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+{"text":"\\section{Experimental Details}\n\nFor the investigation of the redox growth of EuO on \\ce{SrTiO3}(001) without additional oxygen gas we have developed a stepwise deposition method. As substrate we employed Nb:\\ce{SrTiO3}(001) and pure Eu metal was evaporated from a Knudsen cell in an ultra high vacuum molecular beam epitaxy system. Further details are given in the supplementary. Deposition steps of pure Eu-metal alternate with \\textit{in situ} structure analysis (reflection high-energy electron diffraction, RHEED, and low-energy electron diffraction, LEED) and chemical analysis (X-ray photoelectron spectroscopy, XPS). The stepwise deposition was carried out such, that the total deposition time was stopped to enable analysis at the times {$ t =  $\\SIlist{1; 2; 5; 10; 20; 60}{\\minute}}. Between two deposition steps, the sample is cooled from $ T_P $ to room temperature for analysis. This procedure is repeated for different substrate temperatures. In this way, a chemical and structural growth profile of europium oxide on \\ce{SrTiO3} is obtained depending on the deposition time $ t $ and  temperature $ T_P $ as determined from the pyrometer (see supplementary).\n\n\\textit{In situ} XPS is conducted on Ti $ 2p $ and Eu $ 3d $ core-levels with a PHOIBOS-100 hemispherical energy analyzer using Al~K$_{\\alpha}$ radiation from an X-Ray anode (SPECS).\n\nWe characterize the structure \\textit{ex situ} with X-ray diffraction (XRD), X-ray reflection (XRR) and reciprocal space mapping (RSM). Magnetic properties are analyzed with vibrating sample magnetometry and a magnetic property measurement system (See supplementary).\n\n\\section{Results}\n\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip,width=0.47\\textwidth]{fig\/Fig1a_principal_shape}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig1b_XPS}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\n\t\t\tTop: (a) Reference spectra of  metallic, di- and trivalent Eu, EuO and \\ce{Eu2O3} films respectively, obtained on the Eu~$ 3d_{5\/2} $ core-level. (b) Comparison of stoichiometric EuO and $ d \\approx \\SI{2}{\\angstrom} $ Eu metal on a metallic substrate. \n\t\t\n\t\t\tBottom: XPS analysis of the Eu $ 3d_{5\/2} $   for a stepwise EuO redox growth below (c)  and above  (d) the re-evaporation temperature of Eu. For $ t\\geq \\SI{5}{min} $ metal inclusions are observed, while at elevated temperature a decreasing \\ce{Eu^3+} content is present.\n\t\t}\n\t}\n\t\\label{fig:fig1}\n\\end{figure}\n\nOur study combines the redox-driven EuO synthesis with a chemical and structural\\textit{ in situ} (XPS, LEED, RHEED) and magnetic and structural \\textit{ex situ} (SQUID, XRD, XRR, RSM) analysis.\n\nFirst, we acquire reference spectra of the Eu~$3d _{5\/2} $ core-level (Fig.~\\ref{fig:fig1}(a)) as a function of the valence of the Eu cations. As reference systems, we prepare films of pure phases ($ d = \\SI{10}{nm} $) of \\ce{Eu^0} (Eu metal), \\ce{Eu^2+} (EuO) and \\ce{Eu^3+} (\\ce{Eu2O3}, see supplementary). The Eu $ 3d_{5\/2} $ peak of divalent \\ce{Eu^2+} is located at $E_B = \\SI{1125}{\\eV}$~\\cite{caspers_chemical_2011, gerber_thermodynamic_2016, cho_origin_1995}.  This peak is accompanied by a satellite peak (S) at higher binding energy, which is part of the multiplet of the $3d^{9}$ $4f^{7}$ final state~\\cite{cho_origin_1995}. Eu metal is observed at the same binding energy. In order to clearly distinguish the reference spectra of metallic Eu from the divalent state, we use the Doniach-Sunjic inelastic background, which is only present in metallic samples.  At $ E_{B} = \\SI{1135}{eV} $ the Eu $ 3d_{5\/2} $ peak is detected for trivalent \\ce{Eu^3+}. At $ E_{B} = \\SI{1125}{eV} $ an X-ray satellite from the trivalent Eu $ 3d_{5\/2} $ is observed as a consequence of using non-monochromatized Al X-rays.\n\nThe sensitivity of the Doniach-Sunjic background shape as a measure for metallic \\ce{Eu^0} is demonstrated by comparing a stoichiometric EuO film on yttria-stabilized zirconia (YSZ(001)) with $ d \\approx \\SI{2}{\\angstrom} $ Eu metal deposited on a Cu(001) single crystal. The Doniach-Sunjic inelastic background for this sample is clearly observed (Fig.\\ref{fig:fig1}(b)), showing that even mono-layers of Eu metal can be detected by this principle.\n\nKnowing the reference spectra, we analyze in detail the redox-growth of EuO\/\\ce{SrTiO3} at two exemplary temperatures, one being below the re-evaporation temperature of the distillation process and one above that temperature. For the first five minutes of deposition at $ T_P = \\SI{250}{\\degreeCelsius} $ mainly intensity from \\ce{Eu^2+} species are observed (Fig. \\ref{fig:fig1} (c)). Less than 10\\% of \\ce{Eu^3+} is detected and the \\ce{Eu^3+} content reduces over time. However, for $t>$\\SI{5}{\\minute} we observe Doniach-Sunjic inelastic background, indicating Eu metal inclusions in the film.  Further deposition leads to an increase in background and indicates \\ce{Eu^0}. We conclude, that only a small amount of EuO is formed at this temperature for the initial growth. For extended growth the Eu metal inclusions turn the stoichiometry to Eu-rich.\n\nNext, we study a stepwise Eu deposition at $T_P=$\\SI{500}{\\celsius} (Fig.~\\ref{fig:fig1}(d)). \nFor $t=$\\SI{1}{\\minute}, we observe a spectrum with a mixture of Eu$^{2+}$ (85\\% )and Eu$^{3+}$ (15\\%) components, while contributions of Eu metal are absent. For continued growth the spectral weight from Eu$^{3+}$ decreases (1\\% at $t=$\\SI{20}{\\minute}). Already for $t>$\\SI{5}{\\minute}, we find that the XPS spectra of adsorbed Eu-metal are nearly indistinguishable from stoichiometric EuO reference data (compare Fig.\\ref{fig:fig1}(a)). This demonstrates that the Eu metal is oxidized into a Eu$^{2+}$ rich (Eu$^{2+}$,Eu$^{3+}$) mixture at the \\ce{SrTiO3} interface. For extended growth only stoichiometric EuO (Eu$^{2+}$) is observed.\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip,width=0.47\\textwidth]{fig\/Fig2_results_temp}\n\t\t\\includegraphics[width=0.47\\textwidth]{fig\/Fig2b_Temperature_Dependence_Boxes}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\tChemical quantification of the Eu $ 3d_{5\/2} $ core-level for metallic Eu$ ^{0} $  (a) and over-oxidized Eu$ ^{3+} $ (b) content as function of $ t $ and $ T_{P} $. (c) Exponential thickness dependence exemplary shown for a sample grown at $ T_P = \\SI{500}{\\degreeCelsius} $. (d) A reduction in $ d_0 $ and $ \\tau $ is observed at $ t \\geq \\SI{700}{\\degreeCelsius} $ at the same temperature where a \\ce{Eu^3+} rich growth is observed. (e) We summarize the findings in a block diagram depicting the overall trend of the resulting film thickness and stoichiometry as function of $ T_P $.\n\t\t}\n\t}\n\t\\label{fig:fig2}\n\\end{figure}\n\nIn the following, the stoichiometry of the grown film is quantified by fitting a linear combination of the reference spectra to the observed Eu \\textit{3d}$ _{5\/2} $ spectra.  First, we discuss the Eu metal content of the grown films (Fig.~\\ref{fig:fig2}(a)). \nIn addition to the findings presented for $ T_{P} = \\SI{250}{\\degreeCelsius} $, we find for $ T_P\\geq\\SI{350}{\\celsius}$ no \\ce{Eu^0} metal in the spectra. We conclude that now Eu re-evaporation is a dominant process, which is in line with previous reports utilizing the adsorption limited growth mode on YSZ~\\cite{steeneken_new_2002}.\n\nWe observe an exponential decay of the \\ce{Eu^{3+}} fraction with $t$ for films prepared at {$T_P= \\,$\\SI{250}{\\celsius},} \\SI{350}{\\celsius}, \\SI{400}{\\celsius} and \\SI{500}{\\celsius}  (Fig.~\\ref{fig:fig2}(b)).  The Beer-Lambert law predicts an exponential decay of the intensity for the case of a buried layer with the growth over-layer thickness. Therefore, we argue that only the interface is responsible for the \\ce{Eu^{3+}} formation. It is also noted, that the \\ce{Eu^3+} content is higher at $ t = \\SI{1}{\\minute} $ for increasing $ T_{P} $. Yet, the total amount of \\ce{Eu^{3+}} is below one ML of \\ce{Eu2O3}, which is a negligible amount for extended film growth ($ T_{P} = \\SI{500}{\\degreeCelsius} $).\nFor $ T_P=\\SI{600}{\\celsius}$,  the \\ce{Eu^{3+}} content increases with $t$ in the initial growth phase ($ t\\leq \\SI{5}{\\minute}$). For the extended growth ($ t \\geq \\SI{5}{\\minute} $) the content of \\ce{Eu^3+} decreases again and at $ t =\\SI{20}{\\minute} $ only \\ce{Eu^2+} is found in the spectrum. Increasing the temperature further to \\SI{700}{\\celsius}, and \\SI{800}{\\celsius}, the content of \\ce{Eu^{3+}} increases monotonically with time and no EuO growth phase is observed. Therefore, redox growth for stoichiometric EuO on \\ce{SrTiO3} is possible only in the range {$ T_P =\\, $\\SIrange{350}{600}{\\celsius}}.\n\n\\subsection{Thickness dependence}\nExemplary shown for a sample grown at $ T = \\SI{500}{\\degreeCelsius} ,$ we find the temporal dependence of the thickness,  $ d(t) $ (see supplementary), to follow an exponential law of the form $ d(t) = d_0 ( 1 - \\exp( - t \/ \\tau ) ) $ (Fig.~\\ref{fig:fig2}(c)). Here $ d_0 $ is the final thickness and $ \\tau $ is the time constant. We apply this model to all samples (Fig~\\ref{fig:fig2}(d)) and find that both $ d_0 $ and $ \\tau $ increase monotonically for $ T \\leq \\SI{600}{\\degreeCelsius} $. Above this temperature a significant reduction is observed for both $ d_0 $ and $ \\tau $, simultaneous to the transition to \\ce{Eu^3+}-rich growth as observed in the chemical analysis.\n\nWe find that EuO rich films (i.e. $ \\SI{350}{\\degreeCelsius} \\leq T_{P} \\leq \\SI{600}{\\degreeCelsius}$) can be grown with up to $ d=\\SI{25}{nm} $ and $\\tau=$\\SI{25}{\\minute}. The fact that the time constants are in the range of many minutes allows a precise control of the film thickness by stopping the growth at a suitable time. The resulting chemical composition and thickness are compiled in a block diagram as a function of temperature (Fig.~\\ref{fig:fig2}(e)).\n\n\\subsection{Structural profile}\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig5_Structure_in_situ}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t\\textit{In situ} structure determination by LEED of the \\ce{SrTiO3}(001) substrate (a) and the redox-grown EuO film (b). We observe a \\ang{45} rotation of the EuO basis. (c-e) RHEED as a function of time shows a weak island growth mode and (f-h) structural transitions as a function of $ T_P $  from \\ce{Eu^0} to \\ce{Eu^2+} and \\ce{Eu^3+} respectively. The RHEED beam is parallel to \\ce{SrTiO3}(110) using \\SI{20}{keV} electrons.\n\t\t}\n\t}\n\t\\label{fig:fig3}\n\\end{figure}\n\n\nFor the \\textit{in situ} stuctural analysis we first present LEED (Fig.~\\ref{fig:fig3}(a)) for the \\ce{SrTiO3} substrate. We observe clear and sharp reflexes. The LEED pattern is four-fold symmetric, reflecting the symmetry of the perovskite lattice. Indicated by blue arrows is the basis of the reciprocal lattice. The EuO film ($ T_P = \\SI{600}{\\degreeCelsius} $, Fig.~\\ref{fig:fig3}(b)) is well-ordered and exhibits LEED reflexes, with a basis rotated by \\ang{45} to that of \\ce{SrTiO3}, the epitaxial relationship is {EuO(110)\/\\ce{SrTiO3}(100)}. The LEED reflexes are wider indicating a small degree of mosaicity.\n\nIn Fig.~\\ref{fig:fig3}(c-e) we show RHEED of the substrate, Eu deposition for $ t = \\SI{3}{\\minute} $ and $ t = \\SI{20}{\\minute} $ at $ T_{P} = \\SI{600}{\\degreeCelsius}$. The substrate shows RHEED streaks and two sharp spots on the Laue-circle, indicating a flat and well oriented substrate. At three minutes Eu deposition we observe RHEED streaks and a weak transmission pattern. For continued growth the transmission pattern dominates the RHEED reflexes, indicating a island type growth mode.\n\nThe RHEED reflexes as a function of $ T_{P} = $ \\SIlist{250;600;800}{\\degreeCelsius} are shown in Fig.~\\ref{fig:fig3}(f-h). For Eu-rich growth at \\SI{250}{\\degreeCelsius} we observe the hexagonal RHEED reflexes of the Eu-metal crystal. At elevated temperature the expected EuO related reflexes dominate the RHEED pattern, as described above. Finally at $ T_{P} = \\SI{800}{\\degreeCelsius} $ the structure shows the much bigger unit cell, and therefore shorter distances between RHEED reflexes, of the cubic \\ce{Eu2O3} crystal and a complex transmission pattern.\n\nIn conclusion, we observe clear structural transitions that are perfectly in line with the results from the chemical analysis by XPS. The EuO films grow by a redox-driven process epitaxially and are, except for a small surface roughness, flat. \n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig3_Structure}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t\\textit{Ex situ} X-ray analysis of a sample grown at \\SI{600}{\\degreeCelsius}. The reciprocal space map (a) shows the {\\ce{SrTiO3} (113)} and{ EuO (204)} peak in close proximity, a consequence of the epitaxial relation of {EuO(110)\/\\ce{SrTiO3}(100)}. The X-ray reflectometry  shows that the film has a low roughness of \\SI{1.32}{nm} (b).\n\t\t}\n\t}\n\t\\label{fig:fig4}\n\\end{figure}\n\nIn Fig.~\\ref{fig:fig4}(a), we show the \\textit{ex situ} X-ray analysis, starting with a RSM of a sample grown at $ T_{P} = \\SI{600}{\\degreeCelsius} $. The Bragg peaks from the \\ce{SrTiO3}(113) and the EuO (204) peak are shown. The simultaneous observation of both reflexes is in line with the expected rotation of both lattices {EuO(110)$ \\parallel $\\ce{SrTiO3}(100)}, since the reflexes would otherwise be observed at an angle of \\SI{45}{\\degree}. The EuO(204) reflex has a rocking curve width of $ \\delta\\omega = \\SI{1.1}{\\degree} $ in line with the widened LEED reflexes. We calculate the lattice parameters for the in-plane and out-of-plane lattice constants and find that both \\ce{SrTiO3} and EuO ($ a_{\\ce{SrTiO3}} = \\SI{3.901}{\\angstrom} $ and $ a_{\\ce{EuO}} = \\SI{5.14}{\\angstrom} $) are close to the literature values ($ a_{\\ce{SrTiO3}} = \\SI{3.905}{\\angstrom} $ and $ a_{\\ce{EuO}} = \\SI{5.14}{\\angstrom} $ \\cite{karlsruhe_inorganic_2017}).\n\nIn Fig.~\\ref{fig:fig4} (b), we depict the corresponding XRR curve and a fit to the measured data points using the Parrat formalism. The grown film exhibits a total thickness of $ d = \\SI{16.7}{nm} $, while the roughness is $ a = \\SI{1.3}{nm} $. Hence we conclude, that the redox-growth process produces well oriented, i.e. \\SI{45}{\\degree} rotated, EuO films on \\ce{SrTiO3} with a small degree of roughness.\n\n\\subsection{Magnetic Properties}\nThe magnetic properties of samples grown at {$ T_{P} = $~\\SIrange{250}{800}{\\degreeCelsius}} are depicted in Fig~\\ref{fig:fig5}(a) as hysteresis loops and as function of temperature $M(T)$ in Fig.~\\ref{fig:fig5}(b). Since EuO is the only ferromagnetic component in the stack, it is expected, that the saturation magnetization $ M_S $ is proportional to the EuO thickness. Eu metal and \\ce{Eu2O3} exhibit small paramagnetic moments only. In the chemical analysis we showed that the EuO content and the thickness both depend on $ T_{P} $ in a complex manner. Consequently, the magnetic analysis cannot be expected to follow a simple linear temperature dependence\n\nAll $M(H)$ curves of $ T_{P} = $\\SIlist{350;400;500;600}{\\degreeCelsius} increase monotonically in $ M_{S} $, while the coercive field, $ H_{C} $, decreases simultaneously. Unordered and thin films are generally considered to cause an increase of the coercive field~\\cite{muller_thickness_2009}. This finding suggests, that the amount of ferromagnetic EuO increases as a function of the temperature and forms a more ordered lattice at higher temperatures.\n\nThe samples grown at $ T_{P} = $\\SIlist{700;800}{\\degreeCelsius} exhibit no hysteresis, which is in line with the chemical analysis that reveals a \\ce{Eu2O3}-rich (and therefore paramagnetic) composition.\n\nThe temperature dependence of the normalized magnetic moment is shown in Fig~\\ref{fig:fig5}(b). We find for the samples grown at $ T_{P} = $\\,\\SIrange{350}{600}{\\degreeCelsius} a Brillouin-like shape that closely follows a simulation with a Curie temperature of $ T_C = \\SI{69}{K} $, the literature value for EuO \\cite{matthias_ferromagnetic_1961}. Significant deviations are observed for samples grown at high temperature, where a paramagnetic behavior is measured and  for $ T_{P} = \\SI{250}{\\degreeCelsius} $, where a pronounced metallic tail indicates the presence of Eu metal ions included in a EuO film~\\cite{suitsAnnealingStudyEuO1971, altendorf_oxygen_2011}.\n\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig4_Magnetism}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\n\t\t\tMagnetic properties of redox-grown \\ce{Eu}-oxide thin films on \\ce{SrTiO3} as (a) function of applied magnetic field and (b) temperature. EuO rich films, as found in the chemical analysis, exhibit a hysteresis, while high \\ce{Eu^{3+}} contents lead to paramagnetism. The inclusion of \\ce{Eu^{0}} as seen at $ T_{P} = \\SI{250}{\\degreeCelsius} $ causes a magnetic tail in the temperature dependence.\n\t\t\t}\n\t}\n\t\\label{fig:fig5}\n\\end{figure}\n\n\n\\section{Discussion}\n\nUnlike the classical EuO synthesis processes -- for which oxygen gas is supplied during a reactive MBE growth process -- here the oxide substrate itself acts as the supplier of oxygen: The \\ce{SrTiO3} substrate is reduced by the presence of Eu while the reactant oxidizes to EuO, \\ce{Eu3O4} or \\ce{Eu2O3}. These processes can be assessed with an Ellingham analysis (see supplementary). Indeed, we find that at equilibrium the most likely formed oxide is \\ce{Eu2O3}. However, we observe the formation of EuO in the intermediate temperature range. We therefore describe the complex redox growth of EuO\/\\ce{SrTiO3} as an interplay of three factors: (i) the kinetics of the oxygen anion reservoir from the substrate, (ii) the kinetics of the Eu metal on the surface of \\ce{SrTiO3} (and its concomitant re-evaporation) and (iii) the thermodynamics of the interface reactions.\n\n\nAs seen in Fig.~\\ref{fig:fig2}(c) the thickness of a redox-grown Eu oxide film is limited. This is surprising, as a normal diffusion type growth would have lead to a $ d \\propto \\sqrt{t} $ type behavior~\\cite{einsteinUberMolekularkinetischenTheorie1905}. We explain the thickness limit as a consequence of the insulating properties of the Eu oxides: The \\ce{O^{2-}} ions are charged and have to cross an insulating film of already grown Eu-oxides. This is well described in the context of a Mott-Cabrera type growth mode as was also found for the oxidation of Fe~\\cite{cabrera_theory_1949, kruger_room_1964}. In the Mott-Cabrera type growth, the potential that the \\ce{O^{2-}} ion is submitted to depends on the thickness of the film and its ion related resistivity. This explains the sudden reduction of $ d_0 $ and $ \\tau $ at the transition to \\ce{Eu^3+} rich growth in Fig.~\\ref{fig:fig2}(d), as the resistance is larger for the higher Eu oxides. Also, the re-evaporation of Eu from the surface can be expected to be much faster.\n\n\\section{Outlook}\n\\begin{figure}[!tb]\n\t\\begin{center}\n\t\t\\includegraphics[clip, width=0.47\\textwidth]{fig\/Fig6_other_oxides}\n\t\\end{center}\n\t\\caption{\\footnotesize{\n\t\t\t (a) XPS analysis of the Eu$ 3d_{5\/2} $ core-level for redox-grown Eu oxides, (b) calculated thickness $ d $ and oxygen ion conductivity $ \\sigma $ for a selection of oxides grown at $ T_{P} = \\SI{500}{\\degreeCelsius} $ and $ t = \\SI{5}{\\minute} $. A qualitative correlation between $ d $ and $ \\sigma  $ is observed.\n\t\t}\n\t}\n\t\\label{fig:fig6}\n\\end{figure}\n\nIn order to explore the redox-driven growth of EuO more generally, we study the initial growth ($ t = \\SI{5}{\\minute} $) at elevated temperatures ($ T_{P} = \\SI{500}{\\degreeCelsius} $) of Eu(O) on YSZ, \\ce{(LaAlO3){0.3}(Sr2TaAlO6){0.7}} (LSAT),  \\SI{10}{nm} SrO grown on \\ce{SrTiO3}, \\ce{LaAlO3} (LAO),  and MgO and compare it to the previous results of \\ce{SrTiO3}.\n\nAgain we study the Eu \\textit{3d}$ _{5\/2} $ core-level, as shown in Fig.~\\ref{fig:fig6}(a).  In the case for MgO and LAO we observe \\ce{Eu^0} in the film even at this high $ T_P $. For SrO and LSAT, we observe only \\ce{Eu^2+}, whereas for MgO and YSZ we even observe some \\ce{Eu^3+}. This is surprising, as preparations on YSZ are often reported in adsorption limited growth conditions and the interfacial over-oxidation is not mentioned \\cite{sutarto_epitaxial_2009}.\n\n\nWe obtain the film thickness, $ d $, (see supplementary) for all redox-grown films and compare them to estimates for the ionic conductivities of the respective oxide substrate~\\cite{skinnerOxygenIonConductors2003,gunkelInfluenceChargeCompensation2012,rudolphUeberLeitungsmechanismusOxydischer1959,nguyenEffectOxygenVacancy2000,mitoffElectronicIonicConductivity1962,moosDefectChemistryDonorDoped2005}. We find, that $ d $ qualitatively scales with the ionic conductivities of the underlying substrate (Fig.~\\ref{fig:fig6}(b)). We attribute the high discrepancy between SrO conductivity and $ d $ of redox-grown EuO to the fact, that the SrO thin film grown on a \\ce{SrTiO3} will have a higher oxygen mobility than a bulk crystal, due to crystal defects.\n\nFrom this analysis we are able to differentiate substrates into active substrates, with a relevant redox process and passive substrates, where additional oxygen needs to be supplied. We find that \\ce{SrTiO3} is the most active substrate for redox-growth. LAO, SrO thin films, LSAT and YSZ might pose as suitable templates for thin films and are expected to lead to thinner EuO over-layers than would be expected for \\ce{SrTiO3}, while MgO and LAO act as passive substrates. \n\n\\section{Conclusions}\nIn summary, we report a novel route for the synthesis of stoichiometric and single-crystalline EuO films by supplying no gaseous oxygen. Instead, utilizing the oxidic substrate as source of oxygen is the key to a reliable and simplified preparation scheme. We have identified the parameter window in which\nEuO can be grown on \\ce{SrTiO3}(001) and reduce the complexity of the typically applied distillation growth mode. \n\nThe prepared films show the expected chemical, structural and magnetic properties of stoichiometric EuO in the temperature range $ T_{P} = $\\,\\SIrange{350}{600}{\\degreeCelsius}. By changing the growth temperature, the total thickness of the EuO film can be varied from \\SIrange[range-phrase=--]{9}{25}{nm} for $ t = \\SI{60}{\\minute} $, and by stopping the growth earlier the thickness can be varied freely. Thus the redox-driven EuO growth method allows to prepare thicker films compared to a topotractic synthesis mechanism. All redox-grown EuO films grown in the suitable parameter window exhibit bulk ferromagnetic properties with no metal inclusions. With regard to the structural properties, we have observed flat and well oriented films with the epitaxial relationship of {EuO(110)$ \\parallel $\\ce{SrTiO3}(100)}. \n\nFinally, we demonstrate  that the redox-driven EuO growth scheme can be successfully applied not only to \\ce{SrTiO3}, but also to other oxide substrates. We believe, that the universality of a redox-controlled oxide thin film synthesis may also open up exciting perspectives for other topical oxide materials and their integration into complex oxide heterostructures.\n\n\n\\begin{acknowledgments}\nWe thank David~N.~M\u00fcller for fruitful discussions about solid state oxide chemistry. We thank O. Petracic and the J\u00fclich Centre for Neutron science for providing measurement time at the magnetometers.\nM.~M. acknowledges financial support from HGF under contract No. VH-NG-811.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nSpin-exchange pumped NMR gyroscopes~\\cite{Thrasher2019,thrasher2019PRApp,Walker2016,Kornack2005,Jiang2018,Karwacki1980} use the precession of spin-polarized nuclei to measure~rotation.  \nA vapor cell contains one or more isotopes of noble gas atoms and an~alkali-metal~vapor. The alkali-metal vapor is optically pumped using circularly polarized near-infrared laser light. Spin-exchange collisions between the polarized alkali-metal and noble gas atoms polarize the noble gas nuclei.~When subject to a magnetic (or ``bias'') field, the polarized species undergo Larmor precession. By~simultaneously measuring the Larmor precession of each entity, magnetic field correlations can be removed, thereby increasing sensitivity to non-magnetic spin-dependent phenomena, such as inertial~rotation\n\nThe precession of a polarized gas about a constant bias field constitutes an inertial reference frame, i.e., the gas has (ideally) no way of knowing whether or not its container is rotating. If the precession of the polarized gas is measured relative to a fixed point in the laboratory frame, such as with a pick-up~coil, then rotation of the pick-up coil about the bias field will change the measured rate of precession of the polarized gas\n\nThe application of polarized gases as inertial sensors is promising because of their low intrinsic~noise, miniaturize-ability, low power consumption, insensitivity to acceleration, and intrinsic scaling from experimental observable to rotation that is independent of experimental parameters~\\cite{Donley2013,Brinkmann62}. Interest in novel navigation systems which operate in global positioning system (GPS) denied environments has increased as of late. To date, the United States, Russia, China, and India have successfully demonstrated the capability to destroy their own satellites~\\cite{Chaudhury2019}. Such capability could potentially be used to remove GPS satellites. Other GPS denied environments include subterranean and submarine travel. One obstacle to harnessing geothermal energy resources is the limited ``in-hole'' navigation of horizontal drilling rigs. The future development of autonomous vehicles also relies on robust navigation, including in GPS denied environments. NMR gyroscopes will likely first be used to provide long-term corrections to another miniaturized gyroscope whose short term noise performance is superior but whose long term drift is inferior to the NMR gyroscope. A proof-of-concept experiment along these lines was recently performed using a classical and quantum accelerometer~\\cite{Cheiney2018}. \n\nImportant sources of systematic errors in spin-exchange pumped NMR gyroscopes are longitudinal (i.e., parallel to the bias field) spin-exchange fields produced by polarized alkali-metal atoms or noble gas nuclei~\\cite{Thrasher2019,Bulatowicz2013,Walker2016}. Such longitudinal spin-exchange fields are not identical for all noble gas species, and therefore cannot be removed in the same way as classical magnetic field correlations. Longitudinal spin-exchange fields are well suppressed by synchronous spin-exchange optical pumping~\\cite{Thrasher2019,Korver2015}, wherein the alkali-metal vapor is optically pumped transverse to a low duty cycle pulsed bias field. Each bias field pulse produces $2\\pi$ precession of the alkali-metal atoms such~that, despite having a gyromagnetic ratio in excess of $10^3$x that of the noble gas nuclei, the alkali-metal atoms can effectively co-precess with the noble gas nuclei. \n\nThis manuscript describes a $^{131}$Xe-$^{129}$Xe synchronous spin-exchange optically pumped NMR gyroscope which uses modulation of the alkali-metal ($^{85}$Rb) polarization to drive both Xe isotopes' NMR simultaneously. The Rb is also used as an embedded magnetometer to detect the Xe precession. Compared to our recent work~\\cite{Thrasher2019,thrasher2019PRApp} describing NMR excitation by modulating the bias field, polarization modulation (PM) further suppresses the influence of time-averaged longitudinal Rb spin-exchange fields~\\cite{Korver2015} by moving such fields from DC to AC. We study how modulation of the embedded Rb magnetometer causes signal mixing on our detection which leads to an effective change in scale factor~\\cite{ThrasherThesis}.\n\n\\subsection{Bloch Equation} \\label{BEsec}\nThe spin dynamics of the polarized noble gas transverse ($K_+ = K_x+iK_y$) and longitudinal ($K_z$) spins are described by the Bloch equations \n\\begin{subequations}\n\\begin{eqnarray}\n{dK_+\\over dt}=-(\\mp i\\Omega_z^K+\\Gamma_2)K_+ +\\Gamma_S^K S_+ + \\mp i\\Omega_+^K K_z,\\label{BE1}\n\\\\\n{dK_z\\over dt}=\\mp (\\Omega_y^K K_x-\\Omega_x^K K_y)-\\Gamma_1 K_z,\\label{BE2}\n\\end{eqnarray}\n\\end{subequations}\nwhere $\\mathbf{\\Omega}^K$ is the Xe resonance frequency arising from both the Larmor precession $\\gamma^K \\mathbf{B}$ and from rotation $\\omega^R \\mathbf{\\hat{z}}$, $\\Gamma_1$ ($\\Gamma_2$) is the longitudinal (transverse) relaxation rate, $\\mathbf{S}$ is the Rb polarization, and~$\\Gamma_S^K$ is the spin-exchange rate constant. Let superscript $a$ represent $^{129}$Xe and superscript $b$ represent $^{131}$Xe. Since $\\gamma^a <0$ and $\\gamma^b > 0$ we find it useful to write the gyromagnetic ratio $\\gamma$ (and hence $\\Omega$) as a~positive~value. The sign is written explicitly out front (top sign is for $a$, bottom sign is for $b$). \n\nWe null the transverse fields experienced by the Xe, including the spin-exchange field $b_S^K S_+$ (where $b_S^K$ is the spin-exchange coefficient characterizing the influence of the Rb polarization on the Xe)~\\cite{Korver2015}. This suppresses $K_z$ such that $K_+$ is much more sensitive to $\\Omega_z^K$ than $\\Omega_+^K$. Assuming our transverse optical pumping produces negligible $S_z$, the Xe resonance frequencies are then $\\Omega_z^K=\\gamma^K(B_{z0}+B_p)\\mp\\omega^R$, where $B_{z0}$ is the stray field inside our magnetic shields and $B_p$ is the field from the bias pulses. Since the Xe precess on the order of only $2\\pi\/10^3$ radians per pulse, the pulsed field can be approximated as a continuous field with $B_p= \\omega_p\/\\gamma^S$ for pulsing frequency $\\omega_p$.\n\nThe transverse Rb polarization ($S_+$) is used to polarize the Xe via spin-exchange optical pumping, and the longitudinal Rb polarization ($S_z$) is used to detect the Xe precession. For small Rb precession ($\\Omega^S<< \\Gamma'$), the time-average solution to the Bloch equation for the Rb polarization can be expanded~as\n\\begin{equation} \\label{Seq}\n\\mathbf{S} = {\\mathbf{R}\\over \\Gamma'} +{\\mathbf{\\Omega}^S \\times \\mathbf{R} \\over\\Gamma'^2}+{\\mathbf{\\Omega}^S \\times \\mathbf{\\Omega}^S \\times \\mathbf{R}\\over\\Gamma'^3}+..., \n\\end{equation}\nwhere $\\mathbf{R}= \\int d\\nu \\Phi(\\nu) \\sigma(\\nu) \\mathbf{p} \/A$ is the pumping rate, $\\sigma(\\nu)$ is the cross section of the atomic transition, $\\int d\\nu \\Phi(\\nu)=P\/h\\nu$ is the total photon flux for a beam of power P incident on area A, $\\mathbf{p}$ is the photon spin, $\\Gamma'$ is the total relaxation rate (including pumping), and $\\mathbf{\\Omega}^S = \\gamma^S (\\mathbf{B}_0+b_a^S \\mathbf{K}^a+b_b^S \\mathbf{K}^b)$. The~magnetic field experienced by the Rb includes the stray fields $\\mathbf{B}_0$ and the spin-exchange fields ($b_K^S$ is the spin-exchange coefficient characterizing the influence of the Xe polarization on the Rb), but it does not include the bias~field. This is because the bias field is applied as a sequence of low duty cycle pulses, the~area of which correspond to $2\\pi$ precession of the Rb atom. These equations assume negligible back polarization from the Xe to the Rb~\\cite{Limes2018}, and that $\\mathbf{K}$ precesses slowly such that $S_z$ responds adiabatically.  We optically pump along $\\hat{x}$ such that $\\mathbf{R}=R(t)\\hat{x}$, and the solution to Equation~(\\ref{Seq}) is\n\\begin{subequations}  \n\\label{allequations} \n\\begin{eqnarray}\nS_+ = {R(t)\\over\\Gamma'}+i{R(t) \\Omega_z^S\\over \\Gamma'^2} = {R(t) \\over \\Gamma'}e^{i\\epsilon_z},\n\\\\\nS_z = -{R(t)\\over \\Gamma'^2}(\\Omega_y^S -{\\Omega_z^S\\over \\Gamma'} \\Omega_x^S)={-R(t)\\over \\Gamma'^2 }\\text{Im}[\\gamma^S b_K^S K_+e^{-i\\epsilon_z}],\n\\end{eqnarray}\n\\end{subequations}\nwhere \n\\begin{equation} \\label{ezdef}\n\\epsilon_z = \\tan^{-1}({ S_y \\over S_x}) \\equiv \\tan^{-1}({ B_{z0} \\over B_w})<<1\n\\end{equation}\n\nis the magnetometer phase shift. $B_w=\\Gamma'$\/$\\gamma^S$ is the magnetic width of the magnetometer. A stray $B_z$ effectively rotates the quantization axis of the Rb magnetometer thus causing a phase shift ($\\epsilon_z$) in the measurement of a rotating $B_\\perp$.\n\nThe Xe NMR can be driven by modulating the transverse Rb polarization near each isotope's resonance frequency while the bias pulsing frequency is kept fixed. Such an excitation scheme is desirable as it effectively AC couples $S_z$. Hence, we Fourier expand the Rb polarization as  $S_+(t) = e^{i\\epsilon_z} \\sum_{p,q} s_{pq} e^{i(p\\omega_1+q\\omega_2) t}$ where $s_{pq}$ is the Fourier coefficient of $S_+$ at $p\\omega_1+q\\omega_2$. With the substitution $K_{+} = K_{\\perp}e^{\\pm i(\\omega_d t +\\delta)}$, where $\\delta$ is the phase shift of the Xe relative to the phase of $S_+$ and $\\omega_d$ is the drive~frequency, we find the real part of Equation ~(\\ref{BE1}) to be\n\\begin{linenomath}\n\\begin{equation}\n{d K_{\\perp}\\over dt} =-\\Gamma_2 K_{\\perp}+\\Gamma^K_S \\sum_{p,q} s_{pq} \\cos[(\\omega_d+p\\omega_1+q\\omega_2)t + \\delta \\pm \\epsilon_z].\n\\end{equation}\n\\end{linenomath}\n\nThe steady state solution of this equation is $K_{\\perp} = \\Gamma^K_S s_{pq} \/\\Gamma_2$ when $\\omega_d$ is chosen to satisfy the resonance condition $\\omega_d=-p_0\\omega_1-q_0\\omega_2$ for some $p_0$ and $q_0$ and when $\\delta,\\epsilon_z<<1$. Similarly, the~imaginary part of Equation~(\\ref{BE1})  (once again for $\\delta,\\epsilon_z<<1$) is\n\\begin{linenomath}\n\\begin{equation}\n{d \\delta \\over dt}= - \\Delta - \\Gamma_2 (\\delta \\mp \\epsilon_z),\n\\end{equation}\\label{fundgyro}\n\\end{linenomath}\nwhere $\\Delta =\\omega_d- \\gamma B_z\\pm \\omega^R$. The sign in front of $\\epsilon_z$ is isotope dependent because the Xe isotopes precess in opposite directions. We can solve this equation in the Fourier domain to find\n\\begin{linenomath}\n\\begin{equation}\n\\tilde{\\delta} = -{\\tilde{\\Delta} \\mp \\Gamma_2 \\tilde{\\epsilon}_z \\over \\Gamma_2+i\\omega},\\label{fundgyro1}\n\\end{equation}\n\\end{linenomath}\nwhere the notation $\\tilde{f} = f(\\omega)$ is used.   \n\nBy monitoring $S_z$, we can obtain measurements of $\\delta$ for each Xe species. These measurements of $\\delta$ are used to extract the Xe resonance frequencies. We perform comagnetometry using these Xe resonance frequencies to remove magnetic field correlations and arrive at a measurement of inertial~rotation. In~Section~\\ref{apparatus} we detail the implementation of our Xe excitation scheme, and in Section~\\ref{result} we demonstrate the performance of our system.\n\n\\section{Materials and Methods} \\label{apparatus}\nAn 8 mm inner diameter cubic Pyrex cell filled with 68 Torr enriched Xe and 85 Torr N$_2$ with a~hydride coating~\\cite{Kwon1981} is mounted in a ceramic housing with holes for laser light to enter the cell. The~ceramic housing has four symmetric faces which fit together like a jigsaw puzzle (and so we call these ``jig'' heaters). On each face are printed conductive traces through which AC current at $\\sim$150 kHz is passed to heat the ceramic. The conductive traces are arranged to produce minimal stray magnetic fields including gradient magnetic fields. The $\\sim$1 mm gap between the vapor cell and ceramic heating jig is shimmed with a 1.5 mm thermally conductive and slightly compressible gap fill (model: TG~977, manufacturer: T-Global Technology, see Figure~\\ref{cellpics}). The ceramic is wrapped with aerogel (a high temperature insulating material) and secured with Kapton tape and then fitted into a 3D printed (high~temperature nylon) cartridge with holes to allow laser light to enter the cell. The~compressible nature of the aerogel produces a friction fit keeping the ceramic jig structure fixed within the cartridge. The cartridge itself is mounted in a 3D printed (ABS plastic) rig. This rig has support arms which extend out three of the magnetic shield portholes described below. These support arms are secured directly to an optical table on which the entire apparatus is mounted.\n\\vspace{-6pt}\n\\begin{figure}[h\n\\centering\n\\includegraphics[width =0.8\\linewidth]{cellpics.pdf}\n\\caption[Vapor cell heater and cartridge mount photographs.]{Vapor cell heater and cartridge mount photographs. (\\textbf{a}) Four ceramic jig heater sides with aerogel pillows attached to outer faces. AC power enters via MMCX connectors at the bottom of each jig heater. (\\textbf{b}) 1 cm$^3$ vapor cell with stem tucked into fiberglass insulation. A temperature sensor sits between the insulation and outer cell wall. (\\textbf{c}) Cartridge with cell installed. (\\textbf{d}) Gap fill shim on one jig heater face. (\\textbf{e}) View looking into the jig heater with the cell installed. Note the four pieces of gap fill between each outer cell wall and the jig heater inner faces.}\\label{cellpics}\n\\end{figure}\n\nThe three layer $\\mu$-magnetic shield we use is cylindrical with eight access ports (two along the axis of symmetry and six oriented tangentially). Since the tangential access ports are not located halfway between the two ends of the cylindrical shields, and since pump laser light must enter through the~ports, we must place our cell such that it is not equidistant from the end caps. The asymmetry in distance to end caps informs the design of our bias pulsing coil set. To minimize coupling to the shield end caps and maximize uniformity across the volume of the cell, the pulsing coil set consists of two pairs of square Helmholtz coils with differing side lengths wound in series with opposite~polarity. The purpose of the ancillary counter wound coils is to suppress the field produced by the coil set at the nearest end cap. See \\cite{KorverThesis} for specific design details.\n\nThe bias field requires short pulses ($<$5 $\\upmu$sec) of $\\sim$1 Ampere peak current. The circuit used to drive the pulsing coil was custom-made and is described in \\cite{KorverThesis}. The circuit used to drive the shim coils was also custom-made and is described in \\cite{WyllieThesis}.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width =1\\linewidth]{setupv4.pdf}\n\\caption[Experimental setup for PM excitation]{Experimental setup for PM excitation (not to scale). DM: dichroic mirror, Pol: polarizer, HWP: half wave plate, QWP: quarter wave plate, PBS: polarizing beam splitter, WP: Wollaston prism, PD:~photodiode, EOM: electro-optic modulator, TS: two-axis translation stage with lens. The three-axis magnetic shim and pulsing coils are not shown. The setup fits on a four foot square optical table.}\\label{polmodsetup}\n\\end{figure}\n\nTo perform optical pumping of the Rb, the outputs of two distributed feedback laser diodes tuned near the Rb D1 transition (one on either side of resonance) are overlapped (see Figure~\\ref{polmodsetup}). This is accomplished by polarizing pump A so that it is mostly transmitted by a polarizing beam splitter (PBS) and pump B so that it is mostly reflected by the same PBS. The combined beam is then sent through a quarter wave plate and then separated into two beams using a PBS once again. The orientation of the quarter wave plate is chosen such that both pump A and pump B have half their power in each output beam. A telescope is used to couple the beams into individual EOMs. Prior to each EOM is a polarizer and half wave plate. The polarizer ensures that the light incident to the EOM crystal is purely~linear. The half wave plate is used to align the light polarization relative to the EOM crystal~axis. The maximum and minimum voltages of the EOM drive waveform are chosen to produce $\\pm\\lambda\/2$ retardance. The quarter wave plate at the output of the EOM converts the EOM output at $V_{max}(V_{min})$ to be $\\sigma^+(\\sigma^-)$.  The collimated output of each EOM is coupled into the vapor cell from opposing directions. The overall beam resizing is set to somewhat overfill the aperture of the ceramic~heater. Fine tuning of each pump beam's pointing is controlled using long focal length lenses mounted on two axis translation stages just outside the magnetic shield. The position of each steering lens is chosen to optimize the magnetometer gain. The power and detuning of each pump laser is chosen to approximately cancel the AC Stark effect. While we could also suppress the AC Stark effect by pumping with a single laser tuned on resonance, we would not get good spatial uniformity of the Rb polarization without significantly increasing our laser power due to optical thickness effects. To~detect $S_z$, approximately one mW of linearly polarized light from the output of a third distributed feedback laser diode, tuned near the Rb D2 line, is directed through the center of the cell and parallel to $\\hat{z}$ onto a balanced Faraday detector.\n\n\nWe choose to modulate the Rb polarization by switching between just two polarization states, $\\sigma^{\\pm}$. We modulate the x-component of $\\mathbf{R}$ according to\n\\begin{equation}\nR(t) = R_{0}\\;\\text{sign}[\\cos\\left({\\omega_d^a+\\omega_d^b\\over 2}t+2\\cos({\\omega_d^a-\\omega_d^b\\over 2}t)\\right)].\n\\end{equation}\n\nThis waveform is advantageous\nin that the smallest separation between reversals is larger than the finite response times of the EOMs and of the optical pumping of the Rb atoms. Modulating the polarization as a sum of two sine waves would also keep the modulations within the bandwidth of the EOMs and the Rb magnetometer, but we find that producing such a PM waveform in our optically thick vapor cell is very challenging. We use the same modulated cosine waveform to apply a~compensation field, the amplitude of which is set to cancel the spin-exchange field experienced by the Xe from the Rb. Doing so suppresses the production of $K_z$ and narrows the NMR linewidths, $\\Gamma_2$~\\cite{Korver2015}.\n\nFrom Equation~(\\ref{allequations}) we find that the detected longitudinal Rb polarization $S_z$ is \n\\begin{equation}\\label{Szdet1}\nS_z = -{R(t)\\over \\Gamma'^2}[b_a^S K^a_{\\perp} \\sin(\\delta^{a}+\\alpha^a - \\epsilon_z)+ b_b^S K^b_{\\perp} \\sin(\\delta^{b}+\\alpha^b + \\epsilon_z)],\n\\end{equation}\nwhere $\\delta=\\phi-\\alpha$ is the difference between the drive phase $\\alpha=\\int dt \\omega_d$ and the Xe precession phase $\\phi$ for each isotope,  $\\epsilon_z$ is the magnetometer phase shift, and $R\\sim S_+$ is the optical pumping rate of the Rb. Although lock-in detection can be accomplished on $S_z$ as it appears in Eq.~\\ref{Szdet1}, the phase sensitivity is diminished due to the presence of $R(t)$ which effectively mixes some Xe signal to DC. A more effective approach is to \"rectify\" the $S_z$ signal such that $R(t)$ is removed. This is accomplished by multiplying $S_z$ by $R^{-1}(t)$. The resulting signal is then sent into two separate lock-in amplifiers, each referenced to a different isotope's drive frequency. From these demodulations, we arrive at the measured phases $\\delta^a-\\epsilon_z$ and $\\delta^b+\\epsilon_z$.\n\n\\section{Results} \\label{result}\n\\vspace{-6pt}\n\\subsection{NMR Excitation and Detection}\n\n\nFigure~\\ref{polmodsqsigs} shows the time series of the measured $S_z$ with and without rectification when we drive both isotopes near resonance simultaneously using $S_+$. We see that rectification reveals the sinusoidal precession of each isotope. The outlying data on the rectified signal, which occur when the polarization is reversed, are due to the finite response time of the Rb magnetometer. Although rectification collects the many Xe signal sidebands into the two Xe carrier frequencies (see power spectrum on right), it~also maps 1\/$f$ detection noise on $S_{z}$ to the carrier frequencies. The mapping of low-frequency $S_{z}$ noise to the carrier frequencies can be prevented by high-pass filtering $S_z$ with a 1 Hz corner prior to rectification. \n\n\\begin{figure}[t\n\\centering\n\\includegraphics[width=0.75\\linewidth]{myrect2.pdf}\n\\caption[Square wave PM signals]{Measured square wave PM signals. Top: $S_+(t)$. Middle: $S_z(t)$. Bottom: Rectified $S_z(t)$. The plot on the right shows the amplitude spectral density with (red) and without (blue) rectification.}\\label{polmodsqsigs}\n\n\\vspace*{\\floatsep}\n\n\\includegraphics[width=0.75\\linewidth]{polmodnmr5.pdf}\n\\caption[NMR of each species]{Measured NMR lineshapes of each species. Filled circles are $K_x$, open circles are $K_y$, and lines are Lorentzian fits.}\\label{polmodnmr}\n\\end{figure}\n\n\nFigure~\\ref{polmodnmr} shows the detected NMR signals for each isotope. These data were acquired by driving one isotope on resonance while varying the other isotope's drive frequency and recording its $K_x$ and $K_y$ derived using demodulation. We see that the lineshapes are nearly Lorentzian with linewidths (half-width at half-max) of $\\sim$15 mHz and amplitudes of approximately 60 $\\upmu$G. The on-resonance amplitudes are in agreement with estimates similar to those outlined in~\\cite{Walker2016}. The implied $T_2^a$, $T_2^b$ from the fits are in good agreement with independent measurements of each isotope's $T_1$. The $\\sim$15 mHz linewidths are only possible because of two features of our experiment; (i) the use of a Rb hydride cell coating (without which $T_2^b$ would be substantially shorter and $T_2^a$ much longer)~\\cite{Kwon1981}, and (ii) the application of a magnetic compensation field $B_x$ that cancels the Rb spin-exchange field experienced by the Xe. \n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.75\\linewidth]{polmodpn3.pdf}\n\\caption[Phase noise]{Measured phase noise of each species. Traces labeled ``Real'' are recorded when the Xe isotopes are excited on resonance. Traces labeled ``Artificial'' are recorded when the Xe isotopes are both driven off resonance (not excited) and an AC $B_y$ is applied to the magnetometer at the off-resonance drive frequencies.}\\label{polmodpn}\n\\end{figure}\n\nFigure~\\ref{polmodpn} demonstrates the amplitude spectral density of the phase noise measured for each isotope under simultaneous resonant excitation conditions. We see that for frequencies less than 1 Hz the spectra are dominated by $1\/f$ noise which is about $\\rho=\\gamma^a\/\\gamma^b$ greater for isotope $a$ (black traces) than for isotope $b$ (red traces) suggesting the dominant source of $1\/f$ noise is magnetic in~nature. Also shown is the phase noise measured when the Xe isotopes are driven off resonance (not excited) and a so-called ``artificial'' Xe signal is applied as an AC field, $B_y$. The amplitudes of this signal $A^a\\sin(\\omega_d^a t)+A^b\\sin(\\omega_d^b t)$ are chosen to produce the same size magnetometer signal as real Xe. We~note that this artificial Xe signal is planar, unlike the real Xe signal which rotates. The artificial signal allows us to measure the signal-to-noise ratio (SNR) of the Rb magnetometer. These signals do not show $1\/f$ dependence because, unlike the real Xe phase, the SNR of the magnetometer does not depend on the bias magnetic field to first order. The detection phase noise from each artificial Xe measurement is uncorrelated and limits the possible field suppression when performing comagnetometry. The SNR for each isotope is $\\sim$5000$\\sqrt{\\text{Hz}}$. While the detection of artificial Xe is insensitive to $1\/f$ bias magnetic field noise, $1\/f$ $S_z$ noise can still be mapped to the artificial Xe frequencies via rectification, and so the addition of a 1 Hz high-pass filter prior to rectification was essential for realizing such an SNR.\n\n\n\\subsection{Comagnetometry}\n\n\nWe perform comagnetometry by subtracting magnetic field correlations between the two isotope's precession frequencies. Since our device measures phase, we need to know or measure the transfer function from phase to frequency. In Section~\\ref{BEsec} we derived the transfer function (see Equation~(\\ref{fundgyro1})). We measured the transfer function of each isotope by recording the response of the measured phase $\\delta$ to sinusoidal modulation of $B_z$. Figure~\\ref{polmodtf} shows the measured transfer function for isotope $a$. We~use a~chirp waveform to modulate the bias field $B^{mod}_z(t)=B_0\\sin(2\\pi[e^{t\/T_2}-1 - t\/T_2])$ (where~$T_2 = 1\/2\\pi \\Gamma_2$), the time series of which is shown in the inset of Figure~\\ref{polmodtf}. This modulation waveform allows us to measure the transfer function from 0.002 to 0.1 Hz with good SNR in a single data acquisition. The~transfer function is the ratio $\\gamma^K\\tilde{B}_z^{mod}\/\\tilde{\\delta}$. We fit the data according to Equation~(\\ref{fundgyro1}) and find excellent agreement with the linewidth derived from the fits in Figure~\\ref{polmodnmr}.\n\n\\vspace{-6pt}\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.70\\linewidth]{transfunc4.pdf}\n\\caption[Transfer function of $^{129}$Xe]{Transfer function of $^{129}$Xe. Inset depicts the normalized chirp waveform used to modulate the bias field. Black crosses are measured data, while the blue line is a fit of the form $\\sqrt{(\\Gamma_2^{a})^2+f^2}$.}\\label{polmodtf}\n\\end{figure}\n\\unskip\n\nAlthough conversion from phase to frequency for the measured Xe phases is possible using a~measured transfer function, feedback is desirable because (in the high gain limit) the performance of the comagnetometry becomes insensitive to changes in the transfer function. We used the measured precession phase of isotope $a$ to stabilize the bias field and the measured transfer function of isotope $b$~to convert its measured phase noise to frequency noise. Under such conditions the frequency noise of isotope $b$ is proportional to rotation. We apply a feedback field, $B_f$, to hold the measured phase of isotope $a$, $(\\delta^a-\\epsilon_z)$, equal to zero. The field stabilization can then be written $\\tilde{B}_f = \\tilde{G}(\\tilde{\\delta}^a-\\tilde{\\epsilon}_z)$. In the high gain limit, this becomes\n\\begin{linenomath}\n\\begin{equation}\n\\lim_{G \\to \\infty}\\tilde{B}_{tot} = {1\\over \\gamma^a}(\\tilde{\\omega}^R+i\\omega\\tilde{\\epsilon}_z),\n\\end{equation}\n\\end{linenomath}\nwhere $B_{tot}=B_{z0}+B_p+B_f$, and where we made substitutions using Equation~(\\ref{fundgyro1}) and $\\tilde{\\Delta}^a =- \\gamma^a \\tilde{B}_z- \\tilde{\\omega}^R$ (since $\\omega_d^a$ is held constant). The measured phase of isotope $b$, $(\\delta^b+\\epsilon_z)$, converted to frequency is\n\\begin{linenomath}\n\\begin{equation}\n\\tilde{\\omega}^b \\equiv (\\tilde{\\delta}^b+\\tilde{\\epsilon}_z)(\\Gamma_2^b+i\\omega)= (\\rho^{-1}+1)(\\tilde{\\omega}^R+i\\omega\\tilde{\\epsilon}_z).\n\\end{equation}\n\\end{linenomath}\n\nWe see that in the high gain limit, when correcting the bias field to keep the measured phase of isotope $a$ equal to zero, the rotation is simply $\\tilde{\\omega}^R = \\rho\\;\\tilde{\\omega}^b\/(1+\\rho)$ assuming $\\tilde{\\epsilon}_z$ is negligible.  \n\n\n\nThe best performance we observed with bias field feedback activated is shown in Figure~\\ref{polmodcomag}. The~feedback consisted of two analog inverted zero gain stages. The influence of bias field feedback is dramatic from 0.1 to 200 mHz. The servo suppresses $\\tilde{\\omega}^a$ to below 1 $\\upmu$Hz\/$\\sqrt{\\text{Hz}}$ at low frequency, which is nearly $10^4$x less than the open loop noise. Because magnetic noise dominates each isotope's precession, servoing the measured phase of isotope $a$ also greatly suppresses $\\tilde{\\omega}^b$. We observe at least 100$\\times$ improvement in $\\tilde{\\omega}^b$ due to feedback. The modified Allan deviation~\\cite{Allan1981} suggests a rotation ARW sensitivity of $\\sqrt{2}\\;15\\;\\upmu$Hz\/$\\sqrt{\\text{Hz}}{\\rho\\over 1+\\rho}\\sim 16\\; \\upmu$Hz\/$\\sqrt{\\text{Hz}}$ and a rotation bias instability of 1 $\\upmu$Hz ${\\rho\\over 1+\\rho}\\sim 800$~nHz. The size of ARW is within a factor of 3 of the ratio of the measured linewidths divided by the SNRs (shown in Figure~\\ref{polmodpn}). The peaks in the Allan deviation at 15 and 100 s of integration are due to low-frequency narrow-band large-amplitude noise peaks in $\\tilde{\\omega}^b$ which we attribute to the PM~waveform. The bias instability is limited by $\\tau^{1\/2}$ trending noise of unknown origin. We find that feedback causes the measured phase of isotope $b$ to trend linearly in time. The source of this frequency bias is uncertain. Although the trend is very stable over the course of a data run, the trend is not consistent between data runs. The bias instability demonstrated in Figure~\\ref{polmodcomag} was difficult to reproduce. Typically, the bias instability we measured was a few $\\upmu$Hz.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.9\\linewidth]{comag3.pdf}\n\\caption[Comagnetometry noise and stability]{Measured comagnetometry noise and stability. Left: the amplitude spectral density of frequency noise. The cross marks indicate open loop frequency noise. The solid lines are frequency noise when the measured phase noise of isotope $a$ is used to stabilize bias field. Right: modified Allan deviation of $\\omega^b$. Filled circles are measured data. Solid line shows the quadrature sum of $7\\times 10^{-5}\\tau^{-1}$, $15\\;\\upmu\\text{Hz}\/\\sqrt{\\text{Hz}}\\tau^{-1\/2}$, and $30\\;\\text{nHz}\\sqrt{\\text{Hz}}\\tau^{1\/2}$ trends.}\\label{polmodcomag}\n\\end{figure}\n\n\\subsection{Cross Talk}\nOnce we measured the stability of the PM comagnetometer, we desired to know the fidelity with which our detection separated signals from the two Xe isotopes. It is possible that the detection channel designed to measure isotope $a$'s phase was really measuring a linear combination of isotope $a$~and $b$'s phases. If such ``cross talk'' were present then the scale factor (or how we convert the measured precession frequencies to rotation) would change~\\cite{ThrasherThesis}. Suppose there exists cross talk in both channels such that $\\omega^a = \\gamma^a B_z + \\beta \\omega^b - \\omega^R$ and $\\omega^b = \\gamma^b B_z +\\beta' \\omega^a + \\omega^R$ where $\\beta$ and $\\beta'$ represent the cross talk between detection channels. Solving for $\\omega^R$ we find\n\\begin{linenomath}\n\\begin{equation}\n\\omega^R = {\\omega^b (\\rho+\\beta')- \\omega^a (\\rho \\beta +1) \\over 1+\\rho},\n\\end{equation} \\label{polmodcteq}\n\\end{linenomath}\nwhere if $\\beta=\\beta'=0$ we return to the expected expression for rotation without cross talk. Non-zero cross talk is undesirable because the accuracy with which it is known (or measured) limits the accuracy of conversion from measured precession frequencies to rotation (or any other non-magnetic spin-dependent interaction). A measurement of cross talk is vital since an important alleged feature of our comagnetometer is that it has a scale factor which is determined solely by $\\rho$.\n\n\\begin{figure}[h\n\\centering\n\\includegraphics[width=0.65\\linewidth]{pmct3.pdf}\n\\caption[Measurement of cross talk]{Measurement of cross talk for detection with rectification. The Xe drive frequencies are set such that the two isotopes are not driven on resonance at the same effective bias field magnitude. The~in-phase (filled symbols) and out-of-phase (un-filled symbols) components of each isotope's detection channel (black and red correspond to isotopes $a$ and $b$, respectively) are shown as the bias pulsing frequency is varied. We see that when isotope $a$ is resonant ($f_{2\\pi}\\sim 15.8$ kHz) the signal on isotope $b$'s detection channel is not flat despite isotope $b$ being driven many linewidths off resonance. Similarly, when isotope $b$ is resonant ($f_{2\\pi}\\sim 16$ kHz) the signal on isotope $a$'s detection channel is not flat despite isotope $a$ being driven many linewidths off resonance.}\\label{polmodmct}\n\\end{figure}\n\nWe characterize the cross talk present in our detection channels by looking for changes in the detected signal for one isotope when the drive of the other isotope is changed. We do this by detuning one isotope's drive frequency by $\\sim300$ mHz and then scanning the bias pulse repetition~rate. The~effective resonance frequency of each Xe isotope depends on the pulsing frequency ($\\omega_{2\\pi}$) as $\\omega^K = \\omega_{2\\pi}\\gamma^K\/\\gamma^S$. The 300 mHz detuning ensures that both isotopes are not simultaneously on resonance for a given pulsing frequency. When a Xe isotope is far from resonance, we expect the measured signals for that isotope's detection channel to be relatively flat if there is no cross talk. Cross~talk manifests if the signals from the off-resonanct isotope are not flat when the other isotope is scanned through resonance. Figure~\\ref{polmodmct} shows a phase sensitive measurement of cross talk. When~isotope $a$ is excited and isotope $b$ is not, isotope $b$'s detection channel exhibits non-zero signal, and vice versa. We estimate $\\beta = Q_{pp}^b\/Q_{pp}^a = 0.17$ (when isotope $a$ is on resonance) and  $\\beta' =Q^a_{pp}\/Q^b_{pp}=0.07$ (when isotope $b$ is on resonance), where $Q^K_{pp}$ is the peak-to-peak quadrature signal of isotope $K$.\n \n\n\nWe observe cross talk on our detection channels even when Xe is not excited and artificial Xe signals are applied, suggesting that our observed cross talk is not due to physical interactions between Xe isotopes. We believe the measured cross talk stems from imperfect rectification of $S_z$ due to optical pumping transients (i.e., gain reversals) and unaccounted-for phase shifts from high-pass filtering prior to rectifying. The optical pumping transients stem from the few ms finite response time of the magnetometer. Indeed, by implementing a sample-and-hold algorithm to ignore data acquired during optical pumping transients, the cross talk is suppressed. The cross talk is further suppressed when the high-pass filter is removed. Doing so, however, reduces the detection SNR by an order of magnitude since rectification maps $1\/f$ $S_z$ noise to the Xe carrier frequencies. \n\n\\section{Discussion}\n\nWe demonstrated a novel spin-exchange pumped $^{131}$Xe-$^{129}$Xe NMR gyroscope. The~production of longitudinal spin-exchange fields and the systematic uncertainty inherent to them is greatly suppressed by synchronous spin-exchange optical pumping. The simultaneous precession of each transversely polarized noble gas is continuously monitored using the optically pumped Rb~atoms. The measured Larmor resonance frequencies are highly correlated with bias magnetic field fluctuations. \n\nComputing the stationary sensor's perceived inertial rotation when oriented East-West and assuming a scale factor determined solely by $\\rho$ allows for an ARW sensitivity of $\\sim$16\\; $\\upmu$Hz\/$\\sqrt{\\text{Hz}}$ and bias instability of $\\sim$800 nHz. The finite short term sensitivity appears limited by the detection~SNR. What limits the SNR is not known. The photon shot noise limited magnetometer performance should support an SNR more than 100x greater than we measure. It is possible that fluctuations in pump frequency and intensity could limit our SNR by producing noise in the AC Stark shift and the Rb polarization. We confirmed that the fidelity of modulation produced by the EOMs is sufficient to support 4x the measured SNR. The source of drift which limits the bias instability is also unknown. Cell~temperature, which was not stabilized during these measurements, could be an important contribution to long term stability. We see no direct indications of first order quadrupole from $^{131}$Xe interactions with electric field gradients at the cell walls. We showed that cross talk exists between phase sensitive detection channels for each noble gas. Such cross talk influences the device's effective scale factor. Future work will include detecting $S_z$ in a way such that gain modulation is suppressed as demonstrated in~\\cite{Thrasher2019,thrasher2019PRApp}. This device's scale factor can be verified by orienting its sensitive axis along North-South and measuring Earth's rate of rotation. \n\n\n\n\\begin{acknowledgments}\nWe would like to thank Michael Larsen for insightful discussions.\n\nThis research was funded by the National Science Foundation grant numbers PHY-1607439 and PHY-1912543 and Northrop Grumman Mission Systems' University Research Program.\n\nAll authors contributed equally in all respects to this manuscript. All authors have read and agreed to the published version of the manuscript.\n\nThe authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sect:intro}\n\nMaterials that are composed of collections of separate, macroscopic solid grains \nbelong to the general classification of {\\it granular materials}. Examples of such materials are common, including sand, gravel, medicinal pills, coins, and breakfast cereal.\nGranular media are important to numerous industries ranging \nfrom mining to pharmaceuticals. In geophysics, granular \nmaterials are a central problem in understanding\nthe physics of earthquakes and tectonic faulting. \nEarthquake fault zones produce granular wear material continuously as a\nfunction of shear and grinding between the fault surfaces.  The wear \nmaterial, known as fault gouge, varies in thickness from 10's of cm to 1000 m\nand plays a critical role in determining the fault zone frictional\nstrength, the stability of fault slip, and the size of the rupture \nnucleation dimension. \n\n\nGranular media display a variety of complex \nstatic and  dynamic properties that distinguish them from conventional\nsolids and liquids. \nThe complexity of granular media lies primarily in the collective \nproperties of a macroscopic number of grains and how they interact with \neach other. The conditions under which a granular medium is stable \nor flows and the nature of this flow depend critically on the \ndistributions of grain size and shape as well as the interactions \nbetween the grains. The practical importance of granular media combined\nwith the richness of their physical properties has led to a great deal\nof interest from both theoretical and experimental points of view\n\\cite{DEG, jaeger, kadanoff}.\n\n\nAn important class of granular materials consists of nearly rigid particles that possess the following property: if a moderate force is applied, the particles start to move, and only after a substantial increase of the force the particles deform significantly. In other words, for loads which are not very high, the deformations inside a particle\nare small compared with the displacement of the particle center of mass. \nConsequently, the particle shapes change very little, so that\neach particle can be associated with a region of space that is inaccessible\nto any other particle. This gives rise to constraints on the admissible positions of particles. These {\\it impenetrability constraints} are also known as\ngeometric, kinematic, and excluded volume constraints. \n\n\n\nA physical phenomenon related to appearance of constraints is jamming. A particle is jammed when its motion is completely obstructed by the neighbors, so the whole cluster of neighboring particles can only \nmove together as a rigid body. \nThe corresponding mathematical notion of rigidity (\\cite{W}) can be applied to various physical (sphere packings, frameworks (trusses)), as well as mathematical objects. In particular, an important mathematical object associated with any particle packing is a {\\it contact graph} defined as follows:\nvertices of this graph are particle centers of mass, while edges represent interparticle contacts. \n\n\n\n\n\n\nThe simplest physical model that exhibits jamming is a classical hard sphere packing. The particles\nin this model are represented by rigid spheres,  and the only interparticle forces are reactions of constraints.  Rigidity of hard sphere packings is studied in \\cite{Co}. This problem can be formulated as a problem of detecting rigidity of the {\\it cable framework} associated with the contact graph of the packing.  The framework is obtained by replacing edges of the contact graph with the cables, and vertices with flexible hinges. The lengths of the cables can increase but not decrease, which models the impenetrability constraints. Recently, a linear programming algorithm for detecting rigidity in hard sphere packings (equivalently, cable frameworks) was proposed in \\cite{D}.  \n\nIn this work, we also use {\\it bar frameworks}. A bar framework\nis obtained from a graph by replacing the edges with rigid bars, and vertices with hinges.\nA bar framework and the associated graph are called rigid if the only possible vertex motions correspond to rigid body motions of the whole framework.  We note that both bar and cable frameworks can be associated\nto the same graph. To generate the bar framework, the edges of a graph are replaced with rigid bars that can only translate and rotate. In the case of the cable framework, one replaces edges with cables that can either move as rigid bodies or stretch. Thus every motion of a bar framework is also permitted by the cable framework, but the converse is not true in general.  Therefore, it is possible that a bar framework associated to a graph is rigid, while the cable framework corresponding to the same graph is not. Both\ncable frameworks and bar frameworks are special case of the so-called tensegrity frameworks studied in \\cite{CoW}. In a tensegrity framework, properties of edges can vary, e.g. some edges may be bars, others may be cables or struts (that can shrink, but not stretch).\n\n\nIt appears that the currently available mathematical results \\cite{Co, CoW, D} on statics of discrete particle systems with geometric constraints deal only with hard particle packings. \nTo the best of authors' knowledge, there are\nno results on frictional packings, and even elastic frictionless packings have yet not been studied. \nThe present work differs from \\cite{Co, CoW, D} in several respects. First, all these studies\ndeal with rigid particles. We consider a somewhat more realistic situation of geometrically\nconstrained particles with elastic interactions defined by a quadratic potential energy.\nSecond, while \\cite{Co, CoW, D} focus on jamming, we are interested in generic contact patterns of the the energy minimizing configurations.\nThe packings that we study are not jammed. Their contact graphs are such that the associated bar framework is rigid, but the packing can still deform when external boundary conditions are applied.\n\nThe third difference is in the type of the boundary conditions. The conditions in\n\\cite{D} are periodic or hard wall conditions. The periodic conditions are commonly used to\nminimize influence of the boundaries in the problem. However, presence of walls\nis a major factor that determines bulk behavior of granular materials. Therefore it seems better to use boundary conditions corresponding to engineering  and physical experiments, where the walls are rigid and may be moving. \n\n\n\n \nA frequently observed property of granular materials is concentration of the bulk\ndeformation in thin layers called shear bands. Within a band, the contact forces are\nweak, and the relative displacements can be on the order of particle size or larger.\nFor quasi-static flows driven by small shear rates, \nthe corresponding patterns are called micro-bands (\\cite{Kuhn1}). In that paper, shear\nband structures were studied by means of numerical simulations. The simulations\nin \\cite{Kuhn1} show that the typical size and number of bands in quasi-static shear depend on the imposed shear rate. For small shear rates, the bands have length and width comparable with the particle size. The distribution of these micro-bands within the material is rather uniform. As the shear rate increases, the band structure exhibits coarsening: the number of bands becomes smaller, and the length of each band increases. For sufficiently large shear rates, a single macroscopic shear band appears. \n\n\nHere, we are interested in the case of small external boundary conditions. In that case, a micro-band\ncan be formed by weakening of a single contact, or a small group of neighboring contacts. All weak\ncontacts form a subnetwork of the whole contact network. Such networks of weak contacts\n(corresponding to the micro-band patterns in \\cite{Kuhn1}) were studied numerically in \\cite{Radjai}. \nThe goal of this paper is to describe some generic geometric features of micro-band, or weak contact\nnetworks in dense packings of nearly rigid particles. In small deformation, pattern formation may be caused\nby the local jamming (which mathematically amounts to impenetrability constraints), and friction. \nWe are concerned with the role of constraints, while friction in neglected. \nThe notion of high density at this point is rather intuitive. It could mean, for instance, that each particle is\nin contact with at least three other particles, and the packing is jammed in the reference configuration, subject to zero boundary conditions. Below we make the notion of high density more precise (see the second paragraph on p. 4), using the relationship between the contact graph and the Delaunay graph (see e.g. \\cite{Ed}), generated by the set of particle centers. \n\n\nIn two dimensions, particles are represented by disks $D_i$ of radii $a_i$ with centers ${\\bf x}^i$, $i=1, 2,\\ldots, N$. The initial reference configuration is deformed by applying prescribed small displacements to the boundary particles. Assuming that the deformations inside of the individual particles are small, and neglecting rotational degrees of freedom, one can characterize the deformations of  $D_i$ by the displacements ${\\bf u}^i$ of their centers. The elastic interaction forces are modeled as\nin classical mechanics of point particles: the force exerted by $D_j$ on $D_i$ is applied\nat ${\\bf x}^i$, its direction is along the line joining ${\\bf x}^i$ and ${\\bf x}^j$, and its magnitude\ndepends linearly on ${\\bf u}^j-{\\bf u}^i$.\n\nWe further assume that\nthe granular material is pre-stressed (or, equivalently, the material is under confining stress). This means that in the reference state, the particles are squashed into each other as a result of applied external pressure.  \nFurther compression is supposed to be impossible (requires infinite energy), which\nintroduces impenetrability constraints into the problem.\nTo model impenetrability, one can, for instance, require that \n\\begin{equation}\n\\label{0.1}\n|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^j+{\\bf u}^j)|\\geq a_i+a_j,\n\\end{equation}\nfor each pair of particles. Since the the packing is dense, and the displacements are \nexpected to be small, it makes sense to require that a particle cannot escape a cage formed by its neighbors. Therefore, the contacts that exist in the reference configuration may be broken, but no new contacts are created after applying\nexternal  boundary conditions to the reference configuration. \nAn important consequence of this assumption is as follows. If the displacements satisfy ({0.1}) for each pair of particles {\\it in contact}, then for any pair of particles ({0.1}) is automatically satisfied. Indeed, if two particles are not in contact\nin the reference configuration, they cannot come into contact in the deformed configuration, and the distance between\nthem must be larger than the sum of their radii. In the sequel, we use this assumption in the course of proving the main result of this paper (Theorem \\ref{main-theorem}). \n\n\nNext, we introduce a network model which describes a granular material under the above assumptions. The vertices of the network are the particle centers, and the edges represent\nparticle contacts. \nThe collection of vertices ${\\bf x}^i$, $i=1, 2, \\ldots, N$, and edges forms the contact network (graph) $\\Gamma$. We suppose that $\\Gamma$  is a triangulation of a connected, convex polygonal domain $\\Omega$. \nThis assumption is realistic, since, for example, a periodic 2D packing of disks is triangular. \nAnother natural triangulation generated by ${\\bf x}^i, i=1, 2,\\ldots, N$ is the Delaunay graph $G$. In principle, \n$\\Gamma$ and $G$ may be different, since some edges in $G$ may not correspond to contacts. In the present case, we suppose that $\\Gamma$ and $G$ coincide, which corresponds to \"maximally dense\" packings. \n\nFor small displacements ${\\bf u}^i$, the quadratic constraints (\\ref{0.1}) can be approximated by\ntheir linearizations near ${\\bf u}^i={\\bf u}^j=0$, which leads to the linearized impenetrability constraints\n\\begin{equation}\n\\label{i3-intro}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}\\geq 0,\n~~i=1,2,\\ldots, N\n\\end{equation}\nfor each pair of vertices $i,j$ connected by an edge of $\\Gamma$.\nIn (\\ref{i3-intro}), \n$\n{\\bf q}^{ij}=({\\bf x}^j-{\\bf x}^i)\/\\left|{\\bf x}^j-{\\bf x}^i\\right|\n$\nare unit vectors\nthat point from ${\\bf x}^i$ to ${\\bf x}^j$ along the line of centers.\nNote that if the position of $D_i$ is fixed (${\\bf u}^i=0$), then\n${\\bf u}^j$ satisfying (\\ref{i3-intro}) must lie in the half-plane ${\\bf u}\\cdot {\\bf q}^{ij}\\geq 0$, so that $D_j$ would be moving away from $D_i$. \n\nFor certain boundary conditions the deformed packing can become more loose than the reference packing. On the macroscale, this can be observed as swelling of the specimen caused by the increase in the volume of the\nvoid space between the particles. Such swelling is typical in shear deformation, where the overall volume increase, known as dilatation, is observed in experiments.  To increase the void volume, some of the contacts\npresent in the reference configuration must be broken in the deformed configuration. Therefore, \namong all the contacts (satisfying (\\ref{i3-intro})), we further distinguish two types of contacts: broken and solid-like (see Fig. 1). We call a contact broken if\n\\begin{equation}\n\\label{broken}\n({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}>0,\n\\end{equation}\nand solid-like if\n\\begin{equation}\n\\label{solid-intro}\n({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}=0.\n\\end{equation}\n\nThe solid-like contacts correspond to two possible types of pair motions. The first type \nis a rigid motion of a pair, in which case the contact is called {\\it stuck}.\n\\begin{figure}[htbp]\n\\label{contacts}\n\\begin{center}\n\\input{hpackfig7-bis.pstex_t}\n\\caption{Left: a broken contact. Center: a solid-like sheared contact. Right: a solid like stuck contact\n(infinitesimal rigid rotation of a pair). The arrows indicate displacements.}\n\\end{center}\n\\end{figure}\n\nThe second type is a local shear motion.  In the local coordinates of one particle, it is either the motion of the second particle in the direction perpendicular to the line of centers, or an infinitesimal rotation (rolling). The corresponding contacts are called {\\it sheared}. In our idealized model, friction in neglected, and any tangential force would lead to immediate separation of particles, because for disk-shaped particles, the contact surface is a point. We, however, still call these contacts solid-like, because in reality, these contacts are subject to friction forces, the contact surface has a positive area, and the particles in a sheared contact will not separate until the tangential force reaches the static friction threshold. In the case of rolling, the particles stay in contact and the pair is capable of bearing a compressive load.\n\n\nPhysically, vertices of the network can be realized as unit point masses and edges can be realized as elastic springs. Elastic force of the spring $(i,j)$ is determined by the pair potential $H(t_{ij})$, where $t_{ij}=({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}$. \nThe potential is an important ingredient of our model, and therefore\nwe discuss it in detail. To motivate the choice\nof $H$, we first \nrecall the classical hard sphere potential\n$H_{hs}$, which in our notation is defined by\n\\begin{equation}\n\\label{hs}\nH_{hs}(t_{ij})=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\n0 & \\mbox{if}~t_{ij}\\geq 0.\\\\\n\\end{array}\n\\right.\n\\end{equation}\n$H_{hs}$ models the following two options: \n(i) moving non-deformable (hard spheres) particles toward each other requires infinite energy\n(a vertical line at $t_{ij}=0$), \n(ii) moving particles apart requires no energy.  Note that (\\ref{hs}) already incorporates the constraints\n(\\ref{i3-intro}) by requiring infinite potential energy to violate the constraints.\n\\begin{figure}[htbp]\n\\label{potentials}\n\\begin{center}\n\\input{hpackfig4.pstex_t}\n\\caption{\na) Hard sphere potential $H_{hs}$; b) The elastic potential $H(t_{ij})$ combines a vertical wall at $t_{ij}=0$ and a quadratic function with the vertex at $d$. }\n\\end{center}\n\\end{figure}\n\n\nElastic interaction between $D_i$ and $D_j$, together with constraints (\\ref{i3-intro})\ncan be modeled by the following potential\n\\begin{equation}\n\\label{actual}\nH(t_{ij}, d)=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\n\\frac 12 Cd^{-3}(t_{ij}-d)^2 & \\mbox{if}~t_{ij}\\geq 0.\\\\\n\\end{array}\n\\right.\n\\end{equation}\n\\noindent\nHere $d$ characterizes the cut-off distance of the potential, and $C$ determines\nthe magnitude of the pre-stress potential (the value of the potential when $t_{ij}=0$.\nThe potential (\\ref{actual}) is shown on Fig. 2, together with the hard-sphere potential.\nThe formula (\\ref{actual}) describes two options: (i) moving particles toward each other \nrequires infinite energy; (ii) movement of particles apart from each other is caused\nby finite, linear elastic force ${\\bf f}^{ij}=-\\partial H\/\\partial {\\bf u}^i$; This force is repulsive for\nsmall distances ($t_{ij}<d$), since $\\partial H\/\\partial t^{ij}<0$. The magnitude of the force \n$f^{ij}(d)=\\left|\\frac{\\partial H(t_{ij}, d)}{\\partial t_{ij}}\\right|=C\\frac 12d^{-3}|t_{ij}-d|$. \nThe magnitude of the force of pre-stress (or confining stress) is \ngiven by $\\lim_{t_{ij}\\to 0^+} f^{ij}=\\frac 12 Cd^{-2}$, which tends to zero as $d\\to\\infty$\nand $C$ is fixed.\nTherfore, the effect of pre-stress is smaller for larger $d$. Further,\n$H(t_{ij}, d)$ regularizes $H_{hs}$ in the following sense: if $d\\geq d_0>0$, then\n$\\lim_{d\\to\\infty} H(t_{ij}, d)=0$ uniformly\non $(0, d_0]$. In the paper, we do not pass to this limit. Instead we choose $d$ sufficiently large and fix it, so\nthat $H$ is close to $H_{hs}$. Also, for technical simplicity, we set $C=1$ in (\\ref{actual}), which corresponds\nto an appropriate rescaling. \n\n\nIn reality, once the distance between $D_i$ and\n$D_j$ is greater than the sum of their radii $a_i+a_j$, the pair interaction force\nis zero. In our model, we still have a small repulsive force for all $a_i+a_j\\leq t_{ij}\\leq d$. So, the particles in our model\nwould continue accelerating away from each other even after separating. This favors separation of particles, and could lead to\nincrease in the number of broken contacts. Since our goal is estimating the number of solid-like contacts from below, this increase is acceptable.  In fact, our estimate holds \nfor all $d$ sufficiently large.\nWe also mention that elastic contact force predicted by the classical Hertz theory\nis a non-linear function of $t_{ij}$. Our model is chosen for simplicity, and can be viewed\nas an approximation of Hertz theory, valid for sufficiently small displacements.\n\n\nIn general, the cut-off parameter $d$ may be different for different pairs of particles\nin contact. Therefore, we define a pair interaction energy\n\\begin{equation}\n\\label{energy}\nh(t_{ij}, d_{ij})= \\frac 12d^{-3}(t_{ij}-d_{ij})^2,\n\\end{equation}\nand consider \n\\begin{equation}\n\\label{actual-gen}\nH(t_{ij}, d_{ij})=\n\\left\\{\n\\begin{array}{cc}\n\\infty & \\mbox{if}~t_{ij}<0,\\\\\nh(t_{ij}, d_{ij}) & \\mbox{if}~t_{ij}\\geq 0,\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{reg}\nd_{ij}=d\\delta_{ij},~~~~\\delta_{ij}\\in [1\/2, 1].\n\\end{equation}\nThe formula (\\ref{actual-gen}) is more general than  (\\ref{actual}). \nThe choice of $d_{ij}$ in (\\ref{reg}) ensures that for all pairs $(i,j)$, the points where $h(t_{ij}, d_{ij})=0$\nand located in the interval $[1\/2d, d]$. The number 1\/2 is of no particular significance. \nAny number in the interval $(0, 1)$ would work just as well. We only need all $d_{ij}$ to have the same sign and comparable magnitudes controlled by $d$.\nFinally, the total elastic interaction energy $Q$ of the network is obtained by summing up $h(t_{ij}, d_{ij})$\nover all pairs $(i,j)$ corresponding to the edges of the network. \n\n\nThe equilibrium state of a granular material corresponds to a minimum of $Q$ subject to the constraints\n(\\ref{i3-intro}) and the appropriate boundary conditions. Since the functional $Q$ is quadratic, and the\nconstraints and boundary conditions are linear, this is a quadratic programming problem, studied in optimization theory (e.g., \\cite{Hestenes}). In the language of optimization theory, the solid-like contacts (\\ref{solid-intro}) correspond to the so-called {\\it active} constraints, while\nthe constraints corresponding to the broken contacts (\\ref{broken}) are called {\\it inactive}. The question\naddressed in this paper concerns the total number and spatial distribution of each type of constraints in the energy-minimizing configuration of the network. It appears that no general results of this type are currently available in  optimization theory. The present study makes use of the geometric features of the contact graph, in particular its rigidity properties (see above), to investigate the energy minimizer.  We also note the connection between our constrained variational problem  and continuum variational inequalities\n\\cite{Duvaut}, \\cite{Kikuchi}, \\cite{Kinderlehrer}, \\cite{Panagi}. Our problem can be viewed as a discrete\nvariational inequality. \n\n\n\n\nThe main result of the paper is Theorem \\ref{main-theorem} in Section \\ref{sect:optimality}.\nThere we consider a packing whose\ncontact graph in the reference configuration is a triangulation of a convex polygonal domain. The packing is deformed by imposing displacement boundary conditions at the packing boundary. The boundary conditions model the motion of rigid walls in engineering experiments. We prove that for generic contact graphs (precisely defined in Definition \\ref{reg-treg}) and generic pre-stresses (corresponding to the choice of $\\delta_{ij}$ in (\\ref{energy}), \nthe constrained energy minimizer for sufficiently large $d$ provides a packing with at least two solid-like contacts per each particle.  There also some (non-generic) choices of $\\delta_{ij}$ for which theorem does not provide a definite conclusion.\n\nThe network of solid-like contacts is the load-bearing structure. The network of broken contacts can be associated with the micro-bands \\cite{Kuhn1}, \\cite{Kuhn2} that appear during small shear deformations. The result implies that no particle can lose contact with {\\it all} of its neighbors, which eliminates ``micro-avalanches\". Put another way, loss of structural integrity in dense packings is evolutionary rather than catastrophic, so that shearing with a small displacement will first lead to dilatation, during which the packing becomes more loose everywhere, and only then local avalanches may occur.\n\n\nAnother useful consequence of theorem \\ref{main-theorem} is as follows. It provides a lower bound on the order parameter, recently introduced in \\cite{AT, VTA} as one of the main ingredients of the new phenomenological theory of dense granular flows proposed by Aronson and Tsimring. The order parameter \n$\\rho$ characterizes the phase transition from solid to fluidized state. To define $\\rho$ at an arbitrary point ${\\bf x}$ of $\\Omega$, one begins by fixing a mesoscopic averaging volume $V$ of characteristic size $h$\n(e.g., a disk of radius $h$ centered at  ${\\bf x}$).  Then, all solid-like contacts within $V$ should be counted. Next, the obtained number $n_s$ of solid-like contacts is divided by the number $n$ of all contacts within $V$\nto obtain $\\rho$. So, $\\rho$ is a mesoscopic average, which in general depends on $h$.\nIn many systems, such as periodic elastic composites, the results of mesoscopic averaging is practically independent of $h$ for $h$ larger than a certain characteristic length. For disordered granular materials this\nis not necessarily true. Therefore, a rigorous mathematical theory may require study of a family of order parameters parametrized by $h$. In Section 6, we define such a family of order\nparameters using the notion of a $k$-neighborhood of a vertex of\n$\\Gamma$ as a discrete analogue of $V$.  Specifying an integer $k$ in our definition corresponds to\nchoosing $h$ in the continuous case. \n\n\n\nThe paper is organized as follows. In Sect. 2 we formulate the main constrained minimization problem.\nThe problem contains two types of constraints: impenetrability constraints (\\ref{i3-intro}) and the\nboundary constraints (see (\\ref{b1}), (\\ref{b1-bis})), corresponding to the external boundary conditions.  Elimination of these boundary constraints\nleads to a reduced minimization problem. In Sect. 3 we recall some facts concerning first-order rigidity of graphs. In Sect. 4 we show existence of a unique minimizer of the reduced\nproblem. Optimality conditions for the reduced problem are stated and analyzed in Sect. 5, where we also state and prove the main theorem \\ref{main-theorem}. In Sect. 6 we introduce a definition of the order parameter in the spirit of \\cite{VTA} and give a lower bound on the\norder parameter that follows from the main theorem. Finally, conclusions are provided in Sect.\n7.\n\n\n\n\n\n\n   \n   \n\n\n\n\\section{Formulation of the problem}\n\\subsection{Elastic interactions with impenetrability constraints}\n\\label{sect-constraints}\nIn 2D, consider a packing of spheres $D_i$ of radii\n$a_i$ with centers ${\\bf x}^i, i=1,2,\\ldots, N$. (All vectors in this\npaper are column vectors and we use superscript `{\\sf T}' to indicate\ntransposition). The packing fills a bounded region. After an infinitesimal motion, the position of the center of $D_i$ is \n${\\bf y}^i$. We write\n${\\bf y}^i={\\bf x}^i+{\\bf u}^i$ where ${\\bf u}^i$ are displacements.\nThe vertices ${\\bf x}^i, {\\bf x}^j$ are connected by an edge if and only if $D_i, D_j$ are in contact. In this case, we call ${\\bf x}^i$ and ${\\bf x}^j$ neighbors. We denote \nby \n$\n{\\mathcal N}_i\n$\nthe set of $j\\in \\{1, 2,\\ldots, N\\}$ such that ${\\bf x}^j$ is a neighbor of ${\\bf x}^i$.\nOrientation of contacts (equivalently, edges) is prescribed by the unit vectors\n\\begin{equation}\n\\label{orient}\n{\\bf q}^{ij}=\\frac{{\\bf x}^j-{\\bf x}^i}{|{\\bf x}^j-{\\bf x}^i|}.\n\\end{equation}\nThe vertices ${\\bf x}^i$ and edges $(i,j)$ define the contact graph\n$\\Gamma$.  \nLet $E$ denote the number of edges of $\\Gamma$.\nThe edge set ${\\mathcal E}$ of $\\Gamma$ is given by\n$\\{(i,j): j\\in{\\mathcal N}_i, i =1,2, \\ldots, N\\}$. To each edge $(i,j)$\nwe can associate a pair potential energy $h(t_{ij}, d_{ij})$ defined in (\\ref{energy}).\nSumming up all these energies we obtain the total \nelastic interaction energy of the network. It is a quadratic form\n\\begin{equation}\n\\label{q-basic}\nQ({\\bf u}^1,{\\bf u}^2, \\ldots {\\bf u}^N)=\\sum_{i=1}^N\n\\sum_{j\\in {\\mathcal N}_i} h(t_{ij}, d_{ij})=\n\\frac 12 d^{-3}\\sum_{i=1}^N\n\\sum_{j\\in {\\mathcal N}_i}\n\\left (({\\bf u}^j-{\\bf u}^i)\\cdot{\\bf q}^{ij}-d_{ij}\\right)^2,\n\\end{equation}\non the displacements ${\\bf u}^i, i=1,2,\\ldots, N$.\nIn (\\ref{q-basic}), $d, d_{ij}$ are parameters specified by (\\ref{actual-gen}), (\\ref{reg}). \n\nOur objective is to determine the displacements ${\\bf u^i}$, $i=1,2, \\ldots,\nN$ so that the the energy functional $Q$ is minimized subject to two\ntypes of constraints. The first type of constraints consists of\n{\\em linearized impenetrability constraints}. These are obtained by formally\nlinearizing the condition\nthat the distance between two spheres in contact cannot decrease. Consider\ntwo spheres $D_i, D_j$ in contact. In the reference configuration,\n\\begin{equation}\n\\label{i1}\n|{\\bf x}^i-{\\bf x}^j|=a_i+a_j.\n\\end{equation}\nAssuming that $D_i, D_j$ cannot overlap, we have\n\\begin{equation}\n\\label{i2}\n|{\\bf y}^i-{\\bf y}^j|\\geq a_i+a_j.\n\\end{equation}\nThese are the impenetrability constraints.\nWe linearize (\\ref{i2}) by writing\n\\begin{eqnarray*}\n|{\\bf y}^i-{\\bf y}^j|^2 & = & |{\\bf x}^i-{\\bf x}^j+{\\bf u}^i-{\\bf u}^j|^2\\\\\n          & = &|{\\bf x}^i -{\\bf x}^j|^2 +\n            2({\\bf x}^i - {\\bf x}^j) \\cdot ({\\bf u}^i - {\\bf u}^j) +\n            |{\\bf u}^i - {\\bf u}^j|^2 \\\\\n          & = & (a_i + a_j)^2 + \n            2({\\bf x}^i - {\\bf x}^j) \\cdot ({\\bf u}^i - {\\bf u}^j) +\n            |{\\bf u}^i - {\\bf u}^j|^2. \n\\end{eqnarray*}\nNow for for ``small'' $|{\\bf u}^i - {\\bf u}^j|$ we can neglect quadratic term\n$|{\\bf u}^i - {\\bf u}^j|^2$, and (\\ref{i2}) yields $2({\\bf x}^i - {\\bf x}^j)\\cdot ({\\bf u}^i -{\\bf u}^j) \\geq\n0$, which in turn is equivalent to $({\\bf u}^j - {\\bf u}^i)\\cdot\n{\\bf q}^{ij} \\geq 0$ where $ {\\bf q}^{ij}$ is as defined in\n(\\ref{orient}).\nTherefore, the first set of constraints we impose on the displacements\n${\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N$ is\n\\begin{equation}\n\\label{i3}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}\\geq 0, ~~~j\\in {\\mathcal N}_i,\n~~i=1,2,\\ldots, N.\n\\end{equation}\nThe second type of constraints corresponds to the boundary conditions.\nParticles located at the packing boundary have {\\em prescribed} displacements.\nIn the sequel we refer to these particles as {\\it boundary particles}. The corresponding vertices of $\\Gamma$ are called {\\it boundary vertices}.\nOther particles are referred to as  {\\it interior}, or sometimes, {\\it free}, \nand the corresponding vertices of $\\Gamma$ as {\\it interior vertices}. \n\nAll boundary particles are divided into several groups,\nnumbered $1,2,\\ldots,M$. Let $I_m$ denote the set of indices\nof the particles in group $m$ for $m=1,2, \\ldots, M$.\nEach sphere in a certain group is in contact with at least one\nother sphere from the same group. \nEach group moves as a single rigid body. We assume that the prescribed boundary displacements are of the form\n\\begin{equation}\n\\label{b1}\n{\\bf u}^i={\\bf R}^m({\\bf x}^i),~~~i\\in I_m, ~~m=1,2,\\ldots, M,\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{b1-bis}\n{\\bf R}^m({\\bf x}^i)={\\bf c}^m+\\alpha^m K({\\bf x}^i-{\\bf x}^{\\star,m}),\n~~~i\\in I_m, ~~m=1,2, \\ldots, M,\n\\end{equation}\nand ${\\bf c}^m, {\\bf x}^{\\star,m}$ are given vectors, $\\alpha^m$ is a\ngiven scalar, and\n$K$ is the matrix denoting clockwise rotation by $\\pi\/2$.\nThe functions ${\\bf R}^m$ are called {\\em infinitesimal rigid\ndisplacements},\nparametrized by a scalar $\\alpha^m$, and\nvectors ${\\bf c}^m$ and ${\\bf x}^{\\star,m}$. We refer the\nreader to Sect. \\ref{sect:rigidity} for more details\non rigid displacements.\n\nOur description above leads to the\\\\\n\n\\vspace*{0.5\\baselineskip}\n\\noindent\n\\underline{\\bf Main problem:} \n\\begin{eqnarray}\n\\label{mpr1}\n\\mbox{minimize}&~~&  Q({\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N)\\\\\n\\label{mpr2}\n\\mbox{subject to}&~~& \\mbox{linearized impenetrability\n                            constraints (\\ref{i3})}\\\\\n\\label{mpr3}\n                 &~~& \\mbox{and boundary conditions (\\ref{b1}).}\n\\end{eqnarray}\n\n\n\\subsection{Feasible region}\nLet us define the configuration space $U$. Points of this space\nare denoted by ${\\bf U}=(({\\bf u}^1)^{\\mbox{\\scriptsize\\sf T}}, ({\\bf u}^2)^{\\mbox{\\scriptsize\\sf T}}, \\ldots,\n({\\bf u}^N)^{\\mbox{\\scriptsize\\sf T}})^{\\mbox{\\scriptsize\\sf T}}$.\n\n\\noindent\n{\\bf Remark}. To avoid this heavy notation, we simply write\n$$\n{\\bf U}=({\\bf u}^1, {\\bf u}^2, \\ldots, {\\bf u}^N),\n$$\nwhen no confusion can occur.\n\n\nDimension of $U$ is $2N$.  \n{\\it Feasible region} ${\\mathcal F}$ is the subset of $U$\nin which all the constraints (\\ref{i3}) and (\\ref{b1})\nare satisfied. The points satisfying (\\ref{i3}) form a polyhedral\n(not necessarily bounded) region. The boundary of this region\nconsists of parts of the hyperplanes (subspaces of dimension $2N-1$) defined\nby \n\\begin{equation}\n\\label{i3-eq}\n({\\bf u}^j-{\\bf u}^i)\\cdot {\\bf q}^{ij}=0,~~~~j\\in{\\mathcal N}_i,~i=1, 2, \\ldots, N.\n\\end{equation}\nBecause of the close relation to rigidity, we refer to (\\ref{i3-eq}) as\n$R$-{\\it equations}. \nEquations\n(\\ref{b1}) define\n$M$ planes $S_m, m=1, \\ldots, M$. Dimensions\nof $S_m$ depend on the number of the boundary particles in the $m$-th group. \n\nFor each point of ${\\bf U}\\in {\\mathcal F}$, \nsome of the constraints (\\ref{i3}) are satisfied as equations. These constraints\nare called {\\it active}. The corresponding edges of the contact graph $\\Gamma$ \nare called active as well. \nThe rest of (\\ref{i3}) are satisfied as strict inequalities. These are {\\it inactive}\nconstraints (respectively, edges).\n\n\\subsection{Elimination of constraints corresponding to boundary conditions}\n\\label{sect:elimination}\nThe quadratic form $Q$ in (\\ref{q-basic}) can be written in a convenient\nform in terms of a certain matrix $R^r$. To define $R^r$, we\nindex the edges of $\\Gamma$ by $l$,\n$l=1,2, \\ldots, E$. Let $(i_l, j_l) \\in {\\mathcal E}$ be the edge\nof $\\Gamma$ corresponding to $l$ for $l=1,2,\\ldots, E$. \nLet $R^r$ be the $E\\times 2N$ matrix whose $l$-th row is defined by\n\\begin{equation}\n\\label{row}\nR^r_{lm} = \\left\\lbrace\\begin{array}{ll}\n             +({\\bf q}^{i_lj_l})_1 &\\mbox{if $m=2(j_l-1)+1$}\\\\\n             +({\\bf q}^{i_lj_l})_2 &\\mbox{if $m=2(j_l-1)+2$}\\\\\n             -({\\bf q}^{i_lj_l})_1 &\\mbox{if $m=2(i_l-1)+1$}\\\\\n             -({\\bf q}^{i_lj_l})_2 &\\mbox{if $m=2(i_l-1)+2$}\\\\\n             0                     &\\mbox{otherwise}\n             \\end{array}\\right .\n\\end{equation}\nfor $l=1,2, \\ldots, E$. \n\n\n\\noindent\n{\\bf Remarks.} \n1. $R^r$ is the (first-order) rigidity matrix,\na well known object in geometric rigidity theory (see e.g. \\cite{CoW, W}).\n\n\\noindent\n2. Consider\nvertices ${\\bf x}^{i_l}, {\\bf x}^{j_l}$ and the edge $l$ connecting them.\nThe corresponding row ${\\bf r}^l$ of $R^r$ has $2N$ entries. We can view ${\\bf r}^l$\nas a string of $N$ pairs of numbers, the first pair corresponding to ${\\bf x}^1$,\nthe second to ${\\bf x}^2$ and so on. For simplicity, we shall call a pair\nof entries corresponding to a particular vertex ${\\bf x}^i$\na {\\it place corresponding to ${\\bf x}^i$}.\n\nThen we can interpret equation (\\ref{row}) as follows. A row ${\\bf r}^l$ has zeros at all places, except two. The non-zero entries are $-{\\bf q}^{i_l, j_l}$, written as a two-dimensional row vector\nat the place corresponding to ${\\bf x}^{i_l}$; and ${\\bf q}^{i_l, j_l}$,\nwritten as a two-dimensional row\nat the place corresponding to ${\\bf x}^{j_l}$.\n\n\n\\noindent\n3. A row of $R^r$ corresponds to an edge of $\\Gamma$. Therefore it is natural to call a row active (respectively, inactive)\nif a corresponding edge is active (respectively, inactive).\n\n\nNow define the vector ${\\bf d}\\in{\\mathbb R}^E$ by\n\\begin{equation}\n\\label{d}\n{\\bf d}=-(d_{i_1j_1},~d_{i_2j_2},~ \\ldots, ~d_{i_Ej_E}), \n\\end{equation}\nwhere $d_{i_l, j_l}$ are chosen according to (\\ref{reg}).               \nWith these notations the quadratic form $Q$ in (\\ref{q-basic})\ncan be written as\n\\begin{equation}\n\\label{q}\nQ({\\bf U})=d^{-3}\\frac 12 (R^r{\\bf U}+{\\bf d})\\cdot( R^r{\\bf U}+{\\bf d}). \n\\end{equation}\n\n\nWe now eliminate the boundary conditions (\\ref{b1}) from the main\nproblem (\\ref{mpr1}, \\ref{mpr2}, \\ref{mpr3}).\nLet $N_b= \\sum_{m=1}^M \\mbox{card}(I_m)$.\nThen the equations (\\ref{b1}) simply state that the $2N_b$\ncomponents of ${\\bf U}$ corresponding to the $N_b$\nboundary vertices have prescribed\ndisplacements. Without loss of generality assume\nthat the {\\em last} $2N_b$ components of $U$ correspond\nto the boundary vertices.\nLet us partition ${\\bf U}$ as\n\\begin{equation}\n\\label{split1} \n{\\bf U}=[~{\\bf z}~|~{\\bf w}~],\n\\end{equation} \nwhere ${\\bf z}=({\\bf u}^1, {\\bf u}^2, \\ldots,\n{\\bf u}^{N-N_b})$ corresponds\nto displacement vectors of interior vertices, and \n${\\bf w}= ({\\bf u}^{N-N_b+1}, {\\bf u}^{N-N_b+2}, \\ldots,\n{\\bf u}^{N})$ corresponds to the displacements of the\nboundary vertices.\nThe equality constraint (\\ref{b1}) is now simply\n\\begin{equation}\n\\label{b3}\n{\\bf w}={\\bf g}\n\\end{equation} \nwhere $g\\in {\\mathbb R}^{2N_b}$ is the vector of displacements prescribed by\nthe right-hand-sides of (\\ref{b1}).\nThe matrix $R^r$ can be partitioned similarly to (\\ref{split1}):\n\\begin{equation}\n\\label{split2}\nR^r=[~R~|~R^b~],\n\\end{equation}\nwhere dimensions of $R$ and $R^b$ are $E\\times 2(N-N_b)$ and $E\\times\n2N_b$, respectively. \nDenote \n\\begin{equation}\n\\label{a}\n{\\bf a}=R^b{\\bf g}.\n\\end{equation}\nUsing (\\ref{split1})--(\\ref{a}) in (\\ref{q}) and in\n(\\ref{i3}) we can reduce the main problem\n(\\ref{mpr1}, \\ref{mpr2}, \\ref{mpr3})\nto\\\\\n\n\\vspace*{0.5\\baselineskip}\n\\noindent\n\\underline{{\\bf Reduced problem:}}\n\\begin{eqnarray}\n\\label{pr1}\n\\mbox{minimize}&~~&F({\\bf z})=\\frac 12 d^{-3}\\left(R{\\bf z}+{\\bf a}+{\\bf d}\\right)\n\\cdot\\left( R{\\bf z}+{\\bf a}+{\\bf d}\\right)\\\\\n\\label{pr2}\n\\mbox{subject to}&~~&R{\\bf z}+{\\bf a}\\geq 0.\n\\end{eqnarray}\n\\vspace*{0.5\\baselineskip}\nThe minimization in (\\ref{pr1}) is taken over all ${\\bf z}\\in\n{\\mathbb R}^{(N-N_b)}$.\n\n\\section{First-order rigidity}\n\\label{sect:rigidity}\nA rigid motion is a composition of a translation and rotation:\n\\begin{equation}\n\\label{r1}\n{\\bf y}({\\bf x})={\\bf c}+{\\bf x}^\\star+O({\\bf x}-{\\bf x}^\\star),\n\\end{equation}\nwhere $O$ is an orthogonal (rotation) matrix, ${\\bf c}$ is a translation vector,\n${\\bf x}^\\star$ is a center of rotation. If $O$ is close to identity $I$ (infinitesimally small rotation), then\n$$\nO\\approx I+A,\n$$\nwhere $A$ is a skew matrix ($a_{ij}=-a_{ji}$). \n\n\nSuppose that in a two-dimensional rigid motion, the rotation angle $\\alpha$ is close to zero.  \nThen\n$$\nO=\n\\left(\n\\begin{array}{cc}\n\\cos \\alpha & \\sin \\alpha \\\\\n-\\sin \\alpha & \\cos \\alpha\\\\\n\\end{array}\n\\right)\\approx\n\\left(\n\\begin{array}{cc}\n1 & 0 \\\\\n0 & 1\\\\\n\\end{array}\n\\right)\n+\n\\alpha\n\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n-1 & 0\\\\\n\\end{array}\n\\right)=I+\\alpha K,\n$$\nwhere \n$$\nK=\\left(\n\\begin{array}{cc}\n0 & 1 \\\\\n-1 & 0\\\\\n\\end{array}\n\\right)\n$$ \nis a clockwise rotation by $\\pi\/2$. In that case, (\\ref{r1}) becomes\n\\begin{equation}\n\\label{r2}\n{\\bf y}({\\bf x})={\\bf c}+{\\bf x}+\\alpha K({\\bf x}-{\\bf x}^\\star)=\n\\left(\n\\begin{array}{c}\nc_1+x^\\star_1-\\alpha (x-x^\\star)_2\\\\\nc_2+x^\\star_2+\\alpha (x-x^\\star)_1\\\\\n\\end {array}\n\\right).\n\\end{equation}\n\n\nLet ${\\bf u}={\\bf y}({\\bf x})-{\\bf x}$ denote the displacement. We can write\n(\\ref{r2}) as\n\\begin{equation}\n\\label{r3}\n{\\bf u}({\\bf x})={\\bf c}+\\alpha K({\\bf x}-{\\bf x}^\\star).\n\\end{equation}\n\n\\begin{definition}\nWe call (\\ref{r3}) an infinitesimal rigid displacements in 2D.\n\\end{definition}\n\n\n\nNext, let $G$ be a graph. \nConsider all motions of vertices of $G$ that preserve the lengths of the edges.\nIf the only such motions are the rigid body motions of the whole graph, \nthen the graph is called {\\it rigid}.\nA graph $\\Gamma$ is {\\it first-order rigid} \\cite{W} if all solutions\nof the $R$-system (\\ref{i3-eq})\nare infinitesimally rigid displacements.  \nIn addition, $\\Gamma$ is {\\it independent}\nif\nthe rows of the rigidity matrix $R^r$ are linearly independent. Graphs that are both\nfirst-order rigid and independent are called {\\it isostatic} (\\cite{W}). Intuitively, an isostatic graph is minimally rigid, that is removing any edge results in loss of rigidity. Another notion of rigidity\nis {\\it generic rigidity}, see \\cite{W}. According to Thm. 49.1.7 from \\cite{W},\ngeneric rigidity for a neighborhood in a configuration space is equivalent to the first-order rigidity for some specific configuration in that neighborhood.  \n\nWith the definition of $R^r$ and ${\\bf U}$ in Sect. \\ref{sect:elimination}\nthe system (\\ref{i3-eq}) can be written as\n\\begin{equation}\n\\label{r-sys-a}\nR^r{\\bf U}=0.\n\\end{equation}\n\nNote the connection between $R$-system and constraints, \nas well as the functional of the main problem\n(\\ref{mpr1},{\\ref{mpr2},{\\ref{mpr3}).\n\nThe following definition (see \\cite{W} for a $d$-dimensional definition) is useful for verifying rigidity of graphs.\n\\begin{definition}\n\\label{henneberg}\nFor a graph $\\Gamma$, the Henneberg 2-construction in 2D is a sequence of graphs $G_1, G_2, \\ldots, G_n$ such that:\\newline\n\\noindent\n(i) $G_{k+1}$ is obtained from $G_k$ by either vertex addition (attaching a new vertex by 2 edges); or edge splitting (replacing and edge from $G_k$\nwith a new vertex joined to its ends and to 1 other vertex);\\newline\n\\noindent\n(ii) $G_k$ is a complete graph on $k$ vertices, and $G_n=\\Gamma$.\n\\end{definition}\nThe following result\nis stated in (\\cite{W}, thm. 49.1.13):\n\\begin{theorem}\n\\label{hen-thm}\nIf a graph $G\\subset {{\\mathbb R}}^2$ is obtained by a Henneberg $2$-construction, then $G$ is \ngenerically isostatic. \n\\end{theorem}\n\\noindent\n{\\bf Remark 1}.\nIn veiw of the definitions above, Theorem \\ref{hen-thm} implies that a graph obtained by Henneberg $2$-construction is first-order rigid and independent (minimally rigid).\n\n\nIn the present case, the rows of the rigidity matrix $R^r$ are not linearly independent, but the row rank is maximal. This means that we typically have more edges than needed to ensure rigidity of $\\Gamma$. In this situation, the following theorem (thm. 49.1.14 from \\cite{W}) is useful.\n\\begin{theorem}\n\\label{gluing}\nIf two graphs $G_1$ and $G_2$ are generically rigid planar graphs sharing at least $2$ vertices,\nthen the graph $G$ obtained by combining all vertices and edges of $G_1, G_2$ is generically rigid.\n\\end{theorem}\n\n\\noindent\n{\\bf Remark 2}.  Because of the relation between generic rigidity and first-order rigidity, Theorem\n\\ref{gluing} implies that combining first-order rigid graphs $G_1$, $G_2$ as in this Theorem\nyields a first-order rigid graph $G$. We shall use Theorem \\ref{gluing} to obtain first-order rigidity of\ntriangulations. Indeed, one triangle $G_1$ is first-order rigid. Adding another triangle $G_2$ so that\n$G_1$ and $G_2$ share an edge, yields a first-order rigid graph. Then we can proceed sequentially.\nGiven  a first-order rigid triangulation $G_k$, we construct $G_{k+1}$ by combining $G_k$ with a triangle. This new triangle either shares two vertices with $G_k$, or all three vertices. In the first case, $G_{k+1}$ would have\none more vertex and two more edges than $G_k$. In the second case, $G_{k+1}$ has the same number of vertices as $G_k$, and one more edge. By theorem \\ref{gluing}, any planar triangulation obtained by this sequential procedure is first-order rigid.\n\n\n\n\n\n\\section{Existence and uniqueness of minimizers of the reduced problem}\nLet $\\Omega$ be a bounded connected domain in ${\\mathbb R}^2$ with a polygonal boundary.\nFirst we show that, under certain assumptions on geometry of $\\Gamma$, the matrix $R$ has full column rank.\n\n\nWe shall say that $\\Gamma$ is a triangulation if edges of $\\Gamma$ partition $\\Omega$ into a disjoint\nunion of triangles.  \n\nLet $X$ be a set of interior vertices of $\\Gamma$, containing at least two elements. Consider a graph\n$G_X\\subset \\Gamma$ defined as follows. Vertices of $G_X$ are all elements of $X$. Edges of \n$G_X$ are those edges of $\\Gamma$ that join two vertices from $X$. We also assume that $X$ is chosen\nso that $G_X$ is a connected graph.\n\\begin{definition}\n\\label{solid}\nThe contact graph $\\Gamma$ is {\\it cell-connected} if for each $G_X\\subset \\Gamma$ as above, there\nexist two vertices ${\\bf x}^1, {\\bf x}^2$ in $G_X$, and two interior vertices $\\hat{{\\bf x}}^1, \\hat{{\\bf x}}^2$ in $\\Gamma\\backslash G_X$, such that the quadrilateral with vertices \n${\\bf x}^1, {\\bf x}^2, \\hat{{\\bf x}}^1, \\hat{{\\bf x}}^2$ is a union of two {\\it adjacent} triangles of $\\Gamma$.\n\\end{definition}\nAn example illustrating the definition if shown in Fig. 3.\n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig9.pstex_t}\n\\caption{a) --A cell-connected graph. Solid dots indicate vertices of $G_X$;\nb) -- The cell connection with three edges and four vertices is shown separately; \nc)-- A triangulation that is not cell connected. Here the vertices of the small triangle inside\nare connected to other vertices by pairs of collinear edges.}\n\\end{center}\n\\end{figure}\n\n\n\n\\begin{definition}\n\\label{reg-treg}\nWe call $\\Gamma$ a {\\it regular triangulation} if \\newline\n(i) $\\Gamma$ can be obtained by sequential addition of triangles in the way described in the Remark 2;\\newline\n\\noindent\n(ii) every interior vertex, connected to a boundary vertex,  is also connected to at least one other\nboundary vertex, and the corresponding edges are non-collinear;\\newline\n\\noindent\n(iii) $\\Gamma$ is cell-connected.\n\\end{definition}\n\n\n\n\n\\noindent\n{\\bf Remark. } \nNote that i) in Definition \\ref{reg-treg} implies that $\\Gamma$ is first-order rigid.\nThe property ii) states that every edge connecting a boundary vertex \nwith an interior vertex must be a part of the boundary of a triangle, containing two boundary vertices and one  interior vertex. \n\nInformally, Definition  \\ref{solid} (or (iii) in Definition \\ref{reg-treg} )\ncan be interpreted as a strong connectivity property. According to Definition \\ref{solid}, a generic connected subgraph\n$G_X$ is connected to the ``rest of $\\Gamma$\" not just by an edge, but by a \"more robust\" cell structure that consists of three edges, with one edge bracing the other two, (see Fig 3.). Note also that either ${\\bf x}^1$ or ${\\bf x}^2$ are connected to the vertices of $\\Gamma\\backslash G_X$ by a pair of non-collinear edges. This observation is important in the proof of Proposition \\ref{full-rank} below. It is not difficult to see\nthat a periodic triangular planar graph satisfies iii). However, there are triangulations with a mean coordination number four that do not satisfy iii). An example of such a graph is shown in Fig. 3 c).\n\n\n\n\\begin{proposition}\n\\label{full-rank}\nSuppose that $\\Gamma$ is a regular triangulation. Then ${\\mbox rank}~R=2(N-N_b)$.\n\\end{proposition}\n\n\\noindent\n{\\it Proof}. Consider a subgraph $\\Gamma_{max}\\subset \\Gamma$ constructed inductively as follows. \nBegin with $\\Gamma_1$ that consists of all boundary vertices. On the next step, add an interior vertices connected to $\\Gamma_1$ by two or more non-collinear edges. Also, add exactly two non-collinear edges that connect this vertex to $\\Gamma_1$. Call\nthe resulting graph $\\Gamma_2$. Generally, given $\\Gamma_k$, $k\\geq 2$, define\n$\\Gamma_{k+1}=\\Gamma_k\\cup S_k$, where $S_k$ consists of an interior vertex ${\\bf x}^k$, not contained in\n$\\Gamma_k$ but connected to $\\Gamma_k$ by at least two non-collinear edges, together with\na pair of non-collinear edges connecting ${\\bf x}^k$ to $\\Gamma_k$. Since the graph $\\Gamma$ has a finite number of vertices, the\nprocess terminates after a finite number of steps. The resulting graph is $\\Gamma_{max}$. The construction\nis illustrated in  Fig. \n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig15.pstex_t}\n\\caption{The subgraphs $\\Gamma_1$ (left) and $\\Gamma_2$ (right). The boundary vertices are shown\nin gray. The interior vertices of $\\Gamma_1$ and $\\Gamma_2$ are shown in black. The edges of\n$\\Gamma_1$, $\\Gamma_2$ are shown by solid lines. The other vertices of $\\Gamma$ are represented by\nthe unfilled circles. The edges of $\\Gamma$ not included into $\\Gamma_1, \\Gamma_2$ are represented by the dotted lines.}\n\\end{center}\n\\end{figure}\n\\begin{figure}[]\n\\label{non-connected}\n\\begin{center}\n\\input{hpackfig16.pstex_t}\n\\caption{Two different subgraphs $\\Gamma_{max}$ constructed inductively. Both subgraphs\nare constructed starting with $\\Gamma_2$ in Fig. 4.}\n\\end{center}\n\\end{figure}\nWe claim that $\\Gamma_{max}$ contains all vertices of $\\Gamma$. To obtain\na contradiction, suppose that there are vertices not included into $\\Gamma_{max}$.\nDenote the set of these vertices by $X$, and denote by $G_X$ the graph formed by vertices in\n$X$ and all edges of $\\Gamma$ that connect these vertices. Let $G_X^c$ be any connected component\nof $G_X$. If $G_X^c$ is a single point ${\\bf x}_g$, then, since $\\Gamma$ is a triangulation, there must be at least three edges incident at ${\\bf x}_g$, and at least two of these edges must be non-collinear. \nThus  ${\\bf x}_g$ must be in $\\Gamma_{max}$, which gives a contradiction.\nNext suppose that $G_X^c$ contains two or more vertices. By Definition \\ref{solid}, (see also the Remark following that Definition), there must be a pair of vertices of ${\\bf x}^1, {\\bf x}^2$ in $G_X^c$ connected to two vertices in $\\Gamma\\backslash G_X$ by three edges, and at least one of ${\\bf x}^1, {\\bf x}^2$ must\nbe connected to $\\Gamma\\backslash G_X$ by two non-collinear edges. Denote this vertex by ${\\bf x}_g$. It must be included into $\\Gamma_{max}$ which gives a contradiction and proves the claim.\n\n\nNext, we claim that the number of edges in $\\Gamma_{max}$ is $2(N-N_b)$. Indeed, each interior\nvertex in $\\Gamma_2$ has exactly two non-collinear edges incident at it. Then, on each of the next steps, we add an interior vertex together with two non-collinear edges incident at it. Since\nthe number of free vertices in $\\Gamma_{max}$ is $N-N_b$, the claim is proved.\n\n\nFinally, we claim that the rows of $R$ corresponding to the edges of $\\Gamma_{max}$ are linearly independent. Let the matrix of these rows be denoted by\n$R_{max}$. This is a square $2(N-N_b)$ matrix. We claim that an appropriate\nrow-reduction reduces $R_{max}$ to a matrix $R^\\prime_{max}$ that has\nblock-diagonal form: for each\nvertex ${\\bf x}^i$ there are exactly two rows ${\\bf r}^{i,1}, {\\bf r}^{i,2}$ of $R^\\prime_{max}$,\nand two linearly independent unit vectors ${\\bf q}^{ij_1}$ and ${\\bf q}^{ij_2}$,\nsuch that ${\\bf r}^{i,1}$ (${\\bf r}^{i,2}$) contains\n${\\bf q}^{ij_1}$ (${\\bf q}^{ij_2}$) at a place\ncorresponding to ${\\bf x}^i$ while all other entries in these rows are zero.\n\n\nTo see this, consider first a ``basic unit\" of $\\Gamma_2$: an interior vertex ${\\bf x}^i$ and two non-collinear edges\nincident at it. Let the corresponding unit vectors be ${\\bf q}^{i1}$, ${\\bf q}^{i2}$. Recall that these edges connect ${\\bf x}^i$ to two boundary vertices. Consequently,\nthe rows ${\\bf r}^1, {\\bf r}^2$ corresponding to the above pair of edges have zeros\nat all places, except two places corresponding to ${\\bf x}^i$. \nThe non-zero entries of ${\\bf r}^1$ (${\\bf r}^2$) are two components of ${\\bf q}^{i1}$\n(${\\bf q}^{i2}$).\nSince ${\\bf q}^{i1}$, ${\\bf q}^{i2}$ are linearly independent, so are\n${\\bf r}^1, {\\bf r}^2$.  Furthermore, linear combinations\nof ${\\bf r}^1, {\\bf r}^2$ can be used to eliminate non-zero entries in other rows.\nBy adding an appropriate linear combination of ${\\bf r}^1, {\\bf r}^2$ to a row with\nsome unit vector ${\\bf q}^{ij}$ at a place corresponding to ${\\bf x}^i$, we can obtain\nzeros at this place. Hence, by using ${\\bf r}^1, {\\bf r}^2$ as pivots in Gaussian elimination\nwe can obtain rows whose only non-zero entries are at a place corresponding\nto a vertex that was added to $\\Gamma_2$ on the next step of the iteration. These rows, in turn, can be used as pivots. Continuing\nwith row reduction, we can eventually reduce all rows of $R_{max}$ to this form\n(this follows from the fact that $\\Gamma_{max}$ contains all vertices of $\\Gamma$). \nThe proposition is proved.\n\n\\noindent\n{\\bf Remark}. It is interesting to compare $R$ and the rigidity matrix $R^r$. It is \nwell-known that for a first-order rigid graph,  the null space of $R^r$ is non-trivial\nand consists of infinitesimal rigid displacements. The proposition above shows that\nthe null space of $R$ is trivial. The main difference in structure between these two \nmatrices is that $R$ contains special rows, that might be called {\\it broken}. These\nrows correspond to edges connecting an interior vertex to a boundary one. \nA typical row of $R^r$ has four non-zero entries, while each broken row\nhas only two.  These entries occur at a place corresponding to an interior vertex.\nIf an interior vertex is connected to two boundary vertices, then the regular\ntriangulation property of $\\Gamma$ ensures that the corresponding broken rows are non-collinear, and can\nbe used in the sequential Gaussian elimination, as done in the proof. \n\n\n\\begin{proposition}\n\\label{existence}\nConsider the problem (\\ref{pr1},\\ref{pr2}). Suppose that $\\Gamma$ is a regular triangulation,\nthe feasible set of (\\ref{pr1},\\ref{pr2}) is non-empty,\nand that the unconstrained minimizer of $F({\\bf z})$ is not feasible. \nThen\nthe problem (\\ref{pr1},\\ref{pr2}) admits a unique minimizer that is a point\non the boundary of its feasible set.\n\\end{proposition}\n\n\\noindent\n{\\it Proof.} \\vspace{0.1cm}\nThe problem (\\ref{pr1},\\ref{pr2}) has a feasible\npoint $\\bar {\\bf z}$. Then the problem (\\ref{pr1},\\ref{pr2}) has a unique\nminimizer ${\\bf z}^*$ as we now demonstrate.\nLet the set ${\\mathcal L}(\\bar {\\bf z})$\nbe defined by\n\\begin{equation*}\n{\\mathcal L}(\\bar {\\bf z}) = \\{{\\bf z}\\in{\\mathbb R}^{2(N-N_b)}:\nF({\\bf z}) \\leq F(\\bar {\\bf z})\\}.\n\\end{equation*}\nBy Proposition \\ref{full-rank} the matrix $R$ has full column rank. Therefore, the set\n${\\mathcal L}(\\bar {\\bf z})$ is an ellipsoid, which is a closed,\nbounded, convex set.\nThe set\n${\\mathcal H}$ of ${\\bf z} \\in {\\mathbb R}^{2(N-N_b)}$ satisfying the constraint\n(\\ref{pr2})\nis a closed half space. Therefore, ${\\mathcal L}(\\bar {\\bf z}) \\cap {\\mathcal H}$\nis a nonempty, closed, bounded convex set. Indeed, the reduced problem\n(\\ref{pr1},\\ref{pr2}) is equivalent to the problem\n\\begin{eqnarray}\n\\label{preq1}\n\\mbox{minimize}&~~&F({\\bf z})\\\\\n\\label{preq2}\n\\mbox{subject to}&~~&{\\bf z} \\in {\\mathcal L}(\\bar {\\bf z})\\cap{\\mathcal H}.\n\\end{eqnarray}\nNow by the continuity of $F$ and the compactness of\n${\\mathcal L}(\\bar {\\bf z})\\cap{\\mathcal H}$ we see that the problem\n(\\ref{preq1},{\\ref{preq2}) and hence the reduced problem\n(\\ref{pr1},\\ref{pr2}) has a minimizer ${\\bf z}^*$. Since $R$ has full\ncolumn rank, $R^{\\mbox{\\scriptsize\\sf T}} R$ is positive definite. The positive definiteness\nof the matrix $R^{\\mbox{\\scriptsize\\sf T}} R$ implies that $F$ is strictly convex\non ${\\mathcal L}(\\bar {\\bf z})\\cap {\\mathcal H}$, from which we conclude that\n${\\bf z}^*$ must be unique. \n\n\nNow if $R{\\bf z}^\\star+{\\bf a}>0$, then ${\\bf z}^\\star$ must be the\nunconstrained minimizer of $F$. This contradicts the assumption that\nthe unconstrained\nminimizer is not feasible. Therefore, some components of\n$R{\\bf z}^\\star+{\\bf a}$ must be zero, and thus ${\\bf z}^\\star$ must be on the boundary\nof the feasible region.\n\n\n\n\n\n\n\n\\section{Optimality conditions for the reduced problem}\n\\label{sect:optimality}\nThe reduced problem (\\ref{pr1},\\ref{pr2}) has a convex objective\nfunction and linear constraints. For such problems \nit is possible to state optimality conditions \nthat  are both necessary and sufficient \\cite{NW}.\nTo be specific define the Lagrangian $L$ for the problem\n({\\ref{pr1},\\ref{pr2}) by\n\\begin{equation}\n\\label{L}\nL({\\bf z}, \\boldsymbol \\lambda)=\\frac 12 d^{-3}(R{\\bf z}+{\\bf a}+{\\bf d})\\cdot(R{\\bf z}+{\\bf a}+{\\bf d})-\nd^{-3}\\boldsymbol \\lambda\\cdot (R{\\bf z}+{\\bf a}).\n\\end{equation}\n\nThen ${\\bf z}^*$ solves problem (\\ref{pr1},\\ref{pr2}) if and only if\nthere exists $\\boldsymbol\\lambda^*$ such that\n${\\bf z}={\\bf z}^*$ and $\\boldsymbol\\lambda =\\boldsymbol\\lambda^*$\nsatisfy the Karush, Kuhn, Tucker (KKT) conditions\n\\begin{eqnarray*}\n&& \\nabla_{z} L ({\\bf z},\\boldsymbol\\lambda) = 0\\\\\n&& R{\\bf z} +{\\bf a}\\geq 0\\\\\n&& \\boldsymbol\\lambda\\cdot (R{\\bf z}+{\\bf a})=0\\\\\n&& \\boldsymbol\\lambda\\geq 0\n\\end{eqnarray*}\nor equivalently\n\\begin{eqnarray}\n&& R^{^{\\mbox{\\scriptsize\\sf T}}}\\left(R {\\bf z}+{\\bf a}+{\\bf d}-\\boldsymbol \\lambda\\right)=0\\label{cp1}\\\\\n&& R{\\bf z}+{\\bf a}\\geq 0\\label{cp2}\\\\\n&& \\boldsymbol\\lambda\\cdot (R{\\bf z}+{\\bf a})=0\\label{cp3}\\\\\n&& \\boldsymbol\\lambda\\geq 0\\label{cp4}.\n\\end{eqnarray}\nSee \\cite[Chapter 12]{NW}.\n\n\n\n\nBefore stating and proving the main result (Theorem \\ref{main-theorem}), we list all \nthe assumptions, including both new and previously used.\n\n\\noindent\nA1. Consider the problem of  minimizing (\\ref{q-basic}) subject only to the boundary conditions\n(\\ref{b1}, \\ref{b1-bis}), but not the constraints (\\ref{i3-intro}). We assume that the minimizer\nof that problem is not feasible, that is, this minimizer does not\nsatisfy the impenetrability constraints (\\ref{i3-intro}). Further we assume that the feasible region is not\nempty, which means that there is at lest one point ${\\bf z}$ satisfying all the inequality constraints\n(\\ref{cp2}).\n\n\\noindent\nA2. The network $\\Gamma$ is a regular triangulation (as defined in Definition \\ref{reg-treg}).\n\n\\noindent\nA3. The boundary conditions are prescribed so that \n\\begin{equation}\n\\label{upper}\n|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^i+{\\bf u}^j)|\\leq a_i+a_j+\\min_{k=1,2,\n\\ldots N}a_k,\n\\end{equation}\nfor each pair ${\\bf x}^i$, ${\\bf x}^j$ of {\\it boundary vertices in contact}.  \n\n\nLet us provide some comments on the nature  of assumptions A1--A3.\nAssumption A1 means that minimizing the energy of the spring network subject only to boundary conditions leads to a configuration in which at least one spring is compressed (and thus violates the impenetrability constraints). \n\n\nAssumption A2 concerns the contact geometry. The edges of the network split\nthe domain of the problem (a polygon) into elementary cells (triangles). Near the boundary, the cells must be compatible with the geometry of the boundary in the following sense. If an interior vertex is connected to a boundary vertex, then it is also connected with another\nboundary vertex, located next to the first boundary vertex. Hence, every triangular cell adjacent to the exterior boundary must contain one free vertex and two boundary vertices. \n\n\nAssumption A3 means that the boundary conditions (\\ref{b1}, \\ref{b1-bis}) are chosen to prevent particles from escaping through the gaps made by displacing the boundary particles. Clearly, if two boundary particles belong to the same group, then no gap can appear between them, and $|({\\bf x}^i+{\\bf u}^i)-({\\bf x}^i+{\\bf u}^j)|=a_i+a_j$. Formation of gaps would be possible between two boundary\nparticles from different groups which are in contact in the reference configuration. If the parameters of rigid body motions in the boundary conditions (\\ref{b1}, \\ref{b1-bis}) are prescribed arbitrarily, then the two particles may move away from each other, and open a gap large enough for a third particle to slip through. Assumption A3 prohibits formation of such gaps.  \n\n\n\n\n\\begin{theorem}\n\\label{main-theorem}\nSuppose that assumptions A1-A3 hold. Then there exist $d^\\star>0$ and a vector $\\boldsymbol\\delta=\n(\\delta_1, \\delta_2, \\ldots, \\delta_E)\\in {{\\mathbb R}}^E: \\delta_l\\in (-1, -1\/2), l=1, 2, \\ldots, E$, such that for each ${\\bf d}=d\\boldsymbol\\delta$ with $d>d^\\star$,\nthe unique minimizer of (\\ref{pr1},\\ref{pr2}) has the following property. Each\ninterior vertex ${\\bf x}^i$ of $\\Gamma$ has at least\ntwo active edges incident at it. The corresponding unit vectors\n${\\bf q}^{i,j_1}, {\\bf q}^{i,j_2}$ must be linearly independent.\n\\end{theorem}\n\n\\noindent\n{\\it Proof.}\\newline\n\\noindent\n{\\it Step 1}. We claim that A3 implies that there is $c_0>0$, which\ndepends on the boundary conditions, but is independent\nof the choice of $d_{ij}$ in (\\ref{q-basic}), such that each feasible displacement\n${\\bf u}^i, i=1,2, \\ldots, N$, satisfies\n\\begin{equation}\n\\label{up-u}\n|u^i_k|\\leq c_0,\n\\end{equation}\n$k=1,2$. Indeed, first we observe that if\n${\\bf u}^i, {\\bf u}^j$ satisfy the linearized constraint (\\ref{i3-intro}),\nthen they also satisfy the distance constraint (\\ref{0.1}) (the converse is not\ntrue in general). Then any feasible collection of displacements also satisfies\nthe distance constraints (\\ref{0.1}) for each pair of neighboring vertices. Now we recall the assumption made in the introduction to conclude that (\\ref{0.1}) must hold for all pairs of vertices. Fix $l\\in \\{1,2,\\ldots, N\\}$, corresponding to an interior vertex, and consider a smaller packing ${\\mathcal P}$ of particles, containing only $D_l$ and\nall boundary particles. In the reference configuration, $D_l$ is completely surrounded by boundary particles. Then the boundary conditions are prescribed according\nto A3, the boundary particles still completely confine $D_l$, so that\n${\\bf x}^l$ must displace to ${\\bf x}^l+{\\bf w}^l$ that lies inside\na certain bounded domain $\\Omega^\\prime$ that depends only on boundary conditions.\nSince ${\\bf x}^i_k$ are bounded, this implies that the claim is true for\nall displacements ${\\bf w}^l$ which are feasible for the smaller packing ${\\mathcal P}$.\nClearly the set of all such displacements is larger than the set of all ${\\bf u}^l$\nfeasible under all constraints (\\ref{0.1}), and the latter set is larger than the set of all ${\\bf u}^l$ feasible under the linearized constraints (\\ref{i3-intro}).\nThis proves the claim.\n\n\\noindent \n{\\it Step 2}.\nLet \n\\begin{equation}\n\\label{vi}\n{\\bf v}^i=\\sum_{j\\in{\\mathcal N}_i}{\\bf q}^{ij},~~~i=1,2,\\ldots, 2(N-N_b).\n\\end{equation}\nFirst, we prove the theorem under the additional assumption\n\\begin{equation}\n\\label{a4}\n\\mbox{For each}~i=1,2,\\ldots, 2(N-N_b),~~{\\bf v}^i\\ne s{\\bf q}^{ij},\n\\end{equation}\nwhere\n$j\\in {\\mathcal N}_i, s\\in {{\\mathbb R}}$. \n\n\nWe note that (\\ref{a4}) implies that \n\\begin{equation}\n\\label{ii}\n|{\\bf v}^i|\\geq v_0>0\n\\end{equation}\nwith $v_0$ independent of $i$. Indeed, $s$ can be zero, so validity of (\\ref{a4}) means in particular\nthat all ${\\bf v}^i$ are non-zero. Since there is finitely many ${\\bf v}^i$, (\\ref{ii}) holds.\n\n\n \nConsider solutions of the KKT system\n(\\ref{cp1},\\ref{cp2},\\ref{cp3},\\ref{cp4}). From (\\ref{cp3}), (\\ref{cp4})\nit follows that $\\lambda_j=0$ if the $j$-th constraint is inactive.\nLet $\\theta_j=(R{\\bf z}+{\\bf a})_j$.\nIf the $j$-th constraint is active then $\\theta_j=0$,\nwhile $\\lambda_j$ is arbitrary. Suppose that a feasible point\n${\\bf z}^\\star$ is given. Then\n$\\theta_j$ are given. To solve (\\ref{cp1}) we need to find\n$\\boldsymbol \\lambda$. Denote\nby ${\\bf r}^k, k=1,2,\\ldots, E$\nthe rows of $R$ (the columns of $R^{\\mbox{\\scriptsize\\sf T}}$), and suppose\nthat the rows ${\\bf r}^1, {\\bf r}^2\\ldots, {\\bf r}^S$ correspond to\nthe active constraints, and that the rows\n${\\bf r}^{S+1}, {\\bf r}^{S+2},\\ldots, {\\bf r}^{E}$ correspond to the\ninactive constraints. Choose ${\\bf d}=(-d, -d, \\ldots, -d)$. \nThen (\\ref{cp1}) can be written as\n\\begin{equation}\n\\label{cp1-1}\n-\\sum_{l=1}^S {\\bf r}^l \\lambda_l+\\sum_{l=S\n+1}^E{\\bf r}^l\\theta_l+d\\sum_{l=1}^E {\\bf r}^l=0.\n\\end{equation}\n\nPick a vertex ${\\bf x}^i$ of $\\Gamma$ and consider the restriction\nof each ${\\bf r}^l$ in (\\ref{cp1-1}) to the two components\ncorresponding to ${\\bf x}^i$.\nThen we have\n\\begin{equation}\n\\label{cp-local}\n-\\sum_{j\\in {\\mathcal N}_i}^\\prime\\lambda_{ij} {\\bf q}^{ij}+\\sum_{j\\in {\\mathcal N}_i}^{\\prime\\prime}\\theta_{ij} {\\bf q}^{ij}+\nd{\\bf v}^i=0,\n\\end{equation}\nwhere the first sum is taken over active edges incident at ${\\bf x}^i$, while the second sum is over the inactive edges\nincident at ${\\bf x}^i$. \n\nNext, we determine the minimal number of active edges needed for (\\ref{cp-local}) to hold. We can look at\n(\\ref{cp-local}) as a local problem in which ${\\bf u}^i$ may vary, while ${\\bf u}^j, j\\in {\\mathcal N}_i$ are fixed.\nDenote by ${\\mathcal F}_i\\subset {\\mathbb R}^2$ the feasible region of\nthis local problem.\nBy A3, ${\\mathcal F}_i$ is a polygon, each side\nof which corresponds to one or more constraints being active. \n\nIn the generic case, one constraint per side is active. In the non-generic case, two or more\nactive constraints correspond to the same side. Since our goal is estimating the number of\nactive constraints from below, it is sufficient to consider only the generic case,\ncorresponding to the ``worst case scenario\".\nIn the generic case there are only three possibilities.\\newline\n\\noindent\n{\\it Case 1}. ${\\bf u}^i$ is inside ${\\mathcal F}_i$. All edges incident at ${\\bf x}^i$ are inactive.\\newline\n\\noindent\n{\\it Case 2}. ${\\bf u}^i$ belongs to only one of the sides of $\\partial{\\mathcal F}_i$. One edge is active.\\newline\n\\noindent\n{\\it Case 3}. ${\\bf u}^i$ is a vertex of ${\\mathcal F}_i$. Two edges are active.\n\n \nConsider case 1.\nThen (\\ref{cp-local}) cannot hold for $d$ sufficiently large. Indeed, $|{\\bf v}^i|\\geq v_0>0$ by assumption, while\n$|\\sum_{j\\in {\\mathcal N}_i}\\theta_{ij} {\\bf q}^{ij}|$ is bounded from above independent of $d$ in view of (\\ref{up-u}).\n\nConsider case 2.\nLet us number the active edge by $(i,1)$. Then (\\ref{cp-local}) can be written as\n\\begin{equation}\n\\label{cp-loc-1}\n-\\lambda_{i1}{\\bf q}^{i1}+\\sum_{j\\in {\\mathcal N}_i, j>1} \\left(({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}\\right){\\bf q}^{ij}+\nd{\\bf v}^i=0.\n\\end{equation}\nEnlarging $d$, if necessary, we see that (\\ref{cp-loc-1}) can hold only if\n\\begin{equation}\n\\label{crit1}  \n{\\bf v}^i=s{\\bf q}^{i1},\n\\end{equation}\n where\n$s<0$. Since\n(\\ref{crit1}) is not allowed by (\\ref{vi}), (\\ref{cp-local}) cannot hold for sufficiently large $d$.\n\nConsider case 3. Number the two active edges by $(i, 1)$, $(i,2)$. The equation (\\ref{cp-local}) is\n\\begin{equation}\n\\label{cp-loc-2}\n-\\lambda_{i1}{\\bf q}^{i1}-\\lambda_{i2}{\\bf q}^{i2}+\n\\sum_{j\\in {\\mathcal N}_i, j>2} \\left(({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}\\right){\\bf q}^{ij}+\nd{\\bf v}^i=0.\n\\end{equation}\nFor this to hold for large $d$, ${\\bf v}^i$ must be a non-positive linear combination\nof ${\\bf q}^{i1}, {\\bf q}^{i2}$. These two vectors are linearly independent, otherwise\ntheir intersection would not be a vertex of ${\\mathcal F}_i$. So, Case 3 is possible, provided\n${\\bf v}^i$ lies in the negative cone of two active edges.\n\n\\noindent\n{\\it Step 3}. Now we remove the assumption (\\ref{a4}). For each $i=1, 2, \\ldots, (N-N_b)$, and each $\\boldsymbol\\delta\\in {{\\mathbb R}}^E$, define a two-dimensional\nvector\n$\\tilde{\\bf v}^i$ to be the restriction of $R^{^{\\mbox{\\scriptsize\\sf T}}} \\boldsymbol\\delta\\equiv\\sum_{l=1}^E \\delta_l{\\bf r}^l$ to a place $i$. The theorem will be proved is we show that there is\na choice of $\\boldsymbol\\delta$ such that $\\delta_{l}\\in [1\/2, 1]$, and $\\tilde{\\bf v}^i$ has property (\\ref{a4}). \nIndeed, if such $\\boldsymbol\\delta$ is found, we could choose ${\\bf d}=d\\boldsymbol\\delta$, where $d>0$ is sufficiently large, and repeat the arguments made in the first step, using \n$\\tilde{\\bf v}^i$ instead of ${\\bf v}^i$. \n\n\nTo show existence of $\\boldsymbol\\delta$, consider the cube $C_E=\\{{\\bf y}\\in {{\\mathbb R}}^E: y_l\\in (1\/2, 1), l=1, 2, \\ldots, E\\}$. Pick any point ${\\bf y}^\\star\\in C_E$. Since $C_E$ is open, there is a Euclidean open ball\n$B({\\bf y}^\\star, \\rho)\\subset C_E$, with the radius $\\rho>0$. Consider the image of $B({\\bf y}^\\star, \\rho)$\nunder the mapping $R^{^{\\mbox{\\scriptsize\\sf T}}}$. Since $R$ has full rank, $R^{^{\\mbox{\\scriptsize\\sf T}}}$ is surjective, and is therefore an open mapping.\nThus, $R^{^{\\mbox{\\scriptsize\\sf T}}}(B({\\bf y}^\\star, \\rho))$ contains a Euclidean open ball $B(R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star, \\rho^\\star)$ of a positive radius $\\rho^\\star$ depending only on $R^{^{\\mbox{\\scriptsize\\sf T}}}$ and $\\rho$, but not on ${\\bf y}^\\star$. If $R^{^{\\mbox{\\scriptsize\\sf T}}} {\\bf y}^\\star$ has\nproperty (\\ref{a4}), we choose $\\boldsymbol\\delta={\\bf y}^\\star$ and we are done. Otherwise,\nnote that for each $i=1, 2, \\ldots, (N-N_b)$, the ball $B(R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star, \\rho^\\star)$ contains a non-empty two-dimensional\nEuclidean open ball $B_i$ centered at the restriction of $R^{^{\\mbox{\\scriptsize\\sf T}}}{\\bf y}^\\star$ to the place $i$. Since for each $i$\nthe set $\\{{\\bf v}\\in {{\\mathbb R}}^2: {\\bf v}=s{\\bf q}^{ij}, s\\in {{\\mathbb R}}, j\\in {\\mathcal N}^i\\}$ is a union of a finite number of lines, it cannot contain a two-dimensional ball. Therefore, for each $i=1, 2, \\ldots, (N-N_b)$ \nthere must be a vector $\\tilde{\\bf v}^i\\in B_i$ having property (\\ref{a4}). Now we can define \n$\\sum_{l=1}^E \\delta_l{\\bf r}^l$ via its restrictions $\\tilde{\\bf v}^i$. Next, by construction, we can find\na vector $\\boldsymbol\\delta\\in B({\\bf y}^\\star, \\rho)\\subset C_E$ such that $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta=\\sum_{l=1}^E \\delta_l{\\bf r}^l$. \n\nThe theorem is proved.\n\n\\noindent\n{\\bf Remark}. \nThe choice \nof $\\boldsymbol\\delta$ in the proof is based on the following criterion. Consider the vector $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta\\in {{\\mathbb R}}^{2(N-N_b)}$. The proof works if\n\\begin{equation}\n\\label{last}\nR^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta\\ne({\\bf v}_1, {\\bf v}_2, \\ldots, {\\bf v}_{N-N_b}),\n\\end{equation}\nwhere at least one of the two-dimensional vectors ${\\bf v}_i, i=1, 2, \\ldots, (N-N_b)$ is of the form\n${\\bf v}_i=s{\\bf q}^{ij}$, for some real $s$ and $j\\in {\\mathcal N}_i$. In other words, if ${\\bf v}_i$ \nis inadmissible, then it must lie on the line through the origin with direction vector ${\\bf q}^{ij}$. \nFor each fixed $i$, the inadmissible set $V_i=\\{{\\bf v}_i\\in {\\mathbb R}^2: {\\bf v}_i=s{\\bf q}^{ij}\\}$  has Hausdorff dimension one (a finite union of lines in the plane), while the admissible set is the two-dimensional complement of $V_i$ in ${\\mathbb R}^2$. Therefore, the set of inadmissible vectors  in the right hand side of (\\ref{last}) has dimension $2(N-N_b)-1$ while the set of admissible vectors $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta$ is of dimension $2(N-N_b)$. Thus the admissible $R^{^{\\mbox{\\scriptsize\\sf T}}}\\boldsymbol\\delta$ are generic, and the theorem holds for a generic (in the sense of Definition \\ref{reg-treg}) packing under a generic pre-stress (the latter is determined by a generic choice of $\\boldsymbol\\delta$).\n\n\n\n\n\n\\section{Order parameter}\nRecently, a phenomenological theory of slow dense granular flows was proposed in\n\\cite{AT, VTA}. A key quantity in that theory is the order parameter, defined as the ratio\nof the number of solid-like contacts to the number of all contacts within a given control volume.\nIn \\cite{VTA}, a contact is considered solid-like if two particles are jammed together for longer\nthan a characteristic collision time. The relevant characteristic\ntime is $\\tau=a\/v_a$, where $a$ is particle radius and $v_a$ is the speed of sound in\na solid material of the particles. Our model corresponds to the instantaneous material\nresponse, when $\\tau$ is much smaller than other relaxation times in the system, such\nas the ratio of the sample size to a typical particle velocity.\n\nAn obvious type of pair motion leading to a solid-like contact is a rigid displacement\n(a pair of particles infinitesimally moves as a rigid body). We shall call this type of contacts {\\it stuck}. If a contact between $D_i$ and $D_j$ is stuck, then $({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}=0$, which is easy to check using the definition of rigid\ndisplacements. This means that the impenetrability constraint for the corresponding\nedge of $(i,j)$ of the network is satisfied as an equation (the edge is active). \nHowever, not every\nactive edge corresponds to a stuck contact. Another type of a local motion that produces\n$({\\bf u}^i-{\\bf u}^j)\\cdot{\\bf q}^{ij}=0$ is an infinitesimal shear motion when\n${\\bf u}^i-{\\bf u}^j$ is orthogonal to ${\\bf q}^{ij}$. The corresponding contact is\ncalled {\\it sheared}. Note also that infinitesimal shear\nis the same as infinitesimal rotation, so this type of motion includes infinitesimal rolling\nas well as shear sliding. \n\n\nWe consider both sheared and stuck contact as solid-like, because stuck contacts are stable, while sheared contacts in an actual granular material will be subject to friction. Friction can be viewed as partially stabilizing, at least when the shearing force\nis below the static friction threshold. Such non-sliding frictional contacts are considered as solid-like in the simulations performed in \\cite{VTA}. In addition, some heuristic arguments and numerical simulations \npresented in \\cite{GG1, GG2}, suggest that friction enhances elastic behavior of sufficiently large samples. Therefore, it makes sense to think of the network of solid-like contacts as the\nmain load-bearing structure and call this network {\\it strong}. \nIn contrast, a broken contact satisfying (\\ref{broken}) corresponds to\na local weakening in the material because in this case two particles separate completely.\nWe can think of the network of all broken contacts as {\\it weak}. \nMoreover, division of contacts into broken and solid-like corresponds to\nthe division of constraints into active and inactive, as done in optimization theory.\nTherefore, this division is natural mathematically, and also makes sense from the physics point of view. \n\n\nIn addition, the definition in \\cite{VTA} does not sufficiently clarify the nature of averaging. \nThe notion of an order parameter in static problems should not use time averaging. \nThe result of spatial averaging depends on the size of the sample that is being averaged. Thus, if the order\nparameter is obtained by, say, spatial averaging, then it must depend on both location and size of the \n``control volume\". In the discrete situation, the size of the averaging sample can be measured\nby the minimal number of edges connecting a pair of vertices within the sample.\n\n\nThis suggests a definition of the size-dependent order parameter. To state\nthis definition we first define the averaging sample. \n\\begin{definition}\n\\label{k-n}\nA vertex ${\\bf x}^j$ is in the k-th neighborhood of ${\\bf x}^i$\nif $\\Gamma$ contains a path connecting ${\\bf x}^i$ and ${\\bf x}^j$ with\nno more than $k$ edges.\n\\end{definition}\nNow, to each $k$-neighborhood we can associate a value of an order parameter.  \n\\begin{definition}\n\\label{order}\nFor each ${\\bf x}^i$ and each non-negative integer $k\\leq N$, the size dependent\norder parameter $\\rho({\\bf x}^i, k)$ is defined by\n\\begin{equation}\n\\label{op}\n\\rho({\\bf x}^i, k)=\\frac{\\sum_{k} n_s}{\\sum_k n},\n\\end{equation}\nwhere the numerator is the number of active edges in $k$-neighborhood of ${\\bf x}^i$,\nand denominator is the number of all edges in that neighborhood.\n\\end{definition}\n\n\nTheorem \\ref{main-theorem} implies the lower bound \n\\begin{equation}\n\\label{simple}\n\\rho({\\bf x}^i,N)\\geq \\frac{N}{E}\n\\end{equation}\non the order parameter associated with the maximal, $N$-th neighborhood of each interior vertex ${\\bf x}^i$.\nIndeed, counting active edges (two per vertex) gives $2N$ edges, each counted at most twice.\nIn particular, (\\ref{simple}) means that the order parameter $\\rho({\\bf x}^i, N)$\nis bounded from below by the reciprocal of the mean coordination number of the network. \n \n\n\\section{Conclusions}\n\nWe have studied a network model of quasi-static deformation of dense pre-stressed\ngranular materials. \nIn our model, the packing was represented by a network of linear elastic springs.\nEach spring corresponds to a contact between two particles.\nGeometric impenetrability constraints within the packing were modeled by the\nlinearized impenetrability constraints on the displacements of the vertices of the\nnetwork. The constraints have the form of linear inequalities, that can be satisfied either\nas an equality (an active constraint), or as a strict inequality (inactive constraint).\nConstraints are in one-to one correspondence with the interpaticle contacts. An active constraint\ncorresponds to a relatively stable solid-like contact. Inactive constraints represent the relatively weak\nbroken contacts.\nThe question addressed in the paper is to estimate the total number and distribution of the solid-like\ncontacts in the energy-minimizing configuration. We showed that each interior vertex of the network has at least two \nsolid-like contacts corresponding to it. This result qualitatively\nreproduces the micro-band structure obtained in \\cite{Kuhn1, Kuhn2} by numerical simulations. We also discussed the connection between our result and a lower bound on the\norder parameter \\cite{AT, VTA}. In the paper, we proposed a definition of the\norder parameter that is similar to the one introduced in \\cite{VTA}, but differs from it in the interpretation of the so-called solid-like contacts. On the one hand, our definition appears to be in accord with a physical picture of granular statics, recently\nproposed in \\cite{GG1, GG2}. On the other hand, it is a naturally related to optimization theory.   \n\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Derivation of equation (2)}\nIn this appendix we show that eqs.\\ (20)-(22) in \\cite{BrAn98} are\nidentical with the string Bethe equations of Takahashi and that\n(\\ref{centre}) follows from these equations.\n\nLet us first cite the main results of \\cite{BrAn98}: The $n$-complex\nis  parameterized by an $n$-string of the form\n\\begin{equation}\n     \\phi_{a,j}^{(n)} = \\phi_a^{(n)} + (n + 1 - 2j) \\frac{\\text{i} u}{4}\n        \\quad, \\quad j = 1, \\dots, n .\n\\end{equation}\n[Note the slight change of notation compared to \\cite{BrAn98}: We made\nthe replacement $\\phi_0^{(n)} \\rightarrow \\phi_a^{(n)}$ (and\n$\\phi_j^{(n)} \\rightarrow \\phi_{a,j}^{(n)}$) which allows us to include\nscattering of two different $n$ complexes of the same length. $a$\nenumerates the $n$-strings.] The S-matrix of an unbound particle\nwith an $n$-complex is\n\\begin{equation}\n     S_{k \\phi_a^{(n)}}^{u (n)} = \\frac{\\sin k - \\phi_a^{(n)} - \n        n \\textstyle{\\frac{\\text{i} u}{4}}}{\\sin k - \\phi_a^{(n)} +\n        n \\textstyle{\\frac{\\text{i} u}{4}}},\n\\end{equation}\nand the S-matrix of an $m$-complex with an $n$-complex is\n\\begin{equation}\n     S_{\\phi_b^{(m)} \\phi_a^{(n)}}^{(m) (n)}\n        \\: = \\: \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t   |n - m| \\textstyle{\\frac{\\text{i} u}{4}}}\n          {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t   |n - m| \\textstyle{\\frac{\\text{i} u}{4}}} \\:\n          \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t   (n + m) \\textstyle{\\frac{\\text{i} u}{4}}}\n          {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t   (n + m) \\textstyle{\\frac{\\text{i} u}{4}}} \\:\n\t  \\prod_{l=1}^{\\min(m,n) - 1} \\left(\n          \\frac{\\phi_b^{(m)} - \\phi_a^{(n)} -\n\t   (|n - m| + 2l) \\textstyle{\\frac{\\text{i} u}{4}}}\n          {\\phi_b^{(m)} - \\phi_a^{(n)} +\n\t   (|n - m| + 2l) \\textstyle{\\frac{\\text{i} u}{4}}} \\right)^2.\n\\end{equation}\nDiagonalizing the transfer matrix leads to the following set of\nequations\n\\begin{eqnarray} \\label{bac1}\n     e^{\\text{i} k_j L} & = & \\prod_{\\delta = 1}^{M^u}\n          \\frac{\\lambda_\\delta - \\sin k_j - \\textstyle{\n\t  \\frac{\\text{i} u}{4}}}{\\lambda_\\delta - \\sin k_j +\n\t  \\textstyle{\\frac{\\text{i} u}{4}}}\n          \\prod_{(n,a)} S_{\\phi_a^{(n)} k_j}^{(n) u}, \\\\ \\label{bac2}\n     \\prod_{\\delta \\ne \\lambda}^{M^u}\n          \\frac{\\lambda_\\gamma - \\lambda_\\delta -\n\t  \\textstyle{\\frac{\\text{i} u}{2}}}{\\lambda_\\gamma -\n\t  \\lambda_\\delta + \\textstyle{\\frac{\\text{i} u}{2}}}\n\t       & = &\n     \\prod_{j = 1}^{N^u}\n          \\frac{\\lambda_\\gamma - \\sin k_j -\n\t  \\textstyle{\\frac{\\text{i} u}{4}}}{\\lambda_\\gamma -\n\t  \\sin k_j + \\textstyle{\\frac{\\text{i} u}{4}}},\n\t       \\\\ \\label{bac3}\n     e^{\\text{i} q^{(n)}(\\phi_a^{(n)})L} & = &\n          \\prod_{(m,b) \\ne (n,a)} S_{\\phi_b^{(m)}\n\t  \\phi_a^{(n)}}^{(m) (n)}\n          \\prod_{j=1}^{N^u} S_{k_j \\phi_a^{(n)}}^{u (n)},\n\\end{eqnarray}\nwhere\n\\begin{equation}\n     q^{(n)} (\\phi) = - 2 \\text{Re} \\arcsin(\\phi + n \\text{i} u\/4) \n                      + (n + 1) \\pi =\n        - (\\arcsin(\\phi + n \\text{i} u\/4)\n           + \\arcsin(\\phi - n \\text{i} u\/4)) + (n + 1) \\pi\n\\end{equation}\nand\n\\begin{equation}\n     S_{\\phi_a^{(n)} k_j}^{(n) u} =\n          \\left( S_{k_j \\phi_a^{(n)}}^{u (n)} \\right)^{-1}.\n\\end{equation}\n\nEqs. (\\ref{bac1})-(\\ref{bac3}) are the ``new'' equations of Braak and\nAndrei ((20)-(22) or (BAC) in \\cite{BrAn98}). In order to show that\n(\\ref{bac1})-(\\ref{bac3}) agree with Takahashi's string Bethe equations\nlet us now adjust notations. Let\n\\begin{equation}\n     U = u\/4 \\quad, \\quad {\\Lambda'}_a^m = \\phi_a^{(m)} \\quad,\n     \\quad \\alpha = a \\quad, \\quad \\beta = b.\n\\end{equation}\nLet us further introduce the functions \\cite{Takahashi72}\n\\begin{eqnarray}\n     e(x) & = & \\frac{x + \\text{i}}{x - \\text{i}} \\\\\n     E_{nm} (x) & = & \\left\\{ \\begin{array}{l} \n                      {\\displaystyle\n                      e \\left( \\frac{x}{|n - m|} \\right)\n                      e^2 \\left( \\frac{x}{|n - m| + 2} \\right)\n\t\t      \\cdots\n                      e^2 \\left( \\frac{x}{n + m - 2} \\right)\n                      e \\left( \\frac{x}{n + m} \\right)\n\t\t      \\: \\text{for} \\quad n \\ne m,} \\\\[3ex]\n                      {\\displaystyle\n                      e^2 \\left( \\frac{x}{2} \\right)\n                      e^2 \\left( \\frac{x}{4} \\right)\n\t\t      \\cdots\n                      e^2 \\left( \\frac{x}{2n - 2} \\right)\n                      e \\left( \\frac{x}{2n} \\right)\n\t\t      \\: \\text{for} \\quad n = m.}\n\t\t      \\end{array} \\right.\n\\end{eqnarray}\nThen eqs.\\ (\\ref{bac1})-(\\ref{bac3}) turn into\n\\begin{eqnarray} \\label{tbac1}\n     e^{\\text{i} k_j L} & = & \\prod_{\\delta = 1}^{M^u}\n\t  e \\left( \\frac{\\sin k_j - \\lambda_\\delta}{U} \\right)\n          \\prod_{(n,\\alpha)}\n\t  e \\left( \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t  \\\\ \\label{tbac2}\n     \\prod_{j = 1}^{N^u}\n          e \\left( \\frac{\\lambda_\\gamma - \\sin k_j}{U} \\right)\n\t       & = &\n     - \\prod_{\\delta = 1}^{M^u}\n          e \\left( \\frac{\\lambda_\\gamma - \\lambda_\\delta}{2U} \\right),\n\t       \\\\ \\label{tbac3} \\nonumber\n     \\exp \\{ - \\text{i} L\n          [\\arcsin({\\Lambda'}_\\alpha^n + n \\text{i} U) & + &\n\t  \\arcsin({\\Lambda'}_\\alpha^n - n \\text{i} U) +\n\t  (n + 1) \\pi] \\} \\\\ & = &\n          - \\prod_{j=1}^{N^u}\n\t  e \\left( \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right)\n          \\prod_{(m,\\beta)}\n\t  E_{nm} \\left(\n\t  \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U} \\right).\n\\end{eqnarray}\nThe spin rapidities $\\lambda_\\gamma$ in (\\ref{tbac1}) and (\\ref{tbac2})\nmay generally be complex. In order to obtain a set of equations which\ncontains only real unknowns and which transforms into a set of linear\nintegral equations in the thermodynamic limit we have to employ\nTakahashi's string hypothesis for $\\Lambda$ strings: As the number $N$\nof electrons becomes large the spin rapidities are driven to string\npositions characterized by their length $n$ and their real center\n$\\Lambda_\\alpha^n$. Following Takahashi \\cite{Takahashi72} we will\nuse the notation $\\Lambda_\\alpha^{n,j}$ instead of $\\lambda_\\gamma$.\n$\\Lambda_\\alpha^{n,j}$ is the $j$-th spin rapidity involved in an\n$n$-$\\Lambda$ string with center $\\Lambda_\\alpha^n$,\n\\begin{equation}\n     \\Lambda_\\alpha^{n,j} = \\Lambda_\\alpha^n + (n + 1 - 2j) \\text{i}U.\n\\end{equation}\nFollowing again Takahashi let us assume that in the thermodynamic\nlimit all $\\lambda_\\gamma$ are grouped into strings with an accuracy\nof ${\\cal O}(\\exp( - \\delta N))$, where $\\delta$ is some positive\nnumber. Then eqs.\\ (\\ref{tbac1})-(\\ref{tbac3}) lead to\n\\begin{eqnarray} \\label{ttbac1}\n     e^{\\text{i} k_j L} & = & \\prod_{(n,\\alpha)}\n\t  e \\left( \\frac{\\sin k_j - \\Lambda_\\alpha^n}{nU} \\right)\n          \\prod_{(n,\\alpha)}\n\t  e \\left( \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t  \\\\ \\label{ttac2}\n     \\prod_{j = 1}^{N^u}\n          e \\left( \\frac{\\Lambda_\\alpha^n - \\sin k_j}{nU} \\right)\n\t       & = &\n     - \\prod_{(m,\\beta)}\n          E_{nm} \\left( \\frac{\\Lambda_\\alpha^n - \\Lambda_\\beta^m}{U}\n\t  \\right), \\\\ \\label{ttbac3} \\nonumber\n     \\exp \\{ - \\text{i} L\n          [\\arcsin({\\Lambda'}_\\alpha^n + n \\text{i} U) & + &\n\t  \\arcsin({\\Lambda'}_\\alpha^n - n \\text{i} U) +\n\t  (n + 1) \\pi] \\} \\\\ & = &\n          - \\prod_{j=1}^{N^u}\n\t  e \\left( \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right)\n          \\prod_{(m,\\beta)}\n\t  E_{nm} \\left(\n\t  \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U} \\right).\n\\end{eqnarray}\n\nTaking logarithms we arrive at the following form of the string\nBethe equations, which is suitable for considering the thermodynamic\nlimit,\n\\begin{eqnarray} \\label{t1}\n     k_j L & = & 2 \\pi I_j - \\sum_{n=1}^\\infty \\sum_{\\alpha = 1}^{M_n}\n                 \\theta \\left(\n\t\t \\frac{\\sin k_j - \\Lambda_\\alpha^n}{nU} \\right)\n                 - \\sum_{n=1}^\\infty \\sum_{\\alpha = 1}^{M_n'}\n                 \\theta \\left(\n\t\t \\frac{\\sin k_j - {\\Lambda'}_\\alpha^n}{nU} \\right),\n\t\t \\\\ \\label{t2}\n     \\sum_{j=1}^{N - 2M'} \\theta \\left(\n\t\t \\frac{\\Lambda_\\alpha^n - \\sin k_j}{nU} \\right) & = &\n\t\t 2 \\pi J_\\alpha^n +\n\t\t \\sum_{m=1}^\\infty \\sum_{\\beta = 1}^{M_m}\n\t\t \\Theta_{nm} \\left(\n\t\t \\frac{\\Lambda_\\alpha^n - \\Lambda_\\beta^m}{U} \\right),\n\t\t \\\\ \\label{t3}\n     L [\\arcsin({\\Lambda'}_\\alpha^n + \\text{i}nU)\n        + \\arcsin({\\Lambda'}_\\alpha^n - \\text{i}nU)] & = &\n\t         2 \\pi {J'}_\\alpha^n +\n\t\t \\sum_{j=1}^{N - 2M'} \\theta \\left(\n\t\t \\frac{{\\Lambda'}_\\alpha^n - \\sin k_j}{nU} \\right) +\n\t\t \\sum_{m=1}^\\infty \\sum_{\\beta = 1}^{M_m'}\n\t\t \\Theta_{nm} \\left(\n\t\t \\frac{{\\Lambda'}_\\alpha^n - {\\Lambda'}_\\beta^m}{U}\n\t\t \\right).\n\\end{eqnarray}\nHere we assumed $L = 2 \\times \\text{odd}$ to be even.\n$I_j$, $J_\\alpha^n$, and ${J'}_\\alpha^n$ are integer or half-odd\ninteger numbers, $N$ is the total number of electrons, $M' =\n\\sum_{n=1}^\\infty n M_n'$, $\\theta(x) = 2 \\arctan(x)$, and\n\\begin{equation} \\label{defthetas}\n     \\Theta_{nm} (x) = \\left\\{ \\begin{array}{l}\n\t{\\displaystyle\n        \\theta \\left( \\frac{x}{|n - m|} \\right) +\n        2 \\theta \\left( \\frac{x}{|n - m| + 2} \\right) + \\cdots +\n        2 \\theta \\left( \\frac{x}{n + m - 2} \\right) +\n        \\theta \\left( \\frac{x}{n + m} \\right), \\: \\text{if} \\quad\n\tn \\ne m,} \\\\[3ex]\n\t{\\displaystyle\n        2 \\theta \\left( \\frac{x}{2} \\right) +\n        2 \\theta \\left( \\frac{x}{4} \\right) + \\cdots +\n        2 \\theta \\left( \\frac{x}{2n - 2} \\right) +\n        \\theta \\left( \\frac{x}{2n} \\right), \\: \\text{if} \\quad n = m.}\n\t\\end{array} \\right.\n\\end{equation}\nThe branch of $\\arcsin(x)$ in (\\ref{t3}) is fixed as $- \\pi\/2 \\le\n\\text{Re} (\\arcsin (x)) \\le \\pi\/2$. $M_n$ and $M_n'$ are the numbers of\n$\\Lambda$ strings of length $n$, and $\\Lambda'$ strings of length $n$\nin a specific solution of the system (\\ref{t1})-(\\ref{t3}). The\ninteger (half-odd integer) numbers in (\\ref{t1})-(\\ref{t3}) have\nranges\n\\begin{eqnarray} \\label{r1}\n     && - \\frac{L - 1}{2} \\le I_j \\le \\frac{L - 1}{2}, \\\\ \\label{r2}\n     && |J_\\alpha^n| \\le \\frac{1}{2}\n        \\left(N - 2M' - \\sum_{m=1}^\\infty t_{nm} M_m - 1 \\right), \\\\\n\t\\label{r3}\n     && |{J'}_\\alpha^n| \\le \\frac{1}{2}\n        \\left(L - N + 2M' - \\sum_{m=1}^\\infty t_{nm} M_m' - 1 \\right),\n\\end{eqnarray}\nwhere $t_{mn} = 2 \\min (m,n) - \\delta_{mn}$. Each set of numbers\n$\\{I_j\\}, \\{J_\\alpha^n\\}, \\{{J'}_\\alpha^n\\}$ is in one-to-one\ncorrespondence with a set of rapidities $\\{k_j\\}, \\{\\Lambda_\\alpha^n\\},\n\\{{\\Lambda'}_\\alpha^n\\}$, which in turn unambiguously specifies\none Bethe eigenstate of the Hubbard Hamiltonian. Thus the ground state\nand all excited states can be constructed by specifying a set of\nnumbers $\\{I_j\\}, \\{J_\\alpha^n\\}, \\{{J'}_\\alpha^n\\}$ and then taking the\nthermodynamic limit.\n\nIt is our aim to prove eq.\\ (\\ref{centre}) for the charge singlet\nexcitation over the half-filled ground state. The ground state is\ncharacterized  by $N = L$, $M_1 = L\/2$. In this case the inequalities\n(\\ref{r1}) and (\\ref{r2}) lead to unique distributions of the quantum\nnumbers $I_j$ and $J_\\alpha^1$,\n\\begin{equation} \\label{ijg}\n     I_j = - (L + 1)\/2 + j \\quad, \\quad j = 1, \\dots, L \\quad, \\quad\n     J_\\alpha^1 = - (L + 2)\/4 + \\alpha \\quad, \\quad\n                \\alpha = 1, \\dots, L\/2.\n\\end{equation}\nEqs.\\ (\\ref{t1}) and (\\ref{t2}) reduce to\n\\begin{eqnarray} \\label{g1}\n     && L k_j = 2\\pi I_j - \\sum_{\\alpha = 1}^{L\/2}\n                \\theta \\left( \\frac{\\sin k_j - \\Lambda_\\alpha}{U}\n\t\t       \\right) \\quad, \\quad j = 1, \\dots, L,\\\\\n\t\t       \\label{g2}\n     && \\sum_{j=1}^L \\theta \\left( \\frac{\\Lambda_\\alpha - \\sin k_j}{U}\n\t\t       \\right) = 2\\pi J_\\alpha^1 +\n        \\sum_{\\beta = 1}^{L\/2} \\theta \\left(\n\t   \\frac{\\Lambda_\\alpha - \\Lambda_\\beta}{2U} \\right)\n\t   \\quad, \\quad \\alpha = 1, \\dots, L\/2.\n\\end{eqnarray}\nIn the thermodynamic limit eqs.\\ (\\ref{g1}) and (\\ref{g2}) turn\ninto the well known integral equations \\cite{LiWu68}\n\\begin{eqnarray} \\label{gi1}\n     \\rho(k) & = & \\frac{1}{2\\pi} + \\frac{1}{\\pi} \\cos k\n                   \\int_{- \\infty}^\\infty d \\Lambda \\;\n\t\t   \\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t   \\sigma(\\Lambda), \\\\ \\label{gi2}\n     \\sigma(\\Lambda) & = & \\frac{1}{2\\pi^2} \\int_{-\\pi}^\\pi dk \\;\n\t\t   \\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t   - \\frac{1}{\\pi} \\int_{-\\infty}^\\infty d \\Lambda' \\;\n\t\t   \\frac{2U}{4U^2 + (\\Lambda - \\Lambda')^2}\n\t\t   \\sigma(\\Lambda')\n\\end{eqnarray}\nfor the densities $\\rho(k_j) = 1\/(L(k_{j+1} - k_j))$ and\n$\\sigma(\\Lambda_\\alpha) = 1\/(L(\\Lambda_{\\alpha + 1} - \\Lambda_\\alpha))$.\n\nThe charge singlet is  characterized  by $N = L$, $M_1 = L\/2 - 1$ and\n$M_1' = 1$ \\cite{EsKo94a,EsKo94b}. Thus $M' = 1$, and the number of\nunbound electrons is $L - 2M' = L - 2$. We will denote quantities which\ndescribe the charge singlet by a tilde. Eqs.\\ (\\ref{r2}) and (\\ref{r3})\nuniquely determine the set $\\{\\tilde J_\\alpha^1\\}$ and the number\n$\\tilde {J'}^1$,\n\\begin{equation}\n     \\tilde J_\\alpha^1 = - L\/4 + \\alpha \\quad\n                         \\alpha = 1, \\dots, L\/2 - 1 \\quad, \\quad\n     \\tilde {J'}^1 = 0.\n\\end{equation}\nThe set $\\{\\tilde I_j\\}$, however, is not uniquely determined by\nthe inequality (\\ref{r3}). There are $L \\choose 2$ inequivalent such\nsets, which are parameterized  by two vacancies $I_1^h$ and $I_2^h$ in the\ndistribution (\\ref{ijg}) of the numbers $I_j$, which characterize the\nground state. These two vacancies determine two charge rapidities\n$k_1^h$ and $k_2^h$ via eqs.\\ (\\ref{g1}) and (\\ref{g2}). In the\nthermodynamic limit the charge rapidities densely fill the interval\n$[-\\pi,\\pi]$, and $k_1^h$ and $k_2^h$ become the two free parameters of\nthe charge singlet excitation. We see already at this stage, that\nthere can not be a third free parameter, since $\\tilde{J'}^1 = 0$ is\nfixed. For the charge singlet excitation eqs.\\ (\\ref{t1})-(\\ref{t3})\nreduce to\n\\begin{eqnarray} \\label{e1}\n     && L \\tilde k_j = 2\\pi \\tilde I_j - \\sum_{\\alpha = 1}^{L\/2 - 1}\n                \\theta \\left(\n\t\t       \\frac{\\sin \\tilde k_j - \\tilde \\Lambda_\\alpha}{U}\n\t\t       \\right)\n                - \\theta \\left( \\frac{\\sin \\tilde k_j - \\Lambda'}{U}\n\t\t       \\right)\n\t\t       \\quad, \\quad j = 1, \\dots, L - 2,\\\\\n\t\t       \\label{e2}\n     && \\sum_{j=1}^{L -2} \\theta \\left(\n                       \\frac{\\tilde \\Lambda_\\alpha - \\sin \\tilde k_j}{U}\n\t\t       \\right) = 2\\pi \\tilde J_\\alpha^1 +\n        \\sum_{\\beta = 1}^{L\/2 - 1} \\theta \\left(\n\t   \\frac{\\tilde \\Lambda_\\alpha -\n\t                             \\tilde \\Lambda_\\beta}{2U} \\right)\n\t   \\quad, \\quad \\alpha = 1, \\dots, L\/2 - 1, \\\\ \\label{e3}\n     && L(\\arcsin(\\Lambda' + \\text{i} U) +\n                               \\arcsin(\\Lambda' - \\text{i} U)) =\n        \\sum_{j=1}^{L-2} \\theta \\left(\n\t                \\frac{\\Lambda' - \\sin \\tilde k_j}{U} \\right).\n\\end{eqnarray}\nLet us subtract (\\ref{g1}) from (\\ref{e1}) for $I_l = \\tilde I_j$,\n$j = 1, \\dots, L-2$, and (\\ref{g2}) from (\\ref{e2}) for $\\alpha = 1,\n\\dots, L\/2 - 1$. Then, taking the thermodynamic limit, we obtain\na pair of integral equations for the shift functions\n\\begin{equation}\n     F^c (k_j) = \\frac{\\tilde k_j - k_j}{k_{j+1} - k_j} \\quad, \\quad\n     F^s (\\Lambda_\\alpha) = \\frac{\\tilde \\Lambda_\\alpha -\n        \\Lambda_\\alpha}{\\Lambda_{\\alpha + 1} - \\Lambda_\\alpha}.\n\\end{equation}\nThese integral equations are\n\\begin{eqnarray} \\label{f1}\n     F_{CS}^c (k) & = & - \\frac{1}{2} - \\frac{1}{2\\pi}\n                        \\theta \\left( \\frac{\\sin k - \\Lambda'}{U}\n\t\t\t\\right)\n\t\t\t+ \\frac{1}{\\pi} \\int_{-\\infty}^\\infty\n\t\t\td \\Lambda \\;\n\t\t\t\\frac{U}{U^2 + (\\sin k - \\Lambda)^2}\n\t\t\tF_{CS}^s (\\Lambda), \\\\ \\label{f2}\n     F_{CS}^s (\\Lambda) & = & 1 + \\frac{1}{2\\pi} \\sum_{l = 1}^2\n                              \\theta \\left(\n\t\t\t      \\frac{\\Lambda - \\sin k_l^h}{U} \\right) -\n\t\t\t      \\frac{1}{\\pi} \\int_{-\\infty}^\\infty\n\t\t\t      d \\Lambda' \\;\n\t\t\t      \\frac{2U}{4U^2 + (\\Lambda - \\Lambda')^2}\n\t\t\t      F_{CS}^s (\\Lambda').\n\\end{eqnarray}\nHere we have supplied an index ``CS'' to  emphasize  that we are dealing\nwith the charge singlet excitation. The solution $F_{CS}^c (k)$ of\n(\\ref{f1}) can be found on page 526 of \\cite{EsKo94b}. It is a single\nvalued function of $\\sin k$. In the derivation of (\\ref{gi2}) and\n(\\ref{f2}) we made use of the following elementary lemma,\n\\begin{equation} \\label{lemma}\n     \\int_{-\\pi}^\\pi dk \\; f(\\sin k) \\cos k = 0,\n\\end{equation}\nwhich holds for arbitrary single valued functions $f(x)$. This lemma\nfollows from the identities $\\sin (\\pi - k) = \\sin k$ and\n$\\cos (\\pi - k) = - \\cos k$.\n\nNote that there are still three free parameters, $k_1^h$, $k_2^h$\nand $\\Lambda'$, in (\\ref{f1}) and (\\ref{f2}). $\\Lambda'$ becomes\nfixed as a function of $k_1^h$ and $k_2^h$ by considering eq.\\\n(\\ref{e3}), which in the thermodynamic limit turns into\n\\begin{eqnarray} \\nonumber\n     \\lefteqn{\n     L(\\arcsin(\\Lambda' + \\text{i}U) + \\arcsin(\\Lambda' - \\text{i}U))}\n     \\\\ & = &\n     L \\int_{-\\pi}^\\pi dk \\; \\theta \\left( \\frac{\\Lambda' - \\sin k}{U}\n        \\right) \\rho (k) - 2 \\int_{-\\pi}^\\pi dk \\;\n\t\\frac{U}{U^2 + (\\Lambda' - \\sin k)^2} F_{CS}^c (k) \\cos k\n\t- \\sum_{l=1}^2 \\theta \\left(\\frac{\\Lambda' - \\sin k_l^h}{U}\n\t\\right) + {\\cal O} \\left( \\frac{1}{L} \\right) \\nonumber\n     \\\\ & = & \\label{rhs}\n     \\frac{L}{2\\pi} \\int_{-\\pi}^\\pi dk \\;\n        \\theta \\left( \\frac{\\Lambda' - \\sin k}{U} \\right)\n\t- \\sum_{l=1}^2 \\theta \\left(\\frac{\\Lambda' - \\sin k_l^h}{U}\n\t\\right) + {\\cal O} \\left( \\frac{1}{L} \\right).\n\\end{eqnarray}\nIn oder to obtain the second of the above equalities we have used\n(\\ref{gi1}), (\\ref{f1}) and the lemma (\\ref{lemma}). Let us calculate\nthe integral\n\\begin{equation}\n     I (\\Lambda') = \\frac{1}{2\\pi} \\int_{-\\pi}^\\pi dk \\;\n                          \\theta \\left( \\frac{\\Lambda' - \\sin k}{U}\n\t\t\t  \\right) .\n\\end{equation}\non the right hand side of (\\ref{rhs}). First note that $I(0) = 0$.\nThe derivative of $I (\\Lambda')$ can be represented as\n\\begin{equation} \\label{integral}\n     I'(\\Lambda') = \\frac{1}{\\pi} \\int_{-\\pi}^\\pi dk \\;\n                              \\frac{U}{U^2 + (\\Lambda' - \\sin k)^2}\n                        = \\text{Re} \\left\\{ \\frac{1}{2\\pi \\text{i}}\n\t\t\t      \\oint dz \\frac{4}{z^2 + 2z(U - \\text{i}\n\t\t\t      \\Lambda') - 1} \\right\\} ,\n\\end{equation}\nwhere the contour of integration is the unit circle. Let\n\\begin{equation}\n     p(z) = z^2 + 2z(U - \\text{i} \\Lambda') - 1 = (z - z_1)(z - z_2).\n\\end{equation}\nWe see that the poles of the integrand in (\\ref{integral}) are related\nas\n\\begin{equation} \\label{poles}\n     z_1 = - 1\/z_2.\n\\end{equation}\nThus, only one of these poles, say $z_1$, lies inside the unit circle.\nUsing (\\ref{poles}) we obtain $I'(\\Lambda')$ as a function of\n$z_1$,\n\\begin{equation} \\label{phiprime}\n     I'(\\Lambda') = 2 \\text{Re} \\left\\{\n                          \\frac{2}{z_1 + z_1^{-1}} \\right\\} .\n\\end{equation}\nLet us parameterize  the pole at $z_1$ as $z_1 = e^{\\text{i} \\alpha}$.\nSince $z_1$ is located inside the unit circle, $\\text{Im} (\\alpha) > 0$.\nUsing $p(z_1) = 0$ we find that\n\\begin{equation}\n     \\sin \\alpha = \\Lambda' + \\text{i} U \\quad,\n                   \\quad \\text{Im} (\\alpha) > 0.\n\\end{equation}\nThis equation fixes $\\alpha$ modulo $2\\pi$. Now $U > 0$ and $\\text{Im}\n(\\alpha) > 0$, and therefore\n\\begin{equation}\n     \\cos \\text{Re} (\\alpha) = \\frac{U}{\\sinh \\text{Im} (\\alpha)} > 0.\n\\end{equation}\nWe conclude that $- \\pi\/2 < \\text{Re} (\\alpha) < \\pi\/2$. Thus (see the\ndefinition below (\\ref{defthetas})) $\\alpha = \\arcsin (\\Lambda' +\n\\text{i} U)$.  Integrating (\\ref{phiprime}) with respect to $\\Lambda'$\nand using $I (0) = 0$ to fix the integration constant we arrive at\n\\begin{equation}\n     I(\\Lambda') = 2 \\text{Re} (\\alpha)\n                       = 2 \\text{Re} (\\arcsin(\\Lambda' + \\text{i}U))\n\t\t       = \\arcsin(\\Lambda' + \\text{i} U) +\n\t\t         \\arcsin(\\Lambda' - \\text{i} U).\n\\end{equation}\nWe may now insert this result into eq.\\ (\\ref{rhs}). This yields\n\\begin{equation}\n     \\theta \\left(\\frac{\\Lambda' - \\sin k_1^h}{U} \\right) =\n     - \\theta \\left(\\frac{\\Lambda' - \\sin k_2^h}{U} \\right).\n\\end{equation}\nDividing by 2 and taking $\\tan$ gives\n\\begin{equation}\n     2 \\Lambda' = \\sin k_1^h + \\sin k_2^h.\n\\end{equation}\nSo we have accomplished our task to show that (2) follows from eqs.\\\n(20)-(22) of \\cite{BrAn98}.\n\\section{}\nLet us show that for one $k$-$\\Lambda$ string (and several real $k_j$\nand real $\\lambda_\\gamma$) eq.\\ (7) of \\cite{BrAn98} follows from\neqs.\\ (5), (6), (8) of \\cite{BrAn98}. In case of a single\n$k$-$\\Lambda$-two string the numbers $N^u$ and $M^u$ of \\cite{BrAn98}\nare $N^u = N - 2$, $M^u = M - 1$, where $N$ is the total number of\nelectrons and $M$ is the total number of down spins. There are $N - 2$\nreal charge rapidities $k_1, \\dots, k_{N-2}$, which correspond to\nunbound particles. The string is  characterized  by two charge rapidities\n$k^+$ and $k^-$ and by one spin rapidity $\\Lambda$. $k^\\pm$ and\n$\\Lambda$ satisfy\n\\begin{eqnarray}\n     \\sin k^+ = \\Lambda - \\text{i} u\/4 + \\varepsilon^+ \\, \\\\\n     \\sin k^- = \\Lambda + \\text{i} u\/4 - \\varepsilon^- \\,\n\\end{eqnarray}\nwhere $\\varepsilon^+$ and $\\varepsilon^-$ become small as the length\n$L$ of the lattice becomes large. Eqs.\\ (5)-(8) of \\cite{BrAn98} are\n\\begin{eqnarray} \\label{5}\n     e^{\\text{i} k_j L} & = & \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n                                   {\\Lambda - \\sin k_j + \\text{i} u\/4}\n                     \\prod_{\\delta = 1}^{M-1}\n                     \\frac{\\lambda_\\delta - \\sin k_j - \\text{i} u\/4}\n                          {\\lambda_\\delta - \\sin k_j + \\text{i} u\/4},\n                     \\quad j = 1, \\dots, N - 2 , \\\\ \\label{6}\n     \\prod_{\\delta = 1 \\atop \\delta \\ne \\gamma}^{M-1}\n        \\frac{\\lambda_\\gamma - \\lambda_\\delta - \\text{i} u\/2}\n             {\\lambda_\\gamma - \\lambda_\\delta + \\text{i} u\/2}\n     & = & \\prod_{j=1}^{N-2}\n           \\frac{\\lambda_\\gamma - \\sin k_j - \\text{i} u\/4}\n                {\\lambda_\\gamma - \\sin k_j + \\text{i} u\/4},\n                     \\quad \\gamma = 1, \\dots, M - 1 , \\\\ \\label{7}\n     \\prod_{\\delta = 1}^{M-1}\n        \\frac{\\Lambda - \\lambda_\\delta - \\text{i} u\/2}\n             {\\Lambda - \\lambda_\\delta + \\text{i} u\/2}\n     & = & \\frac{\\varepsilon^+}{\\varepsilon^-} \\prod_{j=1}^{N-2}\n           \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n                {\\Lambda - \\sin k_j + \\text{i} u\/4}, \\\\ \\label{8}\n     e^{\\text{i}(k^+ + k^-)L}\n        & = & \\frac{\\varepsilon^+}{\\varepsilon^-}\n\t      \\prod_{\\delta = 1}^{M-1}\n              \\frac{\\lambda_\\delta - \\Lambda - \\text{i} u\/2}\n                   {\\lambda_\\delta - \\Lambda + \\text{i} u\/2}.\n\\end{eqnarray}\nThe ratio\n\\begin{equation}\n     \\frac{\\varepsilon^+}{\\varepsilon^-} = - \\;\n        \\frac{\\Lambda - \\sin k^+ - \\text{i} u\/4}\n             {\\Lambda - \\sin k^- + \\text{i} u\/4}\n\\end{equation}\nis denoted by $e^{\\text{i} \\varphi}$ in \\cite{BrAn98}. We want to show\nthat (\\ref{7}) algebraically follows from (\\ref{5}), (\\ref{6}) and\n(\\ref{8}). First note that momentum conservation implies that\n\\begin{equation} \\label{momentum}\n     \\exp \\left({\\textstyle \\text{i} L \\left(\n        \\sum_{j=1}^{N-2} k_j + k^+ + k^-\n        \\right)} \\right) = 1.\n\\end{equation}\nMultiplying all eqs.\\ (\\ref{6}) we get\n\\begin{equation} \\label{prodisone}\n     \\prod_{j=1}^{N-2} \\prod_{\\delta = 1}^{M-1}\n        \\frac{\\lambda_\\delta - \\sin k_j - \\text{i} u\/4}\n             {\\lambda_\\delta - \\sin k_j + \\text{i} u\/4} =\n     \\prod_{\\delta = 1}^{M-1}\n        \\prod_{\\gamma = 1 \\atop \\gamma \\ne \\delta}^{M-1}\n\t\\frac{\\lambda_\\delta - \\lambda_\\gamma - \\text{i} u\/2}\n\t     {\\lambda_\\delta - \\lambda_\\gamma + \\text{i} u\/2} = 1.\n\\end{equation}\nNow let us multiply all eqs.\\ (\\ref{5}). Taking into account\n(\\ref{prodisone}) we get\n\\begin{equation}\n     e^{\\text{i}L \\sum_{j=1}^{N-2} k_j} =\n        \\prod_{j=1}^{N-2} \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n                               {\\Lambda - \\sin k_j + \\text{i} u\/4}.\n\\end{equation}\nLet us multiply this by (\\ref{8}) and use (\\ref{momentum}). Then\n\\begin{equation}\n     1 = \\exp \\left({\\textstyle \\text{i} L \\left(\n         \\sum_{j=1}^{N-2} k_j + k^+ + k^-\n         \\right)} \\right) =\n         \\frac{\\varepsilon^+}{\\varepsilon^-} \\prod_{\\delta = 1}^{M-1}\n         \\frac{\\lambda_\\delta - \\Lambda - \\text{i} u\/2}\n              {\\lambda_\\delta - \\Lambda + \\text{i} u\/2}\n         \\prod_{j=1}^{N-2}\n\t \\frac{\\Lambda - \\sin k_j - \\text{i} u\/4}\n\t      {\\Lambda - \\sin k_j + \\text{i} u\/4},\n\\end{equation}\nwhich is equivalent to (\\ref{7}) (or (7) in \\cite{BrAn98}).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum chromodynamics (QCD) has emerged as a phenomenologically accurate theory \nthat describes strong interactions \namong the six quark flavors that are bound into two families of hadrons, namely mesons and \nbaryons. From experiments, we understand that strong interactions \nenjoy C (charge conjugation), P (parity), T (time reversal) discrete symmetries of nature. Therefore, \nQCD must also obey such symmetries, both separately and any combinations formed thereof \\citep{KimCarosi2010}. \nHowever, CP symmetry is broken in QCD due to the presence of the following term in the \nQCD Lagrangian \\citep{Wilczek1978}\n\\begin{equation}\n\\mathfrak{L}_{\\text{int}}=\\left(\\frac{\\theta g^2}{32\\pi^2}\\right)\\text{tr}\\ \nG^{\\mu\\nu}_a\\tilde{G}_{a\\mu\\nu}\n\\end{equation}\nwhere $\\theta$ is a periodic parameter, $g$ is the QCD coupling constant, $G_{\\mu\\nu}$ is the \ncolor field strength tensor, and $\\tilde{G}_{\\mu\\nu}$ is its dual. The value of the $\\theta$-parameter \nis not set theoretically, but it can be measured from \nthe electric dipole moment of the neutron ($d_n$), for which many theoretical estimates exist but \nwe only quote one, $\\|d_n\\|\\sim 2.7\\times 10^{-16}\\bar{\\theta}e$ cm \\citep{Baluni1979}. Here $\\bar{\\theta}\n=\\theta+\\text{arg det }m_q$, where $m_q$ is the quark mass matrix. The latest experimental estimate of \n$\\|d_n\\| < 2.9\\times10^{-26}e$ cm \\citep{Bakeretal2006} constrains $\\|\\bar{\\theta}\\| \\lesssim 10^{-11}$ \n\\citep{KimCarosi2010}. This inexplicably small value of $\\bar{\\theta}$ gave rise to \nthe \\emph{strong CP problem}. One of the solutions, also the most favoured, to this problem \nwas envisioned by Peccei \\& Quinn \\citep{PecceiQuinn1977}, whereby the $\\bar{\\theta}$ parameter is driven precisely to \nzero under a global chiral symmetry, later named $U(1)_{\\text{PQ}}$. The pseudo-Nambu-Goldstone \nboson that results upon the spontaneous breakdown of this symmetry was dubbed the axion \n\\citep{Wilczek1978,Weinberg1978}. Not unlike the Higgs boson, the axion has proven to be extremely \ndifficult to observe as it couples only weakly to ordinary matter and radiation.\n\nDespite several attempts to experimentally observe the axion, it remains elusive to this day. \nNevertheless, the experimental efforts have not gone in vain, but have been able to place serious \nconstraints on the coupling strength of the axion to photons $g_{a\\gamma\n\\gamma} < 10^{-10}\\text{ GeV}^{-1}$. Stringent constraints have been placed on the mass of the axion \n$10^{-6} \\lesssim m_a \\lesssim 10^{-3}$ eV with the lower limit arising \nfrom cosmology \\citep{Preskilletal1983} and the upper limit\\footnote{Due to large uncertainties in the axion mass derived for the DFSZ \nmodel \\citep{DineFischlerSrednicki1981,Zhitnitskii1980} from SN 1987A observations \n($0.004 \\lesssim m_a \\lesssim 0.012$ eV), a more relaxed upper limit is \n$m_a \\lesssim 0.01$ eV \\citep{Raffelt2004}} from the neutrino flux recorded \nfor SN 1987A, which placed strong limits on the cooling flux through other channels namely, \nright-handed neutrinos or axions \\citep{Raffelt2004}. If $g_{a\\gamma\\gamma} > 10^{-10}\\text{ GeV}^{-1}$, \nthe production of axions through the Primakoff process will significantly alter the core He burning timescales of \npost main sequence stars, a possibility excluded by the ratio of horizontal branch stars in \nglobular clusters \\citep{Raffelt1996}. Several reviews on the properties of axions have been forthcoming in the past \ndecade, for example see \\citep{Raffelt1999,Asztalosetal2006,KimCarosi2010}, to which we point the \nreader for a more detailed and comprehensive exposition.\n\nStill, there is no denying the fact that none of the laboratory experiments conducted \nthus far have been able to secure a positive detection of this mysterious particle. \nThe \ndetection of very weakly coupled particles demands extremely sensitive laboratory experiments. \nSo far, only a handful of experiments, namely the Cern Axion Solar Telescope (CAST) \n\\citep{Andriamonjeetal2007}, Axion Dark Matter Experiment (ADMX) \\citep{Asztalosetal2004,Asztalosetal2010}, \nand Rochester-Brookhaven-Fermilab \n(RBF) collaboration \\citep{Panfilisetal1987,Wuenschetal1989} have \nbeen able to surpass the astrophysically derived limits on $g_{a\\gamma\\gamma}$ in the above \nquoted axion mass range. Yet, the sensitivity envelope needs to be pushed even further by a \nfew orders of magitude to be able to draw any definitive conclusions about the existence of \nthe axion. Plans are afoot to modifiy the existing experiments and devise new ones to \nimprove upon current limits on $g_{a\\gamma\\gamma}$ (see Section IV).\n\nUnlike the laboratory experiments, the odds are in favour for detecting axions \nin astrophysical systems. This optimism stems from the fact that the \naxion to photon conversion probability scales with large magnetic field strengths and \nlonger coherence lengths \\citep{Cheloucheetal2009}, such that $P_{a\\rightarrow\\gamma}\n\\propto g^2B^2L^2$, where $L$ is the length over which both the photon and axion fields \nare in phase. Thus, there is a very good chance of finding the axion in \nstrongly magnetized compact objects, namely magnetic white dwarfs (mWDs) and neutron \nstars (NSs). The possibility in the latter case has been expounded by many (see for \nexample \\citep{RaffeltStodolsky1988,LaiHeyl2006,Cheloucheetal2009,PshirkovPopov2009}; also see \n\\citep{Jimenezetal2011} where constraints on $g_{a\\gamma\\gamma}$ are derived \nfrom the dimming of radiation by photon-axion conversion in astrophysical sources), however, the \ncase of the mWDs has not been investigated in great detail and warrants further \nstudy. \n\\subsection{Magnetic white dwarfs}\nAfter the discovery of the first mWD by Kemp \\citep{Kempetal1970}, the number \nof white dwarfs with magnetic fields ranging from a few kG to $10^3$ MG has grown to about \n170. The size of this subpopulation is only 3\\% of the total population of known WDs \ncomprising of 5447 objects\\footnote{G.P. McCook and E.M. Sion, web version of the Villanova \nWhite Dwarf Catalog, http:\/\/www.astronomy.villanova.edu\/WDCatalog\/index.html}. The main channel for identifying magnetism \nin WDs is through Zeeman spectropolarimetry, which not only allows one to discern the \nstrength of the field but also the direction of the field lines, and also through \ncyclotron spectroscopy (see for e.g. \\citep{WickramasingheFerrario2000} for a review \non isolated and binary mWDs). Nevertheless, reconstruction of the field topology has \nproven to be very difficult, mainly due to its highly non-dipolar structure. Over the \nlast decade Zeeman tomography of mWDs has enjoyed some success in elucidating the \nunderlying field structure. This technique is based on calculating a database of \nmodel spectra, where different field geometries comprising of single\/multiple dipole, \nand higher multipoles, that may also be off-centered and misaligned with the rotational axis, are \nconsidered. Then a least-squares fit using the pre-calculated synthetic spectra is performed \nthrough a highly optimized algorithm on the phase-resolved Zeeman spectra to obtain the \ncomplex field structures \\citep{Euchneretal2002}. The \ngenerality of the models not only allows greater flexibility but also renders a closer \nfit to the actual field geometry of the source for a given rotational phase.\n\nThe presence of even a small degree of circular polarization in the spectrum of a WD is a \nstrong indicator of the object possessing a magnetic field upwards of $10^6$ G \n\\citep{Kemp1970}. The degree of circular polarization typically reaches up to $\\sim 5\\%$, \nand sometimes beyond that in a few selective objects, near absorption features and also \nin the continuum. Continuum circular polarization stems from the magnetic circular \ndichroism of the atmosphere, \nwhere the left and right circularly polarized waves propagating through a magnetized \nmedium encounter unequal opacities \\citep{Angel1977}. A relatively higher degree of \ncircular polarization also appears near the red and blue shifted wings of the Zeeman \nsplit absorption lines ($\\sigma_+$ and $\\sigma_-$ components, \\citep{WickramasingheFerrario2000}).\n\nOn the other hand, most observations of mWDs indicate that the linear polarization \ncomponent never exceeds that of the circular one, and the spectrum remains dominantly \ncircularly polarized until field strengths $\\geq 10^8$ G are reached \\citep{Angel1977}. \nIn a magneto-active plasma, the plane of linear polarization undergoes many Faraday \nrotations, an effect that arises due to the magnetic birefringence of the medium, so \nthat on average the degree of linear polarization of the emergent radiation is much reduced \n\\citep{SazonovChernomordik1975}.\n\nThe very fact that linear polarization is of the order of a few percent ($\\sim 5\\%$) in the \ncontinuum spectra of most mWDs can be exploited to draw meaningful conclusions on the extent of axion \ninteraction with photons traversing the magnetized plasma of mWDs. We explain how this can be \nimplemented in the next section.\n\\subsection{Plan of this study}\nThe purpose of this study is to conduct a survey of the $m_a-g_{a\\gamma\\gamma}$ parameter \nspace by modelling photon-axion oscillations in the magnetosphere of a mWD. To this end, \nwe model the field structure of a strongly magnetized WD PG 1015+014, for which high \nresolution optical spectropolarimetric observations are available \\citep{Euchneretal2006}. In the \nsame article, the authors also conduct a phase-resolved Zeeman tomographic analysis and derive \na best-fit model of the magnetic field topology. Despite fitting the spectrum with a range of \nfield geometries, they were only able to pin down the field geometry for a single rotational phase \nby fitting it with a superposition of three off-centered and non-aligned dipoles of unequal surface \nfield strengths (see Table \\ref{tab:pg1015} for model parameters). To model the effect of photon-axion oscillation \nin the magnetosphere on the emergent polarization, we propagate an unpolarized photon of a given \nenergy from the photosphere through the encompassing magnetosphere, that has been populated by \na diffuse, cold ionized H gas. The emergent intensity and polarization is then averaged over the \nwhole surface of the star. Finally, we compare the degree of polarization from our model simulation \nto what is observed in mWDs with field strengths in excess of a few $10^6$ G, \nfor example PG 1015+014, and draw conclusions on the strength of the coupling constant for \na given axion mass.\n\\begin{table}\n\\caption{\\textit{Top}: Magnetic field geometry adopted from the spectropolarimetric analysis by \\citep{Euchneretal2006} \nof the the mWD PG 1015+014. The model comprises of three off-centered and non-aligned dipoles \n$D_1,D_2,D_3$ with unequal surface field strengths $B_s$, polar ($\\theta_B$) and azimuthal ($\\phi_B$) angles \nof the magnetic field axes. The center positions of the dipoles relative to the center of the star are \ngiven by ($a_x,a_y,a_z$). \\textit{Bottom}: Collection of parameters used in the study.}\n\\begin{center}\n\\begin{tabular}{l r r r}\n\\multicolumn{4}{c}{\\textbf{Model Parameters}} \\\\\n\\hline\n\\hline\n& $D_1$ & $D_2$ & $D_3$ \\\\\n\\hline\n$B_s$ (MG) & -40 & 92 & -38 \\\\\n$\\theta_B({}^\\circ)$ & 44 & 63 & 63 \\\\\n$\\phi_B({}^\\circ)$ & 339 & 276 & 134 \\\\\n$a_x(R_\\star)$ & 0.04 & -0.012 & 0.27 \\\\\n$a_y(R_\\star)$ & 0.35 & -0.136 & 0.080 \\\\\n$a_z(R_\\star)$ & 0.33 & -0.28 & 0.21 \\\\\n\\hline\n$R_\\star$ & \\multicolumn{3}{r}{$7\\times10^8$ cm} \\\\\n$\\theta_k$ & \\multicolumn{3}{r}{$23^\\circ$} \\\\\n$Y_e$ & \\multicolumn{3}{r}{1} \\\\ \n$B_Q$ & \\multicolumn{3}{r}{$4.413\\times10^{13}$ G} \\\\\n$T$ & \\multicolumn{3}{r}{$10^4$ K} \\\\\n$g_\\star$ & \\multicolumn{3}{r}{$10^8\\text{ cm s}^{-2}$} \\\\\n$\\rho_0$ & \\multicolumn{3}{r}{$10^{-10}\\text{ g cm}^{-3}$} \\\\\n$\\rho_\\infty$ & \\multicolumn{3}{r}{$10^{-20}\\text{ g cm}^{-3}$} \\\\\n\\hline\n\\end{tabular} \n\\end{center}\n\\label{tab:pg1015}\n\\end{table}\n\nIn the following Section, we formulate the key equations describing the interaction of the axion \nwith photons, geometry of the aggregate magnetic field, and structure of the plasma permeating the \nmagnetosphere. The lack of understanding of the density profile of the magnetospheric plasma \nintroduces some level of inaccuracy in any treatment of mWDs. We take the simplest approach and \ndescribe the plasma density by the barometric law for an isothermal atmosphere. In Section III we present \nthe main results of the study along with a comparison to some of the results obtained from \nlab experiments (see Fig. \\ref{fig:exclusion}). A discussion of the results is provided in \nSection IV.\n\\section{Model equations}\nThe interaction of the axion field with an external electromagnetic field is given by the \nfollowing Lagrangian density \\citep{RaffeltStodolsky1988}, in natural units where $\\hbar=c=1$,\n\\begin{eqnarray}\n\\mathfrak{L}=&-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu}+\\frac{1}{2}(\\partial_\\mu a\n\\partial^\\mu a - m_a^2 a^2)-\\frac{1}{4}g_{a\\gamma\\gamma}F_{\\mu\\nu}\n\\tilde{F}^{\\mu\\nu}a& \\nonumber \\\\\n& +\\frac{\\alpha^2}{90m_e^4}\\left[\\left(F_{\\mu\\nu}F^{\\mu\\nu}\n\\right)^2+\\frac{7}{4}\\left(F_{\\mu\\nu}\\tilde{F}^{\\mu\\nu}\\right)^2\\right]&\n\\label{eq:lagrangian}\n\\end{eqnarray}\nwhere the first term describes the external electromagnetic field, with $F_{\\mu\\nu}$ \nas the antisymmetric electromagnetic field strength tensor and $\\tilde{F}^{\\mu\\nu}=\n\\frac{1}{2}\\varepsilon^{\\mu\\nu\\rho\\sigma}F_{\\rho\\sigma}$ as its dual. The second term \nis simply the Klein-Gordon equation for the axion field $a$ where $m_a$ represents its \nmass. The next term is the interaction Lagrangian density, which upon simplification, \nusing the given definitions, yields $\\mathfrak{L}_{\\text{int}}=g_{a\\gamma\\gamma}\na \\mathbf{E}\\cdot\\mathbf{B}$, where $g_{a\\gamma\\gamma}$ is the photon-axion coupling \nstrength, $\\mathbf{E}$ is the polarization vector of the \nphoton field, and $\\mathbf{B}$ is the external magnetic field. Quantum corrections to \nthe classical electromagnetic field, due to the constant creation and annihilation of \nelectron-positron pairs in vacuum, are given in the last term of \\eqref{eq:lagrangian} by \nthe Euler-Heisenberg effective Lagrangian, causing the vacuum to be birefringent.\n\\subsection{Off-centered non-aligned three dipole model}\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig1}\n\\caption{This illustrates the coordinate system used to obtain the aggregate magnetic field. \nHere $\\Sigma_0$ represents the coordinate system centered on the star with the \n$\\mathbf{\\hat{z}}$-axis aligned with the rotational axis. $\\Sigma_i$ represents the coordinate system in which the \ndifferent dipole magnetic field equations are written. This system is misaligned with $\\Sigma_0$ \nby a polar angle $\\theta_{B,i}$ and an azimuthal angle $\\phi_{B,i}$, and then it is displaced from \nthe center by the vector $\\mathbf{a}_i$. For clarity we have chosen the vector $\\mathbf{a}_i$ to \nlie along the $\\mathbf{\\hat{z}}'$-axis (color online).}\n\\label{fig:coord}\n\\end{figure}\nWe start by writing down the aggregate magnetic field $\\mathbf{B}_0$ in the coordinate \nsystem $\\Sigma_0$ where the $\\mathbf{\\hat{z}}$-axis coincides with the rotational axis of the star \n\\begin{equation}\n\\mathbf{B}_0=\\sum_{i=1}^3R_i^T\\mathbf{B}_i'\n\\end{equation}\nThe three off-centered dipole fields $\\mathbf{B}_i'$ are first rotated before they are added \ntogether by operating on each of them with a rotation matrix $R^T=R_{z,i}^TR_{y,i}^T$, \nwhere the superscript $T$ indicates the transpose, and the rotation matrices are \ngiven as follows\n\\begin{eqnarray}\nR_{z,i}=&\\left(\\begin{array}{ccc}\n\\cos \\phi_{B,i} & \\sin \\phi_{B,i} & 0 \\\\\n-\\sin \\phi_{B,i} & \\cos \\phi_{B,i} & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{array}\\right)& \\\\\nR_{y,i}=&\\left(\\begin{array}{ccc}\n\\cos\\theta_{B,i} & 0 & -\\sin\\theta_{B,i} \\\\\n0 & 1 & 0 \\\\\n\\sin\\theta_{B,i} & 0 & \\cos\\theta_{B,i} \\\\\n\\end{array}\\right)&\n\\label{eq:Ry}\n\\end{eqnarray}\nHere the polar and azimuthal angles $\\theta_{B,i}$ and $\\phi_{B,i}$, respectively, are defined \nwith respect to the axis of rotation. In cartesian coordinates, the dipole fields are \nexpressed as\n\\begin{equation}\n\\mathbf{B}'_i=\\frac{B_{s,i}R_\\star^3}{2r'^5_i}(3x'_iz'_i\\mathbf{\\hat{x}'}\n+3y'_iz'_i\\mathbf{\\hat{y}'}+\\{3{z'_i}^2-{r'_i}^2\\}\\mathbf{\\hat{z}'})\n\\end{equation}\nwhere the fields are shifted from the coordinate center, such that $\\mathbf{r}_i'=\n\\mathbf{r}-\\mathbf{a}_i$. In the above equation, $\\mathbf{B}_{s,i}$ is the surface field \nstrength of the $i^{\\text{th}}$ dipole field component, $R_\\star\\simeq7\\times10^8$ cm is the radius of the WD, and $r_i'$ is the \nmagnitude of the radial vector in the coordinate system $\\Sigma_i$. The profile of the aggregate field $B_0$ as a function of distance \nis shown in Fig. \\ref{fig:los}.\n\\subsection{Fully ionized pure H atmosphere}\nThe presence of a magnetic field necessarily introduces anisotropy in the plasma dielectric tensor \n$\\matlabel{\\varepsilon}_p$. In the case of a nonuniform field, none of the dielectric tensor components \nvanish, as compared to the homogeneous case. Below we write all the dielectric components, \nwhich one can easily derive from Maxwell's equations, for completeness.\n\\begin{align}\n\\matlabel{\\varepsilon}_p & = \\left(\\begin{array}{ccc}\n\\varepsilon_{11} & \\varepsilon_{12} & \\varepsilon_{13} \\\\\n\\varepsilon_{21} & \\varepsilon_{22} & \\varepsilon_{23} \\\\\n\\varepsilon_{31} & \\varepsilon_{32} & \\varepsilon_{33} \\\\\n\\end{array}\\right) & \\\\\n\\varepsilon_{11} & = 1-\\sum_{s=e,p}\\hat{\\omega}_{p,s}^2\\left[\\frac{1-\\hat{\\omega}_{c,s}^2\n\\hat{B}_{0x}^2}{1-\\hat{\\omega}_{c,s}^2}\\right] \\nonumber & \\\\\n& \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}_{c,p}^2)\n\\hat{B}_{0x}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] & \\\\\n\\varepsilon_{12} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2\n)}(i\\hat{B}_{0z}+\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0y}) & \\\\\n\\varepsilon_{13} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0y}-\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0z}) & \\\\\n\\varepsilon_{21} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0z}-\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0y}) & \\\\\n\\varepsilon_{22} & \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}\n_{c,p}^2)\\hat{B}_{0y}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] & \\\\\n\\varepsilon_{23} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0x}+\\hat{\\omega}_{c,e}\\hat{B}_{0y}\\hat{B}_{0z}) & \\\\ \n\\varepsilon_{31} & \\approx \\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0y}+\\hat{\\omega}_{c,e}\\hat{B}_{0x}\\hat{B}_{0z}) & \\\\ \n\\varepsilon_{32} & \\approx -\\frac{\\hat{\\omega}_{c,e}\\hat{\\omega}_{p,e}^2}{(1-\\hat{\\omega}_{c,e}^2)}\n(i\\hat{B}_{0x}-\\hat{\\omega}_{c,e}\\hat{B}_{0y}\\hat{B}_{0z}) & \\\\\n\\varepsilon_{22} & \\approx 1-\\hat{\\omega}_{p,e}^2\\left[\\frac{1-\\hat{\\omega}_{c,e}^2(1+\\hat{\\omega}\n_{c,p}^2)\\hat{B}_{0z}^2}{(1-\\hat{\\omega}_{c,e}^2)(1-\\hat{\\omega}_{c,p}^2)}\\right] &\n\\end{align}\nIn the above set of equations, $\\hat{\\omega}_{c,s}=q_sB_0\/\\omega m_s c$ is the normalized cyclotron \nfrequency for species $s=(e,p)$, where $e$ and $p$ signify electrons and protons; $\\hat{\\omega}_{p,s} \n= \\sqrt{4\\pi n_s\/m_s\\omega^2}$ is the normalized plasma frequency, where $n_p = n_e = Y_e\\rho\/m_p$ \nare the electron and proton number densities, $Y_e$ is the electron fraction, and $\\rho$ is the \nproton mass density of the plasma; the normalized magnetic field components are defined as \n$\\hat{B}_{0,i=x,y,z} = B_{0,i}\/B_0$.\n\\subsubsection{Vacuum corrections}\nDue to the polarizability of the vacuum in strong magnetic fields, the plasma dielectric \ntensor $\\matlabel{\\varepsilon}_p$, and the inverse permeability tensor $\\matlabel{\\mu}^{-1}$ \nare modified \\citep{MeszarosVentura1979,LaiHo2003b}, such that $\\matlabel{\\varepsilon}_{p+v} = \n\\matlabel{\\tilde{\\varepsilon}} = \\matlabel{\\varepsilon}_p + \\Delta\\matlabel{\\varepsilon}_v$ \nand $\\matlabel{\\mu}^{-1}_{p+v} = \\matlabel{\\tilde{\\mu}}^{-1} = \\matlabel{I} + \n\\Delta\\matlabel{\\mu}^{-1}_v$, where\n\\begin{eqnarray}\n& \\Delta\\matlabel{\\varepsilon}_v = (a_v-1)\\matlabel{I} + q_v\\hat{\\mathbf{B}}_0\\hat{\\mathbf{B}}_0 & \\\\\n& \\Delta\\matlabel{\\mu}^{-1}_v = (a_v-1)\\matlabel{I} + m_v\\hat{\\mathbf{B}}_0\\hat{\\mathbf{B}}_0 & \\\\\n& a_v = 1-2\\delta_v\\ \\ \\ q_v = 7\\delta_v\\ \\ \\ m_v=-4\\delta_v\\ \\ \\ \\delta_v = \n\\frac{\\alpha}{45\\pi}\\left(\\frac{B_0}{B_Q}\\right)^2 & \\nonumber\n\\end{eqnarray}\nand $B_Q = 4.413\\times10^{13}$ G is the quantum critical field for which the separation in energy \nbetween Landau levels of the electron exceeds its rest mass.\n\\subsubsection{Plasma density profile}\nThat many mWDs are surrounded by hot coronae has been suggested by many to explain the polarized \nflux of those WDs that show comparable degree of linear and circular polarization \n\\citep{SazonovChernomordik1975,InghamBrecherWasserman1976,ZheleznyakovSerber1994}. The thermal electrons in the hot \ntenuous plasma with temperature $T\\sim10^{6-8}$ K radiate at the cyclotron frequency that falls in the \noptical wavelength for field strengths of $B\\sim10^8$ G. This radiation appears \nto be polarized both linearly and circularly, depending on the orientation of the line of sight to the \nmagnetic field, and traverses the corona without any apsorption. Furthermore, slightly polarized \nradiation emanating from the photosphere, with very low degree of linear polarization due \nto Faraday rotation, gets added to that generated in the corona, as a result increasing the amount \nof flux that is polarized linearly. Several hot isolated WDs, with effective temperatures in \nexcess of $\\simeq25,000$ K, emitting X-rays were detected by ROSAT \\citep{Flemingetal1996}, however \nall cases were linked to subphotospheric thermal emission \\citep{Musielaketal2003}. Although the \nnon-detection of any coronal emission may indicate the absence of a hot tenuous corona, it is not \nat all unreasonable to suggest the presence of a tenuous cold plasma of fully ionized H. In this \nstudy, we envisage that the mWDs are encompassed by cold isothermal electron-proton coronae with \nthe following barometric density profile,\n\\begin{equation}\n\\rho(r) = \\rho_0\\exp\\left(-\\frac{r-R_\\star}{H_\\rho}\\right) + \\rho_\\infty\n\\end{equation}\nwhere $\\rho_0$ is the density near the surface of the star, $\\rho_\\infty$ is the density \nthat remains far away from the star as the strength of the magnetic field becomes significantly \nweaker than that at the surface, and $H_\\rho = 2k_BT\/m_pg_\\star \\simeq 1.65\\times10^4$ cm is the density \nscale height with an effective temperature $T\\simeq10^4$ K and surface gravity \n$\\log g_\\star(\\text{cm\/s}^2) = 8$. There is no clear agreement on the surface plasma \ndensity with $10^{-11} \\lesssim \\rho_0 \\lesssim 10^{-6}\\text{ g cm}^{-3}$. Here, we \nassume that the plasma is sufficiently tenuous with $\\rho_0 = 10^{-10}\\text{ g cm}^{-3}$ \nand $\\rho_\\infty = 10^{-20}\\text{ g cm}^{-3}$.\n\\subsection{Axion-photon mode evolution in an inhomogeneous magnetized plasma} \nWe are interested in knowing the evolution of the axion field and the polarization vector \nas the radiation propagates out from the surface of the star, traversing the region with an \ninhomogeneous plasma density and magnetic field. Here we follow the discussion given in \n\\citep{RaffeltStodolsky1988,LaiHeyl2006}, and derive the photon field mode evolution from the \nEM wave equation\n\\begin{equation}\n\\nabla\\times(\\matlabel{\\tilde{\\mu}}^{-1}\\cdot\\nabla\\times\\mathbf{E}) = \\frac{\\omega^2}{c^2}\n\\matlabel{\\tilde{\\varepsilon}}\\cdot\\mathbf{E}\n\\end{equation}\nNext, we assume the ansatz $\\mathbf{E}=\\tilde{\\mathbf{E}}\\exp(ikz)$ where the wave is propagating \nalong the rotational axis of the star, which in this case is also the line of sight direction, \nand the wavenumber $k=\\omega\/c$. Plugging this ansatz into \nthe wave equation, and ignoring second order derivatives, we find\n\\begin{equation}\n\\frac{d}{dz}\n\\left(\\begin{array}{c}\n\\tilde{E}_x \\\\\n\\tilde{E}_y\n\\end{array}\\right) = \\left(\\begin{array}{cc}\n\\chi_{11} & \\chi_{12} \\\\\n\\chi_{21} & \\chi_{22} \n\\end{array}\\right)\\left(\\begin{array}{c}\n\\tilde{E}_x \\\\\n\\tilde{E}_y\n\\end{array}\\right)\n\\label{eq:modeevol}\n\\end{equation}\nwhere the matrix elements are given below\n\\begin{eqnarray}\n\\chi_{11} & = & \\Upsilon^{-1}_3\\left[k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4-\n\\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \\Upsilon_1\n\\Upsilon_5^{-1} \\right. \\nonumber \\\\\n& & \\left. \\times\\left(k^2\\tilde{\\varepsilon}_{21}-\\Upsilon_2-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4\\}\n\\right)\\right] \\\\\n\\chi_{12} & = & \\Upsilon^{-1}_3\\left[k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2-\n\\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \\Upsilon_1\n\\Upsilon_5^{-1} \\right. \\nonumber \\\\\n& & \\left. \\times\\left(k^2\\tilde{\\varepsilon}_{22}-\\Upsilon_6-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2\\}\n\\right)\\right] \\\\\n\\chi_{21} & = & \\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \n\\Upsilon_5^{-1} \\nonumber \\\\ \n& & \\times \\left(k^2\\tilde{\\varepsilon}_{21}-\\Upsilon_2-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{11}-\\Upsilon_4\\}\n\\right) \\\\\n\\chi_{22} & = & \\left(1-\\frac{\\Upsilon_1^2}{\\Upsilon_3\\Upsilon_5}\\right)^{-1} \n\\Upsilon_5^{-1} \\nonumber \\\\ \n& & \\times \\left(k^2\\tilde{\\varepsilon}_{22}-\\Upsilon_6-\n\\frac{\\Upsilon_1}{\\Upsilon_3}\\{k^2\\tilde{\\varepsilon}_{12}-\\Upsilon_2\\}\n\\right) \\\\\n\\Upsilon_1 & = & \\frac{d}{dz}(m_v\\hat{B}_x\\hat{B}_y)+i2km_v\\hat{B}_x\\hat{B}_y \\\\\n\\Upsilon_2 & = & ik\\frac{d}{dz}(m_v\\hat{B}_x\\hat{B}_y)-k^2m_v\\hat{B}_x\\hat{B}_y \\\\\n\\Upsilon_3 & = & -\\frac{d}{dz}(a_v+m_v\\hat{B}_y^2)+i2k(a_v+m_v\\hat{B}_y^2) \\\\\n\\Upsilon_4 & = & -ik\\frac{d}{dz}(a_v+m_v\\hat{B}_y^2)+k^2(a_v+m_v\\hat{B}_y^2) \\\\\n\\Upsilon_5 & = & -\\frac{d}{dz}(a_v+m_v\\hat{B}_x^2)+i2k(a_v+m_v\\hat{B}_x^2) \\\\\n\\Upsilon_6 & = & -ik\\frac{d}{dz}(a_v+m_v\\hat{B}_x^2)+k^2(a_v+m_v\\hat{B}_x^2)\n\\end{eqnarray}\n\\subsubsection{Line of sight geometry}\n\\begin{figure}[t!]\n\\includegraphics[width=0.5\\textwidth]{fig2}\n\\caption{This figure illustrates the coordinate system used to obtain the photon-axion \nmode evolution along a given LOS (top), and the decline of the magnetic field strength with distance \n$s$ from the surface (bottom). Here the LOS vector is represented by $s$ that is tilted \nat an angle $\\theta_k$ to the rotation axis, and is in parallel to $z'$ which is not to be \nconfused with $\\hat{z}'$ in Fig. \\ref{fig:coord}. Several different \npoints on the star's surface with spherical coordinates ($R_\\star,\\theta_R,\\phi_R$) are \nchosen and then averaged to determine the final polarization of the photon leaving the \nmagnetosphere (color online).}\n\\label{fig:los}\n\\end{figure}\n The Zeeman tomography analysis of mWD PG 1015+014 indicates that the line of sight (LOS) \nis inclined at an angle $\\theta_k=23^\\circ$ to the rotational axis of the star. \nFollowing \\citep{DupaysRoncadelli2006}, we modify the matrix Eq.(\\ref{eq:modeevol}) to \nobtain the mode evolution of the photon-axion system in a coordinate system oriented \nalong the LOS (see Fig. \\ref{fig:los}). Again, we assume the ansatz $a\\propto\\exp{(ik's-i\\omega t)}$\n\\begin{equation}\ni\\frac{d}{ds}\\left(\\begin{array}{c}\na \\\\\nE_{x'} \\\\\nE_{y'} \\\\\n\\end{array}\\right) = \\left(\\begin{array}{ccc}\n\\Delta_a-k' & \\Delta_{Mx'} & \\Delta_{My'} \\\\\n\\Delta_{Mx'} & i\\chi_{11}'-k' & i\\chi_{12}' \\\\\n\\Delta_{My'} & i\\chi_{21}' & i\\chi_{22}'-k'\\end{array}\\right)\n\\left(\\begin{array}{c}\na \\\\\nE_{x'} \\\\\nE_{y'} \\\\\n\\end{array}\\right) \n\\label{eq:newmodeevol}\n\\end{equation}\nwhere $\\Delta_a=m_a^2\/2\\omega$, $\\Delta_{Mx'}=-g_{a\\gamma\\gamma}B_x\/2$, \n$\\Delta_{My'}=-g_{a\\gamma\\gamma}B_y\/2$.   \nNotice that Eq.(\\ref{eq:modeevol}) applies to a system for which the LOS vector coincides with the rotational \naxis of the star. For a different LOS vector, such as shown in Fig. \\ref{fig:los}, we perform a rotation of \nthe plasma dielectric tensor around the $\\mathbf{\\hat{y}}$-axis by an angle $\\theta_k$, \n$\\matlabel{\\tilde{\\varepsilon}}' = R_y^T(\\theta_k)\\matlabel{\\tilde{\\varepsilon}}R_y(\\theta_k)$ where $R_y$ is given in \nEq.(\\ref{eq:Ry}). \n\nThe total degree of polarization can be found by integrating Eq.(\\ref{eq:newmodeevol}) from a given point on the surface outwards \nto a distance beyond which the amplitude of photon-axion oscillations and plasma effects become \nnegligible, and then by averaging the Stokes parameters \\citep{RybickiLightman2004} over \nthe whole observable hemisphere. \n\\begin{eqnarray}\nI & = & \\|E_{x'}\\|^2 + \\|E_{y'}\\|^2 \\\\\nQ & = & \\|E_{x'}\\|^2 - \\|E_{y'}\\|^2 \\\\\nU & = & E_{x'} E_{y'}^\\ast + E_{y'} E_{x'}^\\ast \\\\\nV & = & -i (E_{x'} E_{y'}^\\ast - E_{y'} E_{x'}^\\ast)\n\\end{eqnarray}\nThe mode amplitudes\nare in general complex, and in the above set of equations $\\ast$ gives the\ncomplex conjugate. We sample the Stokes vector \nfrom different points on the surface, with coordinates ($R_\\star,\\theta_R,\\phi_R$), that are spread around the LOS vector \n$\\mathbf{\\hat{k}}$ with $\\Delta\\phi_R=30^\\circ$ and $\\Delta\\theta_R=10^\\circ$, where $10^\\circ \\leq \\theta_R \n\\leq 80^\\circ$ and $0^\\circ \\leq \\phi_R \\leq 330^\\circ$. Because the sampling in the azimuthal \nangle is sparse for larger polar angles, we take a weighted average, as shown below for one of the \nStokes parameters, to determine the average degree of polarization of the whole hemisphere\n\\begin{equation}\n\\langle I\\rangle = \\frac{\\sum_{\\theta_R,\\phi_R}I(\\theta_R,\\phi_R)\\sin\\theta_R}{\\sum_{\\theta_R}\\sin\\theta_R}\n\\end{equation}\n\\section{Results}\nIn the following, we look at how an unpolarized photon emitted from the photosphere \nof a mWD gets polarized as it traverses through the magnetosphere. Photon-axion \ninteraction and the intervening plasma make the medium birefringent, consequently, altering \nthe state of polarization of the unpolarized photon. We obtain the degree of polarization from \nthe averaged Stokes parameters\n\\begin{eqnarray}\nP_L & = & \\frac{\\sqrt{\\langle Q\\rangle^2 + \\langle U\\rangle^2}}{\\langle I\\rangle} \\\\\nP_C & = & \\frac{\\langle V\\rangle}{\\langle I\\rangle} \n\\end{eqnarray}\nwhere $P_L$ and $P_C$ represent linear and circular polarization. \n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig3}\n\\caption{Polarization evolution of an unpolarized photon along a given LOS as it starts at \nthe photosphere and propagates through the magnetosphere. In this case, $E_\\gamma = 3$ \neV, $m_a = 10^{-5}$ eV, $g_{a\\gamma\\gamma} = 10^{-9}\\mbox{ GeV}^{-1}$. The magnetic field \ngeometry assumed is that of mWD PG 1015+014 (color online).}\n\\label{fig:stokes}\n\\end{figure}\n\nIn Fig. \\ref{fig:stokes}, we present the evolution of the Stokes vector with distance $s$ \nfrom the surface of the star for the case of radiation with $E_\\gamma = 3$ eV, and axion parameters \n$m_a=10^{-5}\\text{ eV}, g_{a\\gamma\\gamma}=10^{-9}\\text{ GeV}^{-1}$. The oscillations in the solution arise due to the \nmixing of the axion and photon eigenstates, an effect analogous to neutrino oscillations due to the MSW \neffect \\citep{MikheevSmirnov1986,Wolfenstein1978}. However, notice that the interaction is non-resonant because a 50\\% drop in intensity would be \nobserved if the axion and photon modes were to achieve maximal mixing and undergo level crossing. Eventually, \nas the photons travel farther away from the surface, the decline in the magnetic field strength reduces the probability \nof conversion, hence the diminishing of intensity variation. We find that the change in \npolarization is primarily brought about by the axion interaction with the photon. In the event this \ninteraction is made negligible, no significant polarization or change in intensity of the emergent radiation \nis found. The origin of circular polarization in mWDs, as alluded \nto earlier, is understood in terms of the difference in opacities for the two modes of radiation, making \nthe plasma dichroic, in the presence of a magnetic field. Linear polarization, on the other hand, was \nexplained by the cyclotron radiation that emanates from the tenuous corona composed of an ionized plasma. \nIn this study, since the treatment of radiative transfer effects is very simplistic only an upper limit can be\nplaced on how strongly the axion couples to photons, as shown in the next section.\n\\subsection{Constraints on $g_{a\\gamma\\gamma}$}\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig4}\n\\caption{The final state of polarization of radiation after traversal from the WD's magnetosphere for different \n$m_a$ and $g_{a\\gamma\\gamma}$. Here $\\langle I \\rangle$ is the average Stokes intensity, and $P_L$ and $P_C$ are \nthe degrees of linear and circular polarizations. These results apply to the case of mWD PG 1015+014 (color online).}\n\\label{fig:composite}\n\\end{figure}\nAxion production in the mWD magnetosphere can enhance the degree of linear polarization of the \nobserved optical radiation. The goal here is to not determine \nthe precise value of the photon-axion coupling strength but only constrain it from above. To this \nend, we look at the amount of linear polarization that is produced for a given $m_a$ and $g_{a\\gamma\\gamma}$.\nThe underlying assumption here is that all of the observed linear polarization is generated due to \nphoton-axion interaction, and not by the plasma, which effectively yields the absolute upper limit on \n$g_{a\\gamma\\gamma}$. In Fig. \\ref{fig:composite}, we plot the emergent intensity and state of polarization \nfor different axion masses and for photons in the UV - optical waveband with energies between $2 - 5$ eV. \nThe $m_a$ and $g_{a\\gamma\\gamma}$ in Fig. \\ref{fig:composite} were chosen specifically so that $P_L \\gtrsim 0.05$ \nfor all photon energies. \n\nIn Fig. \\ref{fig:exclusion}, we use the same parameters to draw an exclusion region in the $m_a-g_{a\\gamma\\gamma}$ \nparameter space, along with regions excluded by lab experiments and astrophysical considerations. The shaded region \nin red excludes all $m_a$ - $g_{a\\gamma\\gamma}$ values for the case of mWD PG 1015+014, that is for a typical \nsurface field strength $B\\simeq10^8$ G and degree of linear polarization $P_L\\sim 5\\%$. We find that for \nthe range of masses that are of relevance, in particular, to the axion models, the constraints on \n$g_{a\\gamma\\gamma}$ from this study are superseded by that from horizontal branch (HB) stars. Still, the \nlimiting linear polarization criterion used in this study is able to probe smaller $g_{a\\gamma\\gamma}$ \nvalues in comparison to works that only look at radiation dimming (for e.g. see \\citep{Jimenezetal2011}).\n\nThe constraints can be further improved by looking at mWDs with higher magnetic field strengths. The highest \nfield strength that has ever been discovered in a mWD is $B\\simeq1000$ MG in two such objects namely, \nPG 1031+234 and SDSS J234605+385337 \\citep{Jordan2009}. Both objects show linear polarization as low as \n$\\sim1\\%$ for some rotational phases \\citep{PiirolaReiz1992,Vanlandinghametal2005}. Based on these two \nfacts and assuming that the magnetic field geometry of these two mWDs is at least as complex as that found \nin PG 1015+014, we produce two exclusion regions shown in Fig. \\ref{fig:exclusion} with colors blue \nand green. The former corresponds to a surface field strength $B=1000$ MG with the same level of linear polarization \nas before, and the latter studies the case with $P_L\\simeq1\\%$. For these two cases, we have only looked at \n$m_a \\leq 10^{-5}$ eV since higher mass values don't constrain $g_{a\\gamma\\gamma}$ better than limits derived \nfrom HB stars and CAST (Phase-I). On the other hand, we have extended our treatment to smaller particle masses \nwith $m_\\phi\\leq10^{-6}$ eV where $m_\\phi$ should be interpreted as the mass of any light pseudoscalar \nboson that is characteristically very much similar to the axion but isn't a CDM particle.\n\nIt is worth mentioning that the change in $g_{a\\gamma\\gamma}$ is not linear with the change in magnetic \nfield strength, as evident from the comparison between the red and blue regions in Fig. \\ref{fig:exclusion}.\nFor higher field strengths one observes higher degree of polarization of the emerging radiation. Naively, one \nwould expect the plane of polarization to rotate by an amount that is $\\mathcal{O}(g_{a\\gamma\\gamma}^2 \nB^2l^2)$, which is valid strictly in the absence of plasma when the photon and axion are treated as \nmassless particles \\citep{VanBibberetal1987,RaffeltStodolsky1988}. Therefore, for a fixed degree of \npolarization an increase in $B$ should also decrease $g_{a\\gamma\\gamma}$ by the same factor, when \n$l$, the length over which the magnetic field remains homogeneous, is kept constant. However, as \nshown by \\citep{Heyletal2003} in the case of NSs, an increase in magnetic field strength also \nincreases the level of polarization by effectively shifting the polarization-limiting radius, \nthe distance beyond which the two polarization modes couple and which depends weakly on \nthe magnetic field strength $R_{pl}\\propto B^{2\/5}$, farther away from the star. The farther the \npolarization-limiting radius, the more coherently the polarization states from different LOSs add, \nyielding a higher degree of polarization.\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{fig5}\n\\caption{Exclusion plot in the $m_a-g_{a\\gamma\\gamma}$ parameter space. The red region corresponds to \nthe case of mWD PG 1015+014 with $B\\simeq10^8$ G and limiting linear polarization $P_L\\sim5\\%$. The \nblue and green regions correspond to the case with $B\\simeq10^9$ G, but with $P_L\\sim5\\%$ and \n$P_L\\sim1\\%$ respectively. The mass of the \naxion is constrained to $10^{-6} \\lesssim m_a \\lesssim 10^{-2}$ eV from cosmology and SN 1987A measurments.\nThe photon-axion coupling constant is capped from above with $g_{a\\gamma\\gamma} < 10^{-10}\n\\text{ GeV}^{-1}$ by the number of horizontal branch stars in globular clusters. KSVZ \n\\citep{Kim1979,ShifmanVainshteinZakharov1980} and DFSZ \\citep{DineFischlerSrednicki1981,Zhitnitskii1980} \nare two different theoretical models that predict how $g_{a\\gamma\\gamma}$ scales with $m_a$. \nOther exclusion regions are from lab experiments by the CAST experiment \\citep{Andriamonjeetal2007}, \nADMX group \\citep{Asztalosetal2004,Asztalosetal2010} and by Rochester-Brookhaven-Fermilab collaboration \n\\citep{Panfilisetal1987,Wuenschetal1989} (color online).} \n\\label{fig:exclusion}\n\\end{figure}\n\\section{Discussion}\nThis study looks at how the production of axions in mWD magnetospheres can alter the \nstate of polarization of the observed radiation. We find that unpolarized photons of \nphotospheric origin become linearly polarized upon their traversal through the \ninhomogeneous magnetic field of a mWD. We have modeled the magnetospheric plasma, as fully \nionized pure H with a barometric profile. Since the majority of mWDs are strongly \ncircularly polarized and only show a relatively \nsmall degree of linear polarization, at most $5\\%$, we have used this observation to constrain the \ncoupling strength $g_{a\\gamma\\gamma}$ of axions to photons. We find that for the case where \nthe plasma component only contributes negligibly to the state of polarization, the \ncoupling strength $g_{a\\gamma\\gamma}$ increases with the mass of the axion $m_a$. The level \nof linear and circular polarization observed in mWDs is sensitive to the properties of the \nmagnetospheric plasma. The limits on $g_{a\\gamma\\gamma}$ can be improved by modelling all \nthe radiative transfer effects in WD atmospheres and fitting the model spectra to real observations.\n\nMagnetic fields stronger than that of mWDs exist in NSs. Going back to the argument of how \nastrophysical objects, compared to laboratory experiments, benefit from longer coherence \nlengths (see Section I), in comparing mWDs with NSs, one finds that the latter are $\\sim 10^4$ \ntimes more efficient in converting photons to axions and vice-versa. A number of studies have expounded \non the subject of propagation of polarized radiation through the NS magnetosphere, where \nthey have considered IR\/Optical radiation \\citep{ShannonHeyl2006}, and thermal X-rays \n\\citep{LaiHo2003a,LaiHo2003b} produced at the surface of the NS. Unfortunately, no X-ray \npolarimetry observations have been conducted partly due to the very low flux in X-rays \nfrom these objects, and also because none of the high energy telescopes are equipped \nwith a polarimeter. X-ray polarimetry has been neglected for the last 30 years but it \nis hoped that some of the future space missions \\citep{Mulerietal2010}, for example the Gravity and Extreme \nMagnetism Small Explorer (GEMS) \\citep{Jahoda2010}, will fill this \nvoid in X-ray astronomy. In any case, as discussed by \\citep{RaffeltStodolsky1988,\nLaiHeyl2006,Cheloucheetal2009}, NSs are excellent laboratories for the detection of \nany light, weakly coupled pseudoscalar particle. \n\\subsection{Outlook}\nThe ADMX\\footnote{http:\/\/www.phys.washington.edu\/groups\/admx\/experiment.html} \nproject, that employs a microwave cavity to search for cold dark matter axions, \nwill begin its phase II of testing in the year 2012. With the new \nupgrades the ADMX project will be able to exclude $g_{a\\gamma\\gamma}$ up to the DFSZ line in \nthe same mass range as before. Although outside of the range of axion masses probed \nin this study, the modified CAST experiment has been able to exclude axions with \n$g_{a\\gamma\\gamma} \\gtrsim 2.2\\times10^{-10}\\mbox{ GeV}^{-1}$ for $m_a \\lesssim 0.4$ \neV, becoming the first experiment to ever cross the KSVZ line \\citep{Ariketal2009,Cast2011}. \nThe currently running CAST experiment in its phase-II will \nbe able to exclude axions with $m_a\\lesssim 1.15$ eV with unprecedented sensitiviy in \nthis mass range. An improved version of the \\textit{light shining through wall} (LSW) experiment \n(\\citep{VanBibberetal1987}, also see \\citep{RedondoRingwald2011} for a recent review on \nsuch experiments) using Fabry-Perot optical cavities to resonantly enhance photon-axion conversion \nhas been proposed \\citep{HoogeveenZiegenhagen1991,Muelleretal2009,Sikivieetal2007}. The projected limit in sensitivity to \n$g_{a\\gamma\\gamma} \\gtrsim 2.0\\times 10^{-11}\\mbox{ GeV}^{-1}$ typically for \naxion masses $m_a \\lesssim 10^{-4}$ eV achieved using 12 Tevatron superconducting \ndipoles appears quite promising. Further improvements in experiment design and optimization techniques \nyielding increased sensitivity to even smaller coupling strengths have also been suggested by many workers \nin the field, for example the use of the dipole magnets, each providing a field strength of 5 T, \nfrom the Hadron Electron Ring Accelerator (HERA) at DESY in \nHamburg in a 20+20 configuration can potentially exclude \n$g_{a\\gamma\\gamma} \\gtrsim 10^{-11}\\text{ GeV}^{-1}$ for $m_a < 10^{-4}$ eV \\citep{Ringwald2003,Ariasetal2010}. \nAnother proposed line of investigation to search for axion like particles (ALPs) is the use of resonant microwave cavities which are \nmuch similar in design to the optical LSW experiments discussed above \\citep{Hoogeveen1992,Caspersetal2009,JaeckelRingwald2008}. This \nmethod has already been employed to search for hidden sector photons \\citep{Poveyetal2010} and can prove to be a powerful \ntool in the case of axions.\n\nFinally, the simplistic model assumed for the mWD atmosphere only yields an absolute upper bound \non $g_{a\\gamma\\gamma}$. A much tighter constraint can be obtained by adopting a more realistic \natmospheric model and solving the equations of radiative transfer with the photon-axion oscillations \nincluded. Such an analysis is outside the scope of this study, but it is hoped that the novel \nmethod discussed in this work will prove to be extremely useful in better constraining the \nproperties of any ALP.\n\\begin{acknowledgements}\n   We would like to thank the referee for significantly improving the \n   quality of this work. R.G. acknowledges support by the NSERC CGS-D3 scholarship. \n   The Natural Sciences and Engineering Research Council of Canada,\n   Canadian Foundation for Innovation and the British Columbia\n   Knowledge Development Fund supported this work. Correspondence and\n   requests for materials should be addressed to\n   J.S.H. (heyl@phas.ubc.ca). This research has made use of NASA's\n   Astrophysics Data System Bibliographic Services\n\\end{acknowledgements}\n\\bibliographystyle{prsty}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}