\nand the magnetic permeability, this equation has precisely the same form as the wave equation for $p$-polarized EM waves\npropagating in the $xy$ plane. In Table \\ref{tab:table1}, we make a comparison between the pseudospin-1 Dirac equation and the $p$ wave equation in a plasma.\n\n\\begin{table*}\n\t\\caption{\\label{tab:table1} Comparison between the pseudospin-1 Dirac equation and the $p$ wave equation in a plasma.}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{cccccc}\n\t\t\t& & $\\epsilon$ & $\\mu$ & flat band & local oscillation\\\\\n\t\t\t\\hline\n\t\t\t& pseudospin-1 Dirac equation & $1-\\frac{U}{E-M}$ & $1-\\frac{U}{E+M}$ & $E=U+M$ & compact localized states \\\\\n\t\t\t& $p$ wave equation in a plasma & $1-\\frac{{\\omega_p}^2}{\\omega^2}$ & 1 & $\\omega=\\omega_p$ & plasmon \\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table*}\n\nWe solve the wave equation in the presence of an arbitrary potential using\nthe invariant imbedding method \\cite{kly,epl,sk1}. In this method, we first calculate the reflection and transmission coefficients $r$ and $t$ defined\nby the wave functions in the incident and transmitted regions:\n\\begin{eqnarray}\n\\psi_2\\left(x,L\\right)=\\left\\{\\begin{array}{ll}\n  e^{ip\\left(L-x\\right)}+r(L)e^{ip\\left(x-L\\right)}, & x>L \\\\\n  t(L)e^{-ip^\\prime x}, & x<0\n  \\end{array},\\right.\n\\end{eqnarray}\nwhere $p^\\prime$ is the negative $x$ component of the wave vector in the region $x<0$ and $r$ and $t$ are regarded as functions of $L$.\nFollowing the procedure described in \\cite{sk1}, we derive the exact differential equations for $r$ and $t$:\n\\begin{eqnarray}\n&&\\frac{1}{k}\\frac{dr}{dl}=-\\frac{i\\cos\\theta}{2}\\epsilon\\left(r-1\\right)^2\\nonumber\\\\&&~~~~~~~~~\n+\\frac{i}{2\\cos\\theta}\\left(\\mu-\\frac{\\sin^2\\theta}{\\epsilon}\\right)\\left(r+1\\right)^2,\\nonumber\\\\\n&&\\frac{1}{k}\\frac{dt}{dl}=-\\frac{i\\cos\\theta}{2}\\epsilon\\left(r-1\\right)t\\nonumber\\\\&&~~~~~~~~~\n+\\frac{i}{2\\cos\\theta}\\left(\\mu-\\frac{\\sin^2\\theta}{\\epsilon}\\right)\\left(r+1\\right)t.\n\\label{eq:imb1}\n\\end{eqnarray}\nFor any functional form of $U$ and for any values of $kL$ and $\\theta$,\nwe can integrate these equations from $l=0$ to $l=L$ using the initial conditions\n\\begin{eqnarray}\nr(0)=\\frac{\\epsilon_2\\cos\\theta-\\tilde p}{\\epsilon_2\\cos\\theta+\\tilde p},~~t(0)=\\frac{2\\epsilon_2\\cos\\theta}{\\epsilon_2\\cos\\theta+\\tilde p},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&\\tilde p=\\left\\{\\begin{matrix} \\mbox{sgn}(\\epsilon_2)\\sqrt{\\epsilon_2\\mu_2-\\sin^2\\theta} &\\mbox{if }\\epsilon_2\\mu_2\\ge \\sin^2\\theta \\\\\ni\\sqrt{\\sin^2\\theta-\\epsilon_2\\mu_2} & \\mbox{if }\\epsilon_2\\mu_2< \\sin^2\\theta \\end{matrix}\\right.,\\nonumber\\\\\n&&\\epsilon_2=1-\\frac{U_t}{E-M},~~\\mu_2=1-\\frac{U_t}{E+M},\n\\end{eqnarray}\nand obtain $r(L)$ and $t(L)$.\nThe reflectance $R$ and the transmittance $T$ are obtained using\n\\begin{eqnarray}\nR=\\vert r\\vert^2,~~T=\\left\\{\\begin{matrix} \\frac{\\tilde p}{\\epsilon_2\\cos\\theta}\\vert t\\vert^2\n&\\mbox{if }\\epsilon_2\\mu_2\\ge \\sin^2\\theta \\\\ 0 & \\mbox{if }\\epsilon_2\\mu_2< \\sin^2\\theta \\end{matrix}\\right..\n\\end{eqnarray}\nIn the absence of dissipation and mode conversion, the identity $R+T=1$ is satisfied.\n\nThe initial conditions $r(0)$ and $t(0)$ are the reflection and transmission coefficients for the case where there is no inhomogeneous layer (that is, $L=0$),\nand therefore the incident region with $\\epsilon=\\mu=1$ and the transmitted region with $\\epsilon=\\epsilon_2$ and $\\mu=\\mu_2$ have a single interface at $l=0$.\nThey are derived from the continuity of $\\psi_2$ and $\\epsilon^{-1}d\\psi_2\/dx$\nat the interface and are nothing but the well-known Fresnel coefficients. In the simplest case where the incident and transmitted regions have the same potential,\nthe initial conditions are trivially given by $r(0)=0$ and $t(0)=1$.\n\nWe point out that the invariant imbedding equations, Eq.~(\\ref{eq:imb1}), become {\\it singular} at the resonance point $x_r$ where $\\epsilon=0$, which corresponds to $E=M+U(x_r)$.\nThis singularity causes mode conversion in a very similar manner to that\nof transverse EM waves to longitudinal plasma oscillations in an inhomogeneous unmagnetized plasma. When a wave described by $\\psi_2$ with finite group velocity\nis incident obliquely on the inhomogeneous layer in $0\\le x\\le L$, it propagates up to the resonance point $x=x_r$, where the dispersive wave mode is strongly and resonantly coupled to the local flat-band state and the wave energy flows to the latter. Since\nthe group velocity associated with the flat-band state is zero, the energy is accumulated locally and is ultimately converted into heat.\nIn the steady state, a finite fraction of the energy of the incident wave is converted into that of the flat-band state.\n\nIn order to regularize the singularity,\nwe introduce a small imaginary part of $\\epsilon$, $\\epsilon_I$ ($>0$), in Eq.~(\\ref{eq:imb1}) when calculating $r$ and $t$.\nWe find numerically that the absorptance $A$ ($=1-R-T$) converges to a finite value\nin the limit $\\epsilon_I\\rightarrow 0$, if there exists a value of $x$ such that ${\\rm Re}~\\epsilon(x)=0$ in the region $0\\le x\\le L$.\nWe emphasize that this kind of absorption is not due to dissipation but due to the conversion of a propagating wave mode into a local\noscillating mode associated with the flat band. From now on, we will call $A$ as the mode conversion coefficient.\n\nA clear signature of mode conversion is the occurrence of a singularity in the invariant imbedding equations, such as the $(\\sin^2\\theta)\/\\epsilon$ term in Eq.~(\\ref{eq:imb1}).\nIn the systems with no flat band, there appears no singularity in those equations and mode conversion does not occur. In the case of the pseudospin-1\/2\nDirac equation in the presence of inhomogeneous scalar and vector potentials, which describes single-layer graphene and does not have a flat band in its spectrum,\nthe invariant imbedding equations have been derived previously in \\cite{sk11}. It has been verified that there appears no singularity and therefore no mode conversion.\n\n\\begin{figure}\n\\centering\\includegraphics[width=8cm]{fig2.eps}\n\\caption{Mode conversion coefficient $A$ obtained by solving Eq.~(\\ref{eq:imb1}) for the configuration given by Eq.~(\\ref{eq:slow}) plotted versus incident angle $\\theta$,\n(a) when $\\zeta\\equiv U_0L\/(\\hbar v_F)=20$, $M\/U_0=0.2$, and $E\/U_0=0.4$, 0.7, 1 and (b) when $E\/U_0=0.4$, $M\/U_0=0.2$, and $\\zeta=1$, 10, 100, 1000.\nIn all calculations, $\\epsilon_I$ is chosen to be $10^{-8}$.}\n\\label{fig1}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=8.5cm]{fig3.eps}\n\\caption{Color graph of the mode conversion coefficient $A$ obtained by solving Eq.~(7) for the configuration given by Eq.~(8) as a function of $\\theta$\nand $(E-M)\/U_0$, when $M\/U_0=0.2$, $\\zeta=20$, and $\\epsilon_I=10^{-8}$.\n$A$ vanishes for all $\\theta$ if $(E-M)\/U_0> 1$.}\n\\label{sfig1}\n\\end{figure}\n\nTo illustrate the mode conversion phenomenon, we consider a simple linear configuration of the potential\n\\begin{eqnarray}\n  \\frac{U(x)}{U_0}=\\left\\{ \\begin{array}{ll}\n  1,& \\mbox{if } x<0\\\\\n  1-\\frac{x}{L},& \\mbox{if } 0\\le x\\le L\\\\\n  0, & \\mbox{if } x>L\n  \\end{array} \\right..\n\\label{eq:slow}\n\\end{eqnarray}\nThe resonance occurs inside the region $0\\le x\\le L$ if\nthe energy satisfies $M<E<U_0+M$.\nIn Fig.~\\ref{fig1}(a), we plot $A$ versus $\\theta$ when $M\/U_0=0.2$ and $E\/U_0=0.4$, 0.7, 1.\nThe thickness of the inhomogeneous slab satisfies $\\zeta\\equiv U_0L\/(\\hbar v_F)=20$. For the chosen values of the energy,\nthe resonance occurs within the inhomogeneous region and substantial absorption (that is, mode conversion) arises\nin a wide range of the incident angle.\n\nThe efficiency of mode conversion is affected by the rate of the spatial change of the potential near the resonance point, which is measured by the parameter $\\zeta$.\nFor a fixed value of $U_0$, $\\zeta$ is proportional to $L$ and therefore is inversely proportional to the slope of the potential curve.\nIn Fig.~\\ref{fig1}(b), we plot $A$ versus $\\theta$ for the values of $\\zeta$ equal to 1, 10, 100, and 1000, when $E\/U_0=0.4$ and $M\/U_0=0.2$.\nWe find that the mode conversion\nbecomes weaker for both small and large values of $\\zeta$ and is strongest for some intermediate value of $\\zeta$.\n\nIn Fig.~\\ref{fig1}, we observe that $A$ always vanishes at $\\theta=0$. This is because normally incident waves cannot couple to the local flat band mode in the present model.\nIn the equivalent case of mode conversion in an unmagnetized plasma, it has been well-known that normally incident transverse waves cannot excite longitudinal\nplasma oscillations.\nIn order to provide a broader view of the mode conversion in pseudospin-1 systems, we show a color graph of the mode conversion coefficient as a function of the incident angle and the particle energy when $M\/U_0=0.2$ and $\\zeta=20$ in Fig.~\\ref{sfig1}.\nAs it has been expected, when the energy satisfies $0<(E-M)\/U_0<1$, there appears a wide range of the incident angle in which the mode conversion is substantially strong.\n\n\\begin{figure}\n\\centering\\includegraphics[width=\\linewidth]{fig4.eps}\n\\caption{Sketch of the configuration considered in Sec.~\\ref{sec_sp2}. The medium described by the pseudospin-2 Dirac equation is birefringent, and in general\nthere exist two reflected waves and two transmitted waves for each of the incident $a$ and $b$ mode waves.\nFor the parameter values where some of the transmitted waves are evanescent, the corresponding transmittances vanish. When an $a$ mode wave is incident at an angle greater than $30^\\circ$, the reflectance $R_{21}$ also vanishes. }\n\\label{fig_cf}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[width=8cm]{fig5.eps}\n\\caption{Mode conversion coefficients (a) $A_1$ and (b) $A_2$ for massless pseudospin-2 Dirac particles in the configuration given by Eq.~(\\ref{eq:slow}) plotted versus incident angle $\\theta$,\nwhen $\\zeta=15$, $\\epsilon_I=10^{-8}$, and $E\/U_0=0.3$, 0.6, 0.9. $A_1$ ($A_2$) is obtained by calculating the absorptance when the incident wave is the $a$ ($b$) mode.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\\includegraphics[width=8.5cm]{fig6a.eps} \\includegraphics[width=8.5cm]{fig6b.eps}\n\\caption{Color graphs of the mode conversion coefficients (a) $A_1$ and (b) $A_2$ for massless pseudospin-2 Dirac particles as functions of $\\theta$ and $E\/U_0$, when $\\zeta=15$ and $\\epsilon_I=10^{-8}$. $A_1$ ($A_2$) is obtained by calculating the absorptance when the incident wave is the $a$ ($b$) mode.\nBoth $A_1$ and $A_2$ vanish for all $\\theta$ if $E\/U_0> 1$.}\n\\label{f1}\n\\end{figure}\n\n\\section{Pseudospin-2 Dirac equation}\n\\label{sec_sp2}\n\nThe band structure of pseudospin-$N$ Dirac systems with $N$ a positive integer consists of $2N$ dispersive\nbands (that is, Dirac cones) and one flat band \\cite{dora}. Therefore we expect all of these systems to display mode conversion.\nWe consider here the case of pseudospin-2 Dirac systems \\cite{feng}.\nThe Hamiltonian that describes massless pseudospin-2 Dirac particles in 2D\nin a 1D scalar potential $U=U(x)$ has a similar form as Eq.~(\\ref{eq:ham1}), but with $M=0$ and\n$S_x$ and $S_y$ given by\n\\begin{eqnarray}\n&&S_x=\\frac{1}{2}\\begin{pmatrix} 0& 2& 0 &0 &0\\\\ 2& 0& \\sqrt{6} &0 &0\\\\ 0& \\sqrt{6}& 0 &\\sqrt{6} &0\\\\ 0& 0& \\sqrt{6} &0 &2\\\\ 0 &0 &0 &2 &0 \\end{pmatrix},\\nonumber\\\\\n&&S_y=\\frac{i}{2}\\begin{pmatrix} 0& -2& 0 &0 &0\\\\ 2& 0& -\\sqrt{6} &0 &0\\\\ 0& \\sqrt{6}& 0 & -\\sqrt{6} &0\\\\ 0& 0& \\sqrt{6} &0 & -2\\\\ 0 &0 &0 &2 &0 \\end{pmatrix}.\n\\end{eqnarray}\n\nIn the uniform case where the potential $U$ is constant, the eigenvalues of the Hamiltonian are given by\n\\begin{eqnarray}\n&&E=U,\\nonumber\\\\\n&&E=U\\pm \\hbar v_F\\sqrt{{k_x}^2+{k_y}^2},\\nonumber\\\\\n&&E=U\\pm 2\\hbar v_F\\sqrt{{k_x}^2+{k_y}^2}.\n\\end{eqnarray}\nTherefore the spectrum consists of two pairs of Dirac cones with different slopes which are intersected at the common apex by the flat band.\nWe now allow for the $x$ dependence of the potential $U$.\nStarting from the pseudospin-2 Dirac equation for the five-component vector wave function $\\psi$ [$=\\left( \\psi_1, \\psi_2, \\psi_3, \\psi_4, \\psi_5 \\right)^{\\rm T}$],\nwe can eliminate $\\psi_1$, $\\psi_3$,\nand $\\psi_5$ and derive two coupled wave equations for $\\psi_2$ and $\\psi_4$ of the form\n\\begin{eqnarray}\n\\frac{d}{dx}\\left(A\\frac{d\\Psi}{dx} +B\\Psi\\right)+C\\left(A\\frac{d\\Psi}{dx} +B\\Psi\\right)+D\\Psi=0,\\nonumber\\\\\n\\label{eq:s2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&& \\Psi=\\begin{pmatrix} \\psi_2 \\\\ \\psi_4 \\end{pmatrix},~~A=\\frac{1}{\\epsilon}\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix},\\nonumber\\\\ && B=\\frac{k_y}{4\\epsilon}\\begin{pmatrix} 1 & 3 \\\\ -3 & -1 \\end{pmatrix},~~\nC=\\frac{k_y}{8}\\begin{pmatrix} 7 & 9\\\\ -9 & -7 \\end{pmatrix},\\nonumber\\\\ && D=\\frac{\\epsilon {k_0}^2}{8}\\begin{pmatrix} 5 & -3 \\\\ -3 & 5 \\end{pmatrix}+\\frac{3{k_y}^2}{2\\epsilon}\\begin{pmatrix} -1 & 1 \\\\ 1 & -1 \\end{pmatrix},\\nonumber\\\\\n&&\\epsilon=1-\\frac{U}{E},~~k_0=\\frac{E}{\\hbar v_F}.\n\\label{eq:abc}\n\\end{eqnarray}\nIn the uniform region where $U$ is constant, there are four solutions for the $x$ component of the wave vector, $p$, obtained from Eq.~(\\ref{eq:s2}), which are\n\\begin{eqnarray}\np=\\pm\\sqrt{{k_0}^2\\epsilon^2-{k_y}^2},~~p=\\pm\\sqrt{\\frac{1}{4}{k_0}^2\\epsilon^2-{k_y}^2}.\n\\end{eqnarray}\nThe $\\pm$ signs represent the direction of the phase velocity.\nWe notice that there are two orthogonal eigenmodes obtained as linear combinations of $\\psi_2$ and $\\psi_4$, which are associated with the inner and outer cones and called\nhere as $a$ and $b$ modes respectively. These modes are characterized by different effective refractive indices\n$\\epsilon$ and $\\epsilon\/2$ and the system is birefringent. The $a$ and $b$ modes can alternatively be called as $h=1$ and $h=2$ modes respectively, where $h$ refers to the helicity.\nThe effect of the flat band is absorbed into the coefficients of Eq.~(\\ref{eq:s2}).\nIn inhomogeneous media, $a$ and $b$ modes\ninteract with each other and with the local flat band mode.\n\nIn the uniform region, we can show that there exists a linear proportionality relation between $\\psi_2$ and $\\psi_4$, which is different for $a$ and $b$ modes and for the left-moving\nand right-moving waves.\nWe obtain\n\\begin{eqnarray}\n&&{\\psi_4}^{(la)}=\\eta_{la}{\\psi_2}^{(la)},~~{\\psi_4}^{(lb)}=\\eta_{lb}{\\psi_2}^{(lb)},\\nonumber\\\\&& {\\psi_4}^{(ra)}=\\eta_{ra}{\\psi_2}^{(ra)},~~{\\psi_4}^{(rb)}=\\eta_{rb}{\\psi_2}^{(rb)},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&\\eta_{la}=-\\frac{{k_0}^2\\epsilon^2}{\\left(-\\sqrt{{k_0}^2\\epsilon^2-{k_y}^2}-ik_y\\right)^2},\\nonumber\\\\\n&&\\eta_{lb}=\\frac{\\frac{1}{4}{k_0}^2\\epsilon^2}{\\left(-\\sqrt{\\frac{1}{4}{k_0}^2\\epsilon^2-{k_y}^2}-ik_y\\right)^2},\n\\nonumber\\\\ && \\eta_{ra}=-\\frac{{k_0}^2\\epsilon^2}{\\left(\\sqrt{{k_0}^2\\epsilon^2-{k_y}^2}-ik_y\\right)^2},\\nonumber\\\\\n&&\\eta_{rb}=\\frac{\\frac{1}{4}{k_0}^2\\epsilon^2}{\\left(\\sqrt{\\frac{1}{4}{k_0}^2\\epsilon^2-{k_y}^2}-ik_y\\right)^2}.\n\\label{eq:eta}\n\\end{eqnarray}\nThe wave function is expanded in terms of $a$ and $b$ modes as\n\\begin{eqnarray}\n\\Psi=\\begin{pmatrix} \\psi_2 \\\\ \\psi_4 \\end{pmatrix}&=&\\begin{pmatrix} {\\psi_2}^{(la)}+{\\psi_2}^{(lb)}+{\\psi_2}^{(ra)}+{\\psi_2}^{(rb)}\n\\\\ {\\psi_4}^{(la)}+{\\psi_4}^{(lb)}+{\\psi_4}^{(ra)}+{\\psi_4}^{(rb)} \\end{pmatrix}\\nonumber\\\\&=&N_l \\begin{pmatrix} {\\psi_2}^{(la)} \\\\ {\\psi_2}^{(lb)} \\end{pmatrix}\n+N_r \\begin{pmatrix} {\\psi_2}^{(ra)} \\\\ {\\psi_2}^{(rb)} \\end{pmatrix},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nN_l=\\begin{pmatrix} 1 & 1 \\\\ \\eta_{la} & \\eta_{lb} \\end{pmatrix},~~N_r=\\begin{pmatrix} 1 & 1 \\\\ \\eta_{ra} & \\eta_{rb} \\end{pmatrix}.\n\\label{eq:cf}\n\\end{eqnarray}\nSince there are two propagating wave modes, we need to define the reflection and transmission coefficients $r$ and $t$ as $2\\times 2$ matrices.\nIn our notation, $r_{21}$ ($r_{11}$) is the reflection coefficient when the incident wave is $a$ mode and the reflected wave is $b$ ($a$) mode.\nSimilarly, $r_{12}$ ($r_{22}$) is the reflection coefficient when the incident wave is $b$ mode and the reflected wave is $a$ ($b$) mode.\nSimilar definitions are applied to the transmission coefficients.\n\nFollowing the procedure given in \\cite{sk1}, we derive the invariant imbedding equations for $r$ and $t$:\n\\begin{widetext}\n\\begin{eqnarray}\n&&\\frac{dr}{dl}={N_{ri}}^{-1}\\left\\{-A^{-1}B+i\\left(N_{li}+N_{ri}r\\right)\\left(A_iN_{li}P_i+A_iN_{ri}P_i{N_{ri}}^{-1}N_{li}\\right)^{-1}\n\\left[-\\left(iA_iN_{ri}P_i{N_{ri}}^{-1}+B_i\\right)A^{-1}B+D\\right]\\right\\}\\nonumber\\\\\n&&~~~~~~~~~\\times\\left(N_{li}+N_{ri}r\\right)\\nonumber\\\\\n&&~~~~~~+{N_{ri}}^{-1}\\left\\{A^{-1}+i\\left(N_{li}+N_{ri}r\\right)\\left(A_iN_{li}P_i+A_iN_{ri}P_i{N_{ri}}^{-1}N_{li}\\right)^{-1}\n\\left[\\left(iA_iN_{ri}P_i{N_{ri}}^{-1}+B_i\\right)A^{-1}+C\\right]\\right\\}\\nonumber\\\\\n&&~~~~~~~~~\\times\\left[iA_i\\left(-N_{li}P_i+N_{ri}P_ir\\right)+B_i\\left(N_{li}+N_{ri}r\\right)\\right],\\nonumber\\\\\n&&\\frac{dt}{dl}=it\\left(A_i N_{li}P_i+A_i N_{ri}P_i{N_{ri}}^{-1}N_{li}\\right)^{-1}\\Big\\{\\left[-\\left(iA_i N_{ri}P_i{N_{ri}}^{-1}+B_i\\right)A^{-1}B+D\\right]\\left(N_{li}+N_{ri}r\\right)\\nonumber\\\\\n&&~~~~~~+\\left[\\left(iA_iN_{ri}P_i{N_{ri}}^{-1}+B_i\\right)A^{-1}+C\\right]\\left[iA_i\\left(-N_{li}P_i+N_{ri}P_ir\\right)+B_i\\left(N_{li}+N_{ri}r\\right)\\right]\\Big\\},\n\\label{eq:imb2}\n\\end{eqnarray}\n\\end{widetext}\nwhere $A_i$, $B_i$, $N_{li}$, and $N_{ri}$ are the values of $A$, $B$, $N_l$, and $N_r$ in the incident region obtained by setting $\\epsilon=\\epsilon_i=1$ in Eqs.~(\\ref{eq:abc}) and (\\ref{eq:eta}).\nThese equations are integrated using the initial conditions of the form\n\\begin{widetext}\n\\begin{eqnarray}\n&&r(0)=\\left(A_i N_{ri}P_i+A_t N_{lt}P_t{N_{lt}}^{-1}N_{ri}-i B_i N_{ri}+iB_t N_{ri}\\right)^{-1}\n\\left(A_i N_{li}P_i+A_i N_{ri}P_i{N_{ri}}^{-1}N_{li}\\right)-{N_{ri}}^{-1}N_{li},\\nonumber\\\\\n&&t(0)=\\left(A_i N_{ri}P_i{N_{ri}}^{-1}N_{lt}+ A_t N_{lt}P_t-i B_i N_{lt}+i B_t N_{lt}\\right)^{-1}\n\\left( A_i N_{li}P_i+A_i N_{ri}P_i{N_{ri}}^{-1}N_{li}\\right),\n\\label{eq:ic2}\n\\end{eqnarray}\n\\end{widetext}\nwhere $A_t$, $B_t$, and $N_{lt}$ are the values of $A$, $B$, and $N_l$ in the transmitted region obtained by setting $\\epsilon=\\epsilon_t$ in Eqs.~(\\ref{eq:abc}) and (\\ref{eq:eta}).\nThe matrices $P_i$ and $P_t$ in Eqs.~(\\ref{eq:imb2}) and (\\ref{eq:ic2}) are defined by\n\\begin{eqnarray}\nP_i=\\begin{pmatrix} p_{ai} & 0 \\\\ 0 & p_{bi} \\end{pmatrix},~~P_t=\\begin{pmatrix} p_{at} & 0 \\\\ 0 & p_{bt} \\end{pmatrix},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n&&p_{ai}=\\sqrt{{k_0}^2{\\epsilon_i}^2-{k_y}^2},~~p_{bi}=\\sqrt{\\frac{1}{4}{k_0}^2{\\epsilon_i}^2-{k_y}^2},\\nonumber\\\\&&\np_{at}=\\sqrt{{k_0}^2{\\epsilon_t}^2-{k_y}^2},~~p_{bt}=\\sqrt{\\frac{1}{4}{k_0}^2{\\epsilon_t}^2-{k_y}^2}.\n\\label{eq:wns}\n\\end{eqnarray}\nThese initial conditions have been obtained following the procedure and using Eq.~(18) given in \\cite{sk1}.\nIf the incident and transmitted regions have the same potential, they reduce to very simple $2\\times 2$ matrices $r(0)=0$ and $t(0)=I$.\n\nWe assume that the incident waves are propagating waves with a real-valued wave vector. The effective refractive index associated with the $a$ mode is twice as large as that of the $b$ mode. When an $a$ mode wave is incident from the region where $\\epsilon_i=1$, $p_{ai}$ is\nreal and the incident angle $\\theta$ is related to $k_y$ by $k_y=k_0\\sin\\theta$. In this case, if $\\theta$ is greater than $30^\\circ$, we note that $p_{bi}$ becomes imaginary\nand the reflected $b$ wave is evanescent, while the reflected $a$ wave is propagative. However, when a $b$ mode wave is incident, the incident angle $\\theta$ is related to $k_y$ by $k_y=(k_0\\sin\\theta)\/2$. Then $p_{ai}$ is always real regardless of the incident angle and both reflected $a$ and $b$ waves are propagative.\nThe other quantities\n$p_{at}$ and $p_{bt}$ can be either real or imaginary depending on the value of $U_t$. For instance, when $p_{at}$ is imaginary, we use $p_{at}=i\\sqrt{{k_y}^2-{k_0}^2{\\epsilon_t}^2}$ instead of\nthe expression in Eq.~(\\ref{eq:wns}). A sketch of the configuration considered in this section is shown in Fig.~\\ref{fig_cf}.\n\nFinally, from the consideration of the probability currents, we obtain the expressions for the reflectance and transmittance matrices $R_{ij}$ and $T_{ij}$ ($i,j=1,2$),\nwhich are applicable when $p_{bi}$, $p_{at}$, and $p_{bt}$ are real:\n\\begin{eqnarray}\n&&R_{11}=\\vert r_{11}\\vert^2,~~R_{22}=\\vert r_{22}\\vert^2,\\nonumber\\\\&&\nR_{21}=\\frac{4p_{bi}}{p_{ai}}\\vert r_{21}\\vert^2,~~R_{12}=\\frac{p_{ai}}{4p_{bi}}\\vert r_{12}\\vert^2,\\nonumber\\\\\n&&T_{11}=\\frac{\\vert\\epsilon_i\\vert p_{at}}{\\vert\\epsilon_t\\vert p_{ai}}\\vert t_{11}\\vert^2,~~T_{22}=\\frac{\\vert\\epsilon_i\\vert p_{bt}}{\\vert\\epsilon_t\\vert p_{bi}}\\vert t_{22}\\vert^2,\\nonumber\\\\&&\nT_{21}=\\frac{4\\vert\\epsilon_i\\vert p_{bt}}{\\vert\\epsilon_t\\vert p_{ai}}\\vert t_{21}\\vert^2,~~T_{12}=\\frac{\\vert\\epsilon_i\\vert p_{at}}{4\\vert\\epsilon_t\\vert p_{bi}}\\vert t_{12}\\vert^2.\n\\end{eqnarray}\nIf $p_{at}$ ($p_{bt}$) is imaginary, we have to set $T_{11}$ and $T_{12}$ ($T_{21}$ and $T_{22}$) to be identically zero.\nIf $p_{bi}$ is imaginary, we have to set $R_{21}$ to zero and the mode conversion coefficient $A_2$ (see below) is undefined.\nWith these definitions, if there is no dissipation or mode conversion, the law of energy conservation $R_{11}+R_{21}+T_{11}+T_{21}=R_{12}+R_{22}+T_{12}+T_{22}=1$ should\nbe satisfied. When the mode conversion occurs,\nthe mode conversion coefficients $A_1$ and $A_2$ are defined by\n\\begin{eqnarray}\n&&A_1=1-R_{11}-R_{21}-T_{11}-T_{21},\\nonumber\\\\&& A_2=1-R_{12}-R_{22}-T_{12}-T_{22}.\n\\end{eqnarray}\n\nIn Fig.~\\ref{fig3}, we plot the mode conversion coefficients $A_1$ and $A_2$ versus $\\theta$, when $\\zeta=15$ and $E\/U_0=0.3$, 0.6, 0.9. $A_1$ ($A_2$) is obtained by calculating the absorptance when the incident wave is $a$ ($b$) mode.\nWe find that there exists a wide range of the incident angle in both $A_1$ and $A_2$ curves in which the mode conversion is substantially strong.\nSince there are two propagating modes interacting\nwith the local flat band mode, these curves display multiple peaks and cusps associated with various cutoffs.\nInside the inhomogeneous layer, the $a$ and $b$ modes are coupled to each other. When an $a$ mode wave is incident, it can propagate directly to the resonance region and\nconvert to flat-band state or it can take an indirect route, first converting to $b$ wave and then converting to flat-band state at the resonance region.\nSince the mode conversion coefficient obtains the maximum at different parameter values in these two cases, the curves\nof $A_1$ often show two peaks as a function of the incident angle or the energy, as illustrated in Fig.~\\ref{fig3}(a).\nThe case where a $b$ mode wave is incident is significantly different.\nSince the refractive index associated with the $a$ mode is twice as large as that of the $b$ mode, the $a$ wave converted from the incident $b$ wave propagates\nat a much smaller angle with respect to the $x$ axis than the incident angle. Mode conversion is not efficient for small propagation angles and therefore this\nindirect process does not contribute greatly to mode conversion. Therefore, when a $b$ wave is incident, mode conversion is dominated by the direct conversion from $b$ wave to flat-band state and one usually obtains a single peak for $A_2$ as shown in Fig.~\\ref{fig3}(b).\nIn addition, the occurrence of various cutoffs makes cusps to appear in the curves. For example, the cusp at $\\theta=30^\\circ$ in Fig.~\\ref{fig3}(a) comes from the\ncutoff condition that the reflected $b$ wave becomes evanescent.\n\nIn Fig.~\\ref{f1}, we show color graphs of $A_1$ and $A_2$ as a function of the incident angle and the particle energy when $\\zeta=15$.\nWhen the energy satisfies $0<E\/U_0<1$, there appear wide ranges of the incident angle in which the mode conversion is substantially strong in both graphs.\nThe observation that there are two clusters in which the mode conversion is strong in Fig.~\\ref{f1}(a) has the same reason as that explained in the previous paragraph.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{fig7.eps}\n\\caption{Sketches of the (a) stub, (b) sawtooth, (c) 1D pyrochlore, and (d) diamond lattices. In (a), (c), and (d), all hopping parameters between the connected sites are the same. In (b), the hopping parameters associated with the black and green lines are $\\tau$ and $\\sqrt{2}\\tau$, respectively. The 1D pyrochlore and diamond lattices have up-down inversion symmetry, or equivalently, mirror symmetry with respect to the dotted line,\nwhereas the stub and sawtooth lattices have no such symmetry. In one dimension, mode conversion occurs only in the models derived from the lattices with no up-down inversion symmetry.}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\\includegraphics[width=8cm]{fig8.eps}\n\\caption{Mode conversion coefficient $A$ obtained by solving (a) Eq.~(\\ref{eq:stub}) corresponding to the stub lattice when $L\/a=25$ and $U_0\/\\tau=50$ and (b) Eq.~(\\ref{eq:saw}) corresponding to the sawtooth lattice when $L\/a=25$ and $U_0\/\\tau=250$ plotted versus normalized energy $E\/\\tau$. \n\\label{fig5}\n\\end{figure}\n\n\n\n\\section{1D models with flat bands}\n\nThere have been many studies on the lattice models exhibiting one or more flat bands in 1D, including the stub, sawtooth, diamond, 1D pyrochlore, and 1D Lieb lattices \\cite{ley,luck,flach}.\nWe have derived continuum wave equations from these models by linearizing them in the vicinity of the band edge\nand solved the wave equations in the presence of an inhomogeneous scalar potential.\nAll models contain one or more flat bands in addition to dispersive bands.\nIn order for an incident wave to induce mode conversion, it has to couple to the local flat band mode.\nWe have found that this coupling is possible only in the models with no up-down inversion symmetry such as the stub and sawtooth lattices (shown in Fig.~\\ref{fig4}),\nbut is not possible in the symmetric models such as the diamond and 1D pyrochlore lattices. These latter cases have a close similarity to the absence of mode conversion at normal incidence in the pseudospin-1\nDirac equation.\n\nThe tight-binding equations for the stub lattice at energy $E$ can be written as\n\\begin{eqnarray}\n&& E \\psi_n^{\\rm A}= v_n^{\\rm A}\\psi_n^{\\rm A}+\\tau\\psi_{n-1}^{\\rm B}+\\tau\\psi_n^{\\rm B}+\\tau\\psi_n^{\\rm C},    \\nonumber\\\\\n&& E \\psi_n^{\\rm B}= v_n^{\\rm B}\\psi_n^{\\rm B}+\\tau\\psi_{n}^{\\rm A}+\\tau\\psi_{n+1}^{\\rm A},    \\nonumber\\\\\n&& E \\psi_n^{\\rm C}= v_n^{\\rm C}\\psi_n^{\\rm C}+\\tau\\psi_{n}^{\\rm A},\n\\end{eqnarray}\nwhere A, B, and C indicate the sublattice sites A, B, and C shown in Fig.~\\ref{fig4}(a) and $v_n^{\\rm A}$, $v_n^{\\rm B}$, and $v_n^{\\rm C}$ are the potentials at each site.\nAll hopping integrals between the nearest-neighbor sites are assumed to have the same value $\\tau$.\nIn the homogeneous case with no potential, the spectrum of this model is given by the eigenvalues of the matrix\n\\begin{equation}\n\\begin{pmatrix} 0 & 1+e^{-iqa} & 1 \\\\ 1+e^{iqa} &0 &0 \\\\ 1 & 0 & 0 \\end{pmatrix},\n\\end{equation}\nwhere $q$ is the wave vector and $a$ is the lattice constant.\nWe find that the spectrum consists of one flat band at $E=0$ and two dispersive bands satisfying $E\/\\tau=\\pm\\sqrt{3+2\\cos(qa)}$. We linearize this model\nin the neighborhood of the band edge, where $q\\approx \\pi\/a$. Finally, we replace $\\tilde q$ ($=q-\\pi\/a$) with the operator $-id\/dx$\nand reintroduce the scalar potential $U(x)$. We obtain\n\\begin{eqnarray}\n&& E \\psi_{\\rm A}=U\\psi_{\\rm A}+\\tau a\\frac{d\\psi_{\\rm B}}{dx}+\\tau\\psi_{\\rm C},    \\nonumber\\\\\n&& E \\psi_{\\rm B}=U\\psi_{\\rm B}-\\tau a\\frac{d\\psi_{\\rm A}}{dx},    \\nonumber\\\\\n&& E \\psi_{\\rm C}=U\\psi_{\\rm C}+\\tau\\psi_{\\rm A}.\n\\end{eqnarray}\nWe eliminate $\\psi_{\\rm A}$ and $\\psi_{\\rm C}$ and obtain a wave equation of the form\n\\begin{eqnarray}\na^2\\frac{d}{dx}\\left(\\frac{\\varepsilon}{\\varepsilon^2-1}\\frac{d{\\psi_B}}{dx}\\right)+\\varepsilon\\psi_B=0,\n\\label{eq:stub}\n\\end{eqnarray}\nwhere $a$ is the lattice constant and\n$\\varepsilon=[E-U(x)]\/\\tau$. $\\tau$ is the hopping parameter between the neighboring sites and $\\psi_{B}$ describes the wave function at the B sites in Fig.~\\ref{fig4}(a).\nWe apply the invariant imbedding method to the wave equation and derive the equations for the reflection and transmission coefficients of the form\n\\begin{eqnarray}\n&&\\frac{dr}{dl}=2ip\\frac{\\varepsilon_1\\left(\\varepsilon^2-1\\right)}{\\varepsilon\\left({\\varepsilon_1}^2-1\\right)}r\n+\\frac{ip}{2}\\left[\\frac{\\varepsilon}{\\varepsilon_1}-\\frac{\\varepsilon_1\\left(\\varepsilon^2-1\\right)}{\\varepsilon\\left({\\varepsilon_1}^2-1\\right)}\\right]\\left(1+r\\right)^2,\\nonumber\\\\\n&&\\frac{dt}{dl}=ip\\frac{\\varepsilon_1\\left(\\varepsilon^2-1\\right)}{\\varepsilon\\left({\\varepsilon_1}^2-1\\right)}t\n+\\frac{ip}{2}\\left[\\frac{\\varepsilon}{\\varepsilon_1}-\\frac{\\varepsilon_1\\left(\\varepsilon^2-1\\right)}{\\varepsilon\\left({\\varepsilon_1}^2-1\\right)}\\right]\\left(1+r\\right)t,\\nonumber\\\\\n\\label{eq:iest}\n\\end{eqnarray}\nwhere $\\varepsilon_1$ [$=(E-U_1)\/\\tau=E\/\\tau$] is the value of $\\varepsilon$ in the incident region and $p$ ($=\\sqrt{{\\varepsilon_1}^2-1}\/a$) is the wave number of the incident wave.\nThese equations are integrated using the initial conditions\n\\begin{eqnarray}\n&&r(0)=\\frac{p\\varepsilon_1\\left({\\varepsilon_2}^2-1\\right)-p^\\prime\\varepsilon_2\\left({\\varepsilon_1}^2-1\\right)}\n{p\\varepsilon_1\\left({\\varepsilon_2}^2-1\\right)+p^\\prime\\varepsilon_2\\left({\\varepsilon_1}^2-1\\right)},\\nonumber\\\\\n&&t(0)=\\frac{2p\\varepsilon_1\\left({\\varepsilon_2}^2-1\\right)}\n{p\\varepsilon_1\\left({\\varepsilon_2}^2-1\\right)+p^\\prime\\varepsilon_2\\left({\\varepsilon_1}^2-1\\right)},\n\\end{eqnarray}\nwhere $\\varepsilon_2$ [$=(E-U_2)\/\\tau$] is the value of $\\varepsilon$ in the transmitted region and $p^\\prime$ ($=\\sqrt{{\\varepsilon_2}^2-1}\/a$) is the wave number of the transmitted wave. The reflectance $R$ and the transmittance $T$ are given by\n\\begin{eqnarray}\nR=\\vert r\\vert^2,~~T=\\left\\{\\begin{matrix} \\frac{\\vert\\varepsilon_2\\vert\\sqrt{{\\varepsilon_1}^2-1}}{\\vert\\varepsilon_1\\vert\\sqrt{{\\varepsilon_2}^2-1}}\\vert t\\vert^2\n&\\mbox{if }{\\varepsilon_2}^2>1 \\\\ 0 & \\mbox{if }{\\varepsilon_2}^2\\le 1 \\end{matrix}\\right..\n\\end{eqnarray}\nWe notice that the invariant imbedding equations, Eq.~(\\ref{eq:iest}), have\na singularity at $\\varepsilon=0$, which corresponds to $E=U$.\n\nThe tight-binding equations for the sawtooth lattice at energy $E$ are written as\n\\begin{eqnarray}\n&& E \\psi_n^{\\rm A}= v_n^{\\rm A}\\psi_n^{\\rm A}+\\tau\\psi_{n-1}^{\\rm A}+\\tau\\psi_{n+1}^{\\rm A}+\\tau^\\prime\\psi_{n-1}^{\\rm B}+\\tau^\\prime\\psi_{n}^{\\rm B}, \\nonumber\\\\\n&& E \\psi_n^{\\rm B}= v_n^{\\rm B}\\psi_n^{\\rm B}+\\tau^\\prime\\psi_{n}^{\\rm A}+\\tau^\\prime\\psi_{n+1}^{\\rm A},\n\\end{eqnarray}\nwhere A and B indicate the sublattice sites A and B shown in Fig.~\\ref{fig4}(b) and $v_n^{\\rm A}$ and $v_n^{\\rm B}$ are the potentials at each site.\nIn order for this model to have a flat band, the hopping integral $\\tau^\\prime$ should be fine-tuned to satisfy $\\tau^\\prime=\\sqrt{2}\\tau$.\nIn the homogeneous case with no potential, the spectrum of this model consists of a flat band at $E\/\\tau=-2$ and a dispersive band\n$E\/\\tau=2[1+\\cos(qa)]$. Following a similar procedure as that for the stub lattice, it is straightforward to derive\n\\begin{eqnarray}\na^2\\frac{d}{dx}\\left(\\frac{\\varepsilon+2}{\\varepsilon}\\frac{d{\\psi_A}}{dx}\\right)+(\\varepsilon+2)\\psi_A=0,\n\\label{eq:saw}\n\\end{eqnarray}\nwhere $\\psi_A$ describes the wave function at the A sites in Fig.~\\ref{fig4}(b).\nWe apply the invariant imbedding method to it and obtain the equations for the reflection and transmission coefficients:\n\\begin{eqnarray}\n&&\\frac{dr}{dl}=2ip\\frac{\\varepsilon\\left(\\varepsilon_1+2\\right)}{\\varepsilon_1\\left(\\varepsilon+2\\right)}r\n+\\frac{ip}{2}\\left[\\frac{\\varepsilon+2}{\\varepsilon_1+2}-\\frac{\\varepsilon\\left(\\varepsilon_1+2\\right)}{\\varepsilon_1\\left(\\varepsilon+2\\right)}\\right]\\left(1+r\\right)^2,\\nonumber\\\\\n&&\\frac{dt}{dl}=ip\\frac{\\varepsilon\\left(\\varepsilon_1+2\\right)}{\\varepsilon_1\\left(\\varepsilon+2\\right)}t\n+\\frac{ip}{2}\\left[\\frac{\\varepsilon+2}{\\varepsilon_1+2}-\\frac{\\varepsilon\\left(\\varepsilon_1+2\\right)}{\\varepsilon_1\\left(\\varepsilon+2\\right)}\\right]\\left(1+r\\right)t,\\nonumber\\\\\n\\label{eq:iesaw}\n\\end{eqnarray}\nwhere $\\varepsilon_1$ [$=(E-U_1)\/\\tau=E\/\\tau$] is the value of $\\varepsilon$ in the incident region and $p$ ($=\\sqrt{\\varepsilon_1}\/a$) is the wave number of the incident wave.\nThese equations are integrated using the initial conditions\n\\begin{eqnarray}\n&&r(0)=\\frac{p\\varepsilon_2\\left(\\varepsilon_1+2\\right)-p^\\prime\\varepsilon_1\\left(\\varepsilon_2+2\\right)}\n{p\\varepsilon_2\\left(\\varepsilon_1+2\\right)+p^\\prime\\varepsilon_1\\left(\\varepsilon_2+2\\right)},\\nonumber\\\\\n&&t(0)=\\frac{2p\\varepsilon_2\\left(\\varepsilon_1+2\\right)}\n{p\\varepsilon_2\\left(\\varepsilon_1+2\\right)+p^\\prime\\varepsilon_1\\left(\\varepsilon_2+2\\right)},\n\\end{eqnarray}\nwhere $\\varepsilon_2$ [$=(E-U_2)\/\\tau$] is the value of $\\varepsilon$ in the transmitted region and $p^\\prime$ ($=\\sqrt{\\varepsilon_2}\/a$) is the wave number of the transmitted wave. The reflectance $R$ and the transmittance $T$ are given by\n\\begin{eqnarray}\nR=\\vert r\\vert^2,~~T=\\left\\{\\begin{matrix} \\frac{\\sqrt{\\varepsilon_1}\\left(\\varepsilon_2+2\\right)}{\\sqrt{\\varepsilon_2}\\left(\\varepsilon_1+2\\right)}\\vert t\\vert^2\n&\\mbox{if }\\varepsilon_2> 0 \\\\ 0 & \\mbox{if }\\varepsilon_2\\le 0 \\end{matrix}\\right..\n\\end{eqnarray}\nWe notice that the invariant imbedding equations, Eq.~(\\ref{eq:iesaw}), have\na singularity at\n$\\varepsilon=-2$, that is, $E=U-2\\tau$.\n\nIn Fig.~\\ref{fig5}(a), we plot the\nmode conversion coefficient $A$ obtained by solving Eq.~(\\ref{eq:stub}) when $L\/a=25$ and $U_0\/\\tau=50$ versus normalized energy $E\/\\tau$.\nIn Fig.~\\ref{fig5}(b), we plot $A$ obtained by solving Eq.~(\\ref{eq:saw})\nwhen $L\/a=25$ and $U_0\/\\tau=250$. The configuration of the potential is given by Eq.~(\\ref{eq:slow})\nin both cases. In Fig.~\\ref{fig5}(a), $A$ is nonzero in the range $1<E\/\\tau<U_0\/\\tau ~(=50)$, since the wave number of the incident wave is real when $E\/\\tau>1$\nand the resonance point exists when $0<E\/\\tau<U_0\/\\tau$. When $E\/\\tau>49$, the wave number in the transmitted region becomes imaginary and the wave gets strongly reflected, which\ncauses a sharp peak to occur in the region $49<E\/\\tau<50$. Except for this region, $A$ is almost a constant approximately equal to 0.5.\nIn Fig.~\\ref{fig5}(b), $A$ is nonzero in the range $0<E\/\\tau<248$, since the resonance point exists when $0<E\/\\tau<(U_0\/\\tau)-2$. $A$ is approximately equal to 0.49 in the entire nonzero range. We also remind that the continuum models used in our calculation have been derived using the assumption that the wavelength, which is inversely proportional to the energy, is sufficiently larger than the lattice constant $a$. Therefore we expect that our results are quantitatively valid in the low energy region.\n\n\\begin{figure}\n\\centering\\includegraphics[width=8cm]{fig9.eps}\n\\caption{Mode conversion coefficient $A$ for the 2D model described by Eq.~(\\ref{eq:sm}) in the configuration given by Eq.~(\\ref{eq:slow}) plotted versus incident angle $\\theta$, when $L\/a=10$, $U_0\/\\tau=20$, $E\/\\tau=2$, and $\\Delta\/\\tau=0$, 0.1, 1, 1.8.} \n\\label{fig6}\n\\end{figure}\n\n\n\n\\section{2D model with a nearly flat band}\n\nMode conversion can also occur when a dispersive band coexists with a nearly flat band with\na small group velocity. An example in plasma physics is the conversion of a transverse EM wave into an electron plasma wave\nin a warm plasma at nonzero temperature \\cite{yu1}. Here we consider the 2D model with a singular flat band studied in \\cite{im2}, the Hamiltonian for which is given by\n\\begin{eqnarray}\n{\\mathcal H}=\\begin{pmatrix} \\tau a^2 {k_x}^2 +U(x) & \\Delta-i \\tau a^2 k_x k_y\\\\ \\Delta+i\\tau a^2 k_x k_y & \\tau a^2 {k_y}^2 +U(x) \\end{pmatrix},\n\\label{eq:sm}\n\\end{eqnarray}\nwhere $\\tau$ and $a$ are the energy and length scales of the model and $k_x$ ($=-id\/dx$) is a differential operator.\nWhen the parameter $\\Delta$ and the potential $U$ are zero, the model has a quadratic dispersive band touching a flat band with $E=0$ at $k_x=k_y=0$.\nAs $\\Delta$ increases from zero, a gap is opened and\nthe flat band becomes increasingly warped.\n\nWhen the potential $U$ is uniform, the eigenvalues of Eq.~(\\ref{eq:sm}) are given by\n\\begin{eqnarray}\nE=U+\\frac{\\tau a^2k^2}{2}\\pm\\sqrt{\\left(\\frac{\\tau a^2k^2}{2}\\right)^2+\\Delta^2},\n\\label{eq:im}\n\\end{eqnarray}\nwhere $k^2={k_x}^2+{k_y}^2$.\nWhen the parameter $\\Delta$ is zero, the spectrum corresponding to the minus sign in Eq.~(\\ref{eq:im}) is a flat band with $E=U$.\nAs $\\Delta$ increases from zero, this band becomes warped and dispersive, though the group velocity remains small when $\\Delta$ is sufficiently small.\nWe assume that the potential $U$ is a function of $x$ and replace $k_x$ with the operator $k_x=-id\/dx$. We obtain\n\\begin{eqnarray}\n&& (E-U)\\psi_1=-\\tau a^2\\frac{d^2\\psi_1}{dx^2}+\\Delta \\psi_2-\\tau a^2 k_y\\frac{d\\psi_2}{dx},   \\nonumber\\\\\n&& (E-U)\\psi_2=\\Delta \\psi_1+\\tau a^2 k_y\\frac{d\\psi_1}{dx}+\\tau a^2 {k_y}^2\\psi_2,\n\\end{eqnarray}\nfrom which we derive the invariant imbedding equations for $r$ and $t$ of the form\n\\begin{widetext}\n\\begin{eqnarray}\n&&\\frac{dr}{dl}=-\\frac{\\Delta }{E-U}k_y(r+1)+\\frac{E-U-\\tau a^2{k_y}^2}{E-U}\\left[\\frac{E}{E-\\tau a^2{k_y}^2}ip(r-1)+\\frac{\\Delta }{E-\\tau a^2{k_y}^2}k_y(r+1)\\right]\\nonumber\\\\&&~~~~~~+\\left[\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{\\Delta }{E-U}k_y\n-\\frac{E-U}{\\tau a^2}\\frac{E-\\tau a^2{k_y}^2}{E}+\\frac{1}{\\tau a^2}\\frac{\\Delta^2}{E-U}\\frac{E-\\tau a^2{k_y}^2}{E}\\right]\\frac{1}{2ip}(r+1)^2\\nonumber\\\\\n&&~~~~~~+\\left[-\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{E-U-\\tau a^2{k_y}^2}{E-U}+\\frac{\\Delta }{E-U}\\frac{E-\\tau a^2{k_y}^2}{E}k_y\\right]\\frac{1}{2ip}(r+1)\n\\nonumber\\\\&&~~~~~~~~~\\times\\left[\\frac{E}{E-\\tau a^2{k_y}^2}ip(r-1)+\\frac{\\Delta }{E-\\tau a^2{k_y}^2}k_y(r+1)\\right],\\nonumber\\\\\n&&\\frac{dt}{dl}=\\left[\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{\\Delta }{E-U}k_y\n-\\frac{E-U}{\\tau a^2}\\frac{E-\\tau a^2{k_y}^2}{E}+\\frac{1}{\\tau a^2}\\frac{\\Delta^2}{E-U}\\frac{E-\\tau a^2{k_y}^2}{E}\\right]\\frac{1}{2ip}t(r+1)\\nonumber\\\\\n&&~~~~~~+\\left[-\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{E-U-\\tau a^2{k_y}^2}{E-U}+\\frac{\\Delta }{E-U}\\frac{E-\\tau a^2{k_y}^2}{E}k_y\\right]\\frac{1}{2ip}t\n\\nonumber\\\\&&~~~~~~~~~\\times\\left[\\frac{E}{E-\\tau a^2{k_y}^2}ip(r-1)+\\frac{\\Delta }{E-\\tau a^2{k_y}^2}k_y(r+1)\\right],\n\\label{eq:im2}\n\\end{eqnarray}\n\\end{widetext}\nwhere we have assumed that the potential $U$ in the incident region vanishes and the wave number of the incident wave, $p$, is given by\n\\begin{eqnarray}\np=\\sqrt{\\frac{E(E-\\tau a^2{k_y}^2)-\\Delta^2}{\\tau a^2 E}}.\n\\end{eqnarray}\nThese equations are integrated using the initial conditions\n\\begin{widetext}\n\\begin{eqnarray}\n&&r(0)=\\frac{\\left(ip-\\frac{\\Delta }{E}k_y\\right)\\frac{E-U_t-\\tau a^2 {k_y}^2}{E-U_t}+\\left(-ip^\\prime+\\frac{\\Delta }{E-U_t}k_y\\right)\\frac{E-\\tau a^2 {k_y}^2}{E}}{\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{E-U_t-\\tau a^2{k_y}^2}{E-U_t}\n+\\left(ip^\\prime-\\frac{\\Delta }{E-U_t}k_y\\right)\\frac{E-\\tau a^2{k_y}^2}{E}},\\nonumber\\\\\n&&t(0)=\\frac{2ip\\frac{E-U_t-\\tau a^2{k_y}^2}{E-U_t}}{\\left(ip+\\frac{\\Delta }{E}k_y\\right)\\frac{E-U_t-\\tau a^2{k_y}^2}{E-U_t}+\\left(ip^\\prime-\\frac{\\Delta }{E-U_t}k_y\\right)\\frac{E-\\tau a^2{k_y}^2}{E}},\n\\end{eqnarray}\n\\end{widetext}\nwhere $U_t$ is the value of $U$ in the transmitted region and the wave number of the transmitted wave, $p^\\prime$, is given by\n\\begin{eqnarray}\np^\\prime=\\sqrt{\\frac{(E-U_t)(E-U_t-\\tau a^2{k_y}^2)-\\Delta^2}{\\tau a^2 (E-U_t)}}.\n\\end{eqnarray}\nThe reflectance $R$ and the transmittance $T$ are obtained using\n\\begin{eqnarray}\n&&R=\\vert r\\vert^2,\\nonumber\\\\ &&T=\\left\\{\\begin{matrix} \\frac{p^\\prime}{p}\\bigg\\vert\\frac{(E-\\tau a^2{k_y}^2)(E-U_t)}{(E-U_t-\\tau a^2{k_y}^2)E}\\bigg\\vert\\vert t\\vert^2\n&\\mbox{if $p^\\prime$ is real}  \\\\ 0 & \\mbox{if $p^\\prime$ is imaginary} \\end{matrix}\\right..\\nonumber\\\\\n\\end{eqnarray}\nWe notice that the invariant imbedding equations, Eq.~(\\ref{eq:im2}), have\na singularity at\n$E=U$.\n\nIn Fig.~\\ref{fig6}, we plot the mode conversion coefficient versus incident angle, when the potential is given by Eq.~(\\ref{eq:slow}) and the parameters are\n$L\/a=10$, $U_0\/\\tau=20$, $E\/\\tau=2$, and $\\Delta\/\\tau=0$, 0.1, 1, 1.8. When $\\Delta=0$, $A$ is symmetric under the sign change of $\\theta$.\nAs $\\Delta\/\\tau$ increases towards 1, $A$ becomes increasingly asymmetric. As $\\Delta\/\\tau$ increases above 1,\nthe mode conversion becomes less and less efficient and $A$ decreases rapidly towards zero.\nWe find that the overall behavior is quite similar to the mode conversion of extraordinary waves occurring in a magnetized plasma where the external magnetic field\nis perpendicular to the directions of inhomogeneity and wave propagation \\cite{jkps}. The parameter $\\Delta$ plays the role of the magnetic field strength.\nWe conclude that the mode conversion between a strongly dispersive band and a nearly flat band can occur, but its efficiency decreases rapidly as the nearly flat band becomes less and less flat.\n\n\n\n\\section{Conclusion}\n\nIn this paper,\nwe have demonstrated that mode conversion and resonant absorption occur generically in all systems where dispersive bands and flat bands coexist\nin the band structure, unless forbidden by symmetry.\nMode conversion takes place as long as there exists a resonant region inside the system, regardless of the shape of\nthe potential.\nIn electronic materials, the inhomogeneous potential can be easily achieved by applying an external electric field or strain.\nA slowly-varying inhomogeneity in a medium parameter can be introduced into photonic metamaterials by various methods,\nfor instance, by slowly varying the lattice constant in a certain direction.\nThe effects of mode conversion in metamaterials are manifested in many properties, most obviously in the absorption of the incident wave.\nIn electronic systems, the mode conversion effect will substantially influence many physical quantities including the conductance and the shot noise.\nWe will present a detailed analysis on the consequences of mode conversion in experimentally relevant systems in a future work.\n\n\\acknowledgments\nThis research was supported through a National Research\nFoundation of Korea Grant (NRF-2020R1A2C1007655)\nfunded by the Korean Government.\nIt was also supported by the Basic Science Research Program funded by the Ministry of Education (2021R1A6A1A10044950) and by\nthe Global Frontier Program (2014M3A6B3063708).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nGiven a set $S$ of $n$ point sites in the plane, what is the best\nway to connect $S$ into a \\emph{geometric network (graph)}?\nThis question has motivated researchers for a long time, going\nback as far as the 1940s, and beyond~\\cite{F40,Ve51}. Numerous\npossible criteria for a good geometric network have been proposed,\nperhaps the most basic being the \\emph{length}.\nIn 1955, Few~\\cite{Fe55} showed that for any set of $n$ points\nin a unit square, there is a traveling salesman tour of length\nat most $\\sqrt{2n}+7\/4$. This was improved to at most $0.984\\sqrt{2n}+11$\nby Karloff~\\cite{Ka89}. Similar bounds also hold for the shortest\nspanning tree and the shortest rectilinear spanning\ntree~\\cite{CG81,DM18,GP68}. Besides length, two further key factors\nin the quality of a geometric network are the \\emph{vertex dilation}\nand the \\emph{geometric dilation}~\\cite{NS07}, both of which measure\nhow closely shortest paths in a network approximate the\nEuclidean distances between their endpoints.\n\nThe \\emph{dilation} (also called \\emph{stretch factor}~\\cite{MM17}\nor \\emph{detour}~\\cite{AKK+08})\nbetween two points $p$ and $q$ in a geometric graph $G$ is defined as the ratio between\nthe length of a shortest path from $p$ to $q$ and the Euclidean distance $|pq|$.\nThe \\emph{dilation} of the graph $G$ is the maximum dilation over all pairs of vertices in $G$.\nA graph in which the dilation is bounded\nabove by $t \\geq 1$ is also called a \\emph{$t$-spanner}\n(or simply a \\emph{spanner} if $t$ is a constant).\nA complete graph in Euclidean space is clearly a $1$-spanner. Therefore,\nresearchers focused on the dilation of graphs with certain additional constraints,\nfor example, noncrossing (\\ie, plane) graphs. In 1989, Das and Joseph~\\cite{DJ89}\nidentified a large class of plane spanners (characterized\nby two simple local properties).\nBose~\\etal\\cite{BGS05} gave\nan algorithm that constructs for any set of planar sites\na plane $11$-spanner with bounded degree.\nOn the other hand, Eppstein~\\cite{Ep02}  analyzed\na fractal construction showing that \\emph{$\\beta$-skeletons}, a natural class\nof geometric networks, can have arbitrarily large dilation.\n\nThe study of dilation also raises algorithmic questions.\nAgarwal~\\etal~\\cite{AKK+08} described randomized algorithms for computing the dilation of\na given path (on $n$ vertices) in $\\RR^2$ in $O(n \\log n)$ expected time.\nThey also presented randomized algorithms for computing the dilation\nof a given tree, or cycle, in $\\RR^2$ in $O(n\\log^2 n)$ expected time.\nPreviously, Narasimhan and Smid~\\cite{NS00} showed that an $(1+\\eps)$-approximation\nof the stretch factor of any path, cycle, or tree can be computed in $O(n \\log n)$ time.\nKlein~\\etal~\\cite{KKNS09} gave randomized algorithms for a path, tree, or cycle in $\\RR^2$\nto count the number of vertex pairs whose dilation is below a given threshold in $O(n^{3\/2+\\eps})$\nexpected time.\nCheong~\\etal~\\cite{CHL08} showed that it is NP-hard to determine the existence of a\nspanning tree on a planar point set whose dilation is at most a given value.\nMore results on plane spanners can be found in the monograph dedicated to this subject~\\cite{NS07}\nor in several surveys~\\cite{Ep00,BS13,MM17}.\n\nWe investigate a basic question about the dilation of polygonal chains.\nMore precisely, we ask how the dilation between the endpoints of a polygonal chain\n(which we will call the \\emph{stretch factor}, to distinguish it from\nthe more general notion of dilation) is influenced by \\emph{fingerprint} properties of the\nchain, \\ie, by properties that are defined on $O(1)$-size subsets of the vertex set.\nSuch fingerprint properties play an important role in geometry, where classic\nexamples include the \\emph{Carath\\'eodory property}\\footnote{Given a finite set $S$ of points\n  in $d$ dimensions, if every $d + 2$ points in $S$ are in convex position,\n  then $S$ is in convex position.}~\\cite[Theorem~1.2.3]{Mat02}\nor the \\emph{Helly property}\\footnote{Given a finite collection of convex\n  sets in $d$ dimensions, if every $d + 1$ sets have nonempty intersection,\n  then all sets have nonempty intersection.}~\\cite[Theorem~1.3.2]{Mat02}.\nIn general, determining the effect of a fingerprint property may prove elusive:\ngiven $n$ points in the plane,\nconsider the simple property that every $3$ points determine $3$ distinct distances.\nIt is unknown~\\cite[p.~203]{BMP05}  whether this property implies that the total number\nof distinct distances grows superlinearly in $n$.\n\nFurthermore, fingerprint properties appear in  the general study of\n\\emph{local versus global properties of metric spaces} that is highly relevant\nto combinatorial approximation algorithms that are based on\nmathematical programming relaxations~\\cite{ALNRRV12}.\nIn the study of dilation, interesting fingerprint properties have also been\nfound.\nFor example, a (continuous) curve $C$ is said to have the\n\\emph{increasing chord property}~\\cite{CFG91, LM72} if for any points $a$, $b$, $c$, $d$\nthat appear on $C$ in this order, we have $|ad| \\geq |bc|$.\nThe increasing chord property implies that $C$ has (geometric) dilation at most\n$2\\pi\/3$~\\cite{Ro94}. A weaker property is the \\emph{self-approaching property}:\na (continuous) curve $C$ is self-approaching if for any points $a$, $b$, $c$ that appear\non $C$ in this order, we have $|ac| \\geq |bc|$.\nSelf-approaching curves have dilation at most $5.332$~\\cite{IKL99}\n(see also~\\cite{AAI+99}), and they have found interesting applications in the field of\ngraph drawing~\\cite{ACG+12,BKL17,NPR16}.\n\nWe introduce a new natural fingerprint property and see that it can constrain the stretch factor\nof a polygonal chain, but only in a weaker sense than one may expect; we also\nprovide algorithmic results on this property. Before providing details, we give\na few basic definitions.\n\n\\subparagraph*{Definitions.}\nA \\emph{polygonal chain} $P$ in the Euclidean plane is specified by a sequence of $n$ points\n$(p_1, p_2, \\dots, p_n)$, called its \\emph{vertices}.\nThe chain $P$ consists of $n-1$ line segments between consecutive vertices.\nWe say $P$ is \\emph{simple} if only consecutive line segments intersect and they only\nintersect at their endpoints.\nGiven a polygonal chain $P$ in the plane with $n$ vertices and a parameter $c\\geq 1$,\nwe call $P$ a \\emph{$c$-chain} if for all $1\\leq i < j < k\\leq n$,\nwe have\n\\begin{equation} \\label{eq:c-chain}\n|p_ip_j|+|p_jp_k| \\leq c|p_ip_k|.\n\\end{equation}\nObserve that the $c$-chain condition is a fingerprint condition that is not really a local\ndilation condition---it is more a combination between the local chain substructure\nand the distribution of the points in the subchains.\n\n\nThe \\emph{stretch factor} $\\delta_P$ of $P$ is defined as the dilation between\nthe two end points $p_1$ and $p_n$ of the chain:\n\\[\n\\delta_P = \\frac{\\sum_{i = 1}^{n-1} |p_ip_{i+1}|}{|p_1p_n|}.\n\\]\nNote that this definition is different from the more general notion of\ndilation (also called \\emph{stretch factor}~\\cite{MM17}) of a graph\nwhich is the maximum dilation over all pairs of vertices.\nSince there is no ambiguity in this paper,\nwe will just call $\\delta_P$ the stretch factor of $P$.\n\nFor example, the polygonal chain $P=((0,0), (1,0), \\dots ,(n,0))$ is a $1$-chain with\nstretch factor $1$; and\n$Q=((0,0), (0,1), (1,1), (1,0))$ is a $(\\sqrt2+1)$-chain with stretch factor $3$.\n\nWithout affecting the results, the floor and ceiling functions are omitted in our calculations.\nFor a positive integer $t$, let $[t]=\\{1,2,\\dots,t\\}$.\nFor a point set $S$, let $\\conv(S)$ denote the convex hull of $S$.\nAll logarithms are in base 2, unless stated otherwise.\n\n\\subparagraph*{Our results.}\nWe deduce three upper bounds on the stretch factor of a $c$-chain $P$\nwith $n$ vertices (Section~\\ref{sec:upper}).\nIn particular, we have\n(i)~$\\delta_P \\leq c(n-1)^{\\log c}$,\n(ii)~$\\delta_P\\leq c(n-2)+1$, and\n(iii)~$\\delta_P =O\\left(c^2\\sqrt{n-1}\\right)$.\n\nFrom the other direction, we obtain the following lower bound (Section~\\ref{sec:lower}):\nFor every $c\\geq 4$, there is a family $\\P_c=\\{P^k\\}_{k\\in\\NN}$ of simple $c$-chains,\nso that $P^k$ has $n=4^k+1$ vertices and stretch factor $(n-1)^{\\frac{1+\\log (c-2) - \\log c}{2}}$,\nwhere the exponent converges to $1\/2$ as $c$ tends to infinity.\nThe lower bound construction does not extend to the case of $1<c<4$, which remains open.\n\nFinally, we present two algorithmic results (Section~\\ref{sec:algo}):\n(i)~A randomized algorithm that decides, given a polygonal chain $P$ in $\\RR^2$ with\n$n$ vertices and a threshold $c>1$, whether $P$ is a\n$c$-chain in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$\n  expected time and $O(n\\log n)$ space.\n  (ii)~As a corollary, there is a randomized algorithm that finds, for a polygonal chain\n  $P$ with $n$ vertices, the minimum $c\\geq 1$ for which $P$ is a $c$-chain\n  in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$ expected time and $O(n\\log n)$ space.\n\n\n\n\\section{Upper Bounds} \\label{sec:upper}\n\nAt first glance, one might expect the stretch factor of a $c$-chain, for $c\\geq 1$, to be bounded by\nsome function of $c$. For example, the stretch factor of a $1$-chain is necessarily $1$.\nWe derive three upper bounds on the stretch factor of a $c$-chain with $n$ vertices\nin terms of $c$ and $n$ (cf.~Theorems~\\ref{thm:log{c}}--\\ref{thm:1\/2});\nsee Fig.~\\ref{fig:upperbds} for a visual comparison between the bounds.\nFor large $n$, the bound in Theorem~\\ref{thm:log{c}} is the best for $1 \\leq c  \\leq 2^{1\/2}$,\nwhile the bound in Theorem~\\ref{thm:1\/2} is the best for $c > 2^{1\/2}$.\nIn particular, the bound in Theorem~\\ref{thm:log{c}} is tight for $c=1$.\nThe bound in Theorem~\\ref{thm:linear} is the best for $c\\geq 2$ and $n\\leq 111c^2$.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{upperbds}\n\\caption{The values of $n$ and $c$ for which (i) Theorem~\\ref{thm:log{c}}, (ii) Theorem~\\ref{thm:linear},\nand (iii) Theorem~\\ref{thm:1\/2} give the current best upper bound.\\label{fig:upperbds}}\n\\end{figure}\n\nOur first upper bound is obtained by a recursive application of the\n$c$-chain property. It holds for any positive distance\nfunction that\nmay not even\nsatisfy the triangle inequality.\n\\begin{theorem}\\label{thm:log{c}}\nFor a $c$-chain $P$ with $n$ vertices, we have $\\delta_P \\leq c(n-1)^{\\log c}$.\n\\end{theorem}\n\\begin{proof}\nWe prove, by induction on $n$, that\n\\begin{equation}\\label{eq:logc}\n\\delta_P\\leq c^{\\left\\lceil\\log (n-1)\\right\\rceil},\n\\end{equation}\nfor every $c$-chain $P$ with $n \\geq 2$ vertices. In the base case, $n=2$, we\nhave $\\delta_P=1$ and $c^{\\left\\lceil\\log (2-1)\\right\\rceil}=1$.\nNow let $n \\geq 3$, and assume that (\\ref{eq:logc}) holds for every $c$-chain\nwith fewer than $n$ vertices.\nLet $P = (p_1, \\dots, p_n)$ be a $c$-chain with $n$ vertices. Then,\napplying (\\ref{eq:logc}) to the first and second half of $P$, followed\nby the $c$-chain property for the first, middle, and last vertex of $P$, we get\n\\begin{align*}\n\\sum_{i=1}^{n-1}|p_{i}p_{i+1}|\n&\\leq \\sum_{i=1}^{\\lceil n\/2\\rceil-1}|p_{i}p_{i+1}| + \\sum_{i=\\lceil n\/2\\rceil}^{n-1} |p_{i}p_{i+1}|\\\\\n&\\leq c^{\\left\\lceil\\log (\\lceil n\/2\\rceil-1) \\right\\rceil} \\left( |p_1p_{\\lceil n\/2\\rceil}| + |p_{\\lceil n\/2\\rceil}p_n|\\right)\\\\\n&\\leq c^{\\left\\lceil\\log (\\lceil n\/2\\rceil-1) \\right\\rceil}\\cdot  c|p_1p_n|\\\\\n&\\leq  c^{\\left\\lceil\\log (n-1)\\right\\rceil} |p_1p_n|,\n\\end{align*}\nso (\\ref{eq:logc}) holds also for $P$.\nConsequently,\n\\[ \\delta_P \\leq c^{\\left\\lceil\\log (n-1)\\right\\rceil} \\leq c^{\\log (n-1) +1} =c \\cdot c^{\\log (n-1)}\n= c \\, (n-1)^{\\log{c}}, \\]\nas required.\n\\end{proof}\n\nOur second bound interprets the $c$-chain property geometrically and\nmakes use of the fact that $P$ resides in the Euclidean plane.\n\\begin{theorem}\\label{thm:linear}\nFor a $c$-chain $P$ with $n$ vertices, we have $\\delta_P\\leq c(n-2)+1$.\n\\end{theorem}\n\\begin{figure}[htpb]\n\\centering\n\\begin{tikzpicture}[scale=0.45]\n\\draw (5,0) circle [x radius=5, y radius=4];\n\\draw (0,0) --node[above]{$\\frac{c-1}{2}$} (2,0)node[circle, fill, inner sep=1pt, label=below:$p_1$]{}\n-- node[above]{$1$} (8,0)node[circle, fill, inner sep=1pt, label=below:$p_n$]{}\n-- node[above]{$\\frac{c-1}{2}$} (10,0);\n\\draw[dashed] (2,0) -- node[above]{$\\frac{c}{2}$} (5, 4)\n-- node[above]{$\\frac{c}{2}$} (8, 0);\n\\end{tikzpicture}\n\\caption{The entire chain $P$ lies in an ellipse with foci $p_1$ and $p_n$.}\\label{fig:ellipse}\n\\end{figure}\n\n\\begin{proof}\nWithout loss of generality, assume that $|p_1p_n|=1$.\nSince $P$ is a $c$-chain, for every $1<j<n$, we have\n$|p_1p_j|+|p_jp_n|\\leq c|p_1p_n| = c$.\nIf we fix the points $p_1$ and $p_n$, then every $p_j$ lies in an ellipse $E$\nwith foci $p_1$ and $p_n$, for $1 < j < n$,\nsee Figure~\\ref{fig:ellipse}.\nThe diameter of $E$ is its major axis, whose length is $c$.\nSince $E$ contains all vertices of the chain $P$, we have\n$|p_1p_2|, |p_{n-1}p_n|\\leq \\frac{c+1}{2} \\leq c$ and\n$|p_jp_{j+1}|\\leq c$ for all $1<j<n-1$.\nTherefore the stretch factor of $P$ is bounded above by\n\\begin{align*}\n\\delta_P &= \\frac{\\sum_{j=1}^{n-1}|p_{j}p_{j+1}|}{|p_1p_n|}\n= |p_1p_2|+|p_{n-1}p_n|+\\sum_{j=2}^{n-2}|p_jp_{j+1}| \\\\\n&\\leq \\frac{c+1}{2}+\\frac{c+1}{2}+c(n-3) = c(n-2)+1,\n\\end{align*}\nas required.\n\\end{proof}\nOur third upper bound uses a volume argument to bound the number of long edges\nin $P$.\n\\begin{theorem}\\label{thm:1\/2}\n  Let $P=(p_1,\\dots, p_n)$ be a $c$-chain, for some constant $c \\geq 1$,\n  and let $L=\\sum_{i=1}^{n-1}|p_ip_{i+1}|$ be its length.\nThen $L = O\\left(c^2\\sqrt{n-1}\\right) \\, |p_1 p_n|$, hence $\\delta_P =O\\left(c^2\\sqrt{n-1}\\right)$.\n\\end{theorem}\n\\begin{proof}\n  We may assume that $p_1 p_n$ is a horizontal segment of unit length.\n  By the argument in the proof of Theorem~\\ref{thm:linear},\n  all points $p_i$ ($i=1,\\dots,n$) are contained in an ellipse $E$ with foci $p_1$ and $p_n$,\n  where the major axis of $E$ has length $c$.\n  Let $U$ be the minimal axis-aligned square containing $E$; its side is of length $c$.\n\nWe set $x=8c^2\/\\sqrt{n-1}$; and let $L_0$ and $L_1$ be the sum of lengths of all edges in $P$\nof length at most $x$ and more than $x$, respectively. By definition, we have $L=L_0+L_1$ and\n\\begin{equation}\\label{eq:short}\nL_0\\leq (n-1)x = (n-1)\\cdot 8c^2\/\\sqrt{n-1} = 8c^2\\sqrt{n-1}.\n\\end{equation}\nWe shall prove that $L_1\\leq 8c^2\\sqrt{n-1}$, implying $L\\leq 2x(n-1)=O\\left(c^2\\sqrt{n-1}\\right)$.\nFor this, we further classify the edges in $L_1$ according to their lengths:\nFor $\\ell=0,1,\\dots , \\infty$, let\n\\begin{equation}\\label{eq:PL1}\nP_\\ell=\\left\\{p_i: 2^\\ell x< |p_ip_{i+1}|\\leq 2^{\\ell+1}x\\right\\}.\n\\end{equation}\nSince all points lie in an ellipse of diameter $c$, we have $|p_ip_{i+1}|\\leq c$, for all $i=0,\\dots, n-1$.\nConsequently, $P_\\ell=\\emptyset$ when $c\\leq 2^\\ell x$, or equivalently $\\log(c\/x)\\leq \\ell$.\n\nWe use a volume argument to derive an upper bound on the cardinality of $P_\\ell$, for\n$\\ell=0,1,\\dots , \\lfloor\\log (c\/x)\\rfloor$.\nAssume that $p_i,p_k\\in P_\\ell$, and w.l.o.g., $i<k$. If $k=i+1$, then by~\\eqref{eq:PL1}, $2^\\ell x<|p_ip_k|$.\nOtherwise,\n\\[\n2^\\ell x<|p_ip_{i+1}| < |p_ip_{i+1}|+|p_{i+1}p_k| \\leq c|p_ip_k|,\n\\text{  or  } \\frac{2^\\ell x}{c} < |p_ip_k|.\n\\]\nConsequently, the disks of radius\n\\begin{equation}\\label{area:Rad}\nR=\\frac{2^\\ell x}{2c} = \\frac{4\\cdot 2^\\ell c }{\\sqrt{n-1}}\n\\end{equation}\ncentered at the points in $P_{\\ell}$ are interior-disjoint.\nThe area of each disk is $\\pi R^2$. Since $P_\\ell\\subset U$, these disks are contained\nin the $R$-neighborhood $U_R$ of the square $U$, \\ie, the\nMinkowski sum $R+U$.\nFor $\\ell \\leq \\log(c\/x)$,\nwe have $2^\\ell x\\leq c$, hence $R=\\frac{2^\\ell x}{2c} \\leq \\frac{c}{2c}=\\frac12 \\leq \\frac{c}{2}$.\nThen we can bound the area of $U_R$ from above as follows:\n\\begin{equation}\\label{area:UR}\n{\\rm area}(U_R)\n< (c+2R)^2\n\\leq (2c)^2\n=4c^2.\n\\end{equation}\nSince $U_R$ contains $|P_\\ell|$ interior-disjoint disks of radius $R$, we obtain\n\\begin{equation}\\label{eq:PL2}\n|P_\\ell|\n\\leq \\frac{{\\rm area}(U_R)}{\\pi R^2}\n< \\frac{4c^2}{\\pi R^2}\n=\\frac{16c^4}{\\pi 2^{2\\ell} x^2}.\n\\end{equation}\nFor every segment $p_{i-1}p_i$ with length more than $x$, we have that $p_i\\in P_\\ell$,\nfor some $\\ell\\in \\{0,1,\\dots , \\lfloor\\log (c\/x)\\rfloor\\}$.\nThe total length of these segments is\n\\begin{align*}\nL_1&\\leq  \\sum_{\\ell=0}^{\\lfloor\\log (c\/x)\\rfloor} |P_\\ell| \\cdot 2^{\\ell+1}x\n <   \\sum_{\\ell=0}^{\\lfloor\\log (x\/c)\\rfloor} \\frac{16c^4}{\\pi 2^{2\\ell} x^2} \\cdot 2^{\\ell+1}x\n =  \\sum_{\\ell=0}^{\\lfloor\\log (x\/c)\\rfloor} \\frac{32 c^4}{\\pi 2^\\ell x}\\\\\n&<   \\frac{32 c^4}{\\pi x} \\sum_{\\ell=0}^\\infty \\frac{1}{2^\\ell}\n =  \\frac{64 c^4}{\\pi x}\n =   \\frac{8 c^2}{\\pi}\\cdot \\sqrt{n-1},\n\\end{align*}\nas required. Together with \\eqref{eq:short}, this yields $\nL\\leq 8\\left(1+c^2\/\\pi\\right)\\cdot \\sqrt{n-1}$.\n\\end{proof}\n\n\n\n\\section{Lower Bounds} \\label{sec:lower}\n\nWe now present our lower bound construction, showing that the dependence on $n$\nfor the stretch factor of a $c$-chain cannot be avoided.\n\n\\begin{theorem}\\label{thm:lower-bound}\nFor every constant $c\\geq 4$, there is a set $\\P_c=\\{P^k\\}_{k\\in\\NN}$ of simple $c$-chains,\nso that $P^k$ has $n=4^k+1$ vertices and stretch factor $(n-1)^{\\frac{1+\\log (c-2) - \\log c}{2}}$.\n\\end{theorem}\n\nBy Theorem~\\ref{thm:1\/2}, the stretch factor of a $c$-chain in the plane\nis $O\\left((n-1)^{1\/2}\\right)$ for every constant $c\\geq 1$.\nSince\n\\[ \\lim_{c \\to\\infty} \\frac{1+\\log (c-2) - \\log c}{2} =\\frac12, \\]\nour lower bound construction shows that the limit of the exponent cannot be improved.\nIndeed, for every $\\eps>0$, we can set $c=\\frac{2^{2\\eps+1}}{2^{2\\eps}-1}$,\nand then the chains above have stretch factor\n$(n-1)^{\\frac{1+\\log(c-2)-\\log c}{2}}=(n-1)^{1\/2-\\eps}=\\Omega(n^{1\/2-\\eps})$.\n\nWe first construct a family $\\P_c=\\{P^k\\}_{k\\in\\NN}$ of polygonal chains.\nThen we show, in Lemmata~\\ref{lemma:simple} and~\\ref{lemma:c-chain},\nthat every chain in $\\P_c$ is simple and indeed a $c$-chain.\nThe theorem follows since the claimed stretch factor is a consequence of the construction.\n\n\\subparagraph*{Construction of $\\P_c$.}\nThe construction here is a generalization of the iterative construction of the \\emph{Koch curve};\nwhen $c=6$, the result is the original Ces\\`aro fractal (which is a variant of the Koch curve)~\\cite{Ces05}.\nWe start with a unit line segment $P^0$, and for $k=0, 1, \\dots$,\nwe construct $P^{k+1}$ by replacing each segment in $P^k$ by four segments such that\nthe middle three points achieve a stretch factor of $c_*=\\frac{c-2}{2}$ (this choice will be justified\nin the proof of Lemma~\\ref{lemma:c-chain}). Note that $c_*\\geq 1$, since $c\\geq 4$.\n\nWe continue with the details. Let $P^0$ be the unit line segment from $(0,0)$ to $(1,0)$;\nsee Figure~\\ref{fig:p0p1}\\,(left).\nGiven the polygonal chain $P^k$ $(k=0,1,\\dots$), we construct $P^{k+1}$ by replacing\neach segment of $P^k$ by four segments as follows. Consider a segment of $P^k$,\nand denote its length by $\\ell$. Subdivide this segment into three segments of\nlengths $(\\frac{1}{2}-\\frac{a}{c_*})\\ell$, $\\frac{2a}{c_*}\\ell$, and $(\\frac{1}{2}-\\frac{a}{c_*})\\ell$,\nrespectively, where $0<a<\\frac{c_*}{2}$ is a parameter to be determined later.\nReplace the middle segment with the top part of an isosceles triangle of side length $a\\ell$.\nThe chains $P^0$, $P^1$, $P^2$, and $P^4$ are depicted in Figures~\\ref{fig:p0p1} and~\\ref{fig:p2p4}.\n\n\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0.000, 0.000)node[circle, fill, inner sep=1pt, label=below:{$(0,0)$}]{} --\nnode[above]{1} (10.000, 0.000)node[circle, fill, inner sep=1pt, label=below:{$(1,0)$}]{};\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0.000, 0.000)node[circle, fill, inner sep=1pt, label=below:{$(0,0)$}]{}\n-- node[above]{$\\frac{1}{2}-\\frac{a}{c_*}$}\n(4.545, 0.000) node[circle, fill, inner sep=1pt]{}\n-- node[left]{$a$}(5.000, 4.523) node[circle, fill, inner sep=1pt]{}\n-- node[right]{$a$}\n(5.455, 0.000) node[circle, fill, inner sep=1pt]{}\n-- node[above]{$\\frac{1}{2}-\\frac{a}{c_*}$}\n(10.000, 0.000)node[circle, fill, inner sep=1pt, label=below:{$(1,0)$}]{} ;\n\\draw[dashed] (4.545, 0.000) -- node[above]{$\\frac{2a}{c_*}$} (5.455, 0.000);\n\\end{tikzpicture}\n\\caption{The chains $P^0$ (left)  and $P^1$ (right).}\\label{fig:p0p1}\n\\end{figure}\n\nNote that each segment of length $\\ell$ in $P^k$ is replaced by four segments of total\nlength $(1+\\frac{2a(c_*-1)}{c_*})\\ell$.\nAfter $k$ iterations, the chain $P^k$ consists of $4^k$ line segments of total\nlength $\\left(1+\\frac{2a(c_*-1)}{c_*}\\right)^k$.\n\nBy construction, the chain $P^k$ (for $k\\geq 1$) consists of four scaled copies of $P^{k-1}$.\nFor $i=1,2,3,4$, let the \\emph{$i$th subchain of $P^k$} be the\nsubchain of $P^k$ consisting of $4^{k-1}$ segments starting from the $((i-1)4^{k-1}+1)$th segment.\nBy construction, the $i$th subchain of $P^k$ is similar to the chain $P^{k-1}$,\nfor $i=1,2,3,4$.\\footnote{Two geometric shapes are \\emph{similar} if one can be obtained from\n  the other by translation, rotation, and scaling; and are \\emph{congruent} if one can be obtained\n  from the other by translation and rotation.}\nThe following functions allow us to refer to these subchains formally.\nFor $i=1,2,3,4$, define a function $f^k_i: P^k\\to P^k$ as the identity on the $i$th subchain of $P^k$ that\nsends the remaining part(s) of $P^k$ to the closest endpoint(s) along this subchain.\nSo $f^k_i(P^k)$ is similar to $P^{k-1}$. Let $g_i: \\P_c\\setminus\\{P^0\\} \\to\\P_c$ be a piecewise defined\nfunction such that $g_i(C)=\\sigma^{-1}\\circ f^k_i\\circ\\sigma(C)$ if $C$ is similar to $P^k$,\nwhere $\\sigma: C\\to P^k$ is a similarity transformation.\nApplying the function $g_i$ on a chain $P^k$ can be thought of as ``cutting out''\nits $i$th subchain.\n\n\\tikzset{p4\/.pic={code={\n\\draw (0.000, 0.000) -- (0.427, 0.000) -- (0.470, 0.425) -- (0.512,\n0.000) -- (0.939, 0.000) -- (0.982, 0.425) -- (0.563, 0.510) --\n(0.990, 0.510) -- (1.033, 0.934) -- (1.076, 0.510) -- (1.503, 0.510)\n-- (1.084, 0.425) -- (1.127, 0.000) -- (1.554, 0.000) -- (1.597,\n0.425) -- (1.639, 0.000) -- (2.066, 0.000) -- (2.109, 0.425) --\n(1.690, 0.510) -- (2.117, 0.510) -- (2.160, 0.934) -- (1.742, 1.019)\n-- (1.615, 0.612) -- (1.658, 1.036) -- (1.240, 1.121) -- (1.667,\n1.121) -- (1.709, 1.546) -- (1.752, 1.121) -- (2.179, 1.121) --\n(2.222, 1.546) -- (1.803, 1.631) -- (2.230, 1.631) -- (2.273, 2.056)\n-- (2.315, 1.631) -- (2.742, 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(6.018, 0.510) -- (6.445, 0.510) -- (6.488, 0.934) -- (6.530,\n0.510) -- (6.957, 0.510) -- (6.539, 0.425) -- (6.582, 0.000) --\n(7.008, 0.000) -- (7.051, 0.425) -- (7.094, 0.000) -- (7.521, 0.000)\n-- (7.563, 0.425) -- (7.145, 0.510) -- (7.572, 0.510) -- (7.615,\n0.934) -- (7.196, 1.019) -- (7.070, 0.612) -- (7.113, 1.036) --\n(6.694, 1.121) -- (7.121, 1.121) -- (7.164, 1.546) -- (7.206, 1.121)\n-- (7.633, 1.121) -- (7.676, 1.546) -- (7.258, 1.631) -- (7.685,\n1.631) -- (7.727, 2.056) -- (7.770, 1.631) -- (8.197, 1.631) --\n(7.778, 1.546) -- (7.821, 1.121) -- (8.248, 1.121) -- (8.291, 1.546)\n-- (8.333, 1.121) -- (8.760, 1.121) -- (8.342, 1.036) -- (8.385,\n0.612) -- (8.258, 1.019) -- (7.840, 0.934) -- (7.883, 0.510) --\n(8.310, 0.510) -- (7.891, 0.425) -- (7.934, 0.000) -- (8.361, 0.000)\n-- (8.403, 0.425) -- (8.446, 0.000) -- (8.873, 0.000) -- (8.916,\n0.425) -- (8.497, 0.510) -- (8.924, 0.510) -- (8.967, 0.934) --\n(9.010, 0.510) -- (9.437, 0.510) -- (9.018, 0.425) -- (9.061, 0.000)\n-- (9.488, 0.000) -- (9.530, 0.425) -- (9.573, 0.000) -- (10.000,\n0.000);\n}}}\n\n\\begin{figure}[!ht]\n\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0.000, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (2.066, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (2.273, 2.056)node[circle, fill, inner sep=1pt]{}\n -- (2.479, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (4.545, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (4.752, 2.056)node[circle, fill, inner sep=1pt]{}\n -- (2.727, 2.467)node[circle, fill, inner sep=1pt]{}\n -- (4.793, 2.467)node[circle, fill, inner sep=1pt]{}\n -- (5.000, 4.523)node[circle, fill, inner sep=1pt]{}\n -- (5.207, 2.467)node[circle, fill, inner sep=1pt]{}\n -- (7.273, 2.467)node[circle, fill, inner sep=1pt]{}\n -- (5.248, 2.056)node[circle, fill, inner sep=1pt]{}\n -- (5.455, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (7.521, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (7.727, 2.056)node[circle, fill, inner sep=1pt]{}\n -- (7.934, 0.000)node[circle, fill, inner sep=1pt]{}\n -- (10.000, 0.000)node[circle, fill, inner sep=1pt]{};\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}\n\\path (0,0) pic[scale=0.5] {p4};\n\\end{tikzpicture}\n\\caption{The chains $P^2$ (left) and $P^4$ (right).}\\label{fig:p2p4}\n\\end{figure}\n\nClearly, the stretch factor of the chain monotonically increases with the parameter $a$.\nHowever, if $a$ is too large, the chain is no longer simple.\nThe following lemma gives a sufficient condition for the constructed chains to avoid self-crossings.\n\n\\begin{lemma}\\label{lemma:simple}\nFor every constant $c\\geq 4$, if $a\\leq \\frac{c-2}{2c}$, then every chain in $\\P_c$ is simple.\n\\end{lemma}\n\\begin{proof}\n  Let $T=\\conv(P^1)$. Observe that $T$ is an isosceles triangle;\n  see Figure~\\ref{fig:bounding-triangle}\\,(left). We first show the following:\n\n\\begin{claim*}If $a\\leq \\frac{c-2}{2c}$, then $\\conv(P^k) = T$\n    for all $k \\geq 1$.\n\\end{claim*}\n\n\\begin{claimproof}\nWe prove the claim by induction on $k$. It holds for $k=1$ by definition.\nFor the induction step, assume that $k\\geq 2$ and that the claim holds for $k-1$.\nConsider the chain $P^{k}$. Since it contains all the vertices of $P^1$, $T\\subset \\conv(P^{k})$.\nSo we only need to show that $\\conv(P^{k})\\subset T$.\n\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!15] (0.000, 0.000) -- (5.000, 4.623) -- (10.000, 0.000) --\ncycle; \n\\draw (0.000, 0.000) -- node[above]{$\\frac{1}{2}-\\frac{a}{c_*}$} (4.545, 0.000)\n-- node[left]{$a$} (5.000, 4.523)\n-- node[right]{$a$} (5.455, 0.000) -- node[above]{$\\frac{1}{2}-\\frac{a}{c_*}$} (10.000, 0.000);\n\\draw[dashed] (4.545, 0.000) -- node[above]{$\\frac{2a}{c_*}$} (5.455, 0.000);\n\\end{tikzpicture}\n\\hspace{.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!15] (0.000, 0.000) -- (5.000, 4.623) -- (10.000, 0.000) -- cycle;\n\\fill[black!25] (0.000, 0.000) -- (2.273, 2.056) -- (4.545, 0.000) --cycle;\n\\fill[black!25] (4.545, 0.000) -- (2.727, 2.467) -- (5.000, 4.623) --cycle;\n\\fill[black!25] (5.000, 4.623) -- (7.273, 2.467) -- (5.455, 0.000) --cycle;\n\\fill[black!25] (5.455, 0.000) -- (7.727, 2.056) -- (10.000, 0.000) --cycle;\n\\draw (0.000, 0.000) -- (2.066, 0.000)\n-- (2.273, 2.056) -- (2.479, 0.000) -- (4.545, 0.000) -- (4.752, 2.056)\n-- (2.727, 2.467)node[circle, fill, inner sep=1pt, label=above:$p$]{}\n-- (4.793, 2.467)\n-- (5.000, 4.523)node[circle, fill, inner sep=1pt, label=above:$t$]{}\n-- (5.207, 2.467) -- (7.273, 2.467) -- (5.248, 2.056) -- (5.455, 0.000)\n-- (7.521, 0.000) -- (7.727, 2.056) -- (7.934, 0.000) -- (10.000, 0.000);\n\\node (label1) at (0, 3.5) {\\small $a\\left(\\frac{1}{2}-\\frac{a}{c_*}\\right)$};\n\\draw [-latex] (label1.south) to [bend right=20] (2.211, 1.439);\n\\node (label2) at (-0.5, 1) {\\small $\\left(\\frac{1}{2}-\\frac{a}{c_*}\\right)^2$};\n\\draw [-latex] (0.8, 1) to [bend left=20] (1.5, 0.1);\n\\end{tikzpicture}\n\\caption{Left: Convex hull $T$ of $P^1$ in light gray;\n Right: Convex hulls of $g_i(P^2)$, $i=1,2,3,4$, in dark gray, \n are contained in $T$.}\n\\label{fig:bounding-triangle}\n\\end{figure}\n\nBy construction, $P^{k}\\subset \\bigcup_{i=1}^4 \\conv(g_i(P^{k}))$;\nsee Figure~\\ref{fig:bounding-triangle}\\,(right).\nBy the inductive hypothesis, $\\conv(g_i(P^{k}))$ is an isosceles triangle similar to $T$, for $i=1,2,3,4$.\nSince the bases of $\\conv(g_1(P^{k}))$ and $\\conv(g_4(P^{k}))$ are\ncollinear with the base of $T$ by construction, due to similarity, they are contained in $T$.\nThe base of $\\conv(g_2(P^{k}))$ is contained in $T$. In order to show\n$\\conv(g_2(P^{k}))\\subset T$, by convexity, it suffices to ensure that its apex $p$ is also in $T$.\nNote that the coordinates of the top point is $t=\\left(1\/2, a\\sqrt{c_*^2-1}\/c_* \\right)$,\nso the supporting line $\\ell$ of the left side of $T$ is\n\\begin{align*}\ny &=\\frac{2a\\sqrt{c_*^2-1}}{c_*}x, \\text{ and} \\\\\np &=\\left(\\frac{1}{2}-\\frac{a}{2c_*}-\\frac{a^2\\left(c_*^2-1\\right)}{c_*^2},\n\\left(\\frac{a}{2c_*}+\\frac{a^2}{c_*^2}\\right)\\sqrt{c_*^2-1}\\right).\n\\end{align*}\nBy the condition of $a\\leq \\frac{c-2}{2c}=\\frac{c_*}{2(c_*+1)}$ in the lemma,\n$p$ lies on or below $\\ell$.\nUnder the same condition, we have $\\conv(g_3(P^{k})) \\subset T$ by symmetry.\nThen $P^{k}\\subset \\bigcup_{i=1}^4 \\conv(g_i(P^{k}))\\subset T$.\nSince $T$ is convex, $\\conv(P^k)\\subset T$.\nSo $\\conv(P^k) = T$, as claimed.\n\\end{claimproof}\n\nWe can now finish the proof of Lemma~\\ref{lemma:simple} by induction.\nClearly, $P^0$ and $P^1$ are simple. Assume that $k \\geq 2$, and $P^{k-1}$ is simple.\nConsider the chain $P^{k}$. For $i=1,2,3,4$, $g_i(P^{k})$ is similar to $P^{k-1}$,\nhence simple by the inductive hypothesis.\nSince $P^{k}=\\bigcup_{i=1}^4 g_i(P^{k})$, it is sufficient to show that for all $i,j\\in\\{1,2,3,4\\}$,\nwhere $i\\neq j$, a segment in $g_i(P^{k})$ does not intersect any segments in $g_j(P^{k})$,\nunless they are consecutive in $P^{k}$ and they intersect at a common endpoint.\nThis follows from the above claim together with the observation that for $i\\neq j$, the intersection\n$g_i(P^{k})\\cap g_j(P^{k})$ is either empty or contains a single vertex which is the common\nendpoint of two consecutive segments in $P^{k}$.\n\\end{proof}\nIn the remainder of this section, we assume that\n\\begin{equation}\na=\\frac{c-2}{2c}=\\frac{c_*}{2(c_*+1)}.\n\\end{equation}\nUnder this assumption, all segments in $P^1$ have the same length $a$.\nTherefore, by construction, all segments in $P^k$ have the same length\n\\[\na^k=\\left(\\frac{c_*}{2(c_*+1)}\\right)^k.\n\\]\nThere are $4^k$ segments in $P^k$, with $4^k+1$ vertices, and its stretch factor is\n\\[\\delta_{P^k} = 4^k\\left(\\frac{c_*}{2(c_*+1)}\\right)^k=\\left(\\frac{2c_*}{c_*+1}\\right)^k.\\]\nConsequently, $k=\\log_4 (n-1) = \\frac{\\log (n-1)}{2}$, and\n\\[\n\\delta_{P^k} =\\left(\\frac{2c_*}{c_*+1}\\right)^{\\frac{\\log (n-1)}{2}}\n=\\left(\\frac{2c-4}{c}\\right)^{\\frac{\\log (n-1)}{2}}\n=(n-1)^{\\frac{1+\\log (c-2) - \\log c}{2}},\n\\]\nas claimed.\nTo finish the proof of Theorem~\\ref{thm:lower-bound}, it remains to show the\nconstructed polygonal chains are indeed\n$c$-chains.\n\n\\begin{lemma}\\label{lemma:c-chain}\nFor every constant $c\\geq 4$, $\\P_c$ is a family of $c$-chains.\n\\end{lemma}\n\n\nWe first prove a couple of facts that will be useful in the proof of Lemma~\\ref{lemma:c-chain}.\nWe defer an intuitive explanation until after the formal statement of the lemma.\n\\begin{lemma}\\label{lemma:technical}\nLet $k \\geq 1$ and let $P^k=(p_1, p_2, \\dots, p_n)$, where $n=4^k+1$.\nThen the following hold:\n\\begin{enumerate}[{\\rm(i)}]\n\\item\\label{pf:f1}\nThere exists a sequence $(q_1, q_2, \\dots, q_m)$ of $m=2\\cdot 4^{k-1}$ points in $\\RR^2$ such\nthat the chain $R^k=(p_1, q_1, p_2, q_2, \\dots, p_m, q_m, p_{m+1})$ is similar to $P^k$.\n\\item\\label{pf:f2}\n  For $k\\geq 2$, define\n  $g_5: \\P_c\\setminus\\{P^0, P^1\\} \\to \\P_c$ by\n\\[\ng_5(P^k) = \\left(g_3\\circ g_2(P^k)\\right) \\cup \\left(g_4\\circ g_2(P^k)\\right) \\cup\n\\left(g_1\\circ g_3(P^k)\\right) \\cup \\left(g_2\\circ g_3(P^k)\\right).\n\\]\nThen $g_5(P^k)$ is similar to $P^{k-1}$.\n\\end{enumerate}\n\\end{lemma}\n\nPart~(\\ref{pf:f1}) of Lemma~\\ref{lemma:technical} says that given $P^k$,\nwe can construct a chain $R^k$ similar to $P^k$\nby inserting one point between every two consecutive points of the left half of $P^k$,\nsee Figure~\\ref{fig:technical}\\,(left).\nPart~(\\ref{pf:f2}) says that the ``top'' subchain of $P^k$ that consists of the right half of $g_2(P^k)$\nand the left half of $g_3(P^k)$, see Figure~\\ref{fig:technical}\\,(right), is similar to $P^{k-1}$.\n\n\\begin{figure}[ht!]\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\path (0,0) pic[scale=0.5] {p4};\n\\draw[red] (0.000, 0.000) -- (0.213, 0.193) -- (0.427, 0.000) -- (0.256, 0.232) -- (0.470, 0.425) -- (0.683, 0.232) -- (0.512, 0.000) -- (0.726, 0.193) -- (0.939, 0.000) -- (0.768, 0.232) -- (0.982, 0.425) -- (0.734, 0.278) -- (0.563, 0.510) -- (0.777, 0.703) -- (0.990, 0.510) -- (0.820, 0.741) -- (1.033, 0.934) -- (1.246, 0.741) -- (1.076, 0.510) -- (1.289, 0.703) -- (1.503, 0.510) -- (1.332, 0.278) -- (1.084, 0.425) -- (1.298, 0.232) -- (1.127, 0.000) -- (1.340, 0.193) -- (1.554, 0.000) -- (1.383, 0.232) -- (1.597, 0.425) -- (1.810, 0.232) -- (1.639, 0.000) -- (1.853, 0.193) -- (2.066, 0.000) -- (1.895, 0.232) -- (2.109, 0.425) -- (1.861, 0.278) -- (1.690, 0.510) -- (1.904, 0.703) -- (2.117, 0.510) -- (1.947, 0.741) -- (2.160, 0.934) -- (1.912, 0.788) -- (1.742, 1.019) -- (1.863, 0.758) -- (1.615, 0.612) -- (1.445, 0.843) -- (1.658, 1.036) -- (1.410, 0.890) -- (1.240, 1.121) -- (1.453, 1.314) -- (1.667, 1.121) -- (1.496, 1.353) -- (1.709, 1.546) -- (1.923, 1.353) -- (1.752, 1.121) -- (1.965, 1.314) -- (2.179, 1.121) -- (2.008, 1.353) -- (2.222, 1.546) -- (1.974, 1.399) -- (1.803, 1.631) -- (2.017, 1.824) -- (2.230, 1.631) -- (2.059, 1.863) -- (2.273, 2.056) -- (2.486, 1.863) -- (2.315, 1.631) -- (2.529, 1.824) -- (2.742, 1.631) -- (2.572, 1.399) -- (2.324, 1.546) -- (2.537, 1.353) -- (2.367, 1.121) -- (2.580, 1.314) -- (2.794, 1.121) -- (2.623, 1.353) -- (2.836, 1.546) -- (3.050, 1.353) -- (2.879, 1.121) -- (3.092, 1.314) -- (3.306, 1.121) -- (3.135, 0.890) -- (2.887, 1.036) -- (3.101, 0.843) -- (2.930, 0.612) -- (2.683, 0.758) -- (2.804, 1.019) -- (2.633, 0.788) -- (2.385, 0.934) -- (2.599, 0.741) -- (2.428, 0.510) -- (2.642, 0.703) -- (2.855, 0.510) -- (2.684, 0.278) -- (2.437, 0.425) -- (2.650, 0.232) -- (2.479, 0.000) -- (2.693, 0.193) -- (2.906, 0.000) -- (2.735, 0.232) -- (2.949, 0.425) -- (3.162, 0.232) -- (2.992, 0.000) -- (3.205, 0.193) -- (3.418, 0.000) -- (3.248, 0.232) -- (3.461, 0.425) -- (3.214, 0.278) -- (3.043, 0.510) -- (3.256, 0.703) -- (3.470, 0.510) -- (3.299, 0.741) -- (3.512, 0.934) -- (3.726, 0.741) -- (3.555, 0.510) -- (3.769, 0.703) -- (3.982, 0.510) -- (3.811, 0.278) -- (3.564, 0.425) -- (3.777, 0.232) -- (3.606, 0.000) -- (3.820, 0.193) -- (4.033, 0.000) -- (3.862, 0.232) -- (4.076, 0.425) -- (4.289, 0.232) -- (4.119, 0.000) -- (4.332, 0.193) -- (4.545, 0.000) -- (4.375, 0.232) -- (4.588, 0.425) -- (4.341, 0.278) -- (4.170, 0.510) -- (4.383, 0.703) -- (4.597, 0.510) -- (4.426, 0.741) -- (4.639, 0.934) -- (4.392, 0.788) -- (4.221, 1.019) -- (4.342, 0.758) -- (4.095, 0.612) -- (3.924, 0.843) -- (4.137, 1.036) -- (3.890, 0.890) -- (3.719, 1.121) -- (3.932, 1.314) -- (4.146, 1.121) -- (3.975, 1.353) -- (4.189, 1.546) -- (4.402, 1.353) -- (4.231, 1.121) -- (4.445, 1.314) -- (4.658, 1.121) -- (4.487, 1.353) -- (4.701, 1.546) -- (4.453, 1.399) -- (4.282, 1.631) -- (4.496, 1.824) -- (4.709, 1.631) -- (4.539, 1.863) -- (4.752, 2.056) -- (4.504, 1.909) -- (4.334, 2.141) -- (4.455, 1.880) -- (4.207, 1.733) -- (4.037, 1.965) -- (4.250, 2.158) -- (4.002, 2.011) -- (3.832, 2.243) -- (3.953, 1.982) -- (3.705, 1.835) -- (3.977, 1.929) -- (4.098, 1.668) -- (3.851, 1.522) -- (3.680, 1.753) -- (3.801, 1.492) -- (3.554, 1.346) -- (3.383, 1.577) -- (3.596, 1.770) -- (3.349, 1.624) -- (3.178, 1.855) -- (3.392, 2.048) -- (3.605, 1.855) -- (3.434, 2.087) -- (3.648, 2.280) -- (3.400, 2.133) -- (3.229, 2.365) -- (3.351, 2.104) -- (3.103, 1.957) -- (2.932, 2.189) -- (3.146, 2.382) -- (2.898, 2.235) -- (2.727, 2.467) -- (2.941, 2.660) -- (3.154, 2.467) -- (2.983, 2.699) -- (3.197, 2.892) -- (3.410, 2.699) -- (3.240, 2.467) -- (3.453, 2.660) -- (3.666, 2.467) -- (3.496, 2.699) -- (3.709, 2.892) -- (3.462, 2.745) -- (3.291, 2.977) -- (3.504, 3.170) -- (3.718, 2.977) -- (3.547, 3.208) -- (3.760, 3.401) -- (3.974, 3.208) -- (3.803, 2.977) -- (4.016, 3.170) -- (4.230, 2.977) -- (4.059, 2.745) -- (3.812, 2.892) -- (4.025, 2.699) -- (3.854, 2.467) -- (4.068, 2.660) -- (4.281, 2.467) -- (4.110, 2.699) -- (4.324, 2.892) -- (4.537, 2.699) -- (4.367, 2.467) -- (4.580, 2.660) -- (4.793, 2.467) -- (4.623, 2.699) -- (4.836, 2.892) -- (4.588, 2.745) -- (4.418, 2.977) -- (4.631, 3.170) -- (4.845, 2.977) -- (4.674, 3.208) -- (4.887, 3.401) -- (4.640, 3.255) -- (4.469, 3.486) -- (4.590, 3.225) -- (4.343, 3.079) -- (4.172, 3.310) -- (4.385, 3.503) -- (4.138, 3.357) -- (3.967, 3.588) -- (4.180, 3.781) -- (4.394, 3.588) -- (4.223, 3.820) -- (4.437, 4.013) -- (4.650, 3.820) -- (4.479, 3.588) -- (4.693, 3.781) -- (4.906, 3.588) -- (4.735, 3.820) -- (4.949, 4.013) -- (4.701, 3.866) -- (4.530, 4.098) -- (4.744, 4.291) -- (4.957, 4.098) -- (4.787, 4.330) -- (5.000, 4.523);\n\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (2.727, 2.467) -- (5.000, 4.623) -- (7.273, 2.467) --cycle;\n\\path (0,0) pic[scale=0.5] {p4};\n\\end{tikzpicture}\n\\caption{Left: Chain $P^k$ with the scaled copy of itself $R^k$ (in red);\nRight: Chain $P^k$ with its subchain $g_5(P^k)$ marked by its\nconvex hull.}\\label{fig:technical}\n\\end{figure}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:technical}]\nFor~(\\ref{pf:f1}), we review the construction of $P^k$, and show that $R^k$ and $P^k$ can be\nconstructed in a coupled manner.\nIn Figure~\\ref{fig:induced-chain}\\,(left), consider $P^1=(p_1, p_2, p_3, p_4, p_5)$.\nRecall that all segments in $P^1$ are of the same length $a=\\frac{c_*}{2(c_*+1)}$.\nThe isosceles triangles $\\Delta p_1p_2p_3$ and $\\Delta p_1p_3p_5$ are similar.\nLet $\\sigma: \\Delta p_1p_3p_5 \\to \\Delta p_1p_2p_3$ be the similarity transformation.\nLet $q_1=\\sigma(p_2)$ and $q_2 = \\sigma(p_4)$.\nBy construction, the chain $R^1 = (p_1, q_1, p_2, q_2, p_3)$ is similar to $P^1$.\nIn particular, all of its segments have the same length.\nSo the isosceles triangle $\\Delta p_1q_1p_2$ is similar to $\\Delta p_1p_3p_5$.\nMoreover, its base is the segment $p_1p_2$, so $\\Delta p_1q_1p_2$ is precisely\n$\\conv(g_1(P^2))$, see Figure~\\ref{fig:induced-chain}\\,(right).\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0.000, 0.000)node[circle, fill, inner sep=1pt, label=above:$p_1$]{}\n-- (4.545, 0.000)node[circle, fill, inner sep=1pt, label=85:$p_2$]{}\n-- (5.000, 4.523)node[circle, fill, inner sep=1pt, label=left:$p_3$]{}\n-- (5.455, 0.000)node[circle, fill, inner sep=1pt, label=45:$p_4$]{}\n-- (10.000, 0.000)node[circle, fill, inner sep=1pt, label=93:$p_5$]{};\n\\draw[red] (0.000, 0.000)\n-- (2.273, 2.056)node[circle, fill, inner sep=1pt, label=left:$q_1$]{}\n-- (4.545, 0.000)\n-- (2.727, 2.467)node[circle, fill, inner sep=1pt, label=above:$q_2$]{}\n-- (5.000, 4.523);\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[red] (0.000, 0.000)\n-- (2.273, 2.056) -- (4.545, 0.000) -- (2.727, 2.467) -- (5.000, 4.523);\n\\draw (0.000, 0.000)node[circle, fill, inner sep=1pt, label=above:$v_1$]{}\n-- (2.066, 0.000)node[circle, fill, inner sep=1pt, label=135:$v_2$]{}\n-- (2.273, 2.056)node[circle, fill, inner sep=1pt, label=left:$v_{3}$]{}\n-- (2.479, 0.000)node[circle, fill, inner sep=1pt, label=45:$v_{4}$]{}\n-- (4.545, 0.000)node[circle, fill, inner sep=1pt, label=85:$v_{5}$]{}\n-- (4.752, 2.056)node[circle, fill, inner sep=1pt, label=225:$v_{6}$]{}\n-- (2.727, 2.467)node[circle, fill, inner sep=1pt, label=above:$v_{7}$]{}\n-- (4.793, 2.467)node[circle, fill, inner sep=1pt, label=135:$v_{8}$]{}\n-- (5.000, 4.523)node[circle, fill, inner sep=1pt, label=left:$v_{9}$]{}\n-- (5.207, 2.467)node[circle, fill, inner sep=1pt, label=45:$v_{10}$]{}\n-- (7.273, 2.467)node[circle, fill, inner sep=1pt, label=above:$v_{11}$]{}\n-- (5.248, 2.056) -- (5.455, 0.000)\n-- (7.521, 0.000) -- (7.727, 2.056) -- (7.934, 0.000)\n-- (10.000, 0.000)node[circle, fill, inner sep=1pt, label=93:$v_{17}$]{};\n\\end{tikzpicture}\n\\caption{Left: the chains $P^1$ and $R^1$ (red);\nRight: the chains $P^2$ and $R^1$ (red).}\\label{fig:induced-chain}\n\\end{figure}\n\nWrite $P^2=(v_{1}, v_{2}, \\dots, v_{17})$, then $v_{3}=q_1$ by the above argument\nand $v_{7}=q_2$ by symmetry.\nNow $\\Delta v_{1}v_{2}v_{3}$, $\\Delta v_{3}v_{4}v_{5}$, $\\Delta v_{5}v_{6}v_{7}$, and\n$\\Delta v_{7}v_{8}v_{9}$ are four congruent isosceles triangles, all of which are\nsimilar to $\\Delta v_{1}v_{9}v_{17}$, since the angles are the same.\nRepeat the above procedure on each of them to obtain\n$R^2=(v_{1}, u_1, v_{2}, u_2, \\dots, v_{8}, u_8, v_{9})$, which is similar to $P^2$.\nContinue this construction inductively to get the desired chain $R^k$ for any\n$k\\geq 1$.\n\nFor~(\\ref{pf:f2}), see Figure~\\ref{fig:induced-chain}\\,(right).\nBy definition, $g_5(P^2)$ is the subchain $(v_{7}, v_{8}, v_{9}, v_{10}, v_{11})$.\nObserve that the segments $v_{7}v_{8}$ and $v_{10}v_{11}$ are collinear by symmetry.\nMoreover, they are parallel to $v_{1}v_{17}$ since\n$\\angle v_{7}v_{8}v_{9} = \\angle v_{1}v_{5}v_{9}$.\nSo $g_5(P^2)$ is similar to $P^1$; see Figure~\\ref{fig:induced-chain}\\,(left).\nThen for $k\\geq 2$, $g_5(P^k)$ is the subchain of $P^k$ starting at vertex $v_{7}$,\nending at vertex $v_{11}$.\nBy the construction of $P^k$, $g_5(P^k)$ is similar to $P^{k-1}$.\n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:c-chain}]\nWe proceed by induction on $k$ again.\nThe claim is vacuously true for $P^0$.\nFor $P^1$, among all ten choices of $1\\leq i<j<k\\leq 5$,\n$\\frac{|p_2p_3|+|p_3p_4|}{|p_2p_4|}=c_*=\\frac{c-2}{2}<c$ is the largest,\nand so $P^1$ is also a $c$-chain.\nAssume that $m\\geq 2$ and $P^{m-1}$ is a $c$-chain.\nWe need to show that $P^m$ is also a $c$-chain.\nConsider a triplet of vertices $\\{p_i, p_j, p_k\\}\\subset P^m$, where $1\\leq i< j< k\\leq n=4^m+1$.\n\nRecall that $P^m$ consists of four copies of the subchain $P^{m-1}$, namely $g_1(P^m)$, $g_2(P^m)$,\n$g_3(P^m)$, and $g_4(P^m)$, see Figure~\\ref{fig:c-chain}\\,(left).\nIf $\\{p_i, p_j, p_k\\}\\subset g_l(P^{m})$ for any $l=1,2,3,4$, then by the induction hypothesis,\n\\[\n\\frac{|p_ip_j|+|p_jp_k|}{|p_ip_k|}\\leq c.\n\\]\nSo we may assume that $p_i$ and $p_k$ belong to two different $g_l(P^m)$'s.\nThere are four cases to consider up to symmetry:\n\\begin{enumerate}[{Case} 1.]\n\\item\\label{pf:c-chain-case1} $p_i\\in g_1(P^m)$ and $p_k\\in g_2(P^m)$;\n\\item $p_i\\in g_1(P^m)$ and $p_k\\in g_3(P^m)$;\n\\item $p_i\\in g_1(P^m)$ and $p_k\\in g_4(P^m)$;\n\\item\\label{pf:c-chain-case4} $p_i\\in g_2(P^m)$ and $p_k\\in g_3(P^m)$.\n\\end{enumerate}\n\n\\begin{figure}[htbp!]\n \\centering\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (0.000, 0.000) -- (2.273, 2.056) -- (4.545, 0.000) --cycle;\n\\fill[black!25] (4.545, 0.000) -- (2.727, 2.467) -- (5.000, 4.623) --cycle;\n\\fill[black!25] (5.000, 4.623) -- (7.273, 2.467) -- (5.455, 0.000) --cycle;\n\\fill[black!25] (5.455, 0.000) -- (7.727, 2.056) -- (10.000, 0.000) --cycle;\n\\draw[dashed] (4.545, 0.000) -- node[below]{$\\scriptstyle\\frac{1}{c_*+1}$} (5.455, 0.000);\n\\path (0,0) pic[scale=0.5] {p4};\n\\end{tikzpicture}\n\\hspace*{0.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\path (0,0) pic[scale=0.5] {p4};\n\\draw[red] (0.000, 0.000) -- (0.213, 0.193) -- (0.427, 0.000) -- (0.256, 0.232) -- (0.470, 0.425) -- (0.683, 0.232) -- (0.512, 0.000) -- (0.726, 0.193) -- (0.939, 0.000) -- (0.768, 0.232) -- (0.982, 0.425) -- (0.734, 0.278) -- (0.563, 0.510) -- (0.777, 0.703) -- (0.990, 0.510) -- (0.820, 0.741) -- (1.033, 0.934) -- (1.246, 0.741) -- (1.076, 0.510) -- (1.289, 0.703) -- (1.503, 0.510) -- (1.332, 0.278) -- (1.084, 0.425) -- (1.298, 0.232) -- (1.127, 0.000) -- (1.340, 0.193) -- (1.554, 0.000) -- (1.383, 0.232) -- (1.597, 0.425) -- (1.810, 0.232) -- (1.639, 0.000) -- (1.853, 0.193) -- (2.066, 0.000) -- (1.895, 0.232) -- (2.109, 0.425) -- (1.861, 0.278) -- (1.690, 0.510) -- (1.904, 0.703) -- (2.117, 0.510) -- (1.947, 0.741) -- (2.160, 0.934) -- (1.912, 0.788) -- (1.742, 1.019) -- (1.863, 0.758) -- (1.615, 0.612) -- (1.445, 0.843) -- (1.658, 1.036) -- (1.410, 0.890) -- (1.240, 1.121) -- (1.453, 1.314) -- (1.667, 1.121) -- (1.496, 1.353) -- (1.709, 1.546) -- (1.923, 1.353) -- (1.752, 1.121) -- (1.965, 1.314) -- (2.179, 1.121) -- (2.008, 1.353) -- (2.222, 1.546) -- (1.974, 1.399) -- (1.803, 1.631) -- (2.017, 1.824) -- (2.230, 1.631) -- (2.059, 1.863) -- (2.273, 2.056) -- (2.486, 1.863) -- (2.315, 1.631) -- (2.529, 1.824) -- (2.742, 1.631) -- (2.572, 1.399) -- (2.324, 1.546) -- (2.537, 1.353) -- (2.367, 1.121) -- (2.580, 1.314) -- (2.794, 1.121) -- (2.623, 1.353) -- (2.836, 1.546) -- (3.050, 1.353) -- (2.879, 1.121) -- (3.092, 1.314) -- (3.306, 1.121) -- (3.135, 0.890) -- (2.887, 1.036) -- (3.101, 0.843) -- (2.930, 0.612) -- (2.683, 0.758) -- (2.804, 1.019) -- (2.633, 0.788) -- (2.385, 0.934) -- (2.599, 0.741) -- (2.428, 0.510) -- (2.642, 0.703) -- (2.855, 0.510) -- (2.684, 0.278) -- (2.437, 0.425) -- (2.650, 0.232) -- (2.479, 0.000) -- (2.693, 0.193) -- (2.906, 0.000) -- (2.735, 0.232) -- (2.949, 0.425) -- (3.162, 0.232) -- (2.992, 0.000) -- (3.205, 0.193) -- (3.418, 0.000) -- (3.248, 0.232) -- (3.461, 0.425) -- (3.214, 0.278) -- (3.043, 0.510) -- (3.256, 0.703) -- (3.470, 0.510) -- (3.299, 0.741) -- (3.512, 0.934) -- (3.726, 0.741) -- (3.555, 0.510) -- (3.769, 0.703) -- (3.982, 0.510) -- (3.811, 0.278) -- (3.564, 0.425) -- (3.777, 0.232) -- (3.606, 0.000) -- (3.820, 0.193) -- (4.033, 0.000) -- (3.862, 0.232) -- (4.076, 0.425) -- (4.289, 0.232) -- (4.119, 0.000) -- (4.332, 0.193) -- (4.545, 0.000) -- (4.375, 0.232) -- (4.588, 0.425) -- (4.341, 0.278) -- (4.170, 0.510) -- (4.383, 0.703) -- (4.597, 0.510) -- (4.426, 0.741) -- (4.639, 0.934) -- (4.392, 0.788) -- (4.221, 1.019) -- (4.342, 0.758) -- (4.095, 0.612) -- (3.924, 0.843) -- (4.137, 1.036) -- (3.890, 0.890) -- (3.719, 1.121) -- (3.932, 1.314) -- (4.146, 1.121) -- (3.975, 1.353) -- (4.189, 1.546) -- (4.402, 1.353) -- (4.231, 1.121) -- (4.445, 1.314) -- (4.658, 1.121) -- (4.487, 1.353) -- (4.701, 1.546) -- (4.453, 1.399) -- (4.282, 1.631) -- (4.496, 1.824) -- (4.709, 1.631) -- (4.539, 1.863) -- (4.752, 2.056) -- (4.504, 1.909) -- (4.334, 2.141) -- (4.455, 1.880) -- (4.207, 1.733) -- (4.037, 1.965) -- (4.250, 2.158) -- (4.002, 2.011) -- (3.832, 2.243) -- (3.953, 1.982) -- (3.705, 1.835) -- (3.977, 1.929) -- (4.098, 1.668) -- (3.851, 1.522) -- (3.680, 1.753) -- (3.801, 1.492) -- (3.554, 1.346) -- (3.383, 1.577) -- (3.596, 1.770) -- (3.349, 1.624) -- (3.178, 1.855) -- (3.392, 2.048) -- (3.605, 1.855) -- (3.434, 2.087) -- (3.648, 2.280) -- (3.400, 2.133) -- (3.229, 2.365) -- (3.351, 2.104) -- (3.103, 1.957) -- (2.932, 2.189) -- (3.146, 2.382) -- (2.898, 2.235) -- (2.727, 2.467) -- (2.941, 2.660) -- (3.154, 2.467) -- (2.983, 2.699) -- (3.197, 2.892) -- (3.410, 2.699) -- (3.240, 2.467) -- (3.453, 2.660) -- (3.666, 2.467) -- (3.496, 2.699) -- (3.709, 2.892) -- (3.462, 2.745) -- (3.291, 2.977) -- (3.504, 3.170) -- (3.718, 2.977) -- (3.547, 3.208) -- (3.760, 3.401) -- (3.974, 3.208) -- (3.803, 2.977) -- (4.016, 3.170) -- (4.230, 2.977) -- (4.059, 2.745) -- (3.812, 2.892) -- (4.025, 2.699) -- (3.854, 2.467) -- (4.068, 2.660) -- (4.281, 2.467) -- (4.110, 2.699) -- (4.324, 2.892) -- (4.537, 2.699) -- (4.367, 2.467) -- (4.580, 2.660) -- (4.793, 2.467) -- (4.623, 2.699) -- (4.836, 2.892) -- (4.588, 2.745) -- (4.418, 2.977) -- (4.631, 3.170) -- (4.845, 2.977) -- (4.674, 3.208) -- (4.887, 3.401) -- (4.640, 3.255) -- (4.469, 3.486) -- (4.590, 3.225) -- (4.343, 3.079) -- (4.172, 3.310) -- (4.385, 3.503) -- (4.138, 3.357) -- (3.967, 3.588) -- (4.180, 3.781) -- (4.394, 3.588) -- (4.223, 3.820) -- (4.437, 4.013) -- (4.650, 3.820) -- (4.479, 3.588) -- (4.693, 3.781) -- (4.906, 3.588) -- (4.735, 3.820) -- (4.949, 4.013) -- (4.701, 3.866) -- (4.530, 4.098) -- (4.744, 4.291) -- (4.957, 4.098) -- (4.787, 4.330) -- (5.000, 4.523);\n\\node[below, inner sep=2.65mm]{} (5.455, 0.000);\n\\end{tikzpicture}\n\\caption{Left: Chain $P^m$ with its four subchains of type $P^{m-1}$ marked\nby their convex hulls;\nRight: Chain $P^m$ with the scaled copy of itself $R^m$ (in red) constructed in\n   Lemma~\\ref{lemma:technical}\\,(\\ref{pf:f1}).}\\label{fig:c-chain}\n\\end{figure}\n\nBy Lemma~\\ref{lemma:technical}\\,(\\ref{pf:f1}), the vertex set of $g_1(P^m)\\cup g_2(P^m)$\nis contained in the chain $R^m$ shown in Figure~\\ref{fig:c-chain}\\,(right).\nIf we are in Case~\\ref{pf:c-chain-case1}, \\ie, $p_i\\in g_1(P^{m})$ and $p_k\\in g_2(P^{m})$,\nthen $p_i, p_j, p_k$ can be thought of as vertices of $R^m$.\nThe similarity between $R^m$ and $P^m$, maps points $p_i, p_j, p_k$ to\nsuitable points $p_i', p_j', p_k' \\in P^m$ such that\n\\[\n\\frac{|p_i'p_j'|+|p_j'p_k'|}{|p_i'p_k'|} =\\frac{|p_ip_j|+|p_jp_k|}{|p_ip_k|}.\n\\]\nSince $p_i\\in g_1(R^{m}) \\cup g_2(R^{m})$ while $p_k\\in g_3(R^{m}) \\cup g_4(R^{m})$, the triplet\n$(p_i', p_j', p_k')$ does not belong to Case~\\ref{pf:c-chain-case1}.\nIn other words, Case~\\ref{pf:c-chain-case1} can be represented by other cases.\n\nRecall that in Lemma~\\ref{lemma:simple}, we showed that $\\conv(P^m)$ is\nan isosceles triangle $T$ of diameter $1$.\nObserve that if $|p_ip_k|\\geq \\frac{1}{c_*+1}$,\nthen\n\\[\n\\frac{|p_ip_j|+|p_jp_k|}{|p_ip_k|}\\leq \\frac{1+1}{\\frac{1}{c_*+1}}= 2c_*+2=c,\n\\]\nas required.\nSo we may assume that $|p_ip_k|<\\frac{1}{c_*+1}$, therefore only Case~\\ref{pf:c-chain-case4} remains,\n\\ie, $p_i\\in g_2(P^{m})$ and $p_k\\in g_3(P^{m})$.\n\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (2.727, 2.467) -- (5.000, 4.623) -- (7.273, 2.467) --cycle;\n\\path (0,0) pic[scale=0.5] {p4};\n\\node[below, inner sep=2.65mm]{} (5.455, 0.000);\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (4.545, 0.000) -- (2.727, 2.467) -- (4.752, 2.056)--cycle;\n\\fill[black!25] (5.000, 4.623) -- (7.273, 2.467) -- (5.455, 0.000) --cycle;\n\\draw[red] (4.752, 2.056) -- (4.793, 2.467) -- (5.207, 2.467);\n\\draw[dashed] (4.545, 0.000) -- node[below]{$\\scriptstyle\\frac{1}{c_*+1}$} (5.455, 0.000);\n\n\\path (0,0) pic[scale=0.5] {p4};\n\\end{tikzpicture}\n\\caption{Left: Chain $P^m$ with its subchain $g_5(P^{m})$ marked by its convex hull;\nRight: The last case where $p_i$ is in the left shaded subchain and $p_k$ is in the right shaded subchain.\n}\\label{fig:c-chain2}\n\\end{figure}\n\nBy Lemma~\\ref{lemma:technical}\\,(\\ref{pf:f2}), the ``top'' subchain $g_5(P^m)$ of $P^m$\nis also similar to $P^{m-1}$, see Figure~\\ref{fig:c-chain2}\\,(left).\nIf $p_i$ and $p_k$ are both in $g_5(P^{m})$,\n\\ie, $p_i\\in \\left(g_3\\circ g_2(P^m)\\right) \\cup \\left(g_4\\circ g_2(P^m)\\right)$\nand $p_k\\in \\left(g_1\\circ g_3(P^m)\\right) \\cup \\left(g_2\\circ g_3(P^m)\\right)$, then so is $p_j$.\n\nBy the induction hypothesis, we have\n\\[\n\\frac{|p_ip_j|+|p_jp_k|}{|p_ip_k|}\\leq c.\n\\]\nSo we may assume that at least one of $p_i$ and $p_k$ is not in $g_5(P^{m})$.\nWithout loss of generality, let $p_i\\in g_2(P^{m}) \\setminus g_5(P^{m})$.\nThe similarity that maps $P^{m-1}$ to $g_2(P^{m})$ and $g_5(P^{m})$, respectively, have the same\nscaling factor of $a=\\frac{c_*}{2(c_*+1)}$, and they carry the bottom dashed segment in\nFigure~\\ref{fig:c-chain2}\\,(right), to the two red segments.\n\n\\begin{claim*} If $p_i\\in g_2(P^{m}) \\setminus g_5(P^{m})$\nand $p_k\\in g_3(P^{m})$, then $|p_ip_k|>\\frac{c_*}{2(c_*+1)^2}$.\n\\end{claim*}\n\n\\begin{claimproof}\nAs noted above, we assume that $p_i$ is in $\\conv(g_2(P^{m}) \\setminus g_5(P^{m}))=\n\\Delta q_1q_2q_3$ in Figure~\\ref{fig:c-chain-case4}.\nIf $p_k\\in g_5(P^m) \\cap g_3(P^m)=\\Delta q_7q_6q_5$, then the configuration is illustrated in\nFigure~\\ref{fig:c-chain-case4}\\,(left).\nNote that $\\Delta q_1q_2q_3$ and $\\Delta q_7q_6q_5$ are reflections of each other with respect to\nthe bisector of $\\angle q_3q_4q_5$.\nHence the shortest distance between $\\Delta q_1q_2q_3$ and $\\Delta q_7q_6q_5$ is\n$\\min\\{|q_3q_5|, |q_2q_6|, |q_1q_7|\\}$. Since $c_*\\geq 1$, we have\n\\[|q_1q_7|>|q_7q_9|=|q_3q_5|=a^{3\/2}=\\left(\\frac{c_*}{2(c_*+1)}\\right)^{3\/2}\\geq \\frac{c_*}{2(c_*+1)^2}.\\]\nFurther note that $q_2q_4q_6q_8$ is an isosceles trapezoid, so the length of its diagonal is bounded by\n$|q_2q_6|>|q_2q_4|=\\frac{c_*}{2(c_*+1)^2}$.\nTherefore the claim holds when $p_k\\in\\Delta q_7q_6q_5$.\n\nOtherwise $p_k\\in g_3(P^m) \\setminus g_5(P^m)=\\Delta q_9q_8q_7$: see\nFigure~\\ref{fig:c-chain-case4}\\,(right).\nNote that $\\Delta q_1q_2q_3$ and $\\Delta q_9q_8q_7$ are reflections of each other with respect to\nthe bisector of $\\angle q_4q_5q_6$.\nSo the shortest distance between the shaded triangles is $\\min\\{|q_3q_7|, |q_2q_8|, |q_1q_9|\\}$.\nHowever, all three candidates are strictly larger than $|q_4q_6|=\\frac{c_*}{2(c_*+1)^2}$.\nThis completes the proof of the claim.\n\\end{claimproof}\n\\begin{figure}[!ht]\n  \\centering\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (4.545, 0.000) -- (2.727, 2.467) -- (4.752, 2.056)--cycle;\n\\fill[black!25] (5.000, 4.623) -- (5.207, 2.467) -- (7.273, 2.467)--cycle;\n\\draw[red] (4.752, 2.056) -- (4.793, 2.467) -- (5.207, 2.467);\n\\draw (0.000, 0.000) -- (2.066, 0.000)\n-- (2.273, 2.056) -- (2.479, 0.000)\n-- (4.545, 0.000)node[circle, fill, inner sep=1pt, label=135:$q_1$]{}\n-- (4.752, 2.056)node[circle, fill, inner sep=1pt, label=225:$q_2$]{}\n-- (2.727, 2.467)node[circle, fill, inner sep=1pt, label=135:$q_3$]{}\n-- (4.793, 2.467)node[circle, fill, inner sep=1pt, label=135:$q_4$]{}\n-- (5.000, 4.523)node[circle, fill, inner sep=1pt, label=above:$q_5$]{}\n-- (5.207, 2.467)node[circle, fill, inner sep=1pt, label=45:$q_6$]{}\n-- (7.273, 2.467)node[circle, fill, inner sep=1pt, label=45:$q_7$]{}\n-- (5.248, 2.056)node[circle, fill, inner sep=1pt, label=-45:$q_8$]{}\n-- (5.455, 0.000)node[circle, fill, inner sep=1pt, label=45:$q_9$]{}\n-- (7.521, 0.000) -- (7.727, 2.056) -- (7.934, 0.000) -- (10.000, 0.000);\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}[scale=0.5]\n\\fill[black!25] (4.545, 0.000) -- (2.727, 2.467) -- (4.752, 2.056)--cycle;\n\\fill[black!25] (7.273, 2.467) -- (5.248, 2.056) -- (5.455, 0.000)--cycle;\n\\draw[red] (4.752, 2.056) -- (4.793, 2.467) -- (5.207, 2.467);\n\\draw (0.000, 0.000) -- (2.066, 0.000)\n-- (2.273, 2.056) -- (2.479, 0.000)\n-- (4.545, 0.000)node[circle, fill, inner sep=1pt, label=135:$q_1$]{}\n-- (4.752, 2.056)node[circle, fill, inner sep=1pt, label=225:$q_2$]{}\n-- (2.727, 2.467)node[circle, fill, inner sep=1pt, label=135:$q_3$]{}\n-- (4.793, 2.467)node[circle, fill, inner sep=1pt, label=135:$q_4$]{}\n-- (5.000, 4.523)node[circle, fill, inner sep=1pt, label=above:$q_5$]{}\n-- (5.207, 2.467)node[circle, fill, inner sep=1pt, label=45:$q_6$]{}\n-- (7.273, 2.467)node[circle, fill, inner sep=1pt, label=45:$q_7$]{}\n-- (5.248, 2.056)node[circle, fill, inner sep=1pt, label=-45:$q_8$]{}\n-- (5.455, 0.000)node[circle, fill, inner sep=1pt, label=45:$q_9$]{}\n-- (7.521, 0.000) -- (7.727, 2.056) -- (7.934, 0.000) -- (10.000, 0.000);\n\\end{tikzpicture}\n\\caption{$p_i\\in\\Delta q_1q_2q_3$,\n  Left: $p_k\\in\\Delta q_7q_6q_5$;\n  Right: $p_k\\in \\Delta q_9q_8q_7$.}\\label{fig:c-chain-case4}\n\\end{figure}\n\nNow the diameter of $g_2(P^{m}) \\cup g_3(P^{m})$ is $a=\\frac{c_*}{2(c_*+1)}$\n(note that there are three diameter pairs), so\n\\[\n\\frac{|p_ip_j|+|p_jp_k|}{|p_ip_k|}<\\frac{2\\cdot \\frac{c_*}{2(c_*+1)}}{\\frac{c_*}{2(c_*+1)^2}}= 2c_*+2=c,\n\\]\nas required.\nThis concludes the proof of Lemma~\\ref{lemma:c-chain} and Theorem~\\ref{thm:lower-bound}.\n\\end{proof}\n\n\\section{Algorithm for Recognizing $c$-Chains}\n\\label{sec:algo}\n\nIn this section, we design a randomized Las Vegas algorithm to recognize $c$-chains.\nMore precisely, given a polygonal chain $P=(p_1,\\dots, ,p_n)$, and a parameter $c\\geq 1$,\nthe algorithm decides whether $P$ is a $c$-chain, in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$\nexpected time.\nBy definition, $P=(p_1,\\dots, p_n)$ is a $c$-chain if\n$|p_ip_j| + |p_jp_k|\\leq c\\ |p_ip_k|$ for all $1\\leq i<j<k\\leq n$;\nequivalently, $p_j$ lies in the ellipse of major axis $c$\nwith foci $p_i$ and $p_k$. Consequently, it suffices to test,\nfor every pair $1\\leq i<k\\leq n$, whether the ellipse\nof major axis $c|p_ip_k|$ with foci $p_i$ and $p_k$ contains\n$p_j$, for all $j$, $i<j<k$. For this, we can apply recent\nresults from geometric range searching.\n\n\\begin{theorem}\\label{thm:alg}\n  There is a randomized algorithm that can decide, for a polygonal chain $P=(p_1,\\dots , p_n)$ in $\\RR^2$\n  and a threshold $c>1$, whether $P$ is a $c$-chain in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$\n  expected time and $O(n\\log n)$ space.\n\\end{theorem}\n\nAgarwal, Matou\\v{s}ek and Sharir~\\cite[Theorem~1.4]{AMS13} constructed, for a set $S$ of $n$ points\nin $\\RR^2$, a data structure that can answer ellipse range searching queries: it reports the number\nof points in $S$ that are contained in a query ellipse.\nIn particular, they showed that, for every $\\eps>0$, there is a constant $B$\nand a data structure with $O(n)$ space, $O\\left(n^{1+\\eps}\\right)$ expected preprocessing time,\nand $O\\left(n^{1\/2}\\log^B n\\right)$ query time.  The construction was later simplified by\nMatou\\v{s}ek and Pat\\'akov\\'a~\\cite{MP15}. Using this data structure, we can quickly\ndecide whether a given polygonal chain is a $c$-chain.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:alg}.]\n  Subdivide the polygonal chain $P=(p_1,\\dots , p_n)$ into two subchains of equal or almost equal sizes,\n  $P_1=(p_1,\\dots, p_{\\lceil n\/2\\rceil})$ and $P_2=(p_{\\lceil n\/2\\rceil},\\dots, p_n)$;\n  and recursively subdivide $P_1$ and $P_2$ until reaching 1-vertex chains.\n  Denote by $T$ the recursion tree. Then, $T$ is a binary tree of depth $\\lceil \\log n\\rceil$.\n  There are at most $2^i$ nodes at level $i$; the nodes at level $i$ correspond to edge-disjoint subchains of $P$,\n  each of which has at most $n\/2^i$ edges. Let $W_i$ be the set of subchains on level $i$ of $T$;\n  and let $W=\\bigcup_{i\\geq 0}W_i$. We have $|W|\\leq 2n$.\n\n  For each polygonal chain $Q\\in W$, construct an ellipse range searching data structure\n  $\\DS(Q)$ described above~\\cite{AMS13} for the vertices of $Q$, with a suitable parameter $\\eps>0$.\n  Their overall expected preprocessing time is\n\\[\n\\sum_{i=0}^{\\lceil \\log n\\rceil} 2^i\\cdot  O\\left( \\left(\\frac{n}{2^i}\\right)^{1+\\eps} \\right)\n=O\\left(n^{1+\\eps}\\sum_{i=0}^{\\lceil \\log n\\rceil} \\left(\\frac{1}{2^i}\\right)^{\\eps}\\right)\n=O\\left(n^{1+\\eps}\\right),\n\\]\ntheir space requirement is\n$\\sum_{i=0}^{\\lceil \\log n\\rceil} 2^i\\cdot  O\\left(n\/2^i\\right)=O(n\\log n)$,\nand their query time at level $i$ is $O\\left(\\left(n\/2^i\\right)^{1\/2}\\ {\\rm polylog}\\ \\left(n\/2^i\\right)\\right)\n= O\\left(n^{1\/2}\\ {\\rm polylog}\\  n\\right)$.\n\nFor each pair of indices $1\\leq i<k\\leq n$, we do the following. Let $E_{i,k}$ denote the ellipse\nof major axis $c|p_ip_k|$ with foci $p_i$ and $p_k$. The chain $(p_{i+1},\\dots , p_{k-1})$ is subdivided\ninto $O(\\log n)$ maximal subchains in $W$, using at most two subchains from each\nset $W_i$, $i=0,\\dots ,\\lceil \\log n\\rceil$. For each of these subchains $Q\\in W$,\nquery the data structure $\\DS(Q)$ with the ellipse $E_{i,k}$. If all queries are positive\n(\\ie, the count returned is $|Q|$ in \\emph{all} queries), then $P$ is a $c$-chain;\notherwise there exists $j$, $i<j<k$, such that $p_j\\notin E_{i,k}$,\nhence $|p_ip_j|+|p_jp_k|>c|p_ip_k|$, witnessing that $P$ is not a $c$-chain.\n\nThe query time over all pairs $1\\leq i<k\\leq n$ is bounded above by\n\\begin{align*}\n\\binom{n}{2} \\sum_{i=0}^{2\\lceil \\log n\\rceil}\nO\\left(\\left(n\/2^i\\right)^{1\/2}\\ {\\rm polylog}\\ \\left(n\/2^i\\right)\\right)\n&= \\binom{n}{2}\\cdot O\\left(n^{1\/2}\\ {\\rm polylog}\\  n\\right)\\\\\n&=O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right).\n\\end{align*}\nThis subsumes the expected time needed for constructing the structures $\\DS(Q)$, for all $Q\\in W$.\nSo the overall running time of the algorithm is $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$, as claimed.\n\\end{proof}\n\nIn the decision algorithm above, only the construction of the data structures $\\DS(Q)$, $Q\\in W$,\nuses randomization,  which is independent of the value of $c$.\nThe parameter $c$ is used for defining the ellipses $E_{i,k}$, and the queries\nto the data structures; this part is deterministic. Hence, we can find the optimal value\nof $c$ by Meggido's parametric search~\\cite{Meg83} in the second part of the algorithm.\n\nMeggido's technique reduces an optimization problem to a corresponding\ndecision problem at a polylogarithmic factor increase in the running time. An optimization problem\nis amenable to this technique if the following three conditions are met~\\cite{Salowe04}:\n(1) the objective function is monotone in the given parameter;\n(2) the decision problem can be solved by evaluating bounded-degree polynomials, and\n(3) the decision problem admits an efficient parallel algorithm\n(with polylogarithmic running time using polynomial number of processors).\nAll three conditions hold in our case: The area of each ellipse with foci in $S$ monotonically\nincreases with $c$; the data structure of~\\cite{MP15} answers ellipse range counting queries\nby evaluating polynomials of bounded degree; and the $\\binom{n}{2}$ queries can be performed in parallel.\nAlternatively, Chan's randomized optimization technique~\\cite{Chan99} is also applicable. Both techniques\nyield the following result.\n\n\\begin{corollary}\n\\label{cor:alg}\nThere is a randomized algorithm that can find, for a polygonal chain $P=(p_1,\\dots , p_n)$ in $\\RR^2$,\nthe minimum $c\\geq 1$ for which $P$ is a $c$-chain in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$ expected time\nand $O(n\\log n)$ space.\n\\end{corollary}\n\nWe remark that, for $c=1$, the test takes $O(n)$ time: it suffices\nto check whether points $p_3,\\dots, p_n$ lie on the\nline spanned by $p_1p_2$, in that order.\n\n\n\n\\section{Concluding Remarks}\n\\label{sec:con}\n\nWe end with some final observations and  pointers for further research.\n\n\\begin{enumerate} \\itemsep 1pt\n\\item\nFor $k\\geq 1$, let $P^k_* = g_2(P^k)\\cup g_3(P^k)$, see Figure~\\ref{fig:refined-chain}\\,(right).\nIt is easy to see that $P^k_*$ is a $c$-chain with $n=4^k\/2+1$ vertices and has stretch factor\n$\\sqrt{c(c-2)\/8}(n-1)^{\\frac{1+\\log(c-2)-\\log c}{2}}$.\nSince $\\sqrt{c(c-2)\/8}\\geq 1$ for $c\\geq 4$, this improves the result of Theorem~\\ref{thm:lower-bound}\nby a constant factor. Since this construction does not improve the exponent, and the analysis would be\nlonger (requiring a case analysis without new insights), we omit the details.\n\\begin{figure}[htbp!]\n\\centering\n\\begin{tikzpicture}\n\\path (0,0) pic[scale=0.75] {p4};\n\\end{tikzpicture}\n\\hspace*{.1mm}\n\\begin{tikzpicture}[scale=0.75]\n\\draw (4.545, 0.000) -- (4.588, 0.425) -- (4.170, 0.510) -- (4.597,\n0.510) -- (4.639, 0.934) -- (4.221, 1.019) -- (4.095, 0.612) --\n(4.137, 1.036) -- (3.719, 1.121) -- (4.146, 1.121) -- (4.189, 1.546)\n-- (4.231, 1.121) -- (4.658, 1.121) -- (4.701, 1.546) -- (4.282,\n1.631) -- (4.709, 1.631) -- (4.752, 2.056) -- (4.334, 2.141) --\n(4.207, 1.733) -- (4.250, 2.158) -- (3.832, 2.243) -- (3.705, 1.835)\n-- (4.098, 1.668) -- (3.680, 1.753) -- (3.554, 1.346) -- (3.596,\n1.770) -- (3.178, 1.855) -- (3.605, 1.855) -- (3.648, 2.280) --\n(3.229, 2.365) -- (3.103, 1.957) -- (3.146, 2.382) -- (2.727, 2.467)\n-- (3.154, 2.467) -- (3.197, 2.892) -- (3.240, 2.467) -- (3.666,\n2.467) -- (3.709, 2.892) -- (3.291, 2.977) -- (3.718, 2.977) --\n(3.760, 3.401) -- (3.803, 2.977) -- (4.230, 2.977) -- (3.812, 2.892)\n-- (3.854, 2.467) -- (4.281, 2.467) -- (4.324, 2.892) -- (4.367,\n2.467) -- (4.793, 2.467) -- (4.836, 2.892) -- (4.418, 2.977) --\n(4.845, 2.977) -- (4.887, 3.401) -- (4.469, 3.486) -- (4.343, 3.079)\n-- (4.385, 3.503) -- (3.967, 3.588) -- (4.394, 3.588) -- (4.437,\n4.013) -- (4.479, 3.588) -- (4.906, 3.588) -- (4.949, 4.013) --\n(4.530, 4.098) -- (4.957, 4.098) -- (5.000, 4.523) -- (5.043, 4.098)\n-- (5.470, 4.098) -- (5.051, 4.013) -- (5.094, 3.588) -- (5.521,\n3.588) -- (5.563, 4.013) -- (5.606, 3.588) -- (6.033, 3.588) --\n(5.615, 3.503) -- (5.657, 3.079) -- (5.531, 3.486) -- (5.113, 3.401)\n-- (5.155, 2.977) -- (5.582, 2.977) -- (5.164, 2.892) -- (5.207,\n2.467) -- (5.633, 2.467) -- (5.676, 2.892) -- (5.719, 2.467) --\n(6.146, 2.467) -- (6.188, 2.892) -- (5.770, 2.977) -- (6.197, 2.977)\n-- (6.240, 3.401) -- (6.282, 2.977) -- (6.709, 2.977) -- (6.291,\n2.892) -- (6.334, 2.467) -- (6.760, 2.467) -- (6.803, 2.892) --\n(6.846, 2.467) -- (7.273, 2.467) -- (6.854, 2.382) -- (6.897, 1.957)\n-- (6.771, 2.365) -- (6.352, 2.280) -- (6.395, 1.855) -- (6.822,\n1.855) -- (6.404, 1.770) -- (6.446, 1.346) -- (6.320, 1.753) --\n(5.902, 1.668) -- (6.295, 1.835) -- (6.168, 2.243) -- (5.750, 2.158)\n-- (5.793, 1.733) -- (5.666, 2.141) -- (5.248, 2.056) -- (5.291,\n1.631) -- (5.718, 1.631) -- (5.299, 1.546) -- (5.342, 1.121) --\n(5.769, 1.121) -- (5.811, 1.546) -- (5.854, 1.121) -- (6.281, 1.121)\n-- (5.863, 1.036) -- (5.905, 0.612) -- (5.779, 1.019) -- (5.361,\n0.934) -- (5.403, 0.510) -- (5.830, 0.510) -- (5.412, 0.425) --\n(5.455, 0.000);\n\\end{tikzpicture}\n\\caption{The chains $P^4$ (left) and $P^4_*$ (right).}\\label{fig:refined-chain}\n\\end{figure}\n\n\\item\nIf $c$ is used instead of $c_*=(c-2)\/2$ in the lower bound construction, then\nthe condition $c\\geq 4$ in Theorem~\\ref{thm:lower-bound} can be replaced\nby $c\\geq 1$, and the bound can be improved from\n$(n-1)^{\\frac{1+\\log(c-2)-\\log c}{2}}$ to $(n-1)^{\\frac{1+\\log c-\\log(c+1)}{2}}$.\nHowever, we were unable to prove that the resulting $P^k$'s, $k\\in\\NN$,\nare $c$-chains, although a computer program has verified that the first\nfew generations of them are indeed $c$-chains.\n\n\\item\nThe volume argument in Theorem~\\ref{thm:1\/2} easily generalizes to higher dimensions.\nIf $P$ be a $c$-chain in $\\RR^d$ for fixed $c\\geq 1$ and $d\\geq 2$,\nthen $\\delta_P=O\\left(c^2(n-1)^{1-1\/d}\\right)$.\nIt is interesting to find out whether extra dimension(s) allows one to achieve\na larger stretch factor.\n\n\\item\nThe upper bounds in Theorem~\\ref{thm:log{c}}--\\ref{thm:1\/2} are valid regardless of\nwhether the chain is crossing or not.\nOn the other hand, the lower bound in Theorem~\\ref{thm:lower-bound} is given by noncrossing chains.\nA natural question is whether a sharper upper bound holds if the chains are required to be noncrossing.\nMore specifically, can the exponent of $n$ in the upper bound be reduced to $1\/2-\\eps$, where $\\eps>0$\ndepends on $c$?\n\n\\item Our algorithm in Section~\\ref{sec:algo} can recognize $c$-chains with $n$ vertices\n  in $O\\left(n^{2.5}\\ {\\rm polylog}\\ n\\right)$ expected time and $O(n\\log n)$ space,\n  using ellipse range searching data structures.\nIt is likely that the running time can be improved in the future, perhaps at the expense of increased space,\nwhen suitable time-space trade-offs for semi-algebraic range searching become available.\nThe existence of such data structures is conjectured~\\cite{AMS13}, but currently remains open.\n\\end{enumerate}\n\n\\bibliographystyle{plainurl\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor a finite group $G$, let $G$ be the quotient of a free group $F$ by a normal\nsubgroup $R$, then the $c$-nilpotent multiplier $\\mathcal{M}^{(c)}(G)$ is defined as\n\\[R\\cap\\gamma_{c+1}(F)\/\\gamma_{c+1}[R,F],\\]\nin which $\\gamma_{c+1}[R,F]=[\\gamma_{c}[R,F],F]$ for $c\\geq 1$. It is an especial case of the Baer invariant \\cite{ba} with respect to the variety of nilpotent\ngroups of class at most $c$. When $c=1$, the abelian group $\\mathcal{M}(G)=\\mathcal{M}^{(1)}(G)$ is more known as the Schur multiplier of $G$ and it is much more studied, for instance in \\cite{ka, ni1, ni9}.\n\nSince determining the $c$-nilpotent multiplier of groups can be used for the classification of group into\nisoclinism classes$($see \\cite{bay}$)$, there are multiple papers concerning this subject.\n\nRecently, several authors investigated to develop some results on the group theory case to Lie algebra.\nIn \\cite{sal}, analogues to the $c$-nilpotent multiplier of groups, for a given Lie algebra $L$, the $c$-nilpotent multiplier of $L$ is defined as\n\\[\\mathcal{M}^{(c)}(L)=R\\cap F^{c+1}\/[R,F]^{c+1},\\]\nin which $L$ presented as the quotient of a free Lie algebra $F$ by an ideal $R$, $F^{c+1}=\\gamma_{c+1} (F)$ and $[R,F]^{c+1}=\\gamma_{c+1}[R,F]$.\nSimilarly, for the case $c=1$, the abelian Lie algebra $\\mathcal{M}(L)=\\mathcal{M}^{(1)}(L)$ is more studied by the first author and the others\n$($see for instance \\cite{es, es2, bos, el2, har1, har, ni7, ni5, ni6, yan}$)$.\n\nThe $c$-nilpotent multiplier of a finite dimensional nilpotent\nLie algebra $L$ is a new field of interest in literature.\nThe present context is involving the $2$-nilpotent multiplier of a finite dimensional nilpotent Lie algebra $L$.\nThe aim of the current paper is divided into several steps.\nIn \\cite[Corollary 2.8]{sal}, by a parallel result to the group theory result, showed for every finite nilpotent Lie algebra $L$, we have\n\\begin{equation}\\label{e1}\n\\mathrm{dim}(\\mathcal{M}^{(2)}(L))+\\mathrm{dim}(L^3)\\leq\\frac{1}{3}n(n-1)(n+1).\n\\end{equation}\nHere we prove that abelian Lie algebras just attain the bound \\ref{e1}. It shows that always $\\mathrm{Ker}~\\theta=0$ in \\cite[Corollary 2.8 (ii)a]{sal}.\n\nSince Heisenberg algebras $H(m)$ $($a Lie algebra of dimension $2m+1$ with $L^2=Z(L)$ and\n$\\mathrm{dim}~(L^2) = 1)$ have interest in several areas of Lie algebra, similar to the result of\n\\cite[Example 3]{es2} and \\cite[Theorem 24]{mo}, by a quite different way, we give explicit structure of $2$-nilpotent multiplier of these algebras.\nAmong the other results since the Lie algebra which attained the upper bound \\ref{e1} completely described in Lemma \\ref{ab} $($they are just abelian Lie algebras$)$, by obtaining some new inequalities on dimension $\\mathcal{M}^{(2)}(L)$,\nwe reduce bound \\ref{e1} for non abelian Lie algebras as much as possible.\n\nFinally, among the class of Heisenberg algebras, we show that which of them is $2$-capable. It means which of them is isomorphic to\n$H\/Z_2(H)$ for a Lie algebra $H$. For more information about the capability of Lie algebras see \\cite{ni9, sal1}. These generalized the recently results for the group theory case in \\cite{ni10}.\n\n\\section{Further investigation on $2$-nilpotent multiplier of finite dimensional nilpotent Lie algebra }\nThe present section illustrates to obtain further results on $2$-nilpotent multiplier of finite dimensional nilpotent Lie algebra. At first we give basic definitions and known results for the seek of convenience the reader.\n\nLet $F$ be a free Lie algebra on an arbitrary totaly ordered set $X$. Recall from \\cite{sh},\nthe basic commutator on the set $X$, which is defined as follows, is a basis of $F$.\n\nThe elements of $X$ are basic commutators of length one and ordered relative to the total order previously chosen.\nSuppose all the basic commutators $a_i$ of length less than $k \\geq 1$\nhave been defined and ordered. Then the basic commutators of length $k$ to be defined as\nall commutators of the form $[a_i , a_j ]$ such that the sum of lengths of $a_i$ and $a_j$ is $k$, $a_i > a_j$, and if\n$a_i = [a_s, a_t ]$, then $a_j \\geq a_t$.\nAlso the number of basic commutators on $X$ of length $n$, namely $l_d(n)$, is\n\\[\\frac{1}{n}\\sum_{m|n}\\mu(m)d^{\\frac{n}{m}},\\] where $\\mu$ is the M\\\"{o}bius function.\n\nFrom \\cite{el2}, let $F$ be a fixed field, $L, K$ be two Lie algebras and $[ \\ , \\ ]$ denote the Lie bracket. By an action of $L$\non $K$ we mean an $F$-bilinear map \\[(l,k) \\in L\\times K \\mapsto\n~^lk \\in K~\\text{satisfying}~\\]\n\\[^{[l,l']}k= ~^l(~^{l'}k)- ~^{l'}(~^lk)~\\text{and}~^l[k,k']=[~^lk,k']+[k,~^lk'], ~\\text{for all}~ c \\in F, l, l' \\in L, k, k' \\in K.\\]\nWhen $L$ is a subalgebra of a Lie algebra $P$ and $K$ is an ideal in $P$, then $L$\nacts on $K$ by Lie multiplications $~^lk=[l,k]$.\nA crossed module is a Lie homomorphism\n$\\sigma: K\\rightarrow L$ together with an action of $L$ on $K$ such that\n\\[\\sigma(^lk) = [l,\\sigma(k)]~\\text{and}~ ^{\\sigma(k)}k' = [k,k'] ~\\text{for all}~k,k'\\in K ~\\text{and}~ l\\in L.\\]\n\nLet $\\sigma: L\\rightarrow M $ and $\\eta: K\\rightarrow M$ be two crossed modules, $L$ and $K$ act on each other and on themselves by Lie. Then these actions are called compatible provided that\n\\[~^{~^kl}k'=~^{k'}(~^lk)~\\text{and}~^{~^lk}l'=~^{l'}(~^kl).\\]\n\nThe non-abelian tensor product $L\\otimes K$ of $L$ and $K$ is\nthe Lie algebra generated by the symbols $l\\otimes k$ with defining\nrelations\n\\[c(l \\otimes k)=cl \\otimes k = l \\otimes ck,\n(l+l')\\otimes k = l \\otimes k + l' \\otimes k,\\]\n\\[l \\otimes (k+k') = l\\otimes k + l \\otimes k',\n~^ll' \\otimes k = l \\otimes ~^{l'}k -l' \\otimes ~^lk,~ l \\otimes ~^kk'= ~^{k'}l \\otimes k - ~^kl\n\\otimes k',\\]\n\\[[l\\otimes k, l' \\otimes k']=- ~^kl \\otimes ~^{l'}k',~\\text{for all}~c \\in F, l, l' \\in L, k, k' \\in K.\\]\n\n\nThe non-abelian tensor square of $L$ is a special case of tensor product\n$L\\otimes K$ when $K=L$. Note that we denote the usual\nabelian tensor product $L \\otimes_\\mathbb{Z} K$, when $L$ and $K$\nare abelian and the actions are trivial.\n\nLet $L\\square K$ be the submodule of $L\\otimes K$ generated by the elements $l\\otimes k$ such that $\\sigma(l)=\\eta(k)$. The factor Lie algebra $L\\wedge K\\cong L\\otimes K\/L\\square K$ is called the exterior product of $L$ and $K$, and the image of $l\\otimes k$\nis denoted by $l\\wedge k$ for all $l\\in L,k \\in K$. Throughout the paper $\\Gamma$ is denoted the universal quadratic functor $($see \\cite{el2}$)$.\n\nRecall from \\cite{ni9}, the exterior centre of a Lie algebra $Z^{\\wedge}(L)=\\{l\\in L~|~l\\wedge l'=1_{L\\wedge L},~\\forall~l'\\in L \\}$ of $L$.\nIt is shown that in \\cite{ni9} the exterior centre $L$ is a central ideal of $L$ which allows us to decide when\nLie algebra $L$ is capable, that is, whether $L\\cong H\/Z(H)$ for a Lie algebra $H$.\n\nThe following Lemma is a consequence of \\cite[Lemma 3.1]{ni9}.\n\\begin{lem} \\label{ca} Let $L$ be a finite dimensional Lie algebra, $L$ is capable if and only if $Z^{\\wedge}(L)=0$.\n\\end{lem}\nThe next two lemmas are special cases of  \\cite[Proposition 2.1 (i)]{sal} when $c=2$ and that is useful for proving the next theorem.\n\\begin{lem}\\label{ln}Let $I$ be an ideal in a Lie algebra $L$. Then the following sequences are exact.\n\\begin{itemize}\n\\item[(i)]$\\mathrm{Ker}(\\mu_I^2)\\rightarrow \\mathcal{M}^{(2)}(L)\\rightarrow\\mathcal{M}^{(2)}(L\/I)\\rightarrow \\frac {I\\cap L^3}{[[I,L],L]}\\rightarrow 0.$\n    \\item[(ii)]$(I\\wedge  L\/L^3)\\wedge L\/L^3 \\rightarrow \\mathcal{M}^{(2)}(L)\\rightarrow\\mathcal{M}^{(2)}(L\/I)\\rightarrow  I\\cap L^3\\rightarrow 0,$ when $[[I,L],L]=0$.\n\\end{itemize}\n\\end{lem}\n\\begin{lem}\\label{l3}Let $I$ be an ideal of $L$, and put $K=L\/I$. Then\n\\begin{itemize}\n\\item[(i)] $\\mathrm{dim}~\\mathcal{M}^{(2)}(K)\\leq \\mathrm{dim}~\\mathcal{M}^{(2)}(L)+ \\displaystyle\\mathrm{dim}~\\frac{I\\cap L^3}{[[I,L],L]}.$\n\\item[(ii)] Moreover, if $I$ is a $2$-central subalgebra. Then\n\n$(a).$ $(I\\wedge L)\\wedge L\\rightarrow \\mathcal{M}^{(2)}(L)\\rightarrow ~\\mathcal{M}^{(2)}(K)\\rightarrow \\mathrm{dim}~I\\cap L^3\\rightarrow 0$.\n\n$(b).$ $ \\mathrm{dim}~\\mathcal{M}^{(2)}(L)+\\mathrm{dim}~I\\cap L^3 \\leq \\mathrm{dim}~\\mathcal{M}^{(2)}(K)+\\mathrm{dim}~(I\\otimes L\/L^3)\\otimes L\/L^3$.\n\\end{itemize}\\end{lem}\n\\begin{proof} $(i)$. Using Lemma \\ref{ln} (i).\n\n$(ii)(a)$. Since $[I,L]\\subseteq Z(L)$, $\\mathrm{Ker}~\\mu^2_I=(I\\wedge L)\\wedge L$ and $[[I,L],L]=0$ by Lemma \\ref{ln}. It follows the result.\n\n$(ii)(b)$. Since there is a natural epimorphism $(I\\otimes L\/L^3)\\otimes L\/L^3\\rightarrow (I\\wedge L\/L^3)\\wedge L\/L^3 $, the result deduces from\nLemma \\ref{ln} (ii).\n\\end{proof}\nThe following theorem gives the explicit structure of the Schur multiplier of all Heisenberg algebra.\n\\begin{thm}\\cite[Example 3]{es2} $\\mathrm{and}$ \\cite[Theorem 24]{mo}\\label{h}\nLet $H(m)$ be Heisenberg algebra of dimension $2m+1$. Then\n\\begin{itemize}\n\\item[(i)]$\\mathcal{M}(H(1))\\cong A(2)$.\n\\item[(ii)]$\\mathcal{M}(H(m))=A(2m^2-m-1)$ for all $m\\geq 2$.\n\\end{itemize}\n\\end{thm}\nThe following result comes from \\cite[Theorem 2.8]{ni8} and shows the behavior of 2-nilpotent multiplier respect to the direct sum of two Lie algebras.\n\\begin{thm}\\label{ds}\nLet $A$ and $B$ be finite dimensional Lie algebras. Then\n\\[\\begin{array}{lcl}\\mathcal{M}^{(2)}(A\\oplus B)) &\\cong& \\mathcal{M}^{(2)}(A)\\oplus ~\\mathcal{M}^{(2)}(B)\n\\oplus \\big((A\/A^2\\otimes_{\\mathbb{Z}} A\/A^2)\\otimes_{\\mathbb{Z}} B\/B^2\\big )\\vspace{.3cm}\\\\&\\oplus&\\big((B\/B^2\\otimes_{\\mathbb{Z}} B\/B^2)\\otimes_{\\mathbb{Z}} A\/A^2\\big).\\end{array}\\]\n\\end{thm}\nThe following theorem is proved in \\cite{sal} and will be used in the next contribution. At this point, we may give a short proof with a quite different way of \\cite[Proposition 1.2]{sal} as follows.\n\\begin{thm}\\label{ab}Let $L= A(n)$ be an abelian Lie algebra of dimension $L$. Then $\\mathcal{M}^{(2)}(L)\\cong A(\\frac{1}{3}n(n-1)(n+1))$.\n\\end{thm}\n\\begin{proof}\nWe perform induction on $n$. Assume $n=2$. Then Theorem \\ref{ds} allows us to conclude that\n\\[\\begin{array}{lcl}\\mathcal{M}^{(2)}(L) &\\cong& \\mathcal{M}^{(2)}(A(1)) \\oplus\\mathcal{M}^{(2)}(A(1))\n\\oplus \\big(A(1)\\otimes_{\\mathbb{Z}} A(1)\\otimes _{\\mathbb{Z}}A(1)\\big )\\vspace{.3cm}\\\\&\\oplus&\\big(A(1)\\otimes_{\\mathbb{Z}} A(1))\\otimes _{\\mathbb{Z}}A(1)\\big)\\cong A(1)\\oplus A(1)\\cong A(2).\\end{array}\\]\n\nNow assume that $L\\cong A(n)\\cong A(n-1)\\oplus A(1)$. By using induction hypothesis and Theorem \\ref{ds}, we have\n\\[\\begin{array}{lcl}\\mathcal{M}^{(2)}(A(n-1)\\oplus A(1))&\\cong& \\mathcal{M}^{(2)}(A(n-1))\\oplus\\big(A(n-1)\\otimes_{\\mathbb{Z}} A(n-1)\\otimes _{\\mathbb{Z}} A(1)\\big)\n\\vspace{.3cm}\\\\&\\oplus&\\big(A(1)\\otimes_{\\mathbb{Z}} A(1)\\otimes_{\\mathbb{Z}} A(n-1)\\big)\\vspace{.3cm}\\\\&\\cong&\nA(\\frac{1}{3}n(n-1)(n-2))\\oplus A((n-1)^2)\\oplus A(n-1)\\vspace{.3cm}\\\\&\\cong& A(\\frac{1}{3}n(n-1)(n+1)).\\end{array}\\]\n\\end{proof}\n\nThe main strategy, in the next contribution, is to give the similar argument of Theorem \\ref{h} for the $2$-nilpotent multiplier.\nIn the first theorem, we obtain the structure of $\\mathcal{M}^{(2)}(L)$ when $L$ is non-capable Heisenberg algebra.\n\\begin{thm} Let $L=H(m)$ be a non-capable Heisenberg algebra. Then\n\\[\\mathcal{M}^{(2)}(H(m))\\cong A(\\frac{8m^3-2m}{3}).\\]\n\\end{thm}\n\\begin{proof} Since $L$ is non-capable, Lemma \\ref{ca} implies $Z^{\\wedge}(L)=L^2=Z(L)$.\nInvoking Lemma \\ref{l3} by putting $I=Z^{\\wedge}(L)$, we have $\\mathcal{M}^{(2)}(H(m))\\cong \\mathcal{M}^{(2)}(H(m)\/H(m)^2)$.\nNow result follows from Theorem \\ref{ab}.\n\\end{proof}\nThe following theorem from \\cite[Theorem 3.4]{ni9} shows in the class of all Heisenberg algebras which one is capable.\n\\begin{thm}\\label{ca1}$H(m)$ is capable if and only if $m = 1$.\n\\end{thm}\n\\begin{cor} \\label{ca11}$H(m)$ is not $2$-capable for all $m\\geq 2$.\n\\end{cor}\n\\begin{proof} Since every $2$-capable Lie algebra is capable, the result follows from Theorem \\ref{ca1}.\n\\end{proof}\nSince $H(m)$ for all $m\\geq 2$ is not $2$-capable, we just need to discus about the $2$-capability of $H(1)$. Here, we obtain 2-nilpotent multiplier of $H(1)$ and in the next section we show $H(1)$ is $2$-capable.\n\\begin{thm} Let $L=H(1)$. Then\n\\[\\mathcal{M}^{(2)}(H(1))\\cong A(5).\\]\n\\end{thm}\n\\begin{proof}\nWe know that $H(1)$ is in fact the free nilpotent Lie algebra of rank 2 and class 2. That is $H(1)\\cong F\/F^3$ in which $F$ is the free Lie algebra on 2 letters $x,y$. The second nilpotent multiplier of $H(1)$ is $F^4\\cap F^3\/[F^3,F,F]$ which is isomorphic to $F^3\/F^5$ ant the latter is the abelian Lie algebra on the set of all basic commutators of weights 3 and 4 which is the set $\\{[y,x,x],[y,x,y],[y,x,x,x],[y,x,x,y],[y,x,y,y]\\}$. So the result holds.\n\\end{proof}\nWe summarize our result as below\n\\begin{thm}\\label{th1}\nLet $H(m)$ be Heisenberg algebra of dimension $2m+1$. Then\n\\begin{itemize}\n\\item[(i)]$\\mathcal{M}^{(2)}(H(1))\\cong A(5)$.\n\\item[(ii)]$\\mathcal{M}^{(2)}(H(m))=A(\\frac{8m^3-2m}{3})$ for all $m\\geq 2$.\n\\end{itemize}\n\\end{thm}The following Lemma lets us to obtain the structure of the $2$-nilpotent multiplier of all nilpotent Lie algebras with $\\mathrm{dim}~\nL^2=1$.\n\\begin{lem}\\cite[Lemma 3.3]{ni7}\\label{l1} Let $L$ be an $n$-dimensional Lie algebra and\n$\\mathrm{dim}~\nL^2=1$. Then \\[L\\cong H(m)\\oplus A(n-2m-1).\\]\n\\end{lem}\n\\begin{thm}\\label{mt1}Let $L$ be an $n$-dimensional Lie algebra with\n$\\mathrm{dim}~L^2=1$. Then \\[\\mathcal{M}^{(2)}(L) \\cong \\left\\{\\begin{array}{lcl} A(\\frac{1}{3}n(n-1)(n-2)) & if\\ m>1 ,\\\\\n\\\\\nA(\\frac{1}{3}n(n-1)(n-2)+3) & if\\ m=1.\\end{array}\\right.\\]\n\\end{thm}\n\\begin{proof}By using Lemma \\ref{l1}, we have $L\\cong H(m)\\oplus A(n-2m-1)$. Using the behavior of $2$-nilpotent multiplier respect to direct sum\n\\[\\begin{array}{lcl}\\mathcal{M}^{(2)}(L) &\\cong& \\mathcal{M}^{(2)}(H(m)) \\oplus~\\mathcal{M}^{(2)}(A(n-2m-1))\n\\vspace{.3cm}\\\\&\\oplus&\\big((H(m)\/H(m)^2\\otimes_{\\mathbb{Z}} H(m)\/H(m)^2)\\otimes_{\\mathbb{Z}} A(n-2m-1)\\big )\\vspace{.3cm}\\\\&\\oplus&\\big((A(n-2m-1)\\otimes_{\\mathbb{Z}} A(n-2m-1))\\otimes _{\\mathbb{Z}}H(m)\/H(m)^2\\big).\\end{array}\\]\nFirst assume that $m=1$, then by virtue of Theorems \\ref{ab} and \\ref{th1}\n\\[\\mathcal{M}^{(2)}(H(1))\\cong A(5)~\\text{and}~\\mathcal{M}^{(2)}(A(n-3))\\cong A(\\frac{1}{3}(n-2)(n-3)(n-4)).\\]\nThus\n\\[\\begin{array}{lcl}\\mathcal{M}^{(2)}(L) &\\cong& A(5) \\oplus~A(\\frac{1}{3}(n-2)(n-3)(n-4))\n\\vspace{.3cm}\\\\&\\oplus&\\big(A(2)\\otimes_{\\mathbb{Z}} A(2))\\otimes_{\\mathbb{Z}} A(n-3)\\big )\\vspace{.3cm}\\\\&\\oplus&\\big((A(n-3)\\otimes_{\\mathbb{Z}} A(n-3))\\otimes_{\\mathbb{Z}} A(2)\\big)\n\\vspace{.3cm}\\\\&\\cong& A(\\frac{1}{3}n(n-1)(n-2)+3).\\end{array}\\]The case $m\\geq 1$ is obtained by a similar fashion.\n\\end{proof}\n\n\\begin{thm} Let $L$ be a $n$-dimensional nilpotent Lie algebra such that $\\mathrm{dim}~L^2=m (m\\geq 1)$. Then\n\\[\\mathrm{dim}~\\mathcal{M}^{(2)}(L)\\leq \\f13(n-m)\\big((n+2m-2)(n-m-1)+3(m-1)\\big)+3.\\]\nIn particular, $\\mathrm{dim}~\\mathcal{M}^{(2)}(L) \\leq \\f13n(n-1)(n-2)+3$. The equality holds in last inequality if and only if $L\\cong\nH(1)\\oplus A(n-3)$.\n\\end{thm}\n\\begin{proof} We do induction on $m$. For $m=1$, the result follows from Theorem \\ref{mt1}. Let $m\\geq 2$, and taking $I$ a $1$-dimensional central\nideal of $L$. Since $I$ and $L\/L^3$ act to each other trivially we have $(I\\otimes L\/L^3)\\otimes L\/L^3\\cong (I\\otimes_{\\mathbb{Z}} \\frac{L\/L^3}{(L\/L^3)^2})\\otimes_{\\mathbb{Z}} \\frac{L\/L^3}{(L\/L^3)^2})$. Thus by Lemma \\ref{l3} $(ii)(b)$\n\\[\\mathrm{dim}~\\mathcal{M}^{(2)}(L)+\\mathrm{dim}~I\\cap L^3 \\leq \\mathrm{dim}~\\mathcal{M}^{(2)}(L\/I)+\\mathrm{dim}~(I\\otimes_{\\mathbb{Z}} \\frac{L\/L^3}{(L\/L^3)^2})\\otimes_{\\mathbb{Z}} \\frac{L\/L^3}{(L\/L^3)^2}).\\]\nSince \\[\\mathrm{dim}~\\mathcal{M}^{(2)}(L\/I)\\leq \\f13(n-m)\\big((n+2m-5)(n-m-1)+3(m-2)\\big),\\] we have\n\\[\\begin{array}{lcl}\\mathrm{dim}~\\mathcal{M}^{(2)}(L)&\\leq& \\f13(n-m)\\big((n+2m-5)(n-m-1)+3(m-2)\\big)+3+(n-m)^2\n\\vspace{.3cm}\\\\&=&\\f13(n-m)\\big((n+2m-2)(n-m-1)+3(m-1)\\big)+3,\\end{array}\\] as required.\n\\end{proof}\nThe following corollary shows that the converse of \\cite[Proposition 1.2]{sal} for $c=2$ is also true. In fact it proves always\n$\\mathrm{Ker}~\\theta=0$ in \\cite[Corollary 2.8 (ii)a]{sal}.\n\\begin{cor}\\label{lab}Let $L$ be a $n$-dimensional nilpotent Lie algebra. If $\\mathrm{dim}~\\mathcal{M}^{(2)}(L)=\\f13n(n-1)(n+1)$, then $L\\cong A(n)$.\n\\end{cor}\n\\section{2-capability of Lie algebras}\nFollowing the terminology of \\cite{ell2} for groups, a Lie algebra $L$ is said to be $2$-capable provided that\n$L\\cong H\/Z_2(H)$ for a Lie algebra $H$. The concept $Z_2^{*}(L)$ was defined in \\cite{salria} and it was proved that if $\\pi:F\/[R,F,F]\\rightarrow F\/R$ be a natural Lie epimorphism then\n\\[Z^{*}_2(L)=\\pi(Z_2(F\/[[R,F],F])),~ \\text{for}~ c\\geq 0 \\]\n\nThe following proposition gives the close relation between $2$-capability and $Z^{*}_2(L)$.\n\\begin{prop}\nA Lie algebra $L$ is $2$-capable if and only if $Z^{*}_2(L)=0.$ \\end{prop}\n\\begin{proof}Let $L$ has a free presentation $F\/R$, and $Z^{*}_2(L)=0$. Consider the natural epimorphism $\\pi: F\/[[R,F],F]\\twoheadrightarrow F\/R$.\nObviously\\[\\mathrm{Ker}~\\pi=R\/[[R,F],F]=Z_2(F\/[[R,F],F]),\\] and hence $L\\cong F\/[[R,F],F]\/Z_2(F\/[[R,F],F])$, which is a $2$-capable.\n\nConversely, let $L$ is $2$-capable and so $H\/Z_2(H)\\cong L$ for a Lie algebra $H$. Put $F\/R\\cong H$ and $Z_2(H)\\cong S\/R$. There is natural\nepimorphism $\\eta: F\/[[S,F],F]\\twoheadrightarrow F\/S\\cong L$. Since $Z_2(F\/[[R,F],F])\\subseteq \\mathrm{Ker}~\\eta$, $Z^{*}_2(L)=0$, as required.\n\\end{proof}\nThe following Theorem gives an instrumental tools to present the main.\n\\begin{thm}\\label{ti}Let $I$ be an ideal subalgebra of $L$ such that $I\\subseteq Z^{*}_2(L)$. Then the natural Lie homomorphism\n$\\mathcal{M}^{(2)}(L)\\rightarrow \\mathcal{M}^{(2)}(L\/I)$ is a monomorphism.\n\\end{thm}\n\\begin{proof} Let $S\/R$ and $F\/R$ be two free presentations of $L$ and $I$, respectively. Looking the natural homomorphism\n\\[\\phi:\\mathcal{M}^{(2)}(L)\\cong R\\cap F^2\/[[R,F],F]\\rightarrow \\mathcal{M}^{(2)}(L\/I)\\cong R\\cap S^2\/[[S,F],F]\\] and the fact that\n$S\/R\\subseteq Z_2(F\/R)$ show $\\phi$ has trivial kernel. The result follows.\n\\end{proof}\n\\begin{thm} A Heisenberg Lie algebra $H(m)$ is $2$-capable if and only if $m=1$.\n\\end{thm}\n\\begin{proof} Let  $m\\geq 2$, by Corollary \\ref{ca11} $H(m)$ is not capable so it is not $2$-capable as well. Hence we may assume that $L\\cong H(1)$. Let $I$ be an ideal of $L$ od dimension 1. Then $L\/I$ is abelian of dimension 2, and hence $\\mathrm{dim}~\\mathcal{M}^{(2)}(L)=2$. On the other hands, Theorem \\ref{th1} implies $\\mathrm{dim}~\\mathcal{M}^{(2)}(L)=5$, and Theorem\n\\ref{ti} deduces $\\mathcal{M}^{(2)}(L)\\rightarrow \\mathcal{M}^{(2)}(L\/I)$ can not be a monomorphism, as required.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}