diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpdpf" "b/data_all_eng_slimpj/shuffled/split2/finalzzpdpf" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpdpf" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:1}\nThe last decade has seen an explosion of research in topologically-nontrivial states of matter, following the remarkable discovery of topological insulators (TI)~\\cite{Kane05,Zhang06,Balents07,Kane07,Molenkamp07,Hasan09,Kane10,Qi10}. \nOne normally associates topologically nontrivial properties with insulators, the spectral gap providing the rigidity and insensitivity to fluctuations, which are \ncharacteristic of these states of matter. \nOne of the most recent notable developments in this field, however, has been the realization that even gapless metallic and semimetallic states \nmay be topologically nontrivial in much the same sense as gapped insulators~\\cite{Wan11,Ran11,Burkov11,Xu11}.\nThese recent developments have partly been anticipated in the earlier pioneering work of Volovik~\\cite{Volovik}, promoting topological classification \nof all possible fermionic ground states, which shed new light on such wide-ranging phenomena as the robustness of the Fermi liquid and the hierarchy problem in particle physics. \n\nThe recent work focussed on {\\em Weyl semimetals}~\\cite{Wan11,Ran11,Burkov11,Xu11}, which, in Volovik's classification, belong to the Fermi point \nuniversality class of fermionic vacua.\nIn the condensed matter context a Weyl point or node is a point of contact between two nondegenerate bands in the first Brillouin zone (BZ). Such electronic structure features have in \nfact been noticed and studied since the earliest days of the theory of solids~\\cite{Herring,Abrikosov,Landau}, but only recently have their topological properties come into focus. \nTwo important ingredients in the appearance of Weyl node features in the electronic structure are broken time reversal (TR) or inversion (I) symmetries and \ndimensionality of space. The broken symmetry requirement is a consequence of the well-known Kramers theorem. If both TR and I symmetries are present, the band eigenstates \nin a solid must be (at least) doubly degenerate at every value of the crystal momentum in the first BZ. \nContact can then occur only between {\\em pairs} of bands, which is in general impossible without fine-tuning.\nThe contact between two nondegenerate bands is, however, possible generically in three spatial dimensions (3D). \nIndeed, near the point of contact, the momentum-space Hamiltonian must have the following form, dictated by the so-called Atiyah-Bott-Shapiro \nconstruction~\\cite{Horava}\n\\begin{equation}\n\\label{eq:1}\nH = \\pm v_F \\bsigma \\cdot {{\\bf{k}}}, \n\\end{equation}\nwhere the triplet of Pauli matrices $\\bsigma$ describes the pair of touching bands, the sign in front refers to two possible {\\em chiralities} of the band contact point, and we have subsumed any possible spatial anisotropies into the definition of the crystal momentum ${{\\bf{k}}}$. We will use $\\hbar = c =1$ units throughout this paper, except in some of the final results. \nThe point of band degeneracy occurs when all three components of the crystal momentum vanish. \nThis is why three-dimensionality is important: we must have three momentum components available as ``tuning parameters\" to create a point of degeneracy between the two bands. \nA naively analogous degeneracy point in 2D graphene in fact does not exist, as was pointed out in the seminal work of Kane and Mele~\\cite{Kane05}, \nwhich started the field of TI. \nIt only looks like a degeneracy point due the smallness of the spin-orbit (SO) interactions in graphene, which always open up a small gap and destroy the degeneracy. \n\nThe presence of Weyl degeneracy nodes is, in principle, a common feature of the electronic structure of all 3D magnetic or noncentrosymmetric materials. \nIn most cases, however, their presence is of no consequence, since they will generically occur far away from the Fermi energy~\\cite{Doron13}. \n{\\em Weyl semimetal} refers to a situation in which the Fermi level exactly coincides with the Weyl nodes and no other states are present at the Fermi energy. \nThis situation is not generic and occurs only under special, but not entirely uncommon circumstances.\n\nTo identify these, it is convenient to start from the situation in which both TR and I symmetries are present.\nTR symmetry requires that if a Weyl node is present at momentum ${{\\bf{k}}}$, another node with the same chirality must be present at $-{{\\bf{k}}}$. \nI symmetry, on the other hand, requires that if a Weyl node is present at ${{\\bf{k}}}$, a node of opposite chirality must be present at $-{{\\bf{k}}}$. \nThis implies that in the presence of both TR and I, there may exist a pair of opposite-chirality Weyl nodes at the same TR and I-invariant \ncrystal momentum. For concreteness, we will take this momentum to be at the BZ centre, i.e. at the $\\Gamma$ point. \nThe minimal ${{\\bf{k}}} \\cdot {\\bf{p}}$ Hamiltonian, describing this situation, has the form\n\\begin{equation}\n\\label{eq:2}\nH = v_F \\tau^z \\bsigma \\cdot {{\\bf{k}}} + \\Delta \\tau^x, \n\\end{equation}\nwhere $\\btau$ is an additional set of Pauli matrices that describes the two nodes of opposite chiralities, i.e. $H$ has the form of the Hamiltonian \nof a 3D four-component Dirac fermion with ``mass\" $\\Delta$. \nThe mass term $\\Delta$ annihilates the two Weyl nodes, unless it vanishes by symmetry or is fine-tuned to zero. \nTo realize the former situation, the states at the $\\Gamma$ point must either form a four-dimensional irreducible representation of the space group of the crystal, \nas proposed to occur in $\\beta$-cristobalite BiO$_2$~\\cite{Kane12} and, with a few modifications (Dirac Hamiltonian is replaced by Luttinger Hamiltonian in this case), in some of the pyrochlore iridate materials~\\cite{Krempa12,Balents13,Krempa14}. Or they must belong to two distinct two-dimensional representations, so that pairwise crossing, \nprotected by crystal symmetry,\nis possible, as realized in the recently \ndiscovered Dirac semimetal materials $\\textrm{Na}_3 \\textrm{Bi}$ and \n$\\textrm{Cd}_2 \\textrm{As}_3$~\\cite{Fang12,Fang13,Cava13,Shen13,Hasan13}. \n\nThe latter situation, i.e. with $\\Delta$ fine-tuned to zero, occurs at the transition point between an inversion-symmetric 3D TI and a normal insulator (NI)~\\cite{Murakami07,Burkov11,Ando11,Ando14}. In the present review we will focus exclusively on this case. \nOur results, however, do not depend on this choice and are applicable to all realizations of Dirac and Weyl semimetals. \nAs mentioned above, Weyl semimetal may be obtained from Dirac semimetal by breaking either TR or I. We will focus on the TR-breaking case here, \nsince it produces the simplest possible kind of Weyl semimetal, with only a single pair of Weyl nodes. \nGeneralization to many pairs is in most cases straightforward, but it does complicate the technical details of course. \nFormally, this type of Weyl semimetal is obtained by adding a Zeeman spin-splitting term to the Dirac Hamiltonian Eq.~\\ref{eq:2}\n\\begin{equation}\n\\label{eq:3}\nH = v_F \\tau^z \\bsigma \\cdot {{\\bf{k}}} + \\Delta \\tau^x + b \\sigma^z, \n\\end{equation}\nwhere the spin-splitting term may arise either from an external magnetic field, or from an intrinsic magnetic ordering, and we have chosen the \n$z$-direction to be the magnetization direction. \nTo analyze this Hamiltonian, it is convenient to perform the following canonical (i.e. commutation relation preseving) transformation of the $\\bsigma$ and $\\btau$ operators\n\\begin{equation}\n\\label{eq:4}\n\\sigma^{\\pm} \\rightarrow \\tau^z \\sigma^{\\pm},\\,\\, \\tau^{\\pm} \\rightarrow \\sigma^z \\tau^{\\pm}. \n\\end{equation}\nThe Hamiltonian Eq.~\\ref{eq:3} then transforms to\n\\begin{equation}\n\\label{eq:5}\nH = v_F (\\sigma^x k_x + \\sigma^y k_y) + (b + v_F k_z \\tau^z + \\Delta \\tau^x) \\sigma^z, \n\\end{equation}\nDiagonalizing the $v_F k_z \\tau^z + \\Delta \\tau^x$ block of the Hamiltonian we obtain\n\\begin{equation}\n\\label{eq:6}\nH_{\\pm} = v_F (\\sigma^x k_x + \\sigma^y k_y) + m_{\\pm}(k_z) \\sigma^z, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:7}\nm_{\\pm}(k_z) = b \\pm \\sqrt{v_F^2 k_z^2 + \\Delta^2}. \n\\end{equation}\nEq.~\\ref{eq:6} looks like the Hamiltonian of a pair of 2D Dirac fermions with ``masses\" $m_{\\pm}(k_z)$, which depend \non the $z$-component of the crystal momentum as a parameter. \nThe mass $m_+(k_z)$ is always positive, but $m_-(k_z)$ may vanish and change sign if $b > \\Delta$. \nIf this is the case, $m_-(k_z)$ vanishes at $k_z^{\\pm} = \\pm k_0$, where \n\\begin{equation}\n\\label{eq:8}\nk_0 = \\frac{1}{v_F} \\sqrt{b^2 - \\Delta^2}. \n\\end{equation}\nThese points along the $z$-axis in momentum space, where the ``mass\" $m_-(k_z)$ vanishes, are the locations of the Weyl nodes. \nThe two bands, which are the eigenstates of the $H_-$ block of the Hamiltonian, are degenerate at those points. \nSince massive 2D Dirac fermions are associated with half-quantized Hall conductivity~\\cite{Fisher94}\n\\begin{equation}\n\\label{eq:9}\n\\sigma_{xy}^{2D} = \\frac{e^2}{2 h} \\textrm{sign}(m), \n\\end{equation}\nit follows that the Weyl semimetal at $b > \\Delta$ has a Hall conductivity given by\n\\begin{equation}\n\\label{eq:10}\n\\sigma_{xy} = \\frac{e^2}{h} 2 k_0, \n\\end{equation}\ni.e. is equal to the distance between the Weyl nodes in units of $e^2\/h$~\\cite{Klinkhamer}.\n\nIt is easy to show that this Hall conductivity may be associated with chiral edge states. \nIndeed, suppose the sample is finite in the $y$-direction and we will take the sample to\noccupy the $y < 0$ half-space. \nReplacing $k_y \\rightarrow -i \\partial\/ \\partial y$, the $H_-$ block of the Hamiltonian becomes\n\\begin{equation}\n\\label{eq:11}\nH_- = - i v_F \\frac{\\partial}{\\partial y} \\sigma^y + v_F \\sigma^x k_x + m_-(k_z,y) \\sigma^z. \n\\end{equation}\nLet us first set $k_x = 0$ and look for a zero-energy solution of the Schr\\\"odinger equation (SE)\n\\begin{equation}\n\\label{eq:12}\nH_- \\Psi(k_z,y) = 0, \n\\end{equation}\nlocalized at the sample boundary $y = 0$. \nWe look for a solution in the form \n\\begin{equation}\n\\label{eq:13}\n\\Psi(k_z,y) = i \\sigma^y e^{f(k_z,y)} \\phi, \n\\end{equation}\nwhere $f(k_z,y)$ is a scalar function, while $\\phi$ is a two-component spinor. \nPlugging this into the SE Eq.~\\ref{eq:12}, we obtain\n\\begin{equation}\n\\label{eq:14}\n\\left[v_F \\frac{\\partial f(k_z,y)}{\\partial y} + m_-(k_z,y) \\sigma^x \\right] \\phi = 0. \n\\end{equation}\nThe solution of Eq.~\\ref{eq:14}, satisfying our requirements is \n\\begin{equation}\n\\label{eq:15}\nf(k_z,y) = \\frac{1}{v_F} \\int_0^y m_-(k_z, y') d y', \n\\end{equation}\nand $\\phi = | \\sigma^x = -1 \\rangle$, i.e. the eigenstate of $\\sigma^x$ with eigenvalue $-1$. \nThus we finally obtain the following result for the solution of Eq.~\\ref{eq:14}, localized at the \nsample boundary\n\\begin{equation}\n\\label{eq:16}\n\\Psi(k_z,y) = e^{\\frac{1}{v_F} \\int_0^y m_-(k_z,y') dy'} | \\sigma^x = 1 \\rangle. \n\\end{equation}\nThis solution exists as a localized edge state as long as $m_-(k_z, y \\rightarrow - \\infty) \\geq 0$, i.e. \nas long as $-k_0 \\leq k_z \\leq k_0$. \nFor this reason it is called a Fermi arc~\\cite{Wan11}. \nIt is straightforward to see that the dispersion of this state in the $x$-direction is given by $\\epsilon = v_F k_x$, i.e.\nit is chiral. \n\nAt this point we will wrap up the introductory part of this review and move on to a more detailed account of the transport \nproperties of Weyl semimetals and metals, based on a more realistic microscopic model of Weyl semimetal. \nThe rest of the paper is organized as follows. \nIn Section~\\ref{sec:2} we introduce a realistic microscopic model of a Weyl semimetal, based on a magnetically-doped\nTI-NI multilayer heterostructure. \nIn Section~\\ref{sec:3} we discuss basic electromagnetic response properties of this model Weyl semimetal and introduce \ntwo distinct components of the response, which are directly related to the nontrivial topology of Weyl nodes: the Anomalous Hall Effect (AHE)~\\cite{AHE1,AHE2} \nand the Chiral Magnetic Effect (CME). \nIn Section~\\ref{sec:4} we extend the theory of electromagnetic response of Weyl metals to diffusive transport regime. \nWe demonstrate that magnetic Weyl metals are distinguished from ordinary ferromagnetic metals by a lack of impurity-scattering \nand Fermi-surface contributions to their anomalous Hall conductivity. Instead their anomalous Hall conductivity is determined \nby the distance between the Weyl nodes in momentum space and nothing else. \nWe also show that CME manifests in the diffusive regime as a new kind of weak-field magnetoresistance, which is \nnegative and quadratic in the field. This novel magnetoresistance is expected to occur in all types of Weyl metals\nand may be regarded as their smoking-gun transport characteristic. \nWe conclude in Section~\\ref{sec:5} with a brief overview of the main results.\n\n\\section{Heterostructure model of Weyl semimetal}\n\\label{sec:2}\nAs discussed in the Introduction, perhaps the most straightforward (at least theoretically) way to realize a Weyl semimetal phase is to \nbreak TR symmetry in a material, which in its nonmagnetic state is naturally poised near the phase transition between a TI and NI. \nOne way to achieve this is to engineer a composite material, made of alternating thin layers of TI and NI, as shown in Fig.~\\ref{fig:1}. \nThis system may be regarded as a ``hydrogen atom\" of Weyl semimetals: the most elementary yet realistic system, the description of which is simple enough that purely analytical \ntheory of many phenomena is possible, as will be seen below. \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=15cm]{multilayer}\n\\end{center}\n\\vspace{-2cm}\n\\caption{Cartoon of the heterostructure model of Weyl semimetal. Magnetized impurities are shown by arrows.}\n\\label{fig:1}\n\\end{figure}\n\nThis system may be described as a chain of 2D Dirac surface states of the TI layers, which are coupled by tunnelling matrix elements \n$\\Delta_S$ for a pair of surface states, belonging to the same TI layer, and $\\Delta_D$ for a pair belonging to nearest-neighbor TI layers. \nWe will assume, for concreteness, that $\\Delta_{S,D} \\geq 0$. \nThe corresponding Hamiltonian has the form\n\\begin{equation}\n\\label{eq:17}\nH = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot {{\\bf{k}}}_{\\perp} \\delta_{i,j} + \\Delta_S \\tau^x \\delta_{i,j} + \\frac{1}{2}\\ \\Delta_D \\left(\\tau^+ \\delta_{j, i+1} + \\tau^- \\delta_{j, i-1} \\right), \n\\end{equation}\nwhere $i,j$ label the unit cells of the superlattice in the growth ($z$) direction, ${{\\bf{k}}}_{\\perp} = (k_x, k_y)$ are the crystal momentum components, \ntransverse to the growth direction, Pauli matrices $\\btau$ act on the {\\em which surface} pseudospin degree of freedom, while $\\bsigma$ act in the spin degree of freedom. \nDiagonalizing the growth direction hopping part of the Hamiltonian by Fourier transform and performing the canonical transformation of Eq.~\\ref{eq:4}, we obtain \n\\begin{equation}\n\\label{eq:18}\nH = v_F (\\hat z \\times \\bsigma) \\cdot {{\\bf{k}}} + \\hat \\Delta(k_z) \\sigma^z, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:19}\n\\hat \\Delta(k_z) = \\Delta_S \\tau^x + \\frac{\\Delta_D}{2} \\left(\\tau^+ e^{i k_z d} + \\textrm{h.c.} \\right), \n\\end{equation}\n$d$ being the period of the superlattice in the growth direction.\n$\\hat \\Delta(k_z)$ may now be diagonalized separately, as it commutes with the Hamiltonian. \nIts eigenvalues are given by $\\pm \\Delta(k_z)$, where \n\\begin{equation}\n\\label{eq:20}\n\\Delta(k_z) = \\sqrt{\\Delta_S^2 + \\Delta_D^2 + 2 \\Delta_S \\Delta_D \\cos(k_z d)}. \n\\end{equation}\nThe corresponding two-component spinor wavefunctions are\n\\begin{equation}\n\\label{eq:21}\n|u^t(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(1, t \\frac{\\Delta_S + \\Delta_D e^{-i k_z d}}{\\Delta(k_z)} \\right), \n\\end{equation}\nwhere $t = \\pm$ labels the two eigenvalues. \nThe spin block of the Hamiltonian may now be diagonalized as well, giving rise to the following eigenvalues\n\\begin{equation}\n\\label{eq:22}\n\\epsilon_{s t}({{\\bf{k}}}) = s \\epsilon_t({{\\bf{k}}}) = s \\sqrt{v_F^2(k_x^2 + k_y^2) + m^2_t(k_z)}, \n\\end{equation}\nwhere $s = \\pm$ and \n\\begin{equation}\n\\label{eq:23}\nm_t(k_z) = t \\Delta(k_z). \n\\end{equation}\nThe double degeneracy of the band eigenvalues at every ${{\\bf{k}}}$ in Eq.~\\ref{eq:22} is the Kramers degeneracy, arising due to the \npresence of both TR and I symmetries. \nThe corresponding eigenvectors are given by\n\\begin{equation}\n\\label{eq:24}\n|v^{s t}({{\\bf{k}}}) \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m_t(k_z)}{\\epsilon_t({{\\bf{k}}})}}, - i s e^{i \\phi} \\sqrt{1 - s \\frac{m_t(k_z)}{\\epsilon_t({{\\bf{k}}})}} \\right), \n\\end{equation}\nwhere $e^{i \\phi} = \\frac{k_x + i k_y}{\\sqrt{k_x^2 + k_y^2}}$. \nThe full four-component eigenvectors of the Hamiltonian Eq.~\\ref{eq:18} may be represented as a tensor product $|u^t(k_z) \\rangle$ and \n$|v^{s t}({{\\bf{k}}}) \\rangle$:\n\\begin{equation}\n\\label{eq:25}\n|z^{s t}({{\\bf{k}}}) \\rangle = |v^{s t}({{\\bf{k}}}) \\rangle \\otimes |u^t(k_z) \\rangle. \n\\end{equation}\n\nThe heterostructure, described by Eq.~\\ref{eq:17}, can exist in two distinct phases: strong 3D TI and an NI. \nThis can be easily seen by computing the $Z_2$ index, using the method of Fu and Kane~\\cite{Fu07}. \nNamely, we compute the eigenvalues of the parity operator $\\tau^x$ (or $\\tau^x \\sigma^z$ after the canonical transformation of Eq.~\\ref{eq:4}) \nat two TR-invariant momenta in the first BZ: $\\bGamma_1=(0,0,0)$ and $\\bGamma_2=(0,0,\\pi\/d)$. \nTaking the product of the parity eigenvalues over all the occupied bands and over $\\bGamma_{1,2}$, the $Z_2$ index is found to be\n\\begin{equation}\n\\label{eq:26}\n(-1)^{\\nu} = \\textrm{sign}(\\Delta_S - \\Delta_D). \n\\end{equation}\nThus the multilayer is a strong 3D TI when $\\Delta_D > \\Delta_S$ and an NI otherwise. \nThe point $\\Delta_S = \\Delta_D$ marks the transition point between the 3D TI and NI. \nAt this point, the gap at $\\bGamma_2$ closes and the heterostructure is a Dirac semimetal. \n\nTo obtain a Weyl semimetal, we break TR symmetry in the heterostructure, assuming that it is \ntuned close to the TI-NI transition point and the band gap $|\\Delta_S - \\Delta_D|$ is not large. \nThe TR breaking may be accomplished by doping the heterostructure with magnetic impurities. \nAt sufficient concentration, the impurities will form a ferromagnetic (FM) state, and we will assume that \nthe magnetization points along the growth direction of the heterostructure (distinct, non-Weyl nodal states are \nobtained if the magnetization is in the $xy$-plane~\\cite{Burkov11-2}). \nWe will describe the FM state of the heterostructure by adding a term $b \\sigma^z \\delta_{i,j}$ to the Hamiltonian Eq.~\\ref{eq:17}. \nThis splits the degeneracy of the $t = \\pm$ Kramers doublet, modifying Eq.~\\ref{eq:23} as\n\\begin{equation}\n\\label{eq:27}\nm_t(k_z) = b + t \\Delta(k_z). \n\\end{equation}\nTaking the spin splitting $b$ to be always positive, the ``mass\" $m_+(k_z)$ is then always positive, while $m_-(k_z)$ may vanish \nand change sign as a function of $k_z$, if the spin splitting is sufficiently large. \nThis happens when $b \\geq b_{c1} = |\\Delta_S - \\Delta_D|$. \nThe Weyl node locations along the $k_x = k_y = 0$ line in the crystal momentum space are given by the solutions of the equation\n\\begin{equation}\n\\label{eq:28}\nm_-(k_z) = b - \\Delta(k_z) = 0, \n\\end{equation}\nwhich gives $k_{z}^{\\pm} = \\pi\/d \\pm k_0$, where \n\\begin{equation}\n\\label{eq:29}\nk_0 = \\frac{1}{d} \\arccos\\left(\\frac{\\Delta_S^2 + \\Delta_D^2 - b^2}{2 \\Delta_S \\Delta_D} \\right). \n\\end{equation}\nWhen the spin splitting reaches the upper critical value $b_{c2} = \\Delta_S + \\Delta_D$, $k_0 = \\pi\/d$ and the \nWeyl nodes are annihilated again at the BZ edge. The resulting state is a 3D quantum anomalous Hall insulator \nwith a quantized Hall conductivity~\\cite{Halperin92}\n\\begin{equation}\n\\label{eq:30} \n\\sigma_{xy} = \\frac{e^2}{h d}. \n\\end{equation}\n\n\\section{Electromagnetic response of Weyl semimetals}\n\\label{sec:3}\n\\subsection{General remarks}\n\\label{sec:3.1}\nTopologically-nontrivial states of matter, such as 3D TI, and Weyl semimetals, have a distinguishing spectroscopic \nfeature: the presence of metallic edge states. \nThese surface metallic states are unusual and may only exist on the surface of a bulk 3D topological phase. \nAs discussed in the Introduction, in the Weyl semimetal state, of interest to us here, the Fermi surface of this metal is not a closed 2D curve, as it must \nbe in any regular 2D metal, but instead forms an open Fermi arc, with the end points at the locations of the bulk Weyl nodes, \nprojected onto the surface BZ~\\cite{Wan11,Potter14}. \n\nHowever, apart from such spectroscopic distinguishing features, topological phases often have unusual electromagnetic response characteristics, \nwhich are always of particular interest, as these are the manifestations of the unique quantum physics of these phases on macroscopic scales. \nIn 3D TI, this unusual electromagnetic response may be described by the so-called $\\theta$-term in the action of the electromagnetic field~\\cite{Qi10}\n\\begin{equation}\n\\label{eq:31}\nS_{\\theta} = \\frac{e^2 \\theta}{32 \\pi^2} \\int d t \\,\\, d^3 r \\epsilon^{\\mu \\nu \\alpha \\beta} F_{\\mu \\nu} F_{\\alpha \\beta} = \\frac{e^2 \\theta}{4 \\pi^2} \\int d t \\,\\, d^3 r {\\bf{E}} \\cdot {\\bf{B}}. \n\\end{equation}\nThe parameter $\\theta$ is equal to $\\pi$ in 3D TI, the only nonzero value, compatible with TR symmetry~\\cite{Qi10}. \nThe $\\theta$-term in Eq.~\\ref{eq:31}, however, is a full derivative, and thus has no effect on Maxwell's equations in the bulk of the TI. \nIts only real effect is to generate the half-quantized AHE on the sample surfaces in the presence of surface magnetization (e.g. due to magnetic \nimpurities). \n\nWhen TR is broken in the bulk, as it is in the Weyl semimetal state, a $\\theta$-term with non-constant $\\theta$ is now allowed by symmetry\n\\begin{equation}\n\\label{eq:32}\nS_{\\theta} = \\frac{e^2}{32 \\pi^2} \\int d t \\,\\, d^3 r \\theta(\\br) \\epsilon^{\\mu \\nu \\alpha \\beta} F_{\\mu \\nu} F_{\\alpha \\beta}.\n\\end{equation}\nThe only functional form, compatible with translational symmetry, is linear, i.e.\n\\begin{equation}\n\\label{eq:33}\n\\theta(\\br) = 2 {\\bf{b}} \\cdot \\br, \n\\end{equation}\nwhere ${\\bf{b}}$ is a vector, which is odd under TR and even under I and coincides, as will be seen explicitly below, with \nthe bulk spin splitting. \nIf, in addition to TR, I symmetry is violated as well, the ``axion field\" $\\theta$ may also acquire a linear time dependence\n\\begin{equation}\n\\label{eq:34}\n\\theta(\\br, t) = 2 {\\bf{b}} \\cdot \\br - 2 b_0 t, \n\\end{equation}\nwhere $b_0$ is the energy difference between the Weyl nodes, allowed if $I$ is violated. \nUnlike the $\\theta$-term in 3D TI, Eq.~\\ref{eq:32} does modify Maxwell's equations in the bulk of the Weyl semimetal and thus \nhas observable consequences, namely the already mentioned AHE and the Chiral Magnetic Effect (CME)~\\cite{Kharzeev}, both of which will be discussed \nin detail below. \n\nThe action of Eq.~\\ref{eq:32} may be described as being a consequence of chiral anomaly, an important concept in relativistic field theory~\\cite{Adler69,Jackiw69,Nielsen83},\nwhich has recently found its way into condensed matter physics and has been realized to play an important role in the theory of topologically-nontrivial \nstates of matter~\\cite{Ryu12,Nagaosa13,Wen13}. \nTo gain a basic understanding of chiral anomaly, it is useful to go back to the generic low-energy model of Weyl semimetal, given by Eq.~\\ref{eq:3}. \nFor simplicity, we will subsume the Fermi velocity $v_F$ in Eq.~\\ref{eq:3} into the definition of momentum and set the ``mass term\" $\\Delta = 0$. \nThen we obtain\n\\begin{equation}\n\\label{eq:35}\nH = \\tau^z \\bsigma \\cdot {{\\bf{k}}} + b_0 \\tau^z + {\\bf{b}} \\cdot \\bsigma, \n\\end{equation}\nwhere we have included an energy difference between the Weyl nodes, described by the $b_0 \\tau^z $ term, which is allowed when the I symmetry is violated, \nand oriented the TR-breaking vector ${\\bf{b}}$ along an arbitrary direction. \nIt is convenient to represent the system in terms of an imaginary time action, including possible coupling of the electrons to an electromagnetic field\n\\begin{equation}\n\\label{eq:36}\nS = \\int d \\tau d^3 r \\,\\,\\psi^{\\dag} \\left[ \\partial_{\\tau} + i e A_0 + b_0 \\tau^z + \\tau^z \\bsigma \\cdot \\left( -i \\bnabla + e {\\bf{A}} + {\\bf{b}} \\tau^z \\right) \\right] \\psi^{\\vphantom \\dag},\n\\end{equation}\nwhere $A_{\\mu} = (A_0, {\\bf{A}})$ is the electromagnetic gauge potential and $\\psi^{\\dag}, \\psi^{\\vphantom \\dag}$ are the 4-component spinor Grassman fields. We have suppressed all \nexplicit spinor indices in the Grassmann fields for brevity. \nWe now observe that in addition to the charge conservation symmetry, the imaginary time action Eq.~\\ref{eq:36} possesses an extra chiral symmetry\n\\begin{equation} \n\\label{eq:37}\n\\psi \\rightarrow e^{-i \\tau^z \\theta\/2} \\psi, \n\\end{equation} \nwhich expresses an apparent separate conservation of the number of fermions of left and right chirality. This suggests that the terms $b_0 \\tau^z$ and ${\\bf{b}} \\cdot \\bsigma$ in \nEq.~\\ref{eq:36} could be eliminated by a gauge transformation\n\\begin{equation} \n\\label{eq:38}\n\\psi \\rightarrow e^{- i \\tau^z \\theta(\\br, \\tau)\/2} \\psi,\\,\\, \\psi^{\\dag} \\rightarrow \\psi^{\\dag} e^{i \\tau^z \\theta(\\br, \\tau)\/2},\n\\end{equation}\nwhere $\\theta(\\br, \\tau) = 2 {\\bf{b}} \\cdot \\br - 2 i b_0 \\tau$ and one should keep in mind that $\\psi$ and $\\psi^{\\dag}$ are not complex conjugates of each other, but are independent variables in the fermion path integral. \nThe imaginary time action then becomes\n\\begin{equation}\n\\label{eq:39}\nS = \\int d \\tau d \\br \\,\\, \\psi^{\\dag} \\left[ \\partial_{\\tau} + i e A_0 \n + \\tau^z \\bsigma \\cdot \\left( -i \\bnabla + e {\\bf{A}} \\right) \\right] \\psi^{\\vphantom \\dag}, \n\\end{equation}\nwhich describes two Weyl nodes of opposite chirality, existing at the same point in momentum space and in energy. \nThis argument then leads one to the conclusion that the system of Weyl nodes, separated in energy and momentum, is equivalent \nto the system of two degenerate Weyl nodes and thus does not possess any special transport properties, which is incorrect. \nWhat is missing in the above argument is exactly the chiral anomaly: while the imaginary time action Eq.~\\ref{eq:36} does indeed \npossess the chiral symmetry, the gauge transformation of Eq.~\\ref{eq:38} changes not only the action itself, but also the measure of the path integral. \nThis change of the path integral measure is what gives rise to the $\\theta$-term Eq.~\\ref{eq:32}. \nThis may be shown explicitly using the Fujikawa's method~\\cite{Fujikawa,Zyuzin12}. \nWe refer the reader to Ref.~\\cite{Zyuzin12} for technical details. \n\nTo see the physical consequences of the $\\theta$-term, we integrate Eq.~\\ref{eq:32} by parts and eliminate a total derivative term. \nThis leads to the following action, which resembles the Chern-Simons term in $2+1$-dimensions\n\\begin{equation}\n\\label{eq:40}\nS_{\\theta} = - \\frac{e^2}{8 \\pi^2} \\int dt d^3 r \\,\\, \\partial_{\\mu} \\theta \\epsilon^{\\mu \\nu \\alpha \\beta} A_{\\nu} \\partial_{\\alpha} A_{\\beta}. \n\\end{equation}\nVarying Eq.~\\ref{eq:40} with respect to the vector potential, we obtain the following expression for the current\n\\begin{equation}\n\\label{eq:41}\nj_{\\nu} = \\frac{e^2}{2 \\pi^2} b_{\\mu} \\epsilon^{\\mu \\nu \\alpha \\beta} \\partial_{\\alpha} A_{\\beta}, \\,\\,\\, \\mu = 1, 2, 3, \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:42}\nj_{\\nu} = - \\frac{e^2}{2 \\pi^2} b_0 \\epsilon^{0 \\nu \\alpha \\beta} \\partial_{\\alpha} A_{\\beta}. \n\\end{equation}\nEq.~\\ref{eq:41} clearly represents the AHE,\nwhile Eq.~\\ref{eq:42} represents another effect, related to chiral anomaly in Weyl semimetals, the Chiral Magnetic Effect~\\cite{Kharzeev}, whose \nphysical meaning is somewhat subtle and will be discussed in detail later. \n\nHowever, the picture of chiral anomaly and related transport phenomena, presented above, while simple and appealing, is not fully satisfactory, for the following \nreasons. \nChiral anomaly is a sharply-defined concept in the context of relativistic field theory, where \nmassless fermions in unbounded momentum space possess exact chiral symmetry Eq.~\\ref{eq:37}, which may be violated by the anomaly. \nIn the condensed matter context, however, the situation is less clear. Even though chiral symmetry may be formally defined in a low-energy model of a Weyl semimetal \nEq.~\\ref{eq:35}, in which the band \ndispersion is approximated as being exactly linear and unbounded, no real microscopic model of Weyl semimetal actually possesses such an exact symmetry. \nIt may only appear as an approximate low-energy symmetry. \nSince the chiral symmetry is not present to begin with, it is then unclear to what extent is it meaningful to speak of its violation by chiral anomaly. \nThe role of impurity scattering, present in any real condensed matter system, and important even conceptually in any discussion of transport phenomena; as well as the role of \nfinite electron or hole density, present when the Fermi energy does not exactly coincide with the location of the Weyl nodes, \nare also completely unclear in the above field-theory discussion. \nIn the remainder of this paper we will thus develop a fully microscopic theory of chiral anomaly and its manifestations, including the contribution of impurity scattering, and finite \ncharge carrier density, based on the TI-NI heterostructure model, described in Section~\\ref{sec:2}, which does not possess chiral symmetry, yet exhibits both the AHE and CME, described above. \n\n\\subsection{Microscopic theory: Anomalous Hall Effect}\n\\label{sec:3.2}\nWe will now go back to our microscopic model of a Weyl semimetal, described in Section~\\ref{sec:2}. \nWe want to explicitly derive the topological term in the action of the electromagnetic field by integrating out \nelectron variables, coupled to the field. \nWe then start from the imaginary time action of electrons in the Weyl semimetal, coupled to electromagnetic field\n\\begin{equation}\n\\label{eq:43}\nS = \\int d \\tau d^3 r \\left\\{ \\Psi^{\\dag}(\\br, \\tau) \\left[\\partial_{\\tau} - \\mu + i e A_0 (\\br, \\tau) + \\hat H\\right] \\Psi^{\\vphantom \\dag}(\\br, \\tau)\\right\\}, \n\\end{equation}\nwhere $A_0(\\br, \\tau)$ is the scalar potential and \n\\begin{equation}\n\\label{eq:44}\n\\hat H = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot \\left(- i \\bnabla + e {\\bf{A}} \\right) + \\hat \\Delta + b \\sigma^z,\n\\end{equation}\nis the Hamiltonian of noninteracting electrons in Weyl semimetal, minimally coupled to the vector potential ${\\bf{A}}$. \nWe will ignore the $z$-component of the vector potential as it will not play any role in what follows. \n\nUsing the results of Section~\\ref{sec:2} for the eigenvalues and eigenvectors of the Weyl semimetal Hamiltonian in the absence of the electromagnetic \nfield, we can now integrate out electron variables in Eq.~\\ref{eq:43} and obtain an effective action for the electromagnetic field, induced by coupling to the electrons. \nThis action will contain two distinct kinds of contributions. The first kind will contain terms, proportional to ${\\bf{E}}^2$ and ${\\bf{B}}^2$, \nwhere ${\\bf{E}}$ and ${\\bf{B}}$ are the electric and magnetic fields. These terms describe the ordinary electric and magnetic polarizability of the material, and we will not discuss this \npart of the response here. \nThe second kind contains the ``topological\" contribution, which has \nthe form of the ``3D Chern-Simons\" term of Eq.~\\ref{eq:40}. \nAdopting the Coulomb gauge $\\bnabla \\cdot {\\bf{A}} = 0$, we obtain\n\\begin{equation}\n\\label{eq:45}\nS = \\sum_{{\\bf{q}}, i\\Omega}\\epsilon^{z 0 \\alpha \\beta} \\Pi({\\bf{q}}, i\\Omega) A_0(-{\\bf{q}}, -i\\Omega) \\hat q_{\\alpha} A_{\\beta}({\\bf{q}}, i \\Omega),\n\\end{equation} \nwhere $\\hat q_{\\alpha} = q_{\\alpha}\/ q$ and summation over repeated indices is implicit. The $z$-direction in Eq.~\\ref{eq:45} is picked out by the \nmagnetization $b$. \nThe response function $\\Pi({\\bf{q}}, i\\Omega)$ is given by\n\\begin{eqnarray}\n\\label{eq:46}\n\\Pi({\\bf{q}}, i \\Omega)&=&\\frac{i e^2 v_F}{V} \\sum_{{{\\bf{k}}}} \\frac{n_F[\\xi_{s' t'}( {{\\bf{k}}})] - n_F[\\xi_{s t} ({{\\bf{k}}} + {\\bf{q}})]}{i \\Omega + \\xi_{s' t'}({{\\bf{k}}}) - \\xi_{s t} ({{\\bf{k}}} + {\\bf{q}})} \\nonumber \\\\\n&\\times&\\langle z^{s t}_{{{\\bf{k}}} + {\\bf{q}}} | z^{s' t'}_{{{\\bf{k}}}} \\rangle \\langle z^{s' t'}_{{{\\bf{k}}}}| \\bsigma \\cdot \\hat q | z^{s t}_{{{\\bf{k}}} + {\\bf{q}}} \\rangle,\n\\end{eqnarray}\nwhere $\\xi_{s t}({{\\bf{k}}}) = \\epsilon_{s t}({{\\bf{k}}}) - \\epsilon_F$ and summation over repeated band indices $s, t$ is implicit. \n\nTo evaluate $\\Pi({\\bf{q}}, i\\Omega)$ explicitly it is convenient to rotate coordinate axes so that ${\\bf{q}} = q \\hat x$ and assume that $\\epsilon_F > 0$. The $\\epsilon_F < 0$ result is \nevaluated analogously. \nAs seen from Eq.~\\ref{eq:46}, there exist two kinds of contributions to the response function $\\Pi({\\bf{q}}, i \\Omega)$: {\\em interband}, with $s \\neq s'$, \nand {\\em intraband}, with $s = s' = +$. \nBoth in general contribute and need to be taken into account. \nLet us first evaluate the interband contributions. \nIn this case we can set $i \\Omega = 0$ in the denominator of Eq.~\\ref{eq:46} from the start.\nExpanding to first order in $q$ and integrating over the transverse momentum components $k_{x,y}$, we obtain\n\\begin{eqnarray}\n\\label{eq:52}\n&&\\Pi^{inter}({\\bf{q}}, i\\Omega) = \\frac{e^2 q}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)] \\nonumber \\\\\n&\\times& \\left\\{1 - \\left[1 - \\frac{|m_t(k_z)|}{\\epsilon_F} \\right] \\Theta(\\epsilon_F - |m_t(k_z)|)\\right\\}, \n\\end{eqnarray}\nwhere $\\Theta(x)$ is the Heaviside step function. \nAs clear from Eq.~\\ref{eq:52}, $\\Pi^{inter}({\\bf{q}}, i\\Omega)$ consists of two distinct terms. \nThe first term inside the curly brackets in Eq.~\\ref{eq:52} is the contribution of the completely filled $s = -$ bands. The second term is the contribution \nof incompletely filled $s = +$ bands. \n\nTo understand the meaning of Eq.~\\ref{eq:52}, let us now evaluate the {\\em intraband} contribution to $\\Pi({\\bf{q}}, i \\Omega)$. \nIn this case we have $s = s' = +$ in Eq.~\\ref{eq:46}. It is then clear that, unlike in the case of the interband contribution, evaluated \nabove, the value of the intraband contribution depends on the order in which the limits of ${\\bf{q}} \\rightarrow 0$ and $\\Omega \\rightarrow 0$ \nare taken. If the limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$, then one is evaluating the DC limit of a transport quantity, i.e the optical Hall \nconductivity. In this case, the intraband contribution vanishes identically. \nOn the other hand, if the limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, then one is evaluating an equilibrium thermodynamic property, whose \nphysical meaning will become clear below. \nIn this case, the intraband contribution is not zero and is given by \n\\begin{eqnarray}\n\\label{eq:59}\n\\Pi^{intra}({\\bf{q}}, i\\Omega) = \\frac{i e^2 v_F}{V}\\sum_t \\sum_{{{\\bf{k}}}} \\left.\\frac{d n_F(x)}{d x} \\right|_{x=\\epsilon_t({{\\bf{k}}}) - \\epsilon_F} \\langle z^{+ t}_{{{\\bf{k}}} + {\\bf{q}}} | z^{+ t}_{{{\\bf{k}}}} \\rangle \\langle z^{+t}_{{{\\bf{k}}}} |\\bsigma \\cdot \\hat q| z^{+ t}_{{{\\bf{k}}} + {\\bf{q}}} \\rangle. \\nonumber \\\\\n\\end{eqnarray}\nThe derivative of the Fermi distribution function in Eq.~\\ref{eq:59} expresses the important fact that $\\Pi^{intra}({\\bf{q}}, i \\Omega)$ is associated \nentirely with the Fermi surface, unlike $\\Pi^{inter}({\\bf{q}}, i\\Omega)$, to which all filled states contribute, including states on the Fermi surface. \nExpanding to first order in $q$ as before and evaluating the integrals over $k_{x,y}$, we obtain\n\\begin{equation}\n\\label{eq:61}\n\\Pi^{intra}({\\bf{q}}, i\\Omega) = - \\frac{e^2 q}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\, \\, \\frac{m_t(k_z)}{\\epsilon_F} \\Theta(\\epsilon_F - |m_t(k_z)|).\n\\end{equation}\ni.e. the intraband contribution to $\\Pi({\\bf{q}}, i\\Omega)$ is equal to the second term in the square brackets in Eq.~\\ref{eq:52} in magnitude, \nbut opposite in sign. \nCombining the inter- and intraband contributions to $\\Pi({\\bf{q}}, i\\Omega)$ we thus obtain \n\\begin{equation}\n\\label{eq:62}\n\\Pi({\\bf{q}}, i\\Omega) = \\frac{e^2 q}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)] \\left[1 - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right],\n\\end{equation}\ni.e. the last term in Eq.~\\ref{eq:52} cancels out when \nthe low-frequency, long-wavelength limit is taken in such a way that $\\Omega\/v_F |{\\bf{q}}| \\rightarrow 0$. \nOn the other hand, when $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$, the intraband contribution vanishes and \n\\begin{equation}\n\\label{eq:63}\n\\Pi({\\bf{q}}, i\\Omega) = \\Pi^{inter}({\\bf{q}}, i\\Omega).\n\\end{equation}\nThis physical difference in the kind of response the system exhibits is the basis of Streda's separation of contributions to the Hall conductivity into $\\sigma_{xy}^I$ and \n$\\sigma_{xy}^{II}$~\\cite{Streda}.\nOur analysis makes it clear that this is the most physically-meaningful separation of contributions to the intrinsic anomalous Hall conductivity, since it corresponds \nto distinct and, at least in principle, separately measurable, contributions to the response function $\\Pi({\\bf{q}}, i\\Omega)$. \n\n\\begin{figure}[t]\n\\subfigure[]{\n \\label{fig:2a}\n \\includegraphics[width=7cm]{bands1}}\n\\subfigure[]{\n \\label{fig:2b}\n \\includegraphics[width=7cm]{sigmaxy1}}\n \\caption{(a) Plot of the band edges along the $z$-direction in momentum space.\n The parameters are such that two Weyl nodes are present.\n (b) Total intrinsic anomalous Hall conductivity (solid line), $\\sigma_{xy}^I$ (dashed line), and $\\sigma_{xy}^{II}$ (dotted line). Note that the van Hove-like singularities in \n $\\sigma^I_{xy}$ and $\\sigma^{II}_{xy}$, associated with band edges, mutually cancel and the total Hall conductivity $\\sigma_{xy}$ is a smooth function of the Fermi energy.}\n \\label{fig:2}\n\\end{figure}\n\\begin{figure}[t]\n\\subfigure[]{\n \\label{fig:3a}\n \\includegraphics[width=7cm]{bands2}}\n\\subfigure[]{\n \\label{fig:3b}\n \\includegraphics[width=7cm]{sigmaxy2}}\n \\caption{(a) Plot of the band edges along the $z$-direction in momentum space. The spin splitting is not large enough for the \n Weyl nodes to appear (i.e. $b < b_{c1}$) and the spectrum has a full gap. \n (b) Total intrinsic anomalous Hall conductivity (solid line), $\\sigma_{xy}^I$ (dashed line), and $\\sigma_{xy}^{II}$ (dotted line).}\n \\label{fig:3}\n\\end{figure}\n\nGeneralizing the above results to arbitrary sign of $\\epsilon_F$ we finally obtain\n\\begin{equation}\n\\label{eq:64}\nS = - i \\epsilon^{z 0 \\alpha \\beta} \\sigma_{xy} \\int d^3 r d \\tau A_0(\\br, \\tau) \\partial_{\\alpha} A_{\\beta}(\\br, \\tau), \n\\end{equation}\nwhere, if the low-frequency long-wavelength limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow \\infty$:\n\\begin{eqnarray}\n\\label{eq:65}\n\\sigma_{xy}&=&\\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\left\\{\\textrm{sign} [m_t(k_z)] \\left[\\Theta(\\epsilon_F + |m_t(k_z)|) - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right] \\right.\\nonumber \\\\\n&+&\\left. \\frac{m_t(k_z)}{|\\epsilon_F|} \\Theta(|\\epsilon_F| - |m_t(k_z)|) \\right\\}.\n\\end{eqnarray}\nThis expression corresponds to the full DC anomalous Hall conductivity. \nOn the other hand, when the low-frequency, long-wavelength limit is taken so that $\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, we obtain\n\\begin{eqnarray}\n\\label{eq:66}\n\\sigma_{xy}&=&\\sigma_{xy}^{II}=\\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\,\\, \\textrm{sign} [m_t(k_z)] \\nonumber \\\\\n&\\times&\\left[\\Theta(\\epsilon_F + |m_t(k_z)|) - \\Theta(\\epsilon_F - |m_t(k_z)|)\\right].\n\\end{eqnarray}\nThis is precisely the Streda's $\\sigma^{II}_{xy}$ contribution to the Hall conductivity, which is a thermodynamic equilibrium quantity, \nequal to\n\\begin{equation}\n\\label{eq:67}\n\\sigma^{II}_{xy} = e \\left(\\frac{\\partial N}{\\partial B}\\right)_{\\mu}, \n\\end{equation}\nwhere $N$ is the total electron number. This relation follows immediately from Eq.~\\ref{eq:64} and the order of limits\n$\\Omega\/ v_F |{\\bf{q}}| \\rightarrow 0$, which corresponds to thermodynamic equilibrium.\nCorrespondingly, the $\\sigma^{I}_{xy}$ contribution is given by\n\\begin{equation}\n\\label{eq:68}\n\\sigma^I_{xy} = \\sigma_{xy} - \\sigma^{II}_{xy} = \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\frac{m_t(k_z)}{|\\epsilon_F|} \\Theta(|\\epsilon_F| - |m_t(k_z)|).\n\\end{equation}\nAs clear from the above analysis, $\\sigma^I_{xy}$ is the contribution to $\\sigma_{xy}$ that can be associated with \nstates on the Fermi surface. This contribution is nonuniversal, i.e. it depends on details of the electronic structure and \nis a continuous function of the Fermi energy. \n$\\sigma^{II}_{xy}$, on the other hand, is the contribution of all states below the Fermi energy and is a thermodynamic \nequilibrium property of the ferromagnet. It attains a universal value, which depends only on the distance between the \nWeyl nodes, when the Fermi energy coincides with the nodes, i.e. when $\\epsilon_F = 0$\n\\begin{equation}\n\\label{eq:69}\n\\sigma_{xy}^{II} = \\frac{e^2 {\\cal K}}{4 \\pi^2},\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:70}\n{\\cal K} = \\frac{2}{d} \\arccos\\left(\\frac{\\Delta_S^2 + \\Delta_D^2 - b^2}{2 \\Delta_S \\Delta_D} \\right), \n\\end{equation}\nis the distance between the Weyl nodes. \nWhen $b > b_{c2}$, the Weyl nodes annihilate at the edges of the Brillouin zone and a gap opens up. \nIn this case ${\\cal K} = 2 \\pi\/d$, i.e. a reciprocal lattice vector and $\\sigma^{II}_{xy}$ is quantized as long as the Fermi level is in the gap~\\cite{Halperin92}.\nBoth contributions, along with the total anomalous Hall conductivity $\\sigma_{xy}$ are plotted as a function of the Fermi energy in Figs.~\\ref{fig:2},~\\ref{fig:3}\nin two different cases: when Weyl nodes are present and when they are not. \nThe former occurs when $b_{c1} < b < b_{c2}$. \nAs can be seen from Fig.~\\ref{fig:2}, Weyl nodes provide the dominant contribution to $\\sigma^{II}_{xy}$ and to the total Hall conductivity $\\sigma_{xy}$, if the \nFermi level is not too far from the nodes. \n\nIt is interesting to note the following property, which is evident from Fig.~\\ref{fig:2}. \nBoth the total anomalous Hall conductivity $\\sigma_{xy}$ and the two distinct contributions to it, $\\sigma_{xy}^{I,II}$ appear to exhibit a \nquasi-plateau behavior when $\\epsilon_F$ is not too far from the Weyl nodes. To understand the origin of this behavior, consider the derivative \nof $\\sigma^{II}_{xy}$ with respect to the Fermi energy\n\\begin{equation}\n\\label{eq:71}\n\\frac{\\partial \\sigma^{II}_{xy}}{\\partial \\epsilon_F} = - \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m_t(k_z)] \\delta(\\epsilon_F - |m_t(k_z)). \n\\end{equation}\nThis is straightforward to evaluate analytically and we obtain\n\\begin{eqnarray}\n\\label{eq:72}\n\\frac{\\partial \\sigma_{xy}^{II}}{\\partial \\epsilon_F}&=&- \\frac{e^2}{8 \\pi^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\left[\\delta(\\Delta(k_z) - b + \\epsilon_F) - \\delta(\\Delta(k_z) - b -\\epsilon_F) \\right] \\nonumber \\\\ &=&\\frac{e^2}{4 \\pi^2}\\left(1\/\\tilde v_{F+} - 1\/\\tilde v_{F-} \\right), \n\\end{eqnarray}\nwhere we have assumed that $\\epsilon_F$ is sufficiently close to zero, so that only the $t = -$ bands contribute to the integral, and \n\\begin{equation}\n\\label{eq:73}\n\\tilde v_{F\\pm} = \\frac{d}{2 (b \\pm \\epsilon_F)} \\sqrt{[(b \\pm \\epsilon_F)^2 - b_{c1}^2] [b_{c2}^2 - (b \\pm \\epsilon_F)^2]}, \n\\end{equation}\nare the two Fermi velocities, corresponding to two pairs of solutions of the equation $|b - \\Delta(k_z)| = \\epsilon_F$, which \narise from the Fermi level crossing the $s=+, t=-$ band on the two sides of each Weyl node along the $z$-axis in momentum space. \nThe two Fermi velocities are nearly equal when the band dispersion near the nodes is almost perfectly linear, but start to differ significantly \nwhen deviations from linearity become noticeable. \nExplicitly, as long as $b_{c1} \\ll b \\pm \\epsilon_F \\ll b_{c2}$, both Fermi velocities are independent \nof the Fermi energy and thus $\\partial \\sigma_{xy}^{II}\/\\partial \\epsilon_F$ vanishes. \nA useful way to think about this is in terms of an approximate {\\em chiral symmetry}, which emerges in this system in the limit of small $\\epsilon_F$, and which manifests \nin an almost perfectly linear band dispersion near the Weyl nodes. \n \nBy a nearly identical calculation it is easy to show that $\\partial \\sigma^{I}_{xy} \/ \\partial \\epsilon_F$ also vanishes when $\\epsilon_F$ is sufficiently close to zero. \nThis is the origin of the quasi-plateau behavior in Fig.~\\ref{fig:2}.\nThis result implies that the intrinsic anomalous Hall conductivity is equal to its thermodynamic equilibrium part, $\\sigma^{II}_{xy}$, not just when the Fermi \nenergy coincides with the Weyl nodes, but even away from them as long as the band dispersion may be assumed to be linear~\\cite{Burkov14}.\nThis property will be discussed in more detail in Section~\\ref{sec:4}. \n\nWhen attempting to understand the physical meaning of the two contributions to the anomalous Hall conductivity, $\\sigma^I_{xy}$ and $\\sigma^{II}_{xy}$, \nit may be tempting to say that, while $\\sigma^{I}_{xy}$ is clearly associated with states on the Fermi surface, $\\sigma^{II}_{xy}$ might perhaps be associated \nwith the chiral Fermi arc edge states. Unfortunately, such an interpretation is clearly incorrect, at least in the context of the present model, since, as can be \nseen in Fig.~\\ref{fig:3}, $\\sigma^{II}_{xy}$ may be nonzero even when Weyl nodes and thus the Fermi arc edge states are absent. More work is thus needed \nto fully understand the relation between $\\sigma^{II}_{xy}$ and the edge states. \n \n\n\n\\subsection{Microscopic theory: Chiral Magnetic Effect}\n\\label{sec:3.3}\nIn this subsection we consider the second part of the topological response in Weyl semimetals, namely the CME. \nThis arises due to the presence of an energy difference between the Weyl nodes, as in Eq.~\\ref{eq:34}. \nTo induce such an energy difference in our microscopic model, we need to find an operator that acts as an ``axial chemical potential\" term, \nshifting the nodes with different chirality in opposite directions in energy. \nThe expression for this operator may be found based on symmetry considerations. \nWe define an ``axial charge density\" operator, $\\hat n_a$, as a local operator, which is odd under inversion and $z \\rightarrow -z$ reflections, even under \ntime reversal, but odd under time reversal combined with rotation of the spin quantization axis by $\\pi$ around either $x$ or $y$ axis. \nThis leads to the following expression for the axial charge density operator\n\\begin{equation}\n\\label{eq:74}\n\\hat n_a = \\tau^y \\sigma^z. \n\\end{equation}\nAdding a term $- \\mu_a \\hat n_a$, where $\\mu_a$ is the ``axial chemical potential\", to the multilayer Hamiltonian, introduces \nan energy difference between the Weyl nodes of magnitude\n\\begin{equation}\n\\label{eq:75}\n\\Delta \\epsilon = \\frac{2 \\mu_a \\tilde v_F}{\\Delta_S d}, \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:76}\n\\tilde v_F = \\frac{d}{2 b} \\sqrt{(b^2 - b_{c1}^2)(b_{c2}^2 - b^2)}, \n\\end{equation}\nis the $z$-component of the Fermi velocity at the location of the Weyl nodes. \n\nSimilarly to the previous subsection, we now couple the electrons to electromagnetic \nfield and the axial chemical potential and integrate out the electron variables to obtain \nan induced action for the electromagnetic field. \nWe will assume, without loss of generality, that the electromagnetic field consists of a magnetic field in the $z$-direction, \nand a vector potential $A_z$, whose time derivative gives the $z$-component of the electric field $E_z = - \\partial_t A_z$. \nWe will allow for a time and $z$-coordinate dependence of the vector potential $A_z$ and of the axial chemical potential $\\mu_a$, but \nassume that the magnetic field is time-independent and uniform. \n\nFor this calculation it is convenient to use the Landau level (LL) basis of the multilayer placed in a uniform external magnetic field along the growth \ndirection, i.e. the basis of the eigenstates of the following Hamiltonian\n\\begin{equation}\n\\label{eq:77}\n{\\cal H}(k_z) = v_F \\tau^z (\\hat z \\times \\bsigma) \\cdot \\left(- i \\bnabla + e {\\bf{A}} \\right) + \\hat \\Delta(k_z) + b \\sigma^z, \n\\end{equation}\nAdopting Landau gauge for the vector potential ${\\bf{A}} = x B \\hat y$, we obtain the following expressions LL eigenstates, which have the form, typical for LLs in Dirac systems\n\\begin{equation}\n\\label{eq:78}\n| n, a, k_y, k_z \\rangle = \\sum_{\\tau} \\left[z^a_{n \\uparrow \\tau}(k_z) | n -1, k_y, k_z, \\uparrow, \\tau \\rangle + z^a_{n \\da \\tau}(k_z) | n, k_y, k_z, \\da, \\tau \\rangle \\right]. \n\\end{equation} \nHere \n\\begin{equation}\n\\label{eq:79}\n\\langle \\br | n, k_y, k_z, \\sigma, \\tau \\rangle = \\frac{1}{\\sqrt{L_z}} e^{i k_z z} \\phi_{n k_y}(\\br) | \\sigma, \\tau \\rangle, \n\\end{equation}\n$\\phi_{n k_y}(\\br)$ are the Landau-gauge orbital wavefunctions, and $\\sigma, \\tau$ are the spin and pseudospin indices \nrespectively. Finally, the four-component eigenvector $| z^a_{n}(k_z) \\rangle$ may be written as a tensor product of the two-component spin and pseudospin eigenvectors, \ni.e. $| z^a_{n}(k_z) \\rangle = | v^a_{n}(k_z) \\rangle \\otimes | u^a(k_z) \\rangle$, where \n\\begin{eqnarray}\n\\label{eq:80}\n&&|v^{s t}_{n}(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m_t(k_z)}{\\epsilon_{n t}(k_z)}}, - i s \\sqrt{1 - s \\frac{m_t(k_z)}{\\epsilon_{n t}(k_z)}} \\right), \\nonumber \\\\\n&&|u^t(k_z) \\rangle = \\frac{1}{\\sqrt{2}} \\left(1, t \\frac{\\Delta_S + \\Delta_D e^{- i k_z d}}{\\Delta(k_z)} \\right),\n\\end{eqnarray}\nand the eigenstate energies are given by\n\\begin{equation}\n\\label{eq:80.1}\n\\epsilon_{n s t}(k_z) = s \\sqrt{2 \\omega_B^2 n + m_t^2(k_z)} \\equiv s \\epsilon_{n t}(k_z), \n\\end{equation}\nwhere $\\omega_B = v_F\/ \\ell_B$ is the Dirac cyclotron frequency and $\\ell_B = 1\/\\sqrt{e B}$ is the magnetic length. \n\nAs in all Dirac systems, the lowest $n = 0$ LL is special and needs to be considered separately. The $s$ quantum number is absent in this case and \ntaking $B > 0$ for concreteness, we have $\\epsilon_{n t}(k_z) = - m_t(k_z)$, and $|v^t_{0}(k_z) \\rangle = (0,1)$. \n\nThe topological term, of interest to us, is proportional to the product of $\\mu_a$ and $A_z$. \nIntegrating out the electron variables and leaving only this term in the imaginary time action, we obtain\n\\begin{equation}\n\\label{eq:85}\nS = B \\sum_{q, i \\Omega} \\Pi(q, i\\Omega) A_z(q, i\\Omega) \\mu_a(-q, -i \\Omega)\n\\end{equation}\nwhere the response function $\\Pi(q, i\\Omega)$ is given by:\n\\begin{eqnarray}\n\\label{eq:86}\n\\Pi(q, i\\Omega)&=&\\frac{e}{2 \\pi L_z} \\sum_{n, k_z} \\frac{n_F[\\xi_{n a'}(k_z)] - n_F[\\xi_{n a}(k_z + q)]}{i \\Omega + \\xi_{n a'}(k_z) - \\xi_{n a}(k_z + q)} \\nonumber \\\\\n&\\times& \\langle z^a_{n k_z}| \\hat j_z(k_z) | z^{a'}_{n k_z} \\rangle \\langle z^{a'}_{n k_z} | \\tau^y |z^a_{n k_z} \\rangle. \n\\end{eqnarray}\nHere $n_F$ is the Fermi-Dirac distribution function, $\\xi_{n a}(k_z) = \\epsilon_{n a}(k_z) - \\epsilon_F$, and the magnetic field $B$ in Eq.~\\ref{eq:85} arises \nfrom the Landau level orbital degeneracy. \nWe have also ignored the $q$-dependence of the matrix elements in Eq.~\\ref{eq:86}, which is not important for small $q$. \n\nAt this point we will specialize to the case of an undoped Weyl semimetal, i.e. set $\\epsilon_F = 0$. \nThen it is clear from Eq.~\\ref{eq:86} that for Landau levels with $n \\geq 1$, only terms with $s \\neq s'$ contribute due to the difference of Fermi factors \nin the numerator. \nWe are interested ultimately in the zero frequency and zero wavevector limit of the response function $\\Pi(q, i\\Omega)$. \nAs already seen in the previous section in the context of AHE, the value of $\\Pi(0,0)$ depends on the order in which the zero frequency and zero wavevector \nlimits are taken.\nBelow we will consider both possibilities separately and discuss their physical meaning. \n\nBefore we proceed with an explicit evaluation of the $q \\rightarrow 0$ and $\\Omega \\rightarrow 0$ limit, let us note \nan important property of the response function $\\Pi(q, \\Omega)$. \nIf we take into account the following symmetry properties of the matrix elements in Eq.~\\ref{eq:86}\n\\begin{eqnarray}\n\\label{eq:88}\n&&\\langle v^{+ t}_{n k_z} | v^{- t'}_{n k_z} \\rangle = - \\langle v^{+ t'}_{n k_z} | v^{- t}_{n k_z} \\rangle, \\nonumber \\\\\n&&\\langle v^{+ t}_{n k_z} | \\sigma^z| v^{- t'}_{n k_z} \\rangle = \\langle v^{+ t'}_{n k_z} | \\sigma^z| v^{- t}_{n k_z} \\rangle, \n\\end{eqnarray}\nit is easy to see that when the limit $\\Omega \\rightarrow 0$ is taken, independently of the value of $q$, the $n \\geq 1$ Landau levels in fact \ndo not contribute at all, mutually cancelling due to Eq.~\\ref{eq:88}. It follows that $\\Pi(q, i\\Omega)$ at small $\\Omega$ is\ndetermined completely by the contribution of the two $n = 0$ Landau levels, whose energy eigenvalues and the corresponding \neigenvectors are independent of the magnetic field. \nThis means that in the small $\\Omega$ limit $\\Pi(q, i \\Omega)$ becomes independent of the magnetic field and the effective action in Eq.~\\ref{eq:85} \nthen depends linearly on both $\\mu_a$ and $B$, independently of the magnitude of $B$. \n\nLet us now proceed to explicitly evaluate $\\Pi(0,0)$, which determines the low-frequency and long-wavelength \nresponse of our system. \nLet us first look at the situation when we send $q$ to zero before sending $\\Omega$ to zero. \nExplicitly evaluating the matrix elements in Eq.~\\ref{eq:86} and the momentum integrals we obtain the following simple expression for $\\Pi(0,0)$\n\\begin{equation}\n\\label{eq:93} \n\\Pi(0,0) = - \\frac{e^2}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d}.\n\\end{equation}\nThus, after Wick's rotation $\\tau \\rightarrow i t$, $\\Delta \\epsilon \\rightarrow - i \\Delta \\epsilon$, we finally obtain the following result for the electromagnetic field action\n\\begin{equation}\n\\label{eq:94}\nS = - \\frac{e^2 \\Delta \\epsilon}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d} B \\int d^3 r d t \\,\\, A_z(\\br, t) \\mu_a(\\br, t),\n\\end{equation}\nwhich, taking into account Eq.~\\ref{eq:75}, has precisely the form of Eq.~\\ref{eq:40}. \nFunctional derivative of Eq.~\\ref{eq:94} with respect to $A_z$ gives the current that flows in response to magnetic \nfield and axial chemical potential, i.e. the CME\n\\begin{equation}\n\\label{eq:95}\nj_z = - \\frac{e^2 \\Delta \\epsilon}{4 \\pi^2} B. \n\\end{equation}\n\nThe physical interpretation of Eq.~\\ref{eq:95} requires some care. \nThe issue is again the order of limits when calculating $\\Pi(0,0)$, which always arises when calculating \nresponse functions in gapless systems.\nEqs.~\\ref{eq:94},\\ref{eq:95} was obtained by sending $q \\rightarrow 0$ before $\\Omega \\rightarrow 0$. \nLet us now see what happens when $\\Omega$ is sent to zero before taking the limit $q \\rightarrow 0$. \nIn this case, in addition to the contribution to $\\Pi(0,0)$, given by Eq.~\\ref{eq:93}, which arises due to transitions \nbetween the $t = +$ and $t = -$ lowest ($n = 0$) Landau levels, there is an extra contribution due to the intra-Landau-level \nprocesses within the $t = -$ Landau level, which crosses the Fermi energy at the location of the Weyl nodes. \nThis extra contribution is given by\n\\begin{equation}\n\\label{eq:96}\n\\tilde \\Pi(0,0) = \\frac{ e}{2 \\pi L_z} \\sum_{k_z} \\left. \\frac{d n_F(\\epsilon)}{d \\epsilon} \\right|_{\\epsilon = - m_-(k_z)} \\langle z^-_{0 k_z} | \\hat j_z(k_z) | z^-_{0 k_z} \n\\rangle \\langle z^-_{0 k_z} | \\tau^y | z^-_{0 k_z} \\rangle, \n\\end{equation}\nand is easily shown to be equal to Eq.~\\ref{eq:93} in magnitude, but opposite in sign, which means that in this case $\\Pi(0,0)$ vanishes. \nThus, the final result for $\\Pi(0,0)$ depends on the order in which the $q \\rightarrow 0$ and $\\Omega \\rightarrow 0$ limits \nare taken\n\\begin{eqnarray}\n\\label{eq:97}\n&&\\lim_{\\Omega \\rightarrow 0} \\lim_{q \\rightarrow 0} \\Pi(q, \\Omega) = - \\frac{e^2}{4 \\pi^2} \\frac{2 \\tilde v_F}{\\Delta_S d}, \\nonumber \\\\\n&&\\lim_{q \\rightarrow 0} \\lim_{\\Omega \\rightarrow 0} \\Pi(q, \\Omega) = 0. \n\\end{eqnarray}\nWhat is the physical meaning of these two distinct orders of limits in calculating $\\Pi(0,0)$? \nThis is again identical to the AHE case, discussed in the previous section. \nWhen $q$ is taken to zero first, one is calculating the low-frequency limit of response to \na time-dependent external field, in our case magnetic field along the $z$-direction. \nThis response is finite and represents CME, described by Eq.~\\ref{eq:95}. \nIf one takes $\\Omega$ to zero first, however, one is calculating a thermodynamic property, in our \ncase change of the ground state energy of the system in the presence of an additional static vector potential \nin the $z$-direction\n\\begin{equation}\n\\label{eq:98}\nj_z = \\frac{1}{V} \\frac{\\partial E(A_z)}{\\partial A_z} = 0. \n\\end{equation}\nThis could be nonzero in, for example, a current-carrying state of a superconductor, which \npossesses phase rigidity, but vanishes identically in our case~\\cite{Franz13,Burkov13}. \n\n\\section{Diffusive transport in Weyl metals}\n\\label{sec:4}\nIn the previous section we have focused on the basic properties of electromagnetic response \nof clean Weyl semimetals and metals (doped Weyl semimetals), neglecting the effect of impurity scattering. \nIt is, however, always present, and while probably not of significant importance in an undoped Weyl semimetal\nwith density of states at the Fermi energy vanishing as $\\sim \\epsilon_F^2$, it may be expected to play an important role \nin a Weyl metal with $\\epsilon_F$ significantly different from zero. \nIf nothing else, impurity scattering is crucial for establishing a steady state under applied electric field, and thus must be \nincluded in any serious discussion of low-frequency transport phenomena. \nIn this section we will thus generalize the theory of electromagnetic response, presented in Section~\\ref{sec:3}, to the case of diffusive transport in dirty Weyl metals.\nAs will be demonstrated below, this generalization leads to two important new results. \n\nFirst, we will demonstrate that the anomalous Hall conductivity of a Weyl metal is purely intrinsic and universal, \nlacking both the so-called extrinsic contribution due to the impurity scattering and also the part of the intrinsic \nelectronic-structure contribution, coming from incompletely filled bands. Only the purely intrinsic semi-quantized contribution, \n$\\sigma_{xy}^{II}$, arising from completely filled bands and proportional to the separation between the Weyl nodes, survives \nin the Weyl metal, as long as the Fermi energy is close enough to the Weyl nodes, such that the band dispersion at the Fermi energy may be \nassumed to be linear. \n\nSecond, we show that generalization of the chiral magnetic effect to the case of diffusive transport leads to a novel weak-field magnetoresistance \neffect, negative and quadratic in the magnetic field. We argue that this effect may be regarded as a universal smoking-gun transport characteristic \nof all Weyl metals. \n\n\\subsection{Anomalous Hall effect in dirty Weyl metals}\n\\label{sec:4.1}\nWe start again from the heterostructure model of Weyl semimetal described in detail in the previous sections. \nWe would like to evaluate the anomalous Hall conductivity of this model ferromagnetic Weyl metal, in the presence of impurity potential $V(\\br) = V_0 \\sum_a \\delta(\\br - \\br_a)$, which we will assume for simplicity to be gaussian, with only second order correlators present: $\\langle V(\\br) V(\\br ') \\rangle = \\gamma^2 \\delta(\\br - \\br ')$, where $\\gamma^2 = n_i V_0^2$ and $n_i$ is the impurity density. \nThe higher-order correlators, which are known to be important for AHE in principle, as they give rise to the so-called skew-scattering contribution to the Hall conductivity, \ndo not in fact affect our results in a significant way. \nWe will also assume that the impurity potential is diagonal in both the pseudospin $\\btau$ and the $t = \\pm$ index, which labels the eigenstates of the\n$\\hat \\Delta(k_z)$ operator Eq.~\\ref{eq:19}. Again, this assumption \nis used only for computational simplicity and does not affect the essence of our results. \nTo find the anomalous Hall conductivity, we will use the method of Section~\\ref{sec:3.2}, which we find to be the most convenient one for our \npurposes, as it allows to most clearly separate physically distinct contributions to the AHE. \nNamely, we imagine coupling electromagnetic field to the electrons and integrating the electron degrees of freedom out to obtain an \neffective action for the electromagnetic field, which, at quadratic order, describes the linear response of the system. \nThe part of this action we are interested in has the appearance of a Chern-Simons term, which, adopting the Coulomb gauge for the electromagnetic vector potential $\\bnabla \\cdot {\\bf{A}} = 0$, is given by\n\\begin{equation}\n\\label{eq:99}\nS = \\sum_{{\\bf{q}}, i\\Omega}\\epsilon^{z 0 \\alpha \\beta} \\Pi({\\bf{q}}, i\\Omega) A_0(-{\\bf{q}}, -i\\Omega) \\hat q_{\\alpha} A_{\\beta}({\\bf{q}}, i \\Omega),\n\\end{equation} \nwhere $\\hat q_{\\alpha} = q_{\\alpha}\/ q$ and summation over repeated indices is implicit. The $z$-direction in Eq.~\\ref{eq:99} is again picked out by the \nmagnetization $b$. As we will be interested specifically in the zero frequency and zero wavevector limits of the response function $\\Pi({\\bf{q}}, i \\Omega)$, \nwe will assume henceforth that ${\\bf{q}} = q \\hat x$, which does not lead to any loss of generality due to full rotational symmetry in the $xy$-plane. \nThe anomalous Hall conductivity is given by the zero frequency and zero wave vector limit of the response function $\\Pi({\\bf{q}}, i \\Omega)$ as\n\\begin{equation}\n\\label{eq:100}\n\\sigma_{xy} = \\lim_{i \\Omega \\rightarrow 0} \\lim_{q \\rightarrow 0} \\frac{1}{q} \\Pi({\\bf{q}}, i \\Omega).\n\\end{equation}\nA significant advantage of Eq.~\\ref{eq:100}, compared to the more standard Kubo formula for the anomalous Hall conductivity, which relates \nit to the current-current correlation function, is that Eq.~\\ref{eq:100} ties the Hall conductivity to the response of a conserved quantity, \ni.e. the particle density. This means, in particular, that the response function $\\Pi({\\bf{q}}, i \\Omega)$ must satisfy exact Ward identities, which follow from \ncharge conservation, providing a useful correctness check on the results. \n\nThe impurity average of the response function $\\Pi({\\bf{q}}, i\\Omega)$ may be evaluated by the standard methods of diagrammatic perturbation theory. \nDue to our assumption that the impurity potential is diagonal in the band index $t$, we can do this calculation separately for each pair of bands, labeled \nby $t$, and then simply sum the individual contributions. We will thus omit the $t$ index in what follows, until we come to the final results. \nThe retarded impurity averaged one-particle Green's functions have the following general form\n\\begin{equation}\n\\label{eq:101}\nG^R_{\\sigma_1 \\sigma_2}({{\\bf{k}}}, \\epsilon) = \\frac{z^s_{{{\\bf{k}}} \\sigma_1} \\bar z^s_{{{\\bf{k}}} \\sigma_2}}{\\epsilon - \\xi^s_{{\\bf{k}}} + i \/2 \\tau_s}. \n\\end{equation}\nHere $s = \\pm$, as before, labels the positive and negative energy pairs of bands (the sum over $s$ is made implicit above)\n\\begin{equation}\n\\label{eq:101.1}\n\\xi^s_{{\\bf{k}}} = s \\epsilon_{{\\bf{k}}} - \\epsilon_F = s \\sqrt{v_F^2(k_x^2 + k_y^2) + m^2(k_z)} - \\epsilon_F, \n\\end{equation}\nare the band energies, counted from the Fermi energy $\\epsilon_F$, and \n\\begin{equation}\n\\label{eq:101.2}\n| z^s_{{\\bf{k}}} \\rangle = \\frac{1}{\\sqrt{2}} \\left(\\sqrt{1 + s \\frac{m(k_z)}{\\epsilon_{{\\bf{k}}}}}, - i s e^{i \\varphi} \\sqrt{1 - s \\frac{m(k_z)}{\\epsilon_{{\\bf{k}}}}}\\right), \n\\end{equation}\nis the corresponding eigenvector.\nIn what follows we will assume, for concreteness, that $\\epsilon_F > 0$, i.e. the Weyl metal is electron-doped. \nThe impurity scattering rates $1\/\\tau_{\\pm}$ are given, in the Born approximation, by \n\\begin{equation}\n\\label{eq:102}\n\\frac{1}{\\tau_s(k_z)} = \\frac{1}{\\tau} \\left[1 + s \\frac{m(k_z) \\langle m \\rangle}{\\epsilon_F^2} \\right],\n\\end{equation}\nwhere $1\/\\tau = \\pi \\gamma^2 g(\\epsilon_F)$ and \n\\begin{equation}\n\\label{eq:102.1}\ng(\\epsilon_F) = \\int \\frac{d^3 k}{(2 \\pi)^3} \\delta(\\epsilon_{{\\bf{k}}} - \\epsilon_F) = \\frac{\\epsilon_F}{4 \\pi^2 v_F^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\Theta(\\epsilon_F - |m(k_z)|), \n\\end{equation}\nis the density of states at Fermi energy. We have also defined the average of $m(k_z)$ over the Fermi surface as\n\\begin{equation}\n\\label{eq:103}\n\\langle m \\rangle = \\frac{1}{g(\\epsilon_F)} \\int \\frac{d^3 k}{(2 \\pi)^3} m(k_z) \\delta(\\epsilon_{{\\bf{k}}} - \\epsilon_F). \n\\end{equation}\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=12cm]{diagrams}\n\\end{center}\n\\vspace{-2cm}\n \\caption{Graphical representation of the impurity-averaged response function $\\Pi({\\bf{q}}, \\Omega)$.} \n \\label{fig:3.5}\n\\end{figure} \nThe impurity averaged response function, analytically continued to real frequency as $\\Pi({\\bf{q}}, \\Omega) = \\Pi({\\bf{q}}, i \\Omega \\rightarrow \\Omega + i \\eta)$,\nis given, in the self-consistent non-crossing approximation, by the sum of ladder diagrams, as illustrated in Fig.~\\ref{fig:3.5}\n\\begin{equation}\n\\label{eq:104}\n\\Pi({\\bf{q}}, \\Omega) = \\Pi^I({\\bf{q}}, \\Omega) + \\Pi^{II}({\\bf{q}}, \\Omega), \n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:105}\n\\Pi^I({\\bf{q}}, \\Omega) = 2 e^2 v_F \\Omega \\int_{-\\infty}^{\\infty} \\frac{d \\epsilon}{2 \\pi i} \\frac{d n_F(\\epsilon)}{d \\epsilon} P_{0 x}({\\bf{q}}, \\epsilon - i \\eta, \\epsilon + \\Omega + i \\eta), \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:106}\n\\Pi^{II}({\\bf{q}}, \\Omega) = 4 i e^2 v_F \\int_{-\\infty}^{\\infty} \\frac{d \\epsilon}{2 \\pi i} n_F(\\epsilon) \\textrm{Im} P_{0 x}({\\bf{q}}, \\epsilon + i \\eta, \\epsilon + \\Omega + i \\eta). \n\\end{equation} \nThe 4$\\times$4 matrix $P$, whose $0 x$ component we are interested in, is given by \n\\begin{eqnarray}\n\\label{eq:106.1}\nP({\\bf{q}}, - i \\eta, \\Omega + i \\eta)&=&\\gamma^{-2} I^{RA}({\\bf{q}}, \\Omega) {\\cal D}({\\bf{q}}, \\Omega), \\nonumber \\\\\nP({\\bf{q}}, \\epsilon + i \\eta, \\epsilon + \\Omega + i \\eta)&=&I^{RR}(\\epsilon, {\\bf{q}}, \\Omega), \n\\end{eqnarray}\nwhere \n\\begin{equation}\n\\label{eq:106.2}\n{\\cal D} = (1 - I^{RA})^{-1},\n\\end{equation}\nis the diffusion propagator and \n\\begin{eqnarray}\n\\label{eq:106.3}\nI^{RA}_{\\alpha \\beta}({\\bf{q}}, \\Omega)&=&\\frac{\\gamma^2}{2} \\tau^{\\alpha}_{\\sigma_2 \\sigma_1} \\tau^{\\beta}_{\\sigma_3 \\sigma_4} \\int \\frac{d^3 k}{(2 \\pi)^3} G^R_{\\sigma_1 \\sigma_3}({{\\bf{k}}} + {\\bf{q}}, \\Omega) G^A_{\\sigma_4 \\sigma_2}({{\\bf{k}}} , 0), \\nonumber \\\\\nI^{RR}_{\\alpha \\beta}(\\epsilon,{\\bf{q}}, \\Omega)&=&\\frac{1}{2} \\tau^{\\alpha}_{\\sigma_2 \\sigma_1} \\tau^{\\beta}_{\\sigma_3 \\sigma_4} \n\\int \\frac{d^3 k}{(2 \\pi)^3} G^R_{\\sigma_1 \\sigma_3}({{\\bf{k}}} + {\\bf{q}},\\epsilon + \\Omega) G^R_{\\sigma_4 \\sigma_2}({{\\bf{k}}}, \\epsilon). \n\\end{eqnarray}\nThe physical meaning of the two distinct contributions to the response function $\\Pi^{I,II}({\\bf{q}}, \\Omega)$ is clear from Eqs.~\\ref{eq:105}, \\ref{eq:106}. \n$\\Pi^{I}({\\bf{q}}, \\Omega)$ describes the non-equilibrium part of the response that happens at the Fermi surface, as clear from the appearance of the derivative of the Fermi-Dirac distribution function in Eq.~\\ref{eq:105}. This response is \ndiffusive when $\\Omega \\tau \\ll 1$ and ballistic in the opposite limit, and is generally affected significanty by the impurity scattering. We will discuss this in more detail below. \nIn contrast, $\\Pi^{II}({\\bf{q}}, \\Omega)$ is an equilibrium, nondissipative contribution to the overall response, to which all states below \nthe Fermi energy contribute~\\cite{Streda}, and which is essentially unaffected by the impurity scattering, as will be seen below.\n\nWe will start by evaluating the nonequilibrium part of the response function, $\\Pi^I({\\bf{q}}, \\Omega)$. \nComputation of the matrix elements $I^{RA}_{\\alpha \\beta}({\\bf{q}}, \\Omega)$ is easily done in the standard way, assuming $\\epsilon_F \\tau \\gg 1$. \nOne obtains\n\\begin{equation}\n\\label{eq:107}\n\\Pi^I({\\bf{q}}, \\Omega) = i e^2 v_F \\Omega \\tau g(\\epsilon_F) [I^{RA}_{00} {\\cal D}_{0x} + I^{RA}_{0x} {\\cal D}_{xx} + I^{RA}_{0z} {\\cal D}_{zx}], \n\\end{equation}\nwhere we have taken into account that ${\\cal D}_{yx} = 0$ by symmetry. \nThe relevant matrix elements of the diffusion propagator can be found analytically to first order in ${\\bf{q}}$. One obtains\n\\begin{equation}\n\\label{eq:108}\n\\Pi^I({\\bf{q}}, \\Omega) = i e^2 v_F \\Omega \\tau g(\\epsilon_F) \\frac{I^{RA}_{0x} (1 - I^{RA}_{zz}) + I^{RA}_{0z} I^{RA}_{zx}}{\\Gamma (1- I^{RA}_{xx})}, \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:108.1}\n\\Gamma({\\bf{q}}, \\Omega) = (1 - I^{RA}_{00})(1 - I^{RA}_{zz}) - I^{RA}_{0z} I^{RA}_{z0}, \n\\end{equation}\nis the determinant of the $0z$ block of the diffusion propagator, which corresponds to the diffusion of the charge density, \na conserved quantity (this block decouples from the rest of the diffuson when ${\\bf{q}} \\rightarrow 0$). \nThis means, in particular, that $\\Gamma$ must satisfy an exact Ward identity $\\Gamma(0,0) = 0$, which is easily checked to be true. \n\\begin{figure}[t]\n\\subfigure[]{\n \\label{fig:4a}\n \\includegraphics[width=7cm]{bands}}\n\\subfigure[]{\n \\label{fig:4b}\n \\includegraphics[width=7cm]{monopoles}}\n \\caption{(a) Plot of the band edges along the $z$-direction in momentum space for the two bands that touch at the Weyl nodes, using $\\Delta_D\/\\Delta_S = -0.9$. \n (b) Field lines of the Berry curvature \n in the $k_y = 0$ plane, for the same band structure as in (a). Corresponding Fermi surface section is shown by the two contours, enclosing the Weyl nodes.}\n \\label{fig:4}\n\\end{figure} \nExplicitly, the relevant matrix elements of $I^{RA}({\\bf{q}}, \\Omega)$ to first order in ${\\bf{q}}$ are given by\n\\begin{eqnarray}\n\\label{eq:109}\n&&I^{RA}_{00} =\\left\\langle \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+} \\right\\rangle, \\,\\, \nI^{RA}_{0x} = \\frac{i v_F q}{2 \\epsilon_F} \\left\\langle \\frac{m }{\\epsilon_F} \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{0z} = I^{RA}_{z0} = \\left\\langle \\frac{m}{\\epsilon_F} \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+}\\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{zx} = \\frac{i v_F q}{4 \\epsilon_F} \\left\\langle \\left(1+ \\frac{m^2}{\\epsilon_F^2}\\right) \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{zz} =\\left\\langle \\frac{m^2}{\\epsilon_F^2} \\frac{\\tau_+\/\\tau}{1 - i \\Omega \\tau_+} \\right\\rangle, \\nonumber \\\\\n&&I^{RA}_{xx} = \\frac{1}{2} \\left\\langle \\left(1- \\frac{m^2}{\\epsilon_F^2}\\right) \\frac{\\tau_+\/\\tau}{1 - i\\Omega \\tau_+} \\right\\rangle,\n\\end{eqnarray}\nwhere the average over the Fermi surface is defined in the same way as in Eq.~\\ref{eq:103}. \n\nExpanding $\\Gamma(0,\\Omega)$ to first order in $\\Omega$ and taking the limit $\\Omega \\rightarrow 0$ at fixed $\\tau$, which corresponds to the diffusive \nlimit, we obtain\n\\begin{equation}\n\\label{eq:111}\n\\Pi_{dif}^I({\\bf{q}}, 0) = - \\frac{i q \\,e^2 v_F^2 g(\\epsilon_F)}{2 \\epsilon_F}\\left\\langle \\frac{m \\tau_+}{\\epsilon_F \\tau} \\right\\rangle F[m], \n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:112}\nF[m] = \\frac{1 + \\frac{1}{2} \\left\\langle \\left(1 - \\frac{m^2}{\\epsilon_F^2} \\right) \\frac{\\tau_+}{\\tau}\\right\\rangle}\n{1 - \\frac{1}{2} \\left\\langle \\left(1 - \\frac{m^2}{\\epsilon_F^2} \\right) \\frac{\\tau_+}{\\tau}\\right\\rangle}\n\\left[\\left.\\frac{\\partial \\Gamma(0,\\Omega)}{\\partial (\\Omega \\tau)}\\right|_{\\Omega = 0}\\right]^{-1}. \n\\end{equation} \nThe explicit form of the functional $F[m]$ is in fact not that important for our purposes, except for the evenness property,\neasily seen from Eq.~\\ref{eq:112}: $F[m] = F[-m]$. \nAs a consequence, $\\Pi_{dif}^I$ is an odd functional of $m$, which will play an important role below.\nIt is important to note that the charge conservation, whose mathematical consequence is the presence of the diffusion \npole in $\\Pi^I({\\bf{q}}, \\Omega)$, is crucial in obtaining a nonzero result in the diffusive limit in Eq.~\\ref{eq:111}. \nThe analogous quantity in the calculation of the spin Hall conductivity, for example, would vanish in the diffusive limit~\\cite{Sinova04,Halperin04,Molenkamp04,Molenkamp06}.\n\nIt is also of interest to examine the ballistic limit of $\\Pi^I$, which corresponds to the case of a clean Weyl metal. In this case \nwe send both $\\Omega$ and $1\/\\tau$ to zero in such a way that $\\Omega \\tau \\rightarrow \\infty$. \nIn this case we obtain\n\\begin{eqnarray}\n\\label{eq:113}\n\\Pi^I_{bal}({\\bf{q}},0)&=&\\lim_{\\Omega \\rightarrow 0} \\lim_{1\/\\tau \\rightarrow 0} \\Pi^I({\\bf{q}}, \\Omega) = \n\\lim_{\\Omega \\rightarrow 0} \\lim_{1\/\\tau \\rightarrow 0} i e^2 v_F \\Omega \\tau g(\\epsilon_F) I^{RA}_{0x} \\nonumber \\\\\n&=&-\\frac{i q \\, e^2 v_F^2 g(\\epsilon_F)}{2 \\epsilon_F} \\left\\langle \\frac{m}{\\epsilon_F} \\right\\rangle, \n\\end{eqnarray}\nwhich agrees with the clean Weyl metal result Eq.~\\ref{eq:68}. \nA nice feature of the present calculation, compared with the calculation in a clean Weyl metal in Section~\\ref{sec:3.2}, is that here \nit is particularly clear that $\\Pi^I({\\bf{q}}, \\Omega)$ is associated with states on the Fermi surface. \nAs seen from Eqs.~\\ref{eq:111} and~\\ref{eq:113}, the difference between $\\Pi^I_{dif}$ and $\\Pi^I_{bal}$ is \nonly quantitative. In the AHE literature, this difference is said to arise from the \nso-called side-jump processes~\\cite{Molenkamp06,Nagaosa06,MacDonald07,Sinova10,Niu11}. \n\\begin{figure}[t]\n \\includegraphics[width=8cm]{sigma}\n \\caption{Plot of the total anomalous Hall conductivity (solid line), $\\sigma^I_{xy}$ (dashed line) and $\\sigma^{II}_{xy}$ (dotted line) versus the Fermi energy for the same system as in Fig.~\\ref{fig:4}.\n The plateau-like feature in $\\sigma_{xy}$ correlates with the range of the Fermi energies, for which the Fermi surface consists of two separate sheets, each enclosing \n a single Weyl node.}\n \\label{fig:5}\n\\end{figure}\nThe final step of the calculation is to evaluate the equilibrium part of the response function, $\\Pi^{II}({\\bf{q}}, \\Omega)$. \nIn the limit $\\epsilon_F \\tau \\gg 1$ one finds that this part of the response function is unaffected by the impurity scattering and is \ngiven by\n\\begin{eqnarray}\n\\label{eq:114}\n\\Pi^{II}({\\bf{q}}, \\Omega) = e^2 v_F \\int \\frac{d^3 k}{(2 \\pi)^3} \\langle z^{s'}_{{\\bf{k}}} | z^s_{{{\\bf{k}}} + {\\bf{q}}} \\rangle \\langle z^s_{{{\\bf{k}}} + {\\bf{q}}} | \\tau^x | z^{s'}_{{\\bf{k}}} \\rangle \\frac{n_F(\\xi^s_{{{\\bf{k}}} + {\\bf{q}}} - \\Omega) - n_F(\\xi^{s'}_{{\\bf{k}}})}{\\Omega - \\xi^s_{{{\\bf{k}}} + {\\bf{q}}} + \\xi^{s'}_{{\\bf{k}}}}, \\nonumber \\\\\n\\end{eqnarray}\nwhere summation over the band indices $s, s'$ is again implicit. \nEvaluating Eq.~\\ref{eq:114} in the small $\\Omega$ and $q$ limit gives \n\\begin{equation}\n\\label{eq:115}\n\\Pi^{II}({\\bf{q}}, 0) = \\frac{-i q \\, e^2}{8 \\pi^2} \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m(k_z)] \\left\\{1 - \\Theta[\\epsilon_F - |m(k_z)|]\\right\\}.\n\\end{equation}\nThe first term in Eq.~\\ref{eq:115} arises from the completely filled bands, while the second is the contribution of the incompletely filled bands. \n\nWe can now finally evaluate the anomalous Hall conductivity. We will focus on the diffusive limit results, as ballistic limit has already been discussed in Section~\\ref{sec:3}. \nAt this point we will also explicitly include the contribution of both $t = \\pm$ pairs of bands, which simply amounts to restoring the index $t$ in $m_t$, and \nsumming over $t$. \nUsing Eqs.~\\ref{eq:100},\\ref{eq:111} and \\ref{eq:115}, and remembering that $A_0 \\rightarrow i A_0$ upon Wick rotation to the real time, we obtain\n\\begin{equation}\n\\label{eq:116}\n\\sigma^I_{xy} = \\frac{e^2 v_F^2}{2 \\epsilon_F} \\sum_t g_t(\\epsilon_F) \\left\\langle \\frac{m_t \\tau_{+ t}}{\\epsilon_F \\tau_t} \\right\\rangle F[m_t], \n\\end{equation}\nand \n\\begin{equation}\n\\label{eq:117}\n\\sigma^{II}_{xy} = \\frac{e^2}{8 \\pi^2} \\sum_t \\int_{-\\pi\/d}^{\\pi\/d} d k_z \\textrm{sign}[m_t(k_z)] \\left\\{1 - \\Theta[\\epsilon_F - |m_t(k_z)|]\\right\\}.\n\\end{equation}\nSince $m_+(k_z)$ is positive throughout the first BZ, while $m_-(k_z)$ changes sign at the Weyl nodes, the first term \nin Eq.~\\ref{eq:117}, which comes from completely filled bands, gives a universal (almost) quantized contribution\n\\begin{equation}\n\\label{eq:118}\n\\sigma^{quant}_{xy} = \\frac{e^2 {\\cal K}}{4 \\pi^2}, \n\\end{equation}\nwhere ${\\cal K}$ is the distance between the Weyl nodes, which is the same as the clean well metal result, Eq.~\\ref{eq:69}. \nThis equation also describes the cases when the Weyl nodes are absent, \nin which case $\\sigma^{quant}_{xy}$ is truly quantized since ${\\cal K} = 0, G$, where $G= 2\\pi \/d$ is a reciprocal lattice vector. \n\nThe most important new result of this subsection comes from examining the remaining, non-quantized parts of $\\sigma_{xy}$. \nSuppose we have a situation when the Weyl nodes are present and $\\epsilon_F$, while not zero, is not too far from it, \nas shown in Fig.~\\ref{fig:4}.\nRecall that at the location of the Weyl nodes $m_-(k_z) = b - \\Delta(k_z)$ changes sign. \nThis implies that, as long as ${\\cal K} \\left.\\frac{d \\Delta}{d k_z}\\right|_{k_z = k_{z0}} \\gg \\epsilon_F$, \nwhere $k_{z0}$ is the location of a given Weyl node, the average of any odd function of $m_-(k_z)$ over the Fermi surface will vanish. \nThis means that in such a situation, which we call {\\em Weyl metal}, all contributions to the anomalous Hall conductivity, \nassociated with incompletely filled bands, will vanish and $\\sigma_{xy}$ attains a universal value, characteristic \nof Weyl semimetal $\\sigma_{xy} = \\sigma_{xy}^{quant}$, where $\\sigma_{xy}^{quant}$ is given by Eq.~\\ref{eq:118}. \nThis may be seen explicitly in Fig.~\\ref{fig:5}. \nNote that the linear dispersion sufficiently close to Weyl nodes is a topological property, \nin the sense that it follows directly and exclusively from the existence of a nonzero topological charge by the so-called Atiyah-Bott-Shapiro construction~\\cite{Horava}. \n\nTo understand this result physically, recall that the Weyl nodes are monopole sources of the Berry curvature $\\bOmega_{{\\bf{k}}}$. \nIn a clean metal, the anomalous Hall conductivity $\\sigma_{xy}$ is given by the integral of the $z$-component of the Berry curvature over all \noccupied states $\\sigma_{xy} = e^2 \\int \\frac{d^3 k}{(2 \\pi)^3} n_F(\\epsilon_{{\\bf{k}}}) \\Omega^z_{{\\bf{k}}}$.\nHowever, as clear from Fig.~\\ref{fig:5}, when the Fermi surface breaks up into disconnected sheets, enclosing individual nodes, \nthe contribution of the states, enclosed by the Fermi surface, to this integral will always be very small, vanishing exactly in the limit when the band dispersion away from the nodes may be taken to be exactly linear.\nA useful analogy here is with the electric field of a dipole. A pair of Weyl nodes is like a dipole of two topological charges. Its field has a well-defined and nonzero on average \n$z$-component at large distances from the dipole. At short distances, however, the field is that of individual charges, which winds around the location \nof each charge and thus any particular component of it averages to zero. \n\nWe have thus demonstrated that the AHE in Weyl metals has a purely intrinsic origin and can be associated entirely with the Weyl nodes, just \nas in the case of a Weyl semimetal, when the Fermi energy coincides with the nodes and the Fermi surface is absent~\\cite{Burkov14-1}. \nThis is in contrast to an ordinary ferromagnetic metal, in which the anomalous Hall conductivity always has both a significant Fermi surface contribution and \nan extrinsic contribution. This property of magnetic Weyl metals may be thought of as being a consequence of emergent chiral symmetry, as discussed in \nSection~\\ref{sec:3.2}. \n\n\\subsection{Magnetoresistance in Weyl metals}\n\\label{sec:4.2}\nIn this section we will discuss measurable consequences of the Chiral Magnetic Effect, discussed in Section~\\ref{sec:3.3}, in the diffusive transport regime. \nAs we will demonstrate, CME in this regime leads to a novel type of magnetoresistance: negative and quadratic in the magnetic field, first discovered by Son and Spivak~\\cite{Spivak12,Son12} in TR-symmetric Weyl semimetals. As we will show below, this novel magnetoresistance is in fact a universal feature of all types of Weyl semimetals and \nmay be regarded as their smoking-gun transport characteristic~\\cite{Burkov14-2}. \n\nWe start from the axial charge density operator, given by Eq.~\\ref{eq:74}.\nWe now ask the following question: does $\\hat n_a$ represent a conserved quantity, as it would in a low-energy model of Weyl semimetal? \nOnly when $n_a$ is conserved, or nearly conserved, may we expect it to contribute significantly to observable phenomena, at least at long times on \nlong length scales. \nTo answer this we need to evaluate the commutator of $\\hat n_a$ with the Hamiltonian ${\\cal H}({{\\bf{k}}})$. \nAs before, it is convenient at this point to apply the following canonical transformation to all the operators: $\\sigma^{\\pm} \\rightarrow \\tau^z \\sigma^{\\pm},\\,\\, \\tau^{\\pm} \\rightarrow\n\\sigma^z \\tau^{\\pm}$. Evaluating the commutator at the Weyl node locations, we now obtain\n\\begin{equation}\n\\label{eq:119}\n\\left[{\\cal H}({{\\bf{k}}}), \\hat n_a\\right]_{k^z_{\\pm}} = i \\frac{b^2 - \\Delta_D^2 + \\Delta_S^2}{\\Delta_S} \\tau^z \\sigma^z. \n\\end{equation}\nThis means that $n_a$ may indeed be a conserved quantity in the Weyl semimetal or weakly-doped Weyl metal, provided $\\Delta_D \\geq \\Delta_S$ and $b = \\sqrt{b_{c1} b_{c2}}$, \ni.e. the magnitude of the spin splitting is exactly the geometric mean of its lower- and upper-critical values, at which the transitions out of the Weyl semimetal phase occur. \nOtherwise, the commutator is nonzero and $n_a$ is not conserved. However, as will be shown below, the relevant relaxation time may in fact still be long, even when the above \ncondition is not exactly satisfied, in which case the axial charge density is still a physically meaningful quantity. \nThe near-conservation of the axial charge density $n_a$ may be viewed as a consequence of an emergent low-energy {\\em chiral symmetry}, which is an important \ncharacteristic feature of Weyl semimetals. Both the quadratic negative magnetoresistance, and the fully intrinsic disorder-independent AHE, discussed above, are consequences \nof this emergent symmetry. \n\nWe now want to derive hydrodynamic transport equations for both the axial charge density $n_a(\\br, t)$ and the total charge density $n(\\br, t)$. \nAs will be shown below, what is known as chiral anomaly will be manifest at the level of these hydrodynamic equations as a coupling between \n$n_a$ and $n$ in the presence of an external magnetic field. This coupling leads to significant observable magnetotransport effects, provided the \naxial charge relaxation time, calculated below, is long enough. As we will show explicitly below, long axial charge relaxation time is a direct \nconsequence of the emergent chiral symmetry, which exists in a Weyl metal at low energies. \n\nTo proceed with the derivation, we add a constant uniform magnetic field in the $\\hat z$ direction ${\\bf{B}} = B \\hat z$ and a scalar impurity potential $V(\\br)$, \nwhose precise form will be specified later. \nAdopting Landau gauge for the vector potential ${\\bf{A}} = x B \\hat y$, the second-quantized Hamiltonian of our system may be written as\n\\begin{eqnarray}\n\\label{eq:120}\nH&=&\\sum_{n a k_y k_z} \\epsilon_{n a}(k_z) c^{\\dag}_{n a k_y k_z} c^{\\vphantom \\dag}_{n a k_y k_z} \\nonumber \\\\\n&+&\\sum_{n a k_y k_z, n' a' k_y' k_z'} \\langle n, a, k_y, k_z | V | n', a', k_y', k_z' \\rangle c^{\\dag}_{n a k_y k_z} c^{\\vphantom \\dag}_{n' a' k_y' k_z'}. \n\\end{eqnarray}\nHere $\\epsilon_{n a}(k_z)$ are Landau-level (LL) eigenstate energies of a clean multilayer in magnetic field, $n=0,1,2,\\ldots$ is the main LL\nindex, $k_y$ is the Landau-gauge intra-LL orbital quantum number, $k_z$ is the conserved component of the crystal momentum \nalong the $z$-direction, and $a = (s, t)$ is a composite index (introduced for \ncompactness of notation), consisting of $s = \\pm$, which labels the electron- ($s = +$) and hole- ($s = -$) like sets of Landau levels, and $t = \\pm$, which \nlabels the two components of a Kramers doublet of LLs, degenerate at $b = 0$.\nThe LL eigenstate energies and the corresponding eigenvectors have been derived in Section~\\ref{sec:3.3}. \n\nTo proceed, we will make the standard assumption, which we also made in the previous subsection, that the impurity potential obeys Gaussian distribution, with $\\langle V(\\br) V(\\br') \\rangle = \\gamma^2 \\delta(\\br - \\br ')$.\nTo simplify calculations further we will also assume that the momentum transfer due to the impurity scattering is smaller than the size of the BZ, i.e. $| k_z - k_z' | d \\ll 1$. \nIn this case $\\langle u^t(k_z)| u^{t'}(k_z') \\rangle \\approx \\delta_{t t'}$, i.e. the $t$ quantum number may be assumed to be approximately preserved during the impurity scattering. \n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=12cm]{scba}\n\\end{center}\n\\vspace{-2cm}\n \\caption{Graphical representation of the impurity-averaged Green's function within SCBA and of the diffusion propagator.} \n \\label{fig:3.6}\n\\end{figure} \nWe treat the impurity scattering in the standard self-consistent Born approximation (SCBA), illustrated in Fig.~\\ref{fig:3.6}. \nThe retarded SCBA self-energy satisfies the equation\n\\begin{equation}\n\\label{eq:122}\n\\Sigma^R_{n a k_y k_z }(\\omega) = \\frac{1}{L_z} \\sum_{n' a' k_y' k_z'} \\langle |\\langle n, a, k_y, k_z| V |n', a', k_y', k_z'\\rangle|^2 \\rangle G^R_{n' a' k_y' k_z'}(\\omega), \n\\end{equation}\nWe will assume that the Fermi energy $\\epsilon_F$ is positive, i.e. the Weyl semimetal is electron-doped, and large enough that the impurity-scattering-induced broadening of the density of states is small on the scale of the Fermi energy $\\epsilon_F$~\\cite{Biswas14}. We can then restrict ourselves to the electron-like states with $s=+$ (we will drop the $s$ index henceforth for brevity), and easily solve the SCBA equation analytically. We obtain\n\\begin{equation}\n\\label{eq:123}\n\\textrm{Im} \\Sigma^R_{n t k_z} \\equiv -\\frac{1}{2 \\tau_t(k_z)} = - \\frac{1}{2 \\tau} \\left[1 + \\frac{m_t(k_z) \\langle m_t \\rangle}{\\epsilon_F^2} \\right], \n\\end{equation}\nwhere $1\/ \\tau = \\pi \\gamma^2 g(\\epsilon_F)$ and \n\\begin{equation}\n\\label{eq:124}\ng(\\epsilon_F) = \\frac{1}{2 \\pi \\ell_B^2} \\int_{-\\pi\/d}^{\\pi\/d} \\frac{d k_z}{2 \\pi} \\sum_{n t} \\delta[\\epsilon_{n t}(k_z) - \\epsilon_F],\n\\end{equation}\nis the density of states at Fermi energy. \nWe have also introduced the Fermi-surface average of $m_t(k_z)$ as\n\\begin{equation}\n\\label{eq:125}\n\\langle m_t \\rangle = \\frac{1}{2 \\pi \\ell_B^2 g(\\epsilon_F)} \\int_{-\\pi\/d}^{\\pi\/d} \\frac{d k_z}{2 \\pi} \\sum_{n t} m_t(k_z) \\delta[\\epsilon_{n t}(k_z) - \\epsilon_F]. \n\\end{equation}\n\nWe are interested in hydrodynamic, i.e. long-wavelength, low-frequency density response of our system. \nAs is well-known~\\cite{Altland}, the relevant information is contained in the diffusion propagator, or diffuson ${\\cal D}$, given \nby the sum of ladder impurity-averaging diagrams.\nThis is evaluated in the standard manner and we obtain\n\\begin{equation}\n\\label{eq:126}\n{\\cal D}^{-1} ({\\bf{q}}, \\Omega)= 1 - I({\\bf{q}}, \\Omega), \n\\end{equation}\nwhere $I$ is a $16 \\times 16$ matrix, given by\n\\begin{eqnarray}\n\\label{eq:127}\nI_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4}({\\bf{q}}, \\Omega) = \\frac{\\gamma^2}{L_x L_y L_z} \\int d^3 r d^3 r' e^{-i {\\bf{q}} \\cdot (\\br - \\br')} G^R_{\\alpha_1 \\alpha_3}(\\br, \\br'| \\Omega) G^A_{\\alpha_4 \\alpha_2}(\\br', \\br | 0), \\nonumber \\\\\n\\end{eqnarray}\nwhere we have introduced a composite index $\\alpha = (\\sigma, \\tau)$ to simplify the notation. \nThe impurity-averaged Green's functions $G^{R,A}$ are given by\n\\begin{equation}\n\\label{eq:128}\nG^{R,A}_{\\alpha \\alpha'}(\\br, \\br' |\\Omega) = \\sum_{n t k_y k_z} \\frac{\\langle \\br, \\alpha | n,t,k_y,k_z \\rangle \\langle n,t,k_y,k_z | \\br', \\alpha' \\rangle}\n{\\Omega - \\xi_{n t}(k_z) \\pm i \/ 2 \\tau_t(k_z)}, \n\\end{equation}\nwhere $\\xi_{n t}(k_z) = \\epsilon_{n t}(k_z) - \\epsilon_F$. \n\nIn general, the evaluation of Eq.~\\ref{eq:127} is a rather complicated task, primarily due to the fact that the impurity scattering will mix different LLs. \nAt this point we will thus specialize to the case of transport along the $z$-direction only, as this is where we can expect \nchiral anomaly to be manifest. In this case the contributions of different LLs to Eq.~\\ref{eq:127} decouple. \nSetting ${\\bf{q}} = q \\hat z$, we obtain\n\\begin{eqnarray}\n\\label{eq:129}\n&&I_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4} (q, \\Omega) = \\frac{\\gamma^2}{2 \\pi \\ell_B^2 L_z} \\sum_{n t t' k_z} \\frac{\\langle \\alpha_1| z^{t}_{n}(k_z + q\/2) \\rangle \\langle z^{t}_{n}(k_z + q\/2) | \\alpha_3 \\rangle}\n{\\Omega - \\xi_{n t}(k_z + q\/2) + i\/ 2 \\tau_t(k_z + q\/2)} \\nonumber \\\\\n&\\times&\\frac{\\langle \\alpha_4 | z^{t'}_{n}(k_z - q\/2) \\rangle \\langle z^{t'}_{n}(k_z - q\/2) | \\alpha_2 \\rangle}{-\\xi_{n t'}(k_z - q\/2) - i\/ 2 \\tau_t'(k_z - q\/2)}, \n\\end{eqnarray}\nwhich is much easier to evaluate.\n\nAs mentioned above, $I$ and ${\\cal D}^{-1}$ are large $16 \\times 16$ matrices, which contain a lot of information of no interest to us. \nWe are interested only in hydrodynamic physical quantities, with long relaxation times. All such quantities need to be identified, \nif they are expected to be coupled to each other. \nOne such quantity is obviously the total charge density $n(\\br, t)$, which has an infinite relaxation time due to the exact conservation of particle number. \nAnother is the axial charge density $n_a(\\br, t)$, which, as discussed above, may be almost conserved under certain conditions. \nOn physical grounds, we expect no other hydrodynamic quantities to be present in our case. We are thus only interested in the $2 \\times 2$ block of the matrix ${\\cal D}^{-1}$, which corresponds to the coupled evolution of the total and the axial charge densities. \nTo separate out this block, we apply the following transformation to the inverse diffuson matrix\n\\begin{equation}\n\\label{eq:130}\n{\\cal D}^{-1}_{a_1 b_1, a_2 b_2} = \\frac{1}{2} (\\sigma^{a_1} \\tau^{b_1})_{\\alpha_2 \\alpha_1} {\\cal D}^{-1}_{\\alpha_1 \\alpha_2, \\alpha_3 \\alpha_4}\n(\\sigma^{a_2} \\tau^{b_2})_{\\alpha_3 \\alpha_4}, \n\\end{equation}\nwhere $a_{1,2}, b_{1,2} = 0, x, y, z$. The components of interest to us are $a_{1,2} = b_{1,2} = 0$ which corresponds to the total charge density, $a_{1,2} = 0, b_{1,2} = y$, \nwhich corresponds to the axial charge density, and the corresponding cross-terms. \n\nWe will be interested, as mentioned above, in the hydrodynamic regime, which corresponds to low frequencies and long wavelengths, i.e. $\\Omega \\tau \\ll 1$ and \n$v_F q \\tau \\ll 1$. We will also assume that the magnetic field is weak, so that $\\omega_B \\ll \\epsilon_F$. \nFinally, we will assume that the Fermi energy is close enough to the Weyl nodes, so that only the $t = -$ states participate in transport and \n$\\langle m_- \\rangle \\approx 0$, since $m_-(k_z)$ changes sign at the nodes~\\cite{Burkov14}. \n\nIn accordance with the above assumptions, we expand the inverse diffusion propagator to leading order in $\\Omega \\tau$, $v_F q \\tau$ and $\\omega_B\/ \\epsilon_F$ \nand obtain after a straightforward but lengthy calculation \n\\begin{eqnarray}\n\\label{eq:131}\n{\\cal D}^{-1}(q, \\Omega) = \\left(\n\\begin{array}{cc}\n -i \\Omega \\tau + D q^2 \\tau & -i q \\Gamma \\tau \\\\\n - i q \\Gamma \\tau & -i \\Omega \\tau + D q^2 \\tau + \\tau\/ \\tau_a\n \\end{array}\n \\right). \\nonumber \\\\\n \\end{eqnarray} \nHere $D = \\tilde v_F^2 \\tau \\langle m_-^2 \\rangle\/ \\epsilon_F^2$ is the charge diffusion constant, associated with the diffusion in the $z$-direction, \n$\\Gamma = e B \/ 2 \\pi^2 g(\\epsilon_F)$ is the total charge-axial charge coupling coefficient and \n\\begin{equation}\n\\label{eq:132}\n\\frac{1}{\\tau_a} = \\frac{1 - (\\tilde v_F\/ \\Delta_S d)^2}{(\\tilde v_F\/ \\Delta_S d)^2 \\tau}, \n\\end{equation}\nis the axial charge relaxation rate. \nSeveral comments are in order here. First, an important thing to note is that the off-diagonal matrix elements in ${\\cal D}^{-1}$, proportional \nto $B$ and responsible for the total charge to axial charge coupling, come entirely from the contribution of the $n = 0$ LL. \nThe remaining matrix elements arise from the contribution of all the $n \\geq 1$ Landau levels and we have taken the limit $B \\rightarrow 0$ \nafter summing over the LLs, i.e. left only the leading term in the $\\omega_B\/ \\epsilon_F$ expansion. The next-to-leading term results in \na negative correction to the diffusion coefficient, proportional to $B^2$, which corresponds to the well-known classical positive magnetoresistance. \nWe have ignored this correction here, but will comment on its effects later. \nSecond, we note that the axial charge relaxation rate $1\/\\tau_a \\geq 0$, as it should be, and vanishes when $\\tilde v_F = \\Delta_S d$. \nIt is easy to see that this is identical to the condition of the vanishing of the commutator of the axial charge operator with the Hamiltonian Eq.~\\ref{eq:119}, \nagain as it should be.\nHenceforth we will assume that this condition is nearly satisfied so that $\\tau_a \\gg \\tau$. \nFinally, the situation when $\\tilde v_F = \\Delta_S d$ and thus $1\/\\tau_a$ appears to vanish, actually needs to be treated with some care. Namely, \nthe condition $\\tilde v_F = \\Delta_S d$ may be satisfied exactly only in the limit $\\epsilon_F \\rightarrow 0$. The Fermi velocity depends on the Fermi energy as~\\cite{Pesin14}\n\\begin{equation} \n\\label{eq:132.1}\n\\tilde v_F(\\epsilon_F) = \\frac{d}{2 (b + \\epsilon_F)} \\sqrt{[(b + \\epsilon_F)^2 - b_{c1}^2] [b_{c2}^2 - (b + \\epsilon_F)^2]}. \n\\end{equation}\nWhen $b = \\sqrt{b_{c1} b_{c2}}$ and thus $\\tilde v_F(0) = \\Delta_S d$, the Fermi energy dependence of $\\tilde v_F$ needs to be taken into account. \nExpanding to leading non vanishing order in $\\epsilon_F$ we obtain in this case\n\\begin{equation}\n\\label{eq:132.2}\n\\frac{1}{\\tau_a} = \\frac{\\epsilon_F^2}{\\Delta_S^2 \\tau}, \n\\end{equation}\ni.e. $1\/\\tau_a$ is in fact always finite, but may be very small. We can estimate the minimal value of the axial charge relaxation rate by \nsetting $\\epsilon_F \\approx 1\/\\tau$ in Eq. ~\\ref{eq:132.2}, which gives $(\\tau\/\\tau_a)_{min} \\approx 1\/ (\\Delta_S \\tau)^2$. \n\nWe may now write down the coupled diffusion equations for the total and axial charge densities, which correspond to the propagator Eq.~\\ref{eq:131}. \nThese equations read\n\\begin{eqnarray}\n\\label{eq:133}\n\\frac{\\partial n}{\\partial t}&=&D \\frac{\\partial^2 n}{\\partial z^2} + \\Gamma \\frac{\\partial n_a}{\\partial z}, \\nonumber \\\\\n\\frac{\\partial n_a}{\\partial t}&=&D \\frac{\\partial^2 n_a}{\\partial z^2} - \\frac{n_a}{\\tau_a} + \\Gamma \\frac{\\partial n}{\\partial z}. \n\\end{eqnarray}\nManifestation of chiral anomaly in these equations is the coupling between the total and the axial charge densities, proportional to the \napplied magnetic field. \nSince the total particle number is conserved, the right-hand side of the first of Eqs.~\\ref{eq:133} must be equal to minus the divergence of the total particle current. \nThen we obtain the following expression for the density of the charge current in the $z$-direction\n\\begin{equation}\n\\label{eq:134}\nj = - \\frac{\\sigma_0}{e} \\frac{\\partial \\mu}{\\partial z} - \\frac{e^2 B}{2 \\pi^2} \\mu_a, \n\\end{equation}\nwhere $\\sigma_0 = e^2 g(\\epsilon_F) D$ is the zero-field diagonal charge conductivity, $\\mu$ and $\\mu_a$ are the total and axial electrochemical potentials \nand we have used $\\delta n = g(\\epsilon_F) \\delta \\mu$, $\\delta n_a = g(\\epsilon_F) \\delta \\mu_a$. The last relation is valid when $\\tilde v_F\/ \\Delta_S d$ is close\nto unity, as seen from Eq.~\\ref{eq:75}. \nThus chiral anomaly manifests in an extra contribution to the charge current density, proportional to the magnetic field and the axial electrochemical potential. \nThis is known as chiral magnetic effect (CME) in the literature~\\cite{Kharzeev}. \nIt is important to realize that the second term in Eq.~\\ref{eq:134} does not by any means imply that an equilibrium current may be driven by an applied \nmagnetic field, despite appearances. The axial chemical potential $\\mu_a$, appearing in Eq.~\\ref{eq:134}, is a purely nonequilibrium quantity. \nIf an equilibrium energy difference, $\\mu_{a0}$, exists between the Weyl nodes due to explicitly broken inversion symmetry~\\cite{Zyuzin12-2}, then it is the difference $\\mu_a - \\mu_{a0}$ \nthat enters in Eq.~\\ref{eq:134}. We have explicitly considered an inversion-symmetric Weyl metal here, in which case $\\mu_{a0} = 0$. \n\nTo find measurable consequences of the CME contribution to the charge current, we consider a steady-state situation, with a fixed current density $j$ flowing through the \nsample in the $z$-direction. We want to find the corresponding electrochemical potential drop and thus the conductivity. \nAssuming the current density is uniform, we obtain from the second of Eqs.~\\ref{eq:133}\n\\begin{equation}\n\\label{eq:135}\nn_a = \\Gamma \\tau_a \\frac{\\partial n}{\\partial z}, \n\\end{equation} \nwhich is the nonequilibrium axial charge density, induced by the current and the corresponding electrochemical potential gradient. \nSubstituting this into the expression for the charge current density Eq.~\\ref{eq:134}, we finally obtain the following result\nfor the conductivity\n\\begin{equation}\n\\label{eq:136}\n\\sigma = \\sigma_0 + \\frac{e^4 B^2 \\tau_a}{4 \\pi^4 g(\\epsilon_F)}. \n\\end{equation}\nIn the limit when $\\epsilon_F$ is not far from the Weyl nodes, such that the dispersion may be assumed to be linear, \nwe have $g(\\epsilon_F) = \\epsilon_F^2\/ \\pi^2 v_F^2 \\tilde v_F$, which gives\n\\begin{equation}\n\\label{eq:137}\n\\Delta \\sigma = \\sigma - \\sigma_0 = \\frac{e^2 \\tilde v_F \\tau_a}{(2 \\pi v_F)^2} \\left(\\frac{e^2 v_F^2 B}{\\epsilon_F} \\right)^2,\n\\end{equation}\nwhich agrees with the Son and Spivak result~\\cite{Spivak12,Son12}. \nThus we see that a measurable consequence of CME is a positive magnetoconductivity, proportional to $B^2$ in the \nlimit of a weak magnetic field. \nThis of course needs to be compared with the classical negative magnetoconductivity, which is always \npresent and arises from the $B^2$ corrections to the diffusion constant $D$, which we have neglected\n\\begin{equation}\n\\label{eq:138}\n\\frac{\\Delta \\sigma_{c \\ell}}{\\sigma_0} \\sim - (\\omega_c \\tau)^2, \n\\end{equation} \nwhere $\\omega_c = e v_F^2 B\/ \\epsilon_F$ is the cyclotron frequency. \nThis gives \n\\begin{equation}\n\\label{eq:139}\n\\left|\\frac{\\Delta \\sigma}{\\Delta \\sigma_{c \\ell}}\\right| \\sim \\frac{\\tau_a\/\\tau}{(\\epsilon_F \\tau)^2}. \n\\end{equation}\nThus the CME-related positive magnetoconductivity will dominate the classical negative magnetoconductivity, provided \n$\\tau_a$ is long enough. \n\nWe have so far ignored the Zeeman effect due to the applied magnetic field. \nIts effect is to modify the spin-splitting parameter $b$ as $b \\rightarrow b + g \\mu_B B\/2$. \nIn principle, the dependence on $b$ does enter into our final results through the dependence \nof the Fermi velocity $\\tilde v_F$ on $b$. Naively, this will then generate an additional {\\em linear} magnetoconductivity, which \nmay be expected to dominate the quadratic one at small fields.\nHowever, the condition of large $\\tau_a$, which is the same as $\\tilde v_F\/ \\Delta_S d \\approx 1$, is equivalent to \nthe condition $b_{c1} \\ll b \\ll b_{c2}$, in which case the dependence of $\\tilde v_F$ on $b$ becomes negligible. \nThus, in the regime in which the positive magnetoconductivity dominates the negative classical one, and is thus observable, \none may also expect a negligible linear magnetoconductivity in any type of Weyl metal. \n\\section{Conclusions}\n\\label{sec:5}\nIn this paper we have provided an overview of transport phenomena in Weyl metals, which may be attributed to chiral anomaly, \nin particular the semi-quantized intrinsic Anomalous Hall Effect and the Chiral Magnetic Effect, which manifests in negative quadratic in the \nmagnetic field longitudinal magnetoresistance. \n\nThe AHE is generally present in any ferromagnetic metal. \nIn a generic FM metal this is a complicated phenomenon, with many physically distinct sources contributing to the \nfinal observed effect. In particular, {\\em extrinsic} contribution to AHE, arising from impurity scattering, is always at least \nof the same order of magnitude, and often significantly larger, than the more theoretically appealing and universal {\\em intrinsic} one. \nIn a Weyl metal, however, as we have demonstrated in this paper, AHE has a purely intrinsic origin. As long as the Fermi energy is \nnot too far from the Weyl nodes, such that the band dispersion may be taken to be linear and the chiral symmetry is present to a good approximation, \nthe anomalous Hall conductivity of a Weyl metal \nturns out to be given exactly by the semi-quantized expression of a pure undoped Weyl semimetal, where it simply measures separation \nbetween the Weyl nodes in momentum space in units of $e^2\/h$. \nBoth the extrinsic, and even the part of the intrinsic contribution, associated with the Fermi surface (i.e. incompletely filled bands), vanish identically. \nWe have connected this property with the geometry of the Berry curvature field in the vicinity of the Weyl nodes, which in turn is a direct consequence\nof their topology, through the Atiyah-Bott-Shapiro construction. \nFM Weyl metal is thus distinguished from other FM metals by the fact that its AHE is of purely intrinsic origin. \n\nPerhaps even more important in the context of Weyl metals is the Chiral Magnetic Effect and the associated negative quadratic \nmagnetoresistance. This may be expected to occur in any type of Weyl metal, either characterized by broken time reversal symmetry\nor broken inversion (or both). It is thus of particular importance for the experimental characterization of Weyl metals. \nThe negative quadratic longitudinal magnetoresistance should in fact be observed even in Dirac semimetals, which have already \nbeen realized experimentally. The theory, presented in Section~\\ref{sec:4.2} is directly applicable to this case, if $\\Delta_S = \\Delta_D$ \nlimit is taken and the spin-splitting $b$ is identified with the Zeeman splitting $b = g \\mu_B B \/2$. \nIn fact, this effect appears to have been observed very recently in a 3D Dirac semimetal material ZrTe$_5$~\\cite{Kharzeev14}. \n\nWhat has been presented in this paper is by no means a complete story of the possible observable manifestations of chiral anomaly \nin Weyl metals. Other related phenomena, such as linear high-field magnetoresistance~\\cite{Aji12}, plasmon-magnon coupling~\\cite{Liu13},\nnonlocal transport~\\cite{Pesin13}, anomalous density response~\\cite{Pesin14}, anomalous thermoelectric response~\\cite{Fiete14}, and others~\\cite{Parameswaran12,Grushin12,Balatsky13,Goswami13,Qi13,Xiao14,Zyuzin14,Chang14,Abanin14,Miransky14,Nomura14,Hughes14}, have been discussed in the literature. \nOne of the possible promising directions for future research in this area is the interplay of chiral anomaly and superconductivity in Weyl metals~\\cite{Balents12,Moore12,Aji14}. \n\n\\section{Acknowledgments}\nWe would like to thank L. Balents, Y. Chen, M.D. Hook, I. Panfilov, D.A. Pesin, S. Wu, and A.A. Zyuzin for collaboration on the topics, covered \nin this review, and closely related ones. We also thank Y. Ando and B.Z. Spivak for useful discussions. Financial support was provided by NSERC of Canada.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The First Section}\n\n\\section{Introduction}\nThe original Camassa-Holm equation was introduced by Fuchssteiner and Fokas \\cite{Fokas} through the method of recursion operators in 1981 and derived from physical principles by Camassa and Holm in 1999 \\cite{Camassa}.\n\nOur purpose is to investigate the modified Camassa-Holm equation (mCH)\n\t\t\t\\begin{eqnarray}\n\t\t\t\t\\label{mCH}\n\t\t\t\tu_{t}-u_{xxt} = uu_{xxx}+2u_{x}u_{xx}-3u^{2}u_{x},\\ \\ x\\in\\mathbb{R},\\ t>0\\\\\n\t\t\t\tu(x,0)=u_0(x),\\label{dado}\n\t\t\t\n\t\t\t\t\\end{eqnarray}\nobtained from modified Dullin-Gottwald-Holm (mDGH) equation \\cite{Yin}\n\t\t\t\t\t$$\\tilde{u}_{t} + \\kappa\\tilde{u}_{x} - \\tilde{u}_{xxt} - \\kappa \\tilde{u}_{xxx} = \\tilde{u}\\tilde{u}_{xxx} + 2\\tilde{u}_{x}\\tilde{u}_{xx} - 3\\tilde{u}^{2}\\tilde{u}_{x},\\ \\ x\\in\\mathbb{R},\\ t>0,$$\nby transformation $u(x,t)\\mapsto \\tilde{u}(x+\\kappa t, t)$. The original DGH equation was obtained by Dullin, Gottwald and Holm \\cite{Dullin} for a unidirectional water wave with fluid velocity $u(x,t)$, where the constant $\\kappa\\neq 0$ is the linear wave speed for undisturbed water at rest at spatial infinity. \n\n\t\tThe mCH equation can also be obtained from the $ab$-family of equations \\cite{H1,Hakkaev,H}\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{eq qntds conservadas}\n\t\t\t\tu_{t} + (a(u))_{x} - u_{xxt} = \\left(b^{'}(u)\\frac{u_{x}^{2}}{2} + b(u)u_{xx}\\right)_{x}\n\t\t\t\t\\end{equation}\nwhere $a,b:\\mathbb{R}\\longrightarrow\\mathbb{R}$ are smooth function and $a(0)=0$, by considering $a(u)=u^{3}$ and $b(u)=u$. From \\cite{H1,H} it is also known that (\\ref{mCH}) has three natural invariants:\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{qntds conservadas}\n\t\t\t\tE(u)=-\\int_{\\mathbb R}\\left[\\frac{u^{4}}{8}+\\frac{uu_{x}^{2}}{2}\\right]\\ dx, \\ \\ F(u)=\\frac{1}{2}\\int_{\\mathbb R}[u^{2} +u_{x}^{2}]\\ dx\\ \\ \\ \\text{and}\\ \\ \\ V(u)= \\int_{\\mathbb R}u\\ dx.\n\t\t\t\t\\end{equation}\n\t\t\t\n\n\t\n\t\n\t\n\t\n\t\n\n\t\t\tEquation (\\ref{mCH}) has been investigated in recent years. Tian and Song in \\cite{Tian} obtained peakons composed of hyperbolic functions. In \\cite{Wazwaz,Wazwaz2}, Wazwaz employed this modified form with $k=0$ as a vehicle to explore the change in the physical structure of the solution from peakons to bell-shaped solitary wave solutions and showed that mCH equation has a $sec\\ h$ like solitary wave solution. In \\cite{Yousif}, a variational homotopy perturbation method (VHPM) has been studied to obtain solitary wave solutions of the mCH equation. More recently in \\cite{Daros}, we guarantee results about the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period.\n\n\nIn this paper, we prove steepening at inflection points, i.e., we consider an initial condition in $H^3(\\mathbb R)$ that has an inflection point to the right of its maximum and get that the time dependent slope at the inflection point becomes vertical in finite time. This property means that wave breaking holds. Next we classify all traveling wave solutions, $u(x, t) = \\phi(x -ct)$, $c \\in \\mathbb{R}$, for mCH equation $(\\ref{mCH})$ using a weak formulation in $H^{1}_{loc}(\\mathbb{R})$ and obtain explicit formulas for peakons in terms of Jacobian elliptic functions. As for the mDP equation \\cite{Ding}, we also obtain some very interesting types of solutions such as kinks, cuspons, composite waves and stumpons, in addition to the more familiar smooth waves and peakons. To get an idea: the composite waves are obtained by combining cuspons and peakons into new traveling waves and the waves called stumpons are obtained by inserting intervals where $\\phi$ equals a constant at the crests of suitable cusped waves. \n\n\n\n\nIn the course of this work, we denote $\\star$ the convolution on $\\mathbb R$. We also use $(\\cdot, \\cdot)$ to represent the standard inner product in $L^2_{per}(\\mathbb R)$. For $1\\leq p \\leq \\infty$, the norm in the $L^p_{per}(\\mathbb R)$ will be denoted by $||\\cdot ||_{L^p}$, while $||\\cdot||_{s}$ will stand for the norm in the classical Sobolev spaces $H^s_{per}(\\mathbb R)$ for $s\\geq 0$. \n\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\nFormally, problem (\\ref{mCH})-(\\ref{dado}) is equivalent to the hyperbolic-elliptic system \n\\begin{eqnarray}\nu_t+\\partial_x\\left(\\frac{u^2}{2}\\right)+P_x =0,\\ \\ (x,t)\\in \\mathbb{R} \\times \\mathbb R_+, \\label{H}\\\\\nP-P_{xx}=u^3+\\frac{u^2}{2}+\\frac{{u_x}^2}{2}, \\ \\ (x,t)\\in \\mathbb{R}\\times \\mathbb R_+,\\label{E}\\\\\nu(x,0)=u_0(x),\\ \\ x\\in \\mathbb R\\label{idata}\n\\end{eqnarray}\n\n \n\nThe operator $(1-\\partial_{xx}^2)^{-1}$ has a convolution structure: \n \\begin{equation}\n (1-\\partial_{xx}^2)^{-1}(f)(x)=(G\\star f)(x),\n \\end{equation}\n where $G(x)$ is the Green function\n \\begin{eqnarray*}\n G(x)=\\frac{e^{-|x|}}{2}, \\ \\ x\\in \\mathbb R.\n \\end{eqnarray*}\n\n Hence we have \n\\begin{eqnarray}\\label{P}\nP(x,t)=G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)(x,t), \n\\end{eqnarray} \nand (\\ref{H})-(\\ref{idata}) can be rewritten as a conservation law with a non-local flux function $F$: \n\\begin{eqnarray}\nu_t+F(u)_x =0,\\ \\ (x,t)\\in \\mathbb R\\times \\mathbb R_+, \\label{CL}\\\\\nu(x,0)=u_0(x),\\ \\ x\\in \\mathbb R\\label{idata1},\n\\end{eqnarray}\t\t\nwhere $F(u)=\\frac{u^2}{2}+G\\star (u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2})$.\n\n\\begin{definition}\\label{ws} Let $u_0\\in H^1(\\mathbb R)$ be given. A function $u:[0,T]\\times \\mathbb{R} \\rightarrow \\mathbb{R} $ is called a weak solution to (\\ref{mCH}), if $u\\in L^{loc}_{\\infty}([0,T]; H^1)$ satisfies the identity\n$$ \\int_0^T \\int_{\\mathbb R} (u\\psi_t+F(u)\\psi_x)dxdt+\\int_{\\mathbb R}u_0(x)\\psi_0(x)dx=0$$\nfor all $\\psi\\in C^{\\infty}_0([0,T]\\times \\mathbb{R})$ that are restrictions to $[0,T)\\times \\mathbb{R}$\nof a continuously differentiable function on $\\mathbb{R}^2$ with compact support contained in $(-T,T)\\times \\mathbb{R}.$\n\\end{definition}\n\t\t\n\\section{Steepening lemma}\nThe local well posedness of the Cauchy problem of Equation (\\ref{mCH}) with initial data $u_0\\in H^s(\\mathbb R)$, $s>\\frac{3}{2}$ can be obtained by applying Kato's semigroup theory \\cite{Kato}. More precisely, we have the following well-posedness result. \n\n\\begin{theorem}\n\\label{teorema local}\nGiven $u_{0}\\in H^{s}(\\mathbb R)$, $s>\\frac{3}{2}$, there exists a maximal $t_0> 0$ and a unique solution $u(x,t)$ to problem (\\ref{mCH})-(\\ref{dado}) such that\n\t\t\t\t$$u\\in C([0,t_{0}), H^{s}(\\mathbb R))\\cap C^{1}([0,t_{0}), H^{s-1}(\\mathbb R)).$$\nMoreover, the solution depends continuously on the initial data. For $u_{0}\\in H^{3}(\\mathbb R)$ the solution posseses the aditional regularity \n$$u\\in C([0,t_{0}), H^{3}(\\mathbb R))\\cap C^{1}([0,t_{0}), H^{2}(\\mathbb R)). $$\n \n\\end{theorem}\n\\begin{proof}\nSee Hakkaev, Iliev and Kirchev in \\cite{H}.\n\\end{proof}\n\\begin{remark} The solutions obtained in Theorem \\ref{teorema local} are called strong solutions to Equation (\\ref{mCH}).\n\\end{remark}\n\n\n\nAssume now that $u_0\\in H^3(\\mathbb{R})$ and let $u\\in C([0,t_0);H^3(\\mathbb{R}) )\\cap C^1([0,t_0);H^2(\\mathbb{R}) )$ be the corresponding strong solution of (\\ref{mCH})-(\\ref{dado}). \n\n\\begin{remark}\nIf $u_0\\in H^3(\\mathbb{R})$, then the $|| u(\\cdot,t)||_1$ norm is conserved in time as long as the solution exists, which implies by the Sobolev embedding $H^1(\\mathbb R)\\subset L_{\\infty}(\\mathbb R)$ that \n\\end{remark}\n\\begin{equation}\\label{tbound}\nM:=\\sup_{t\\in [0,\\infty)} ||u(\\cdot,t) ||_{L_{\\infty}} <\\infty.\n\\end{equation}\n\n\n\n\nThe operator $(1-\\partial_{xx}^2)^{-1}$ can be represented as a convolution operator:\n$$((1-\\partial_{xx}^2)^{-1} )f(x)=\\int_{\\mathbb R} G(x-y)f(y)dy,\\ \\ f\\in L^2(\\mathbb R),$$ where \n$$ G(x)=\\frac{e^{-|x|}}{2},\\ \\ x\\in\\mathbb{R}.$$\nFrom equation (\\ref{CL}) we get\n\\begin{eqnarray}\\label{advective}\nu_t+uu_x&=&-\\partial_x\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\nonumber\\\\\n&=&-\\partial_x\\int_{\\mathbb R}\\frac{1}{2}\\exp(-|x-y|)\\left(u^3(y,t)+\\frac{u^2}{2}(y,t)+\\frac{u_y^2}{2}(y,t) \\right)dy\n\\end{eqnarray}\n in $C([0,T);H^1(\\mathbb R) )$.\n\n\nBy differentiation with respect to $x$ we get \n\\begin{eqnarray*}\nu_{tx}+u_x^2+uu_{xx}&=&-\\partial_x^2\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\\\\n&=& (Q^2-Id)\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\\\\\n&=& \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)-\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right),\n\\end{eqnarray*}\nand therefore \n\\begin{equation}\\label{eb}\n u_{tx}+uu_{xx}=u^3+\\frac{u^2}{2}-\\frac{u_x^2}{2}-\\left(G\\star \\left(u^3+\\frac{u^2}{2}+\\frac{u_x^2}{2}\\right)\\right)\n\\end{equation}\nin the space $C([0,T);L^2(\\mathbb R) ).$\n\n\nWe prove now the following blow-up result for (\\ref{mCH})-(\\ref{dado}):\n\n\\begin{lemma}(Steepening Lemma \\cite{Camassa,Crisan}) Suppose the initial profile of velocity $u_0\\in H^{3}(\\mathbb R)$, has an inflection point at $x=\\overline{x}$ to the right of its maximum. Moreover we assume that $u_x(\\overline x,0)<-\\sqrt{2(M^2+2M^3)}$ , where $M $ is the constant defined in (\\ref{tbound}).Then, the negative slope at the inflection point will become vertical in finite time.\n\\end{lemma}\n\\begin{proof}\nConsider the evolution of the slope at the inflection point $t \\rightarrow \\overline{x}(t)$ that starts at time $0$ from an inflection point $x=\\overline{x}$ of $u_0(x)$ to the right of its maximum so that\n$$ \\rho_0=u_x(\\overline{x}(0),0)< \\infty.$$\n\nDefine $\\rho_t=u_x(\\overline{x}(t),t), \\ \\ t\\geq 0$. \n\nEquation (\\ref{eb}) yields an equation for the evolution of $t\\rightarrow \\rho_t$. Namely, by using that $u_{xx}(\\overline{x}(t),t)=0$ $(u(t)\\in H^3(\\mathbb R)\\subset C^2(\\mathbb R))$ and (\\ref{tbound}) one finds \n\n\\begin{eqnarray}\n\\frac{d{\\rho}_t}{dt} &=& -\\frac{{\\rho}_t^2}{2}+u^3(\\overline{x}(t),t)+\\frac{u^2}{2}(\\overline{x}(t),t)\\nonumber\\\\ &-&\\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)\\left(u^3(y,t)+\\frac{u^2}{2}(y,t)+\\frac{u_y^2}{2}(y,t) \\right)dy\\nonumber\\\\ \n&\\leq & -\\frac{{\\rho}_t^2}{2} +M^3+ M^2+ \\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)(u_{-}^3(y,t))dy\\nonumber\\\\ \n& \\leq &-\\frac{{\\rho}_t^2}{2} + M^2+2M^3,\n\\end{eqnarray}\nwhere $u_-$ stands for the negative part of $u$. Here, we used that $ \\int_{\\mathbb R}\\frac{1}{2}\\exp(-|\\overline{x}(t)-y|)dy=1. $\n\nLet $\\tilde{\\rho}_t$ be the solution of the equation \n\\begin{equation}\\label{edo}\n\\frac{d{\\tilde{\\rho}_t}}{dt}=-\\frac{\\tilde{\\rho}_t^2}{2} + \\overline{M}, \\ \\ \\tilde{\\rho}_0=\\rho_0,\n\\end{equation}\nwhere $\\overline{M}= M^2+2M^3.$\n\nObserve that \n\\begin{eqnarray*}\n\\frac{d}{dt}\\left((\\rho_t-\\tilde{\\rho}_t)e^{\\frac{1}{2}\\int_0^t(\\rho_s+\\tilde{\\rho}_s)ds}\\right)\\leq 0, \\ \\ \\rho_0-\\tilde{\\rho}_0=0,\n\\end{eqnarray*}\ntherefore $\\rho_t\\leq \\tilde{\\rho}_t$ for all $t>0$ for which both are well defined. \n\nIntegrating (\\ref{edo}) we obtain\n\\begin{eqnarray}\n\\coth^{-1}\\left(\\frac{\\tilde{\\rho}_t}{\\sqrt{2\\overline{M}}}\\right)=\\frac{1}{2} \\ln \\left| \\frac{\\tilde{\\rho}_t+\\sqrt{2\\overline{M}}}{\\tilde{\\rho}_t-\\sqrt{2\\overline{M}}}\\right|=\\sqrt{2\\overline{M}}\\frac{t}{2}+k,\n\\end{eqnarray} \nwhere $k= \\ln \\left| \\frac{{\\rho}_0+\\sqrt{2\\overline{M}}}{{\\rho}_0-\\sqrt{2\\overline{M}}}\\right|<0$. Since $ \\lim_{t\\rightarrow -2k\/ \\sqrt{2\\overline{M}}} \\tilde{\\rho}_t=-\\infty$ it follows that there is a time $\\tau\\leq -2k\/ \\sqrt{2\\overline{M}}$ by which the slope $\\rho_t = u_x(\\overline{x}(t),t)$ becomes negative and vertical.\n\\end{proof}\n\n\n\n \n \n \n\n\n\n\n\n\\section{Weak formulation}\\label{weak formulation}\n\n\t\tFor a traveling wave $u(x,t)=\\phi(\\xi)$, with $\\xi=x-ct$, the equation (\\ref{mCH}) is equivalent to\n\t\t\t\\begin{equation}\n\t\t\t\\label{169}\n\t\t\t-c\\phi^{'} +c\\phi^{'''} = \\phi\\phi^{'''} + 2\\phi^{'}\\phi^{''} - 3\\phi^{2}\\phi^{'},\n\t\t\t\\end{equation}\nwhere $'$ denotes the derivative with respect to the variable $\\xi$. \n\t\t\t\n\n\t\t\tAfter an integration with respect to the variable $\\xi$ we can rewrite (\\ref{169}) as follows\n\t\t\t\\begin{equation}\n\t\t\t\\label{171}\n\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a.\n\t\t\t\\end{equation}\nfor some constant of integration $a\\in\\mathbb{R}$. We see that for (\\ref{171}) to make any sense it is sufficient to require $\\phi\\in H^{1}_{loc}(\\mathbb{R})$. Thus, the following definition is plausible.\n\n\\begin{definition}\n\\label{205}\nA function $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ is a traveling wave of the mCH equation if $\\phi$ satisfies (\\ref{171}) in distribution sense for some $a\\in\\mathbb{R}$.\n\\end{definition}\n\n\n\\section{Classification of traveling waves}\\label{classification}\n\nTheorems \\ref{classificacao1} and \\ref{classificacao2} classify all bounded traveling waves $\\phi\\in H^{1}_{loc}(\\mathbb{R})$, of (\\ref{mCH}). The bounded traveling waves are parametrized by their maxima, minima and speeds of propagation.\n\n\\begin{theorem}\n\\label{classificacao1}\nConsider $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ such that $\\Re(z)=-(m+M)\/2$ satisfying the equation $m^{2} + M^{2} -|z|^{2} + mM -2c = 0$. Any bounded traveling wave of the mCH equation belongs to one of the following categories:\n\n$(a)$ Smooth periodic: If $mc \\geqslant M\\}$ there is a corresponding cuspon or peakon according to $(b)$ or $(c)$ and $(b^{'})$ or $(c^{'})$. A countable number of cuspons and peakons corresponding to points $(m,M,c)$ that belong to the same hyperboloid of two sheet, may be joined at their crests to form a composite wave $\\phi$. If the Lebesgue measure $\\mu(\\phi^{-1}(c)) = 0$, then $\\phi$ is a traveling wave of (\\ref{mCH}).\n\n$(e)$ Stumpons: For $a=2c^{3} -2c^{2}$ the equation (\\ref{303}) jointly with (\\ref{304}) contains the points $(m,M,\\Im(z)) = (c,c, \\pm\\sqrt{4c^{2}-2c})$. These equations correspond only to cuspons. Let $\\phi$ be a composite wave obtained by joining countably many of these cuspons with each other and with intervals where $\\phi\\equiv c$. Then $\\phi$ is a traveling wave of (\\ref{mCH}) even if $\\mu(\\phi^{-1}(c)) >0$. \n\\end{theorem}\n\n\\begin{theorem}\n\\label{classificacao2}\nConsider $z= -m-r-M$ with $r\\in\\mathbb{R}$ satisfying the equation $r^{2} + m^{2} + M^{2} +rm +rM + mM -2c = 0$. Any bounded traveling wave of the mCH equation belongs, less than an alternation of $z$ for $r$, to one of the following categories:\n\n$(a)$ Smooth periodic: If $z0$, the equation\n\t\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\t\\label{203}\n\t\t\t\t\t\t\t\t\t2a = -(M+m)r^{2} - (m+r)M^{2} - (M+r)m^{2} -2mrM\n\t\t\t\t\t\t\t\t\t\\end{equation}\ndescribes three planes intersecting in space $(m,M,r)$ with $r\\in\\mathbb{R}$ satisfying, less than a change of variables, the ellipsoid\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{204}\n\t\t\t\t\t\\frac{m^{2}}{\\alpha^{2}} + \\frac{M^{2}}{\\beta^{2}} + \\frac{r^{2}}{\\gamma^{2}} = 1,\n\t\t\t\t\t\\end{equation}\nwhere $\\alpha =\\beta = 2\\sqrt{c}$ and $\\gamma = \\sqrt{c}$. For any $(m,M,r,c)\\in \\{zr\\geqslant m >c \\geqslant M\\}$ there is a corresponding cuspon or peakon according to $(d)-(g)$ and $(d^{'})-(g^{'})$. A countable number of cuspons and peakons corresponding to points $(m,M,r,c)$ that belong to the same ellipsoid, may be joined at their crests to form a composite wave $\\phi$. If the Lebesgue measure $\\mu(\\phi^{-1}(c)) = 0$, then $\\phi$ is a traveling wave of (\\ref{mCH}).\n\n$(i)$ Stumpons: For $a=2c^{3} -2c^{2}$ the equation (\\ref{203}) jointly with (\\ref{204}) contains the points $(m,M,r) = (c,c,-c\\pm\\sqrt{-2c^{2}+2c})$. These equations correspond only to cuspons. Let $\\phi$ be a composite wave obtained by joining countably many of these cuspons with each other and with intervals where $\\phi\\equiv c$. Then $\\phi$ is a traveling wave of (\\ref{mCH}) even if $\\mu(\\phi^{-1}(c)) >0$. \n\\end{theorem}\n\n\t\t\tIn addition to Theorems \\ref{classificacao1} and \\ref{classificacao2} is possible to characterize the existence or no of some unbounded traveling waves of the mCH equation.\n\t\t\t\n\\begin{theorem}\n\\label{classificacao3}\nIf the polynomial $P(\\phi)$ in (\\ref{188}) has only a real zero, say $m$, then the mCH equation has not bounded solutions $\\phi$ to $c0$, then $a=2c^{3} -2c^{2}$.\n\n\n\\noindent\\noindent $(TW3)$ $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$.\n\\end{lemma}\n\\begin{proof}\n\t\t\t\tApplying Lemma \\ref{lema1}, with $v=\\phi -c$ and $p(v) = 2\\phi^{3} -2c\\phi -a$, we see that \n\t\t\t\t\t\t\t\t$$(\\phi -c)^{k}\\in C^{j}(\\mathbb{R})\\ \\ \\text{para}\\ \\ k\\geqslant 2^{j}.$$\n\n\t\t\tDefine $C=\\phi^{-1}(c)$. Since $\\phi$ is continuous, $C$ is a closed set. Since every open set is a countable union of disjoint open intervals, there are disjoint open intervals $E_{i}$, $i\\geqslant 1$, such that $\\mathbb{R}\\setminus C = \\bigcup_{i=1}^{\\infty}E_{i}$. By construction, it follows that $(TW1)$ is satisfied.\n\t\t\t\n\t\t\tNow let's prove $(TW2)$. To get this, we consider $E_{i}$ one of these open intervals. Since $\\phi$ is $C^{\\infty}$ in $E_{i}$, we infer that (\\ref{171}) holds pointwise in $E_{i}$. So, multiplying (\\ref{171}) by $\\phi^{'}$ we obtain\n\t\t\t$$-2c\\phi\\phi^{'} +2c\\phi^{'}\\phi^{''} +2\\phi^{3}\\phi^{'} = 2\\phi\\phi^{'}\\phi^{''} + (\\phi^{'})^{3} + a\\phi^{'}.$$\nEquivalently, we can rewrite the above equation as\n\t\t\t\t\t\t$$-c(\\phi^{2})^{'} + c[(\\phi^{'})^{2}]^{'} + \\left(\\frac{\\phi^{4}}{2}\\right)^{'} = [(\\phi^{'})^{2}\\phi]^{'} +a\\phi^{'}$$\nand from an integration with respect to the parameter $\\xi$, we deduce\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{180}\n\t\t\t\t(\\phi^{'})^{2}(c-\\phi) = \\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d_{i},\\ \\xi\\in E_{i}\n\t\t\t\t\\end{equation}\nfor some constants of integration $d_{i}$. Dividing (\\ref{180}) by $c-\\phi$ we have $(\\ref{178})$. That $\\phi\\rightarrow c$ at the finite\nendpoints of $E_{i}$ it follows from the continuity of $\\phi$ and $(TW1)$. This proves $(i)$ of $(TW2)$.\n\n\t\t\t\tThe left-hand side of $(\\ref{171})$ is in $L^{1}_{loc}(\\mathbb{R})$. Hence, $((\\phi -c)^{2})^{''}\\in L^{1}_{loc}(\\mathbb{R})$ and so follows $(TW3)$.\n\t\t\t\t\n\t\t\t\tTo show $(ii)$ of $(TW2)$, let us assume $\\mu(C)>0$. Since $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ and $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we obtain from Lemma \\ref{lema2} and Lemma \\ref{lema3} respectively,\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{181}\n\t\t\t\t\\phi^{'}(\\xi) = 0\\ \\ \\text{and}\\ \\ ((\\phi -c)^{2})^{''}(\\xi) = 0\\ \\text{a.e. in}\\ C.\n\t\t\t\t\\end{equation}\nBy the fact that $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we have that (\\ref{171}) holds a.e. in $\\mathbb{R}$, i.e., \n\t\t\t\t$$(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{a.e. in}\\ \\mathbb{R} .$$\nIn particular, the above equation occurs a.e. in $C$. Then, from (\\ref{181}) it follows that\n\t\t\t\t$$2\\phi^{3} -2c\\phi = a\\ \\ \\text{a.e. in}\\ C.$$\nSince $\\mu(C)>0$ and $\\phi\\equiv c$ in $C$, we conclude that $a=2c^{3} -2c^{2}$. This shows that all the traveling waves of equation (\\ref{mCH}) satisfy $(TW1)-(TW3)$.\n\n\n\t\tReciprocally, note that from the differentiation of (\\ref{178}) we have\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{182}\n\t\t\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{in}\\ E\\ \\dot{=}\\ \\bigcup_{i=1}^{\\infty} E_{i}.\n\t\t\t\t\t\\end{equation}\nIf $\\mu(C) = 0$, then (\\ref{182}) implies that\n\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\\label{183}\n\t\t\t\t\t(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a\\ \\ \\text{a.e. in}\\ \\mathbb{R} .\n\t\t\t\t\t\\end{equation}\nSince $((\\phi -c)^{2})^{''}\\in L^{1}_{loc}(\\mathbb{R})$ by $(TW3)$, (\\ref{183}) implies (\\ref{171}), from which we conclude that $\\phi$ is a traveling wave solution of (\\ref{mCH}).\n\t\t\n\t\t\tTo complete the demonstration it remains to show that (\\ref{183}) also occurs in the case where $\\mu(C)>0$. Suppose $\\mu(C)>0$. Since $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ and $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$, we obtain from Lemmas \\ref{lema2} and \\ref{lema3} that equation (\\ref{181}) holds. From $(ii)$ of $(TW2)$ we have $a=2c^{3}-2c^{2}$. So, since $\\phi\\equiv c$ in $C$, we deduce for a.e. $\\xi$ in $C$,\n\t\t\n\t\t\t$$(\\phi^{'})^{2} + 2\\phi^{3} -2c\\phi = ((\\phi -c)^{2})^{''} +a.$$\nJointly with (\\ref{182}), what we have just shown implies (\\ref{183}) and the result follows.\n\\end{proof}\n\t\t\t\t\t\t\n\t\t\tTo prove Theorems \\ref{classificacao1} and \\ref{classificacao2}, we will show that the set of bounded functions satisfying $(TW1)-(TW3)$ in Lemma \\ref{lema4} consists exactly of the waves presented in the statements of these theorems.\n\t\t\t\t\t\t\n\t\t\tSuppose that $\\phi$ satisfies $(TW1)-(TW2)$. From $(TW1)$ and $(TW2)$ there is a countable number of $C^{\\infty}$ wave segments separated by a closed set $C$ such that each wave segment $\\phi$ satisfies\n\t\t\t\t\\begin{eqnarray}\n\t\t\t\t\\label{184}\n\t\t\t\t&& (\\phi^{'})^{2} = F(\\phi)\\ \\text{for}\\ \\xi\\in E,\\ \\ F(\\phi) = \\frac{\\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d}{c-\\phi}\\nonumber\\\\\n\t\t\t\t&&\\text{and}\\ \\ \\phi\\rightarrow c\\ \\text{at any finite endpoint of}\\ E\n\t\t\t\t\\end{eqnarray}\nfor some interval $E$ and constants $a,d$. If we are able to find all solutions of $(\\ref{184})$ for different intervals $E$ and different values of $a$ and $d$, then we can join solutions defined at intervals in which the union is $\\mathbb{R}\\setminus C$ for some closed set $C$ of null measure. We will have the function that we constructed defined in $\\mathbb{R}$, satisfying $ (TW1) $ and $ (TW2) $ if, and only if, all the wave segments satisfy (\\ref{184}) with the same constant $a$. In addition, if for $a=2c^{3}-2c^{2}$ we allow $\\mu(C)>0$, this procedure will give us all the functions satisfying $(TW1)$ and $(TW2)$. Let us show that these functions we have just constructed belong to $H^{1}_{loc}(\\mathbb{R})$, satisfy $(TW3)$ and are exactly the waves in Theorems \\ref{classificacao1} and \\ref{classificacao2}.\n\nTo study (\\ref{184}), we first observe that for general equations of the form\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{185}\n\t\t\t\t(\\phi^{'})^{2} = F(\\phi),\n\t\t\t\t\\end{equation}\nwhere $F:\\mathbb{R}\\longrightarrow\\mathbb{R}$ is a rational function, Lenells in \\cite{Lenells} explored the qualitative behavior of solutions to (\\ref{185}) at points where $F$ has a zero or a pole. In resume, if $F(\\phi)$ has a simple zero at $\\phi = m$, so that $F^{'}(m) >0$, then the solution $\\phi$ of (\\ref{185}) satisfies\n\t\t\t\\begin{equation}\n\t\t\t\\label{eq polinomial}\n\t\t\t\\phi(\\xi) = m+\\frac{1}{4}(\\xi-\\eta)^{2}F^{'}(m) + O((\\xi-\\eta)^{4})\\ \\ \\text{at}\\ \\ \\xi\\uparrow\\eta,\n\t\t\t\\end{equation}\nwhere $\\phi(\\eta)=m$. Assuming that $F(\\phi)$ has a double zero at $m$, so that $F^{'}(m) = 0$ and $F^{''}(m)>0$, we get\n\t\t\t\\begin{equation}\n\t\t\t\\label{eq exp}\n\t\t\t\\phi(\\xi) - m \\approx \\alpha\\textit{exp}(-\\xi\\sqrt{F^{''}(m)})\\ \\ \\text{as}\\ \\ \\xi\\rightarrow \\infty\n\t\t\t\\end{equation}\nfor some constant $\\alpha$, then $\\phi \\downarrow m$ exponentially as $\\xi\\rightarrow \\infty$. So, whenever $F$ has two simple zeros $m, M$ and $F(\\phi)>0$ for $m<\\phi 0$ for $m<\\phi 0$ for $m<\\phi$, then no bounded solution $\\phi$ exists.\n\nIf we include all the functions $\\phi\\in H^{1}_{loc}(\\mathbb{R})$ that are solutions of (\\ref{185}), we will expand the class of solutions, so that they will have another qualitative behavior. If $\\phi$ approaches a simple pole $\\phi =c$ of $F$, then, if $\\phi(\\xi_{0}) = c$,\n\t\t\t\\begin{equation}\n\t\t\t\\label{funcao com polo}\n\t\t\t\\phi(\\xi) -c = \\alpha |\\xi-\\xi_{0}|^{\\frac{2}{3}} + O((\\xi-\\xi_{0})^{\\frac{4}{3}})\\ \\ \\text{as}\\ \\ \\xi\\rightarrow \\xi_{0}\n\t\t\t\\end{equation}\nfor some constant $\\alpha$, so cusped solutions occur. Also, peakons occur when the evolution of $\\phi$ according to (\\ref{185}) suddenly changes direction, that is, $\\phi^{'}\\mapsto -\\phi^{'}$.\n\n\t\t\tApplying the above discussion to the particular case\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{188}\n\t\t\t\tF(\\phi) = \\frac{P(\\phi)}{c-\\phi},\\ \\ \\text{with}\\ \\ P(\\phi) = \\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi +d , \n\t\t\t\t\\end{equation}\nwe can classify all the bounded solutions of (\\ref{184}). The proof is basically an inspection of all possible distributions of zeros and poles of $F$. Note that $P(\\phi)$ is a fourth-degree polynomial with real coefficients, then it has one, two or four real zeros. Moreover,\n\n\t\t\tTo prove Theorem \\ref{classificacao3} suppose that $P(\\phi)$ has only a real zero. This zero has double multiplicity because the polynomial $P(\\phi)$ is fourth-degree. Let $m\\in\\mathbb{R}$ this double zero and $z,\\overline{z}$ the others zeros in $\\mathbb{C}\\setminus\\mathbb{R}$. For this distribution of zeros, we can write\n\t\t\t$$P(\\phi) = -\\frac{1}{2}(\\phi -m)^{2}(\\phi -z)(\\phi -\\overline{z})$$\nand so, we obtain $F^{'}(m) = 0$ and\n\t\t\t\t$$F^{''}(m) = - \\frac{|m-z|^{2}}{c-m}.$$\nThus, we can see that the solution $\\phi$ of the mCH equation satisfies $(\\ref{eq exp})$ to $c0 $ and a similar result is obtained.\n\n\t\t\tSuppose that there are two simple real zeros, say $m$ and $M$, and we going to prove Theorem \\ref{classificacao1}. We write for $z\\in\\mathbb{C}\\setminus\\mathbb{R}$\n\t\t\t\t\t$$P(\\phi) = \\frac{1}{2}(M-\\phi)(\\phi -m)(\\phi -z)(\\phi -\\overline{z})$$ \nand comparing the coefficients of (\\ref{188}) we obtain that $\\Re(z)= -(M+m)\/2$ with $z\\in\\mathbb{C}\\setminus\\mathbb{R}$ satisfying the hyperboloid of two sheet in $(m, M, \\Im(z))$,\n\t\t\t\t\\begin{equation}\n\t\t\t\t\\label{hiperboloide}\n\t\t\t\t\\frac{5}{4}m^{2} +\\frac{5}{4}M^{2} -\\Im(z)^{2} + \\frac{3}{2}mM -2c = 0.\n\t\t\t\t\\end{equation}\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\nIn addition, we have\n\t\t\t\t\t\t$$F^{'}(m) = \\frac{1}{2}\\cdot\\frac{(M-m)|m-z|^{2}}{c-m}$$\nand, from (\\ref{eq polinomial}), we deduce from that there are periodic solutions $C^{\\infty}$ for (\\ref{184}) if and only if $m0$ for $m<\\phi 0$. Gluing of all possible ways the solutions corresponding to the same ellipsoid we obtain that all bounded functions satisfying $(TW1)$ and $(TW2)$ for this case are the waves of Theorem \\ref{classificacao2}.\n\t\t\t\n\t\t\tTo finalize the proof of Theorems \\ref{classificacao1} and \\ref{classificacao2} is enough to show that these waves belong to $H^{1}_{loc}(\\mathbb{R})$ and satisfy $(TW3)$. This fact is ensured by Lemma \\ref{lema importante} but in order to prove this lemma we need to establish some technical results and notation.\n\t\t\t\t\t\t\n\t\t\t\\begin{definition}\nA function $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is said to be of bounded variation if\n\t\t\t\t\t\t$$\\sup\\sum_{i=1}^{N}|f(t_{i})-f(t_{i-1})|<+\\infty ,$$\nwhere the supremum is taken over all natural number $N$ and all choices of partitions $\\{t_{i}\\}_{i=1}^{N}$ such that $a=t_{0}< t_{1}< \\cdot\\cdot\\cdot < t_{N} = b$. $BV(I)$ denote the set of functions $f:I\\longrightarrow\\mathbb{R}$ of bounded variation on $I$. $BV_{loc}(\\mathbb{R})$ denote the space of functions $f:\\mathbb{R}\\longrightarrow\\mathbb{R}$ of bounded variation for all compact intervals $I\\subset\\mathbb{R}$.\n\\end{definition}\n\n\\begin{definition}\nA function $f:X\\longrightarrow\\mathbb{R}$ defined on some set $X\\subset\\mathbb{R}$ is an $N$-function if $f$ maps sets of measure zero to sets of measure zero.\n\\end{definition}\n\n\\begin{lemma}\n\\label{lema0}\nA function $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is absolutely continuous if and only if $f\\in W^{1,1}_{loc}(I)$.\n\\end{lemma}\n\n\\begin{lemma}\n\\label{lema6}\nLet $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, be a continuous function of bounded variation. Then, $f$ is absolutely continuous if and only if $f$ is an $N$-function.\n\\end{lemma}\n\t\t\n\\begin{lemma}\n\\label{lema7}\nIf $f:I\\longrightarrow\\mathbb{R}$, $I=[a,b]\\subset\\mathbb{R}$, is continuous and $|f|$ is absolutely continuous, the $f$ is absolutely continuous.\n\\end{lemma}\t\t\t\n\t\t\n\t\tThe proof of Lemmas \\ref{lema0}$-$\\ref{lema7} can be found on book \\cite{Rudin}.\n\t\t\n\\begin{lemma}\n\\label{lema importante}\nAny bounded function $\\phi$ satisfying (TW1) and (TW2) belongs to $H^{1}_{\\text{loc}}(\\mathbb{R})$ and satisfies (TW3).\n\\end{lemma}\n\\begin{proof}\n\t\t\tConsider $E_{i}$ and $C$ as in $(TW1)$. We initially prove that $\\phi$ is an $N$-function. To get this, we write\n\t\t\t\t\t\t$$\\phi(N) = \\phi(N\\cap C) \\cup \\left(\\bigcup_{i=1}^{+\\infty} \\phi(N\\cap E_{i})\\right)$$\nfor any set $N$ of null measure and then we use Lemma \\ref{lema6} for $\\phi$ restricted to $N\\cap E_{i}$ and also observe that $\\phi(N\\cap C) = \\{c\\}$.\n\n\t\t\tNow, we will show that $\\phi\\in BV_{\\text{loc}}(\\mathbb{R})$. Consider $\\{i_{k}\\}_{k=1}^{+\\infty}$, or simply $\\{j\\}_{j=1}^{+\\infty}$, a subsequence and $I$ a compact subinterval such that\n\t\t\t$$I\\subset C \\cup \\left(\\bigcup_{j=1}^{+\\infty} E_{j}\\right)\\ \\ \\text{and}\\ \\ \\sum_{j=1}^{+\\infty}|E_{j}|\\ <+\\infty,$$\nwhere $|E_{j}|$ denote the length of the interval $E_{j}$.\n\n\t\t\tSuppose, for example, the polynomial $P_{j}(\\phi) = \\phi^{2}(c-\\phi^{2}\/2) +a\\phi + d_{j}$ with four real zeros, say $z_{j}, r_{j}, m_{j}$ and $M_{j}$, and the case $z_{j} < r_{j} \\leqslant m_{j}\\leqslant \\phi < c \\leqslant M_{j}$. From $(\\ref{178})$, we see that\n\t\t\t\\begin{eqnarray}\n\t\t\t\\label{equacao 30}\n\t\t\t|E_{j}| = 2\\displaystyle\\int_{m_{j}}^{c} \\frac{\\sqrt{c-\\phi}}{\\sqrt{\\Big{|}\\phi^{2}\\left(c-\\frac{\\phi^{2}}{2}\\right) +a\\phi + d_{j}\\Big{|}}}\\ d\\phi.\n\t\t\t\\end{eqnarray}\n\t\t\t\n\t\t\tNote that $P_{j}(c)\\geqslant 0$ for all $j$. Moreover, the derivative of $P_{j}$ is independent of $j$ and since $P_{j}$ has four zeros we have that the $b_{j}$'s form a bounded set. Therefore, we obtain\n\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\\label{equacao 31}\n\t\t\t\t\t\t0\\leqslant P_{j}(\\phi) \\leqslant D(c-m_{j}),\\ \\ m_{j}\\leqslant \\phi\\leqslant c,\\ j\\geqslant 1,\n\t\t\t\t\t\t\\end{equation}\nfor some positive real constant $D$ independent of $j$.\n\n\t\tSo, using $(\\ref{equacao 30})$ and $(\\ref{equacao 31})$ it follows that\n\t\t\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\t\t\\label{equacao 32}\n\t\t\t\t\t\t\t\t|E_{j}| \\geqslant D_{0}(c-m_{j}),\\ \\ j\\geqslant 1,\n\t\t\t\t\t\t\t\t\\end{equation}\nwhere $D_{0}=4\/3\\sqrt{D}$. Thus, $\\phi\\in BV_{\\text{loc}}(\\mathbb{R})$ because the total variation of $\\phi$ in $I$ is bounded above by\n\t\t\t\t\t\t\t$$2\\sum_{j=1}^{+\\infty}\\left( c- \\min_{\\xi\\in E_{j}}\\phi(\\xi)\\right) = 2\\sum_{j=1}^{+\\infty} (c-m_{j}) \\leqslant \\frac{2}{D_{0}} \\sum_{j=1}^{+\\infty}|E_{j}|\\ <+\\infty .$$\n\t\t\t\t\t\t\t\n\t\tMoreover, $\\phi$ is continuous on $I$ (see \\cite{Lenells}, page 417) and so $\\phi$ is continuous in $\\mathbb{R}$. Hence, since $\\phi$ is an $N$-function of bounded variation, we have by Lemma \\ref{lema6} that $\\phi$ is absolutely continuous. This means by Lemma \\ref{lema0} that the distributional derivative of $\\phi$ is in $L^{1}_{\\text{loc}}(\\mathbb{R})$, $\\phi$ is differentiable in the classical sense at almost all points and the two derivatives coincide. \n\t\t\n\t\tTo conclude that $\\phi \\in H^{1}_{loc}(\\mathbb{R})$, we will show that $\\phi^{'}\\in L^{2}_{loc}(\\mathbb{R})$. Using $(\\ref{178})$ and $(\\ref{equacao 31})$, we get\n\t\t\t\t\t$$ \\int_{E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\sqrt{D}\\sqrt{c-m_{j}} \\int_{m_{j}}^{c} \\displaystyle\\frac{1}{\\sqrt{c-\\phi}}\\ d\\phi = 2\\sqrt{D}(c-m_{j}),\\ \\ j\\geqslant 1$$\nand so, by $(\\ref{equacao 32})$,\n\t\t\t$$\\int_{E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\frac{3}{2}D|E_{j}|,\\ \\ j\\geqslant 1.$$\n\t\t\t\n\t\tThus, we obtain\n\t\t\t\t\t$$\\int_{\\bigcup_{j=1}^{+\\infty}E_{j}} (\\phi^{'})^{2}\\ d\\xi \\leqslant \\frac{3}{2}D \\sum_{j=1}^{+\\infty} |E_{j}|\\ <+\\infty.$$\nAs Lemma \\ref{lema2} implies that $\\phi^{'}=0$ a.e. on $C=\\phi^{-1}(c)$, it follows from the above inequality that $\\phi \\in H^{1}_{loc}(\\mathbb{R})$.\n\t\t\n\t\tFinally, it remains to prove that $(\\phi - c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$. To get this, we will show that $|(\\phi-c)\\phi^{'}|$ is a bounded variation continuous $N$-function.\n\t\t\n\t\tSince $\\phi^{'}\\in L^{2}_{loc}(\\mathbb{R})$ and $\\phi^{'}= 0$ a.e. on $C$, we have that $(\\phi - c)\\phi^{'} = 0$ a.e. on $C$. Therefore, from $(\\ref{178})$,\n\t\t\t\t\t\t\\begin{equation}\n\t\t\t\t\t\t\\label{equacao 35}\n\t\t\t\t\t\t|(\\phi -c)\\phi^{'}| = \\sqrt{P_{j}(\\phi)}\\sqrt{c-\\phi},\\ \\ c\\in E_{j},\\ \\ j\\geqslant 1\n\t\t\t\t\t\t\\end{equation}\nand, then, $|(\\phi -c)\\phi^{'}|$ is smooth on any interval $E_{j}$ and $|(\\phi -c)\\phi^{'}| \\rightarrow 0$ at finite endpoint of $E_{j}$. So, on each $E_{j}$, $|(\\phi -c)\\phi^{'}|$ is a symmetric function with respect to the midpoint of $E_{j}$ with two humps. Thus, by $(\\ref{equacao 31})$ and $(\\ref{equacao 32})$, the total variation of $|(\\phi -c)\\phi^{'}|$ in $I$ is bounded by\n\t\t$$4\\sum_{j=1}^{+\\infty} \\sup_{\\xi\\in E_{j}} |(\\phi(\\xi) -c)\\phi^{'}(\\xi)| = 4\\sum_{j=1}^{+\\infty} \\sup_{\\xi\\in E_{j}}\\sqrt{P_{j}(\\phi)} \\sup_{\\xi\\in E_{j}}\\sqrt{c-\\phi} \\leqslant \\frac{8}{3} \\sum_{j=1}^{+\\infty} |E_{j}|\\ < +\\infty.$$\n\n\t\t\tWe see from $(\\ref{equacao 35})$ that $|(\\phi -c)\\phi^{'}|$ is continuous. Now writing,\n\t\t\t\t\t$$|(\\phi -c)\\phi^{'}|(N) = |(\\phi -c)\\phi^{'}|(N\\cap C) \\cup \\left(\\bigcup_{j=1}^{+\\infty} |(\\phi -c)\\phi^{'}|(N\\cap E_{j})\\right)$$\nand using Lemma \\ref{lema6} with $|(\\phi -c)\\phi^{'}|$ restricted to $N\\cap E_{j}$, we obtain that $|(\\phi -c)\\phi^{'}|$ is an $N$-function.\n\t\t\t\n\t\t\tAgain, we apply Lemma \\ref{lema6} to get that $|(\\phi -c)\\phi^{'}|$ is absolutely continuous. Moreover, since $(\\phi -c)\\phi^{'}$ is smooth on $E_{j}$ and, for $\\xi\\in C$, $(\\phi(\\xi) -c)\\phi^{'}(\\xi) = |(\\phi -c)\\phi^{'}|(\\xi) = 0$, we have that $(\\phi -c)\\phi^{'}$ is continuous. Thus, Lemma \\ref{lema7} implies that $(\\phi -c)\\phi^{'}$ is absolutely continuous.\n\t\t\t\n\t\t\tWe conclude, since $(\\phi - c)^{2}$ is absolutely continuous, that\n\t\t\t\t\t\t\t$$\\left[(\\phi -c)^{2}\\right]^{'} = 2(\\phi - c)\\phi^{'}\\ \\in W^{1,1}_{loc}(\\mathbb{R})$$\nand, consequently, $(\\phi -c)^{2}\\in W^{2,1}_{loc}(\\mathbb{R})$.\n\\end{proof}\t\n\t\t\n\\begin{remark}\nIn Theorems \\ref{classificacao1} and \\ref{classificacao2} the solutions are defined on the whole real line. Here we restrict the solution to the interval between two crests in the case of periodic waves, and to the part left or right of the crest in the case of decaying waves. The interval $E$ is accordingly defined to be a finite or half-infinite interval.\n\\end{remark}\t\t\t\n\t\t\t\n\t\t\t\n\\section{Dependence on parameters}\\label{regularity}\n\n\t\t\tIn addition to Theorems \\ref{classificacao1} and \\ref{classificacao2}, it is possible to characterize the relation between the traveling waves of (\\ref{mCH}) and the parameters $m$, $M$ and $c$. Namely, such traveling waves are continuously dependents on these parameters. \n\n\\begin{theorem}\n\\label{teorema2}\nLet $(m_{i}, M_{i}, c_{i})$, $i\\geqslant 1$, and $(m,M,c)$ be such that there are corresponding traveling waves of (\\ref{mCH}) according to $(a)-(c)$ of Theorem \\ref{classificacao1} or $(a)-(g)$ of Theorem \\ref{classificacao2}. Let $\\phi_{i}$, $i=1,2,...$ and $\\phi$ be these traveling waves translated so that they all have crests at $\\xi =0$. If $(m_{i}, M_{i}, c_{i})\\rightarrow (m,M,c)$, then $\\phi_{i}\\rightarrow \\phi$ in $H^{1}_{loc}(\\mathbb{R})$. In particular, we have uniform convergence on compact sets.\n\\end{theorem}\n\\begin{proof}[Sketch of proof]\n\t\t\tAssume $(m_{i}, M_{i}, c_{i})$, $i\\geqslant 0$ and $(m, M, c)$ as in the statement of Theorem \\ref{classificacao1} or \\ref{classificacao2} and $\\phi_{i}$ and $\\phi$ the correspond traveling waves with crests at zero.\n\t\t\t\n\t\t\tNote that to the mCH equation we obtain from (\\ref{178}) and (\\ref{179}) that $\\phi$ and $\\phi_{i}$ are given implicitly by\n\t\t\t\\begin{equation}\n\t\t\t\\label{6.2}\n\t\t\t\\xi=\\left\\{\\begin{array}{ll}\n\t\t\t\\xi_{0} + \\int_{\\phi_{0}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\phi^{'} >0&\\\\\n\t\t\t\\xi_{0} - \\int_{\\phi_{0}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\phi^{'} <0,&\n\t\t\t\\end{array}\\right.\n\t\t\t\\end{equation}\nwhere $\\phi(\\xi_{0})=\\phi_{0}$. Moreover, by choosing $\\xi_{0}=0$, we have $\\phi_{0} = \\max_{\\xi\\in\\mathbb{R}}\\phi(\\xi) = \\min\\{M, c\\}$ and\n\t\t\t$$\\xi=\t- \\int_{\\min\\{M, c\\}}^{\\phi}\\frac{dy}{\\sqrt{F(y)}},\\ \\ 0<\\xi<\\frac{L}{2},$$\nwhere $L$ is the period of $\\phi$ given by\n\t\t\t$$L=2\\int_{m}^{\\min\\{M, c\\}}\\frac{dy}{\\sqrt{F(y)}} = 2\\int_{m}^{\\min\\{M, c\\}}\\frac{\\sqrt{c-y}}{\\sqrt{P(y)}} \\ dy .$$\n\n\t\t\tThe rest of proof follows a similar line to that of Theorem 2 in \\cite{Lenells}, by considering the different distributions of zeros to the polynomial $P(\\phi) = \\phi^{2}\\left(c-\\phi^{2}\/2\\right) +a\\phi +d$ instead of $P(\\phi)=\\phi^{2}(c-\\phi) +a\\phi +d$ in the equations (6.4), (6.13) and (6.17) of \\cite{Lenells}.\n\\end{proof}\n\n\t\t\tAlso, Theorem \\ref{teorema2} is true for waves satisfying $(a^{'})-(c^{'})$ and $(a^{'})-(g^{'})$ of Theorems \\ref{classificacao1} and \\ref{classificacao2}, respectively.\n\n\n\\section{Explicit formulas for peakons}\\label{explicit peakons}\n\t\t\tSuppose that $\\phi$ is a wave corresponding to $(d)-(e)$ of Theorem \\ref{classificacao2}. In this section we determine explicit formulas for the peaked traveling waves.\n\t\t\t\n\t\t\tFrom (\\ref{6.2}), we have\n\t\t\t\\begin{equation}\n\t\t\t\\label{7.1}\n\t\t\t|\\xi-\\xi_{0}| = \\displaystyle\\int_{\\phi_{0}}^{\\phi} \\frac{dy}{\\sqrt{F(y)}} = \\sqrt{2}\\displaystyle\\int_{\\phi_{0}}^{\\phi} \\frac{\\sqrt{c-y}}{\\sqrt{(M-y)(y-m)(y-z)(y-r)}}\\ dy.\n\t\t\t\\end{equation}\n\t\t\t\nMaking $\\phi = m+ (M-m)\\sin^{2}(\\theta)$, we rewrite (\\ref{7.1}) by\n\t\t\t\\begin{equation}\n\t\t\t\\label{7.2}\n\t\t\t|\\xi-\\xi_{0}| =\\frac{2\\sqrt{2}}{\\sqrt{M-m}} \\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\sqrt{A -\\sin^{2}(t)}}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt\n\t\t\t\\end{equation}\nwhere $$A=\\frac{c-m}{M-m},\\ \\ B=\\frac{m-r}{M-m}\\ \\ \\text{and}\\ \\ C=\\frac{m-z}{M-m}.$$\n\nThe period of $\\phi$ is given by\n\\begin{eqnarray}\n\\label{eq periodo}\nL &=& 2\\sqrt{2}\\displaystyle\\int_{m}^{\\min\\{M,c\\}} \\frac{\\sqrt{c-y}}{\\sqrt{(M-y)(y-m)(y-z)(y-r)}}\\ dy\\nonumber\\\\\n&=& 4\\frac{\\sqrt{2}}{\\sqrt{M-m}}\\displaystyle\\int_{\\theta_{0}}^{\\theta_{\\text{max}}} \\frac{\\sqrt{A -\\sin^{2}(t)}}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt .\n\\end{eqnarray}\n\nAssume $\\phi$ a periodic peakon solution as in item $(d)$ of Theorem \\ref{classificacao1}. So, $A=1$ and $B,C>0$, with $B\\neq C$. From (\\ref{7.2}), we have\n\t\t\t\\begin{eqnarray*}\n\t\t\t|\\xi-\\xi_{0}| &=& \\frac{2\\sqrt{2}}{\\sqrt{M-m}}\\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\cos(t)}{\\sqrt{B+\\sin^{2}(t)}\\sqrt{C+\\sin^{2}(t)}}\\ dt\\\\\n\t\t\t&=& \\frac{2\\sqrt{2}}{\\sqrt{M-m}}\\left[- \\frac{i}{\\sqrt{C}}\\cdot F\\left( \\text{arcsin}\\left( \\frac{i}{B}\\sin(t)\\right)\\ ; \\sqrt{\\frac{B}{C}}\\right) \\right]\\Bigg{|}_{\\theta_{0}}^{\\theta},\n\t\t\t\\end{eqnarray*}\nwhere $F$ denotes the elliptic integral of the first kind (see Appendix), since $0 < B\/C< 1$.\n\nChoosing $\\xi_{0}$ such that $\\phi(\\xi_{0}) = m$, that is, $\\xi_{0}$ is a trough, we get $\\theta_{0} = 0$. Moreover, since $M=c$ we obtain $\\sin(\\theta) = \\sqrt{\\phi -m}\/\\sqrt{c-m}$. Thus\n\t\t\t$$|\\xi-\\xi_{0}| = -\\frac{i}{D_{1}} \\cdot F\\left( i\\cdot\\text{arcsinh}\\left( \\frac{1}{B} \\sqrt{\\frac{\\phi -m}{c-m}} \\right)\\ ; k\\right),$$\nwhere $D_{1} = \\sqrt{C}\\sqrt{c-m}\/2\\sqrt{2}$ and $k=\\sqrt{B\/C}$ is the module of the elliptical integral (see Appendix).\n\nSo, by properties of the elliptic integral of the first kind \\cite{BF}, we obtain\n\t\t\t\t$$\\sin\\left(i\\cdot\\text{arcsinh}\\left( \\frac{1}{B} \\sqrt{\\frac{\\phi -m}{c-m}} \\right)\\right) = \\text{sn}(D_{1}|\\xi-\\xi_{0}|i\\ ;k),$$\nwhere $sn$ is a Jacobian elliptic function (see Appendix).\n\nTherefore, solving this equation we conclude that\n\t\t\t\t\t$$\\phi(\\xi) = m + D_{2}\\text{tn}^{2}(D_{1}|\\xi-\\xi_{0}|\\ ;k'),\\ \\ |\\xi -\\xi_{0}| \\leqslant \\frac{L}{2},$$\nwhere\n\t\t\t\t\t$$L= D_{3}\\cdot F\\left(\\text{arcsin}\\left(\\sqrt{\\frac{1\/B^{2}}{1+ 1\/B^{2}}}\\right)\\ ; k'\\right)$$\nis the period of $\\phi$ obtained from (\\ref{eq periodo}), $D_{2} = B^{2}(c-m)$, $D_{3}= 2\/D_{1}$, $k'^{2}= 1 -k^{2}$ and $tn$ is a Jacobian elliptic function (see Appendix). This is the explicit formula to the periodic peakons of the mCH equation. \n\t\t\t\t\nIf $\\phi$ is a peakon with decay as in $(e)$ of Theorem \\ref{classificacao2}, then $A=1$, $B=0$ and $C>0$. So, (\\ref{7.2}) gives us\n\t\t\t\t$$|\\xi-\\xi_{0}| =\\frac{2\\sqrt{2}}{\\sqrt{M-m}} \\displaystyle\\int_{\\theta_{0}}^{\\theta} \\frac{\\cos(t)}{\\sin^{2}(t)\\sqrt{C+\\sin^{2}(t)}}\\ dt = \\frac{2\\sqrt{2}}{\\sqrt{C}\\sqrt{M-m}}\\Big{[}-\\text{arccsch}|t|\\Big{]}\\Big{|}_{\\frac{\\sin(\\theta_{0})}{\\sqrt{C}}}^{\\frac{\\sin(\\theta)}{\\sqrt{C}}}.$$\n\nChoosing $\\xi_{0}$ such that $\\phi(\\xi_{0}) = c$, that is, $\\xi_{0}$ at the peak, we get $\\sin(\\theta_{0}) = \\sin(\\theta_{\\text{max}}) = 1$. Moreover, since $M=c$ we obtain $\\sin(\\theta) = \\sqrt{\\phi -m}\/\\sqrt{c-m}$. Thus\n\t\t\t\t$$|\\xi-\\xi_{0}| = -\\frac{2\\sqrt{2}}{\\sqrt{C}\\sqrt{c-m}} \\left[ \\text{arccsch}\\left|\\frac{\\sqrt{\\phi -m}}{\\sqrt{C}\\sqrt{c-m}}\\right| - \\text{arccsch} \\left(\\frac{1}{\\sqrt{C}}\\right) \\right].$$\n\t\t\t\t\nNow, we use the identity\n\t\t\t$$\\text{arccsch}(x) = \\ln\\left(\\frac{1}{x} + \\sqrt{\\frac{1}{x^{2}} +1}\\right) = \\ln\\left(\\frac{1 +\\sqrt{1 + x^{2}}}{x}\\right);\\ \\ x\\in\\mathbb{R},\\ x\\neq 0$$\nin the above equation to obtain\n\t\t\t\t\t\t$$ -\\frac{\\sqrt{C}\\sqrt{M-m}|\\xi-\\xi_{0}|}{2\\sqrt{2}} = \\ln\\left(\\frac{\\frac{\\sqrt{C}\\sqrt{c-m}}{\\sqrt{\\phi -m}}\\left( 1+ \\sqrt{1+ \\frac{\\phi -m}{C(c-m)}}\\right) }{\\sqrt{C}\\left(1+ \\sqrt{1+\\frac{1}{C}} \\right)}\\right).$$\n\nThus,\n\t\t\t\t$$\\phi(\\xi) = m +\\frac{D_{4}e^{-D_{5}|\\xi-\\xi_{0}|}}{(D_{6}e^{-D_{5}|\\xi-\\xi_{0}|} -1)^{2}}$$\nis an explicit expression for the peakons with decay, where $D_{4} = 4C^{2}(c-m)(1+\\sqrt{1+1\/C})^{2}$, $D_{5} = \\sqrt{C}\\sqrt{c-m}\/\\sqrt{2}$ and $D_{6}= C(1+\\sqrt{1+1\/C})^{2}$.\n\n\n\\section{Acknowledgment}\nThe first author was supported by CAPES\/Brazil.\n\n\\section{Appendix}\nIn this appendix we will talk about some concepts used so far without further explanation. In accordance with \\cite{BF}, we start setting the \\textit{normal elliptic integral of the first kind}\n\t\t\t$$F(\\phi , k) = F(\\phi\\ |\\ k^{2} ) = F_{k}(\\phi) = \\int_{0}^{y} \\ \\frac{dt}{\\sqrt{(1-t^2)(1-k^2 t^2)}} = \\int_{0}^{\\phi} \\frac{d\\varphi}{\\sqrt{(1 - k^{2}\\sin^{2}\\varphi)}},$$\nwhere $y=\\sin\\phi$.\n\n\t\t\tThe parameter $k$ is called the \\textit{modulus of the elliptic integral} and $k'^{2} = 1-k^{2}$ its \\textit{complementary modulus}, and both may take any real or imaginary value. Here we wish to take $0 < k^{2} < 1$. Moreover, the variable $\\phi$ is called \\textit{argument} and it is usually taken belonging to $\\left[\\frac{-\\pi}{2} , \\frac{\\pi}{2}\\right]$.\n\n\t\t\tIn his algebraic form, the elliptic integral above is finite for all real (or complex) values of $y$, including infinity. When $\\phi= \\frac{\\pi}{2}$, the integral $K(k) \\dot{=} F \\left(\\frac{\\pi}{2}, k \\right)$ is said to be \\textit{complete}.\n\t\t\t\n\t\t\tWe define the \\textit{Jacobian elliptic functions} using the inverse function of the elliptic integral of the first kind. This inverse function exists because that \n$$ u(y_{1}, k) \\equiv u = \\int_{0}^{y_{1}} \\frac{dt}{\\sqrt{(1-t^{2})(1-k^{2}t^{2})}} = \\int_{0}^{\\phi} \\frac{d\\varphi}{\\sqrt{1 - k^2 \\sin^2 \\varphi}} = F(\\phi, k),$$\nis a strictly increasing function of the real variable $y_{1}$ and, in its algebraic form, this integral has the property of being finite for all values of $y_{1}$. This inverse $\\phi = \\mathrm{am}(u,k) = \\mathrm{am}u$ is called \\textit{amplitude function}.\n\n\t\tThere are several Jacobian elliptic functions that can be seen in \\cite{BF}, but here we will only define the functions $\\textit{sn}$, $\\textit{cn}$ and \\textit{tn} as follows\n\t\t\t$$\n\t\t\t\\begin{array}{lll}\n\t\t\t\\text{sn}(u,k) = \\text{sin}\\ \\mathrm{am}(u,k) = \\text{sin} \\phi,&&\\\\\n\t\t\t\\text{cn}(u,k) = \\text{cos}\\ \\mathrm{am}(u,k) = \\text{cos} \\phi,&&\\\\\n\t\t\t\\text{tn}(u,k) = \\displaystyle\\frac{\\text{sin}\\ \\mathrm{am}(u,k)}{\\text{cos}\\ \\mathrm{am}(u,k)} = \\frac{\\text{sin} \\phi}{\\text{cos} \\phi}.&&\n\t\t\t\\end{array}\n\t\t\t$$\nThese functions have a real period, namely $4 K(k)$, and some important properties summarized by the formulas given below.\n$$\n\\begin{array}{lll}\n\\text{sn}(-u) = -\\text{sn}(u), &&\\ \\ \\text{sn}^2u + \\text{cn}^2u = 1,\\\\\n \\text{cn}(-u)=\\text{cn}(u), &&\\ \\ -1 \\leqslant \\text{sn} u , \\text{cn} u \\leqslant 1,\\\\\n \\text{tn}(-u) = -\\text{tn}(u), &&\\ \\ \\text{sn}(iu,k) = i\\text{tn}(u,k').\n\\end{array}\n$$\n\\\\\n\n\n\n{\n\\fontsize{10pt}{\\baselineskip}\\selectfont\n\n\\textbf{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nFrom the groundbreaking work of BPST \\cite{Brower:2006ea} till today, many articles have shown how the theoretical ideas of $AdS\/CFT$ can be used to model experimental data where pomeron exchange dominates~\\cite{Hatta:2007he, Cornalba:2008sp, Brower:2008ix, Levin:2009vj, Hatta:2009ra, Kovchegov:2009yj, Avsar:2009xf, Cornalba:2009ax, Cornalba:2010vk, Kovchegov:2010uk, Levin:2010gc, Brower:2010wf, Ballon-Bayona:2017vlm, Costa:2012fw, Costa:2013uia, Brower:2012mk, Anderson:2014jia, Nally:2017nsp, Ballon-Bayona:2015wra, Amorim:2018yod, Hatta:2008st, Brower:2009bh, Domokos:2009hm, Domokos:2010ma, Stoffers:2012zw, Koile:2014vca, Kovensky:2018xxa, Lee:2018zud, Kovensky:2018gif, Mamo:2019mka, FolcoCapossoli:2020pks}.\n These works are tailored for a specific process and\/or kinematic regime. On the other hand, we have shown in~\\cite{Amorim:2021ffr} that a holographic model with few free parameters that describes successfully several processes dominated by pomeron exchange is indeed possible. The processes analysed were sensitive to the forward scattering amplitude, i.e. for Mandelstam $t = 0$. So it remains an open question if it is possible to extend previous results to processes where amplitudes with non-zero values of $t$ are necessary. One such example is Deeply Virtual Compton Scattering (DVCS), which is the focus of this letter.\n\nDVCS is an exclusive Compton scattering process where the incoming photon has high virtuality. It has been studied throughly by the H1 and ZEUS collaborations as well at JLAB. While DIS allows the determination of nucleon parton distribuition functions (PDFs), e.g. the proton PDFs, DVCS data is important for the study of the generalized parton distribution functions (GDPs) \\cite{ji_deeply_1997,radyushkin_nonforward_1997}, which are related to the correlation between the transverse and longitudinal components of the momentum of quarks and gluons inside the nucleons.\n\nHolographic techniques have been already employed to describe DVCS data. In \\cite{Costa:2012fw} the conformal and hard wall pomeron models have been used, while in \\cite{Stoffers:2012ai} a holographic description of dipole-dipole scattering has been studied. DIS is connected to the forward Compton scattering amplitude via the optical theorem. Hence, our analysis of DVCS in this work is close to the one presented in \\cite{Ballon-Bayona:2017vlm}. We will show that in order to include the description of DVCS data one needs to include extra parameters that are related to the holographic wave function of the proton and to the coupling dependence with the spin of the exchanged Reggeons.\n\n\\section{Holographic computation of DVSC amplitude}\n\nWe start by deriving holographic expressions for the differential cross-section $d \\sigma \/ dt$ and total cross-section~$\\sigma$ of the\n$ \\gamma^* p \\to \\gamma p $ DVCS process. In~\\cite{Amorim:2021ffr} we computed the proton structure functions $F_2^p$ and $F_L^p$ and the total cross-section $\\sigma\\left(\\gamma p \\to X \\right)$ by determining the forward scattering amplitude of the process $\\gamma^{*} p \\to \\gamma^{*} p$. Thus, the only difference between the DIS and the DVCS computation is that the outgoing photon is \\textit{on-shell}. The associated Witten diagram is show in figure \\ref{fig:DVCS Witten diagram}.\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.5]{Witten_diagram}\n \\caption[Tree level Witten diagram for the DVCS scattering amplitude]{Tree level Witten diagram associated with the computation of the amplitude $\\mathcal{A}_J^{\\lambda_1\\lambda_3}$ of the DVCS process $\\gamma^{*}p\\to \\gamma p$. The bottom lines represent the proton modeled by a scalar $\\Upsilon$.}\n \\label{fig:DVCS Witten diagram}\n\\end{figure}\nHence we will consider the general case of $\\gamma^* p \\to \\gamma^* p$ where the incoming photon has virtuality $Q_1^2$ while the outgoing photon has virtuality $Q_3^2$. At the end of the calculation we take $Q^3 \\equiv k_3^2 \\to 0$ and use the identity\n\\begin{equation}\n\\lim_{Q \\rightarrow 0} f_Q (z) = 1\\,, \\quad \\lim_{Q \\rightarrow 0} \\frac{\\dot{f_Q}}{Q} = 0\\,,\n\\label{eq:nonnormalizable_Q_0}\n\\end{equation}\nwhere $f_Q$ is the non-normalizable mode of the bulk U(1) gauge field dual to the current $J^\\mu = \\bar{\\psi} \\gamma^\\mu \\psi$.\nWe take for the large $s$ kinematics of $\\gamma^*p\\to \\gamma^*p$ scattering the following momenta\n\\begin{align}\n \\label{eq:NMC:kinematics}\n&k_1=\\left(\\!\\sqrt{s},-\\frac{Q_1^2}{\\sqrt{s}} ,0\\right),\n\\qquad\n k_3=-\\left(\\!\\sqrt{s},\\frac{ q_\\perp^2 -Q_3^2}{\\sqrt{s}} , q_\\perp \\right),\n\\\\\n&k_2=\\left(\\frac{M^2}{\\sqrt{s}},\\sqrt{s} ,0\\right),\n\\qquad\n k_4=-\\left(\\frac{M^2+ q_\\perp^2}{\\sqrt{s}},\\sqrt{s} ,-q_\\perp \\right),\n\\nonumber\n\\end{align}\nwhere $k_1$ and $k_3$ are, respectively, the incoming and outgoing photon momenta and $k_2$ and $k_4$ are, respectively, the incoming and outgoing proton momenta. \nThe incoming and outgoing off-shell photons have the following polarization vectors\n\\begin{align}\n &n_1=\n \\begin{cases}\n \\left(0,0,\\epsilon_\\lambda \\right), & \\lambda=1, 2 \\\\\n \\frac{1}{Q_1} \\left( \\sqrt{s}, \\frac{Q_1^2}{\\sqrt{s}}, 0, 0 \\right), & \\lambda = 3\n \\end{cases}\\,,\\\\\n &n_3=\n \\begin{cases}\n \\left(0,\\frac{2 q_\\perp \\cdot \\epsilon_\\lambda}{\\sqrt{s}},\\epsilon_\\lambda\\right), & \\!\\lambda=1,2 \\\\\n \\frac{1}{Q_3} \\left(\\sqrt{s}, \\frac{Q_3^2+q_\\perp^2}{\\sqrt{s}}, q_\\perp \\right), & \\!\\lambda = 3\n \\end{cases}\\,,\n \\label{eq:inPolarization}\n\\end{align}\nwhere $\\epsilon_1 = (1,0)$ and $\\epsilon_2 = (0,1)$. \n\nTo compute the Witten diagram in figure \\ref{fig:DVCS Witten diagram}, consider \nthe minimal coupling between the gauge field dual to the current $J^\\mu = \\bar{\\psi} \\gamma^\\mu \\psi$ \nand the higher spin field $h_{a_1 \\dots a_J}$\n\\begin{align}\n\\label{eq:gauge_field_spin_j_coupling}\n k_J \\int d^5 X \\sqrt{-g} e^{-\\Phi} F_{b_1 a} D_{b_2} \\dots D_{b_{J-1}} F^{a}_{b_J} h^{b_1 \\dots b_J},\n\\end{align}\nwhere the field strength $F_{ab}$ is computed from the non-normalizable mode\n\\begin{equation}\nA_a (X; k,\\lambda)= n_a(\\lambda) f_Q(z) \\, e^{ik\\cdot x}\\,.\n\\end{equation}\nThe coupling between the higher spin field and the scalar field is\n\\begin{align}\n\\label{eq:scalar_field_spin_j_coupling}\n \\bar{k}_J \\int d^5 X \\sqrt{- g} e^{- \\Phi} \\Upsilon D_{b_1} \\dots D_{b_J} \\Upsilon h^{b_1 \\dots b_J},\n\\end{align}\nwhere \n\\begin{equation}\n \\Upsilon(X;k) = \\upsilon(z)\\,e^{ik\\cdot x}\\,,\n\\end{equation}\nis the normalizable mode describing the proton state.\nThe computation of the scattering amplitude follows the same steps as the ones presented in~\\cite{Amorim:2021ffr}, so we will not repeat it here. The scattering amplitude for $\\gamma^* p \\to \\gamma p$ is\n\\begin{equation}\n \\label{eq:amplitude gamma* gamma}\n \\mathcal{A}^{\\gamma^* p \\to \\gamma p}(s,t) = \\sum_n g_n(t) s^{j_n(t)}\\int dz \\,e^{-(j_n(t)-\\frac{3}{2})A} f_Q \\,\\psi_n(j_n(t))\\,,\n\\end{equation}\nwhere $A$ is the string frame warp factor and\n\\begin{equation}\n g_n (t) = H(j_n(t)) \\left[i + \\cot \\left(\\frac{\\pi j_n(t)}{2}\\right)\\right] \\frac{d j_n}{d t} \\int dz \\,e^{(-j_n(t)+\\frac{7}{2})A} \\upsilon^2 {\\psi_n^{*} (j_n(t))} \\, , \\label{eq:dvcs_gn_def}\n \\end{equation}\n with\n \\begin{equation}\n H(J) = \\frac{\\pi}{2} \\frac{k_{J} \\bar{k}_{J}}{2^{J}} \\, .\n\\end{equation}\nThe amplitude (\\ref{eq:amplitude gamma* gamma}) is the one where both photons have transverse polarizations, other combinations are subleading in $s$. To obtain this expression we used (\\ref{eq:nonnormalizable_Q_0}).\nThe wavefunctions $\\psi_n(j_n(t))$ are eigenfunctions of the effective Schr\\\"odinger potential~\\cite{Ballon-Bayona:2017vlm}\n\\begin{align}\nV_J(z) = \\frac{3}{2} \\left( \\ddot{A} - \\frac{2}{3} \\ddot{\\Phi} \\right) + & \\frac{9}{4} {\\left( \\dot{A} - \\frac{2}{3} \\dot{\\Phi} \\right)}^2 + e^{2A} (J-2) \\left[ \\frac{2}{l_s^2} \\left(1+\\frac{d}{\\sqrt{\\lambda}}\\right) + \\right. \\notag \\\\\n&\\left.+ e^{-2A} \\left( a \\ddot{\\Phi} + b\\left( \\ddot{A} - \\dot{A}^2 \\right) + c \\dot{\\Phi}^2 \\right) + \\frac{J+2}{\\lambda^{4\/3}} \\right] ,\n\\end{align}\nwhere $\\lambda = e^\\Phi$ is the 't Hooft coupling. The constants $l_s$, $a$, $b$, $c$ and $d$ are phenomenological parameters that will be fixed later by reproducing the best match with DVCS data. These parameters describe the analytic continuation of the equation of motion for the AdS spin $J$ field dual to twist 2 gluonic operators, in an expansion around $J=2$. This approximation gives the description of the graviton\/pomeron Regge trajectory in the strong coupling approximation, in contrast with the weak coupling BFKL pomeron, whose expansion starts at $J=1$. The details of the derivation are given in \\cite{Ballon-Bayona:2017vlm}.\n\nTo proceed we need to know the proton wavefunction in order to compute the $z$ integral in (\\ref{eq:dvcs_gn_def}). In AdS\/QCD, it is expected that baryons are dual to a configuration where three open strings are attached to a D-brane~\\cite{witten_baryons_1998, Polchinski:2000uf}. So far it is not known how to derive the baryon spectrum from such a configuration. An acceptable holographic description of the proton would consist of a spectrum that matches the mass of the proton as well of the other hadrons in the same trajectory. Here we follow a phenomenological approach by approximating the combination $e^{3 A - \\Phi}\\upsilon^2$ by the delta function $\\delta(z-z^*)$. The $z^*$ parameter is related, by dimensional analysis, to the inverse of the mass of the proton and will be used as a fitting parameter. By making this approximation we are assuming that $e^{3 A - \\Phi}\\upsilon^2$ is null in the UV and the IR, and that it has a global maximum. This is expected if the spectrum of the baryons is associated with a Schrodinger problem whose ground state is the proton. This approach, as an example, has successfully described DVCS data in \\cite{Costa:2012fw}. Another issue is the unknown functional form of the function $H(J)$ and its analytic continuation. In the next section we motivate a good ansatz for this expression, at least in the range $1 \\lesssim J \\lesssim 1.2$ that is relevant for the process here considered.\n\n\\section{AdS local coupling to the graviton trajectory}\n\nTo find a good ansatz for $H(J)$ let us first recall that in~\\cite{Amorim:2021ffr}, based on the forward scattering amplitude for the process $\\gamma^*p \\to \\gamma^* p$, the holographic formula for the proton structure function $F_2^p$ is given by\n\\begin{align}\n&F_2^p(x, Q^2) = \\sum_{n} \\frac{ {\\rm Im} \\, g_n}{4 \\pi^2 \\alpha} \\, x^{1-j_n} Q^{2 j_n} \\int dz \\,e^{-\\left(j_n-\\frac{3}{2}\\right)A} \\left( f_Q^2 + \\frac{\\dot{f}_Q^{2}}{Q^2} \\right) \\psi_n(z) \\, ,\n\\end{align}\nwhere $j_n$ and \n\\begin{equation}\n{\\rm Im} \\, g_n = H(j_n(0)) \\frac{d j_n}{d t} \\int d z \\,e^{(-j_n(0)+\\frac{7}{2}) A} \\upsilon^2 {\\psi}_n^{*} (j_n(0))\n\\end{equation}\nare computed at $t=0$.\nUsing (\\ref{eq:dvcs_gn_def}) and the approximation $e^{3 A - \\Phi}\\upsilon^2 \\sim \\delta(z-z^*)$, we can compute $H(J)$ for any value of $j_n(t)$ with the result\n\\begin{equation}\n \\label{eq:dvcs:H(J) vs g_n(t)}\n H(j_n (t)) = \\frac{{\\rm Im} \\, g_n(t)}{e^{(-j_n (t) + \\frac{1}{2})A(z^*)}\\psi^*_n(j_n (t)) \\frac{dj_n (t)}{dt}} \\, .\n\\end{equation}\nWe can now plot $H(J)$ as a function of $J$ using\nthe $t=0$ values of ${\\rm Im} \\, g_n$, $j_n$, $\\frac{dj_n}{dt}$ and $\\psi_n$ for the different trajectories found in \\cite{Ballon-Bayona:2017vlm} and choosing $z^{*}$ to be around 1, since by dimensional analysis it should be of the order of the inverse mass of the proton (in GeV units). \nThe black dots shown in figure \\ref{fig:dvcs:H(J)} represent the actual reconstructed values of $H(J)$ for each $n$.\nIn a logarithmic plot, $\\log H (J)$ can be well approximated by a quadratic curve in $J$, i.e. $\\log H(J) = h_0 + h_1 (J-1) + h_2 (J-1)^2$. We can then choose $z^{*}$ and the $h_i$ to get the best quadratic fit. For the best fit value of $z^* = 0.565$ we obtained the result presented in figure \\ref{fig:dvcs:H(J)}.\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.5]{H_reconstructed.pdf}\n \\caption[Reconstruction of the $H(J)$ function using the best parameters found in \\cite{Ballon-Bayona:2017vlm} and $z^*=0.565$]{Reconstruction of the $H(J)$ function using the best parameters found in \\cite{Ballon-Bayona:2017vlm} and $z^*=0.565$. The solid line represents the function $\\exp \\left(h_0 + h_1 (J-1) + h_2 (J-1)^2\\right)$ with $h_0=3.70$, $h_1=-30.3$ and $h_2=89.1$.}\n \\label{fig:dvcs:H(J)}\n\\end{figure}\n\nWe can check if the resulting curve has physical meaning by considering the dependence with $J$ of the\nAdS local couplings $\\kappa(J)$ and $\\bar \\kappa (J)$, that define $H(J)$,\nin the range $1 \\lesssim J \\lesssim 1.2$. According to the gauge\/gravity duality such couplings are related to the OPE coefficient of a spin $J$ operator \nwith two spin 1 operators (for the coupling with the EMG current operator) or spin 0 operators (for the coupling with a scalar, as we model the unpolarized proton). \nIn particular, in the UV fixed point the OPE coefficient of two EMG current operators with the spin $J$ glueball operator associated with the pomeron \ntrajectory vanishes, as only Wick contractions contribute.\nOf course, QCD is not a free theory and perturbative corrections should be taken into account. Instead, to check if the proposed curve for $H(J)$ \nis reasonable let us consider the $O(N)$ vector model, which is also free in the UV where the corresponding OPE coefficient is non-vanishing \\cite{Sleight:2017fpc}. In this model we can retrieve a similar shape of $H (J)$ as the one in figure \\ref{fig:dvcs:H(J)} from the three-point bulk vertex coupling between massless higher spin fields of spin $s_1$, $s_2$ and $s_3$ in type A minimal higher-spin theory in $AdS_{d+1}$. The coupling, as a function of $s_1$, $s_2$ and $s_3$, is given by\n\\begin{equation}\ng_{s_1,s_2,s_3} = \\frac{\\pi^{\\frac{d-3}{4}} 2^{\\frac{3d-1+s_1+s_2+s_3}{2}}}{\\sqrt{N}\\Gamma(d+s_1+s_2+s_3-3)} \\sqrt{\\frac{\\Gamma(s_1 + \\frac{d-1}{2})}{\\Gamma(s_1 + 1)}\\frac{\\Gamma(s_2 + \\frac{d-1}{2})}{\\Gamma(s_2 + 1)}\\frac{\\Gamma(s_3 + \\frac{d-1}{2})}{\\Gamma(s_3 + 1)}} \\, .\n\\end{equation}\n After approximating the gamma functions in the resulting expression with the Stirling formula and expanding up to quadratic order around $s_3=J = 1$, we obtain a \n function $H(J)$ consistent with our ansatz. Because we are comparing different theories caution should be taken. However, our point is that the overall function shapes does not change, and beyond the fixed point, corrections may be well captured in suitable redefinitions of the $h_i$ parameters in our ansatz. In the next section we test our hypothesis against DVCS experimental data.\n\n\n\\section{Data analysis and results}\nNow that we have a reasonable parameterisation of $H(J)$ we proceed to find the best values for the pomeron kernel parameters $l_s$, $a$, $b$, $c$ and $d$,\n and the constants $h_0$, $h_1$ and $h_2$ in the $H(J)$ ansatz. The optimal set of parameters is found by minimising the $\\chi^2$ statistic\n\\begin{equation}\n \\label{eq:global chi2 definition}\n \\chi_g^2=\\chi_{\\sigma}^2 +\\chi_{\\frac{d \\sigma}{dt}}^2 \\, ,\n\\end{equation}\nwhich is the sum of the $\\chi^2$ for $\\sigma\\left(\\gamma^* p \\to \\gamma p\\right)$ and the $\\chi^2$ for $\\frac{d\\sigma\\left( \\gamma^* p \\to \\gamma p \\right)}{dt}$. \nAs usual, for a given observable $O \\in \\{\\sigma\\left(\\gamma^* p \\to \\gamma p\\right), \\frac{d\\sigma\\left( \\gamma^* p \\to \\gamma p \\right)}{dt} \\}$, the respective $\\chi^2$ function is defined as\n\\begin{equation}\n \\chi^2_{O} \\equiv \\sum_n \\left(\\frac{O_n^{\\text{pred}} - O_n}{\\delta O_n}\\right)^2 \\, ,\n\\end{equation}\nwith $O^{\\text{pred}}$ being the predicted theoretical value and $\\delta O$ the experimental uncertainty. The sum goes over the available experimental points.\nThe total and differential cross-section data used is the combined one from H1-ZEUS available in \\cite{aaron_deeply_2009,chekanov_measurement_2009}.\n\nThe DVCS differential cross-section is given by\n\\begin{align}\n \\frac{d\\sigma}{dt} = \\frac{1}{16 \\pi s^2} \\frac{1}{2} \\sum_{\\lambda_1,\\lambda_3 = 1}^2 \\left|A^{\\lambda_1,\\lambda_3}\\left( s, t \\right) \\right|^2 =\\frac{1}{16 \\pi^2 s^2} \\vert \\mathcal{A}^{\\gamma^*\\gamma}(s,t) \\vert^2,\n\\end{align}\nwhere we average over the incoming photon polarization and the scattering amplitude is given by equation (\\ref{eq:amplitude gamma* gamma}).\nThe total cross-section is just the integral of the above\n\\begin{equation}\n\\label{eq:dvcs_sigma}\n \\sigma = \\int_{-1}^0 dt \\, \\frac{d \\sigma (t)}{dt} \\,,\n\\end{equation}\nwhere the integration range $-1 \\leq t \\leq 0$ comes from the data. \n\nTo solve the $\\chi^2$ minimisation problem we developed the $\\tt{HQCDP}$ $\\tt{R}$ package.\nThe Schrodinger problem associated to the gluon kernel is solved with $N=400$ Chebyshev points. To compute the differential and total cross-sections efficiently we divided the interval $-1 \\leq t \\leq 0$ in 20 pieces of length 0.05 and computed the differential cross-section for each point. From these values we created a spline interpolation function that can be used to predict the differential cross-section values and the total cross-section through equation (\\ref{eq:dvcs_sigma}). We also make use of the $\\tt{REDIS}$ \\textit{in memory} database to avoid redoing expensive computations. The code also makes use of multiple cores, if available, in order to compute in parallel the integrals that appear in (\\ref{eq:amplitude gamma* gamma}) for different kinematical points. The present results were found using a node in a High Performance Computing (HPC) cluster with 16 cores.\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics[scale=0.5]{DVCSDSigma}\n \\caption[Predicted vs. experimental values of the differential cross-section $\\frac{d \\sigma(t)}{dt}$ for DVCS]{Predicted vs. experimental values of the differential cross-section $\\frac{d \\sigma(t)}{dt}$ for DVCS. Different gray levels correspond to different combinations of $Q^2$ and $W$ as described in the legends. Here $Q^2$ and $t$ are in $\\text{GeV}^2$, $W$ in $\\text{GeV}$ and $\\frac{d \\sigma}{d t}$ is in $\\frac{\\text{nb}}{\\text{GeV}^2}$. }\n \\label{fig:dvcs:DVCSDSigma}\n\\end{figure}\n\n\nThe best fit to the data we have found has a $\\chi^2_{dof} \\sim 1.5$ with the parameter values of table~\\ref{tab:DVCS DIS best values}. The individual values of $\\chi^2_{dof}$ for the total and differential cross-section experimental data are $1.8$ and $1.3$ respectively, meaning that the model offers a good description of both processes. The comparison of the theoretical predictions against the experimental data can be seen in figures \\ref{fig:dvcs:DVCSDSigma} and \\ref{fig:dvcs:DVCSSigma}. Figure~\\ref{fig:regge_trajectories} plots the first four Regge trajectories obtained with the kernel parameters of table~\\ref{tab:DVCS DIS best values}. For the data range $0 \\lesssim-t \\lesssim1\\ ({\\rm GeV}^2)$ the Reggeon spin \nis in the range $1 \\lesssim j_n(t) \\lesssim 1.2$.\n \n\\begin{table}[h!]\n \\centering\n \\begin{tabular}{|c||c||c||c||c||c|}\n \\hline \n Kernel parameters & Extra parameters & Intercepts\\\\\n \\hline \n \\hline \n $a = -4.55$ & $h_0 = 4.74$ & $j_0=1.24$ \\\\\n \\hline \n $ b = 0.980$ & $h_1 = -35.9$ & $j_1=1.13$ \\\\\n \\hline \n $ c = 0.809$ & $h_2 = 142$ & $j_2=1.08$\\\\\n \\hline \n $d = -0.160$ & $z^* = 0.296$ & $j_3=1.05$\\\\\n \\hline \n $l_{s} = 0.153$ &-- &-- \\\\\n \\hline \n \\end{tabular}\n \\caption[Best fit parameters for the DVCS data fit]{The 9 parameters for our best fit and the intercept of the first four pomeron trajectories. All parameters are dimensionless except $z^*$ and $l_s$ which are in $\\rm{GeV}^{-1}$.}\n \\label{tab:DVCS DIS best values}\n\\end{table}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.5]{DVCSSigma}\n \\caption[Predicted vs. experimental values of the total cross-section $\\sigma$ of DVCS]{Predicted vs. experimental values of the total cross-section $\\sigma$ of DVCS. Small numbers attached to lines and points of the same grey level indicate the respective value of $Q^2$ in $\\text{GeV}$.}\n \\label{fig:dvcs:DVCSSigma}\n\\end{figure}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[scale=0.5]{regge_trajectories_dvcs}\n \\caption{Regge trajectories obtained from our model and their intercepts. They were obtained using the kernel parameters of table~\\ref{tab:DVCS DIS best values}. The plot includes the kinematical range\n $0 \\lesssim-t \\lesssim1\\ ({\\rm GeV}^2)$ as in the data used in this work.}\n \\label{fig:regge_trajectories}\n\\end{figure}\n\n\nTo conclude, we have shown that the holographic model presented in~\\cite{Amorim:2021ffr} can be extended to include a quantitative description of total and differential cross-sections of DVCS data from H1-ZEUS. One should also include the kernel of the twist 2 fermion operators, as suggested by Donnachie and Landshoff~\\cite{Donnachie:1998gm} and study the importance\nof the exchange of meson trajectories on the results above. A first step towards the inclusion of the twist 2 fermion operators has been done~\\cite{Amorim:2021gat}.\n\n\n\\section*{Acknowledgments}\n\n\nThis research received funding from the Simons Foundation grant 488637 (Simons collaboration on the Non-perturbative bootstrap). \nCentro de F\\'\\i sica do Porto is partially funded by Funda\\c c\\~ao para a Ci\\^encia e a Tecnologia (FCT) under the grant\nUID-04650-FCUP.\n AA is funded by FCT under the IDPASC doctorate programme with the fellowship PD\/BD\/114158\/2016.\n\n\\bibliographystyle{elsarticle-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{section1}\n\nSuperfluids are extraordinary fluids characterised by the absence of \nviscosity.\nThey are irrotational everywhere except at vortex lines\nwhose circulation is quantised in units of $\\kappa=h\/m$, where \n$h$ is Planck's constant and $m$ is the \nmass of the boson which composes the fluid \\cite{Annett2004,Barenghi2016}.\nFirst discovered and studied in liquid helium-4 and, decades later,\nin helium-3, superfluidity has since been observed in ultracold gases and \nphotonic systems. The constraint of quantised\nvorticity is a consequence of quantum mechanics - vorticity can only arise as $2 \\pi$ topological defects of the macroscopic single-particle wavefunction of the quantum many-body system. These defects manifest as vortex lines through the fluid. As well as possessing a circulating flow, the vortex lines have a core of depleted density about their axis, out to a core radius $a_0$ which is of the order of the superfluid healing length. In helium-4 and helium-3 the vortex core size is around $10^{-10}$m and $10^{-8}$m, respectively.\n\nThe textbook paradigm of superfluidity is a cylindrical\nbucket of superfluid helium\nrotating at constant angular frequency $\\Omega$. \nClassical solid-body \nrotation is forbidden by the irrotational nature of the superfluid. At sufficiently small values of\n$\\Omega$, the fluid remains quiescent. However, if $\\Omega$\nin increased past a critical\nvalue $\\Omega_{\\rm c}$, the presence of a vortex line\nis energetically favourable. Using hydrodynamic arguments and up to \na logarithmic correction, it is estimated \\cite{Barenghi2016} that this\ncritical angular frequency is\n\n\\begin{equation}\n\\Omega_{\\rm c} = \\frac{\\hbar}{mR^2} \\ln \\left( \\frac{R}{a_0} \\right),\n\\label{eqn:omega_c}\n\\end{equation}\n\n\\noindent\nwhere $m$ is the mass of a helium atom and $R$ is the radius of the bucket. At larger values of\n$\\Omega$, two vortices become favourable, and so on. \nFor $\\Omega \\gg \\Omega_{\\rm c}$ the stationary state of the fluid is \nthe famed vortex lattice, an array of vortex lines aligned along the\naxis of rotation with areal density\n\\begin{equation}\nn_{\\rm v}=\\frac{2\\Omega}{\\kappa},\n\\label{eq:Feynman}\n\\end{equation}\nknown as Feynman's rule. The vortex lattice was \nfirst imaged in superfluid helium by Packard {\\it et al} \\cite{Yarmchuk1979} \nand more recently by Bewley {\\it et al} \\cite{Bewley2006}. \nThe lattice has also been observed in ultracold gaseous superfluids\ntrapped by smooth confining \npotentials \\cite{Madison2000,AboShaeer2001}. (At much higher rotation frequencies where centrifugal effects dominate, a giant or macroscopic vortex carrying many quanta of circulation can become formed in both superfluid helium \\cite{Tsakadze1964,Josserand2004} and gaseous superfluids \\cite{Kasamatsu2002,Fischer2003,Kavoulakis2003,Engels2003}; however, this regime is outside the scope of this work.)\n\n\n\nThe process in which the vortices enter the superfluid in the first\nplace is called vortex nucleation. \nBeing associated with a $2\\pi$ phase singularity of the macroscopic \nwavefunction, a vortex line is topologically protected.\nThus, starting from some initially vortex-free state, vortices must enter\nthe superfluid from the boundary. It is believed that\nvortex lines are nucleated\neither {\\it intrinsically} by the flow of the superfluid past the \nmicroscopic roughness of the bucket wall (overcoming a critical velocity)\nor {\\it extrinsically} by stretching some pre-existing vortex lines \ncalled ``remanent vortices\" which,\nunder suitably conditions, can spool additional \nvortices \\cite{Schwarz-mill}. Remanent vortices are thought \nto arise when cooling the helium sample through the superfluid transition, \nand can be avoided by using careful\nexperimental protocols \\cite{Yano-2007}. \n\nIndividual vortex nucleation in a rotating bucket,\neither intrinsic or extrinsic, has never been visualised in detail. \nExperimentally, it remains challenging to image the flow in \nthe vicinity of a boundary, despite progress in flow visualisation \nin the bulk \\cite{Bewley2006,Zmeev2015b,Duda2015}, more so because\nthe microscopic scale of the vortices themselves. \nTheoretically, the nucleation problem has been addressed using\nenergy arguments \\cite{Fetter1966,StaufferFetter1968} with no insight\nin the dynamics. With few exceptions \\cite{Stagg2017},\nthe effect of microscopic boundary roughness on the vortex nucleation has not been studied. \nA related and better understood nucleation process\ntakes place when an ion bubble is driven in liquid helium by an applied \nelectric field; compared to the bucket, the nucleation is more \ncontrolled in terms of geometry (the shape of the bubble can be\ndetermined theoretically)\nand velocity (experimentally determined by time of flight measurements).\nVortex nucleation by the ion bubble has thus received much detailed \nexperimental and theoretical attention\n\\cite{MuirheadVinenDonnelly1984,McClintockBowley1995,BerloffRoberts2000,\nWiniecki2000,Villois2018} than nucleation by the walls of the bucket \nwhich contains the helium sample.\n\nIn this work we are not concerned with the vortex nucleation as such,\nbut rather with the intermediate state between the nucleation and\nthe final vortex lattice. This intermediate stage is still unexplored, \nbut, given that the length scales and the time scales involved depend on\nthe vortex separation rather the vortex core size (i.e. they are\nmesoscopic rather than microscopic), there is prospect of \nexperimental visualisation in the near future.\nThe focus of attention is therefore not individual vortex dynamics at \nnucleation but the collective dynamics of many vortex lines in the \npresence of a boundary which is not smooth. For simplicity we consider \nthe problem at sufficiently low temperature that the normal fluid does\nnot play an important role.\n\nThe traditional method to model the dynamics of superfluid vortices is\nthe Vortex Filament Method (VFM) \\cite{Schwarz1988}, \nwhich models vortex lines as infinitesimally thin filaments \ninteracting with themselves, their neighbours and the boundary \n(via suitable images). However, this approach is not applicable\nto our problem.\nFirstly, if the boundary varies on atomic length scales comparable \nto the vortex core (which is likely to be the case for any metal or glass \nbucket containing liquid helium),\nthen the core lengthscale can no longer be ignored compared to\nother relevant lengthscale, invalidating the \nassumptions behind the VFM.\nSecondly, the implementation of the boundary\ncondition is cumbersome to set up and\nnot simple to change from one boundary shape to another; indeed, the VFM \nhas been implemented for plane \\cite{Schwarz1985}, \nsemi-spherical \\cite{Schwarz1985,Tsubota1993}, \nspherical \\cite{Schwarz1974,Kivotides2006} and cylindrical\n\\cite{Hanninen2005,HanninenBaggaley2014} boundaries, but never\nfor irregular boundaries relevant to our problem. Thirdly, the VFM does\nnot describe vortex nucleation, but requires to initialise the calculation\nwith arbitrary seeding vortex lines.\nAn alternative approach is through the Gross-Pitaevskii equation (GPE) \n\\cite{Pitaevskii,Barenghi2016}. This is a formal description \nof a dilute weakly-interacting gas of bosons, and is equivalent \nto a continuity equation and an Euler-like equation for an inviscid \nfluid (the modification being the presence\nof a quantum pressure term). While the GPE is an excellent quantitative \ndescription of Bose gas superfluids, it is limited to being a qualitative \ndescription of superfluid helium due to the stronger interactions\ntaking place in a liquid rather than in a gas. \nNevertheless, its capability to describe the microscopic detail of \nsuperfluid dynamics - the finite-sized core, \nvortex interactions and reconnections, even the intrinsic nucleation - makes \nit a useful model to study superfluid flows at a boundary. \nAn important feature is that the GPE can easily implement\nirregular boundaries.\nIndeed, recent GPE simulations have predicted the occurrence of a \nturbulent boundary layer when the superfluid flows past a locally\nrough surface \\cite{Stagg2017}: above a critical imposed flow speed, \nvortices are nucleated from the surface features, interact and become \nentwined in a layer adjacent to the surface. \n\nReturning to the rotating bucket of superfluid helium, \nit is natural to ask if some kind of boundary layer may similarly \nform at the boundary of the rotating bucket\nin the transient evolution to the vortex lattice. Whether disordered\nor laminar, this layer will certainly involve vortex interactions.\nIt is in fact unlikely that the vortex lines which nucleate extend\nfrom the top to the bottom of the bucket, as if the process were\nessentially two-dimensional (2D). More likely, the first vortex lines\nwhich nucleate are small, and become long only after a sequence of\ninteractions and reconnections.\nTo qualitatively explore these interactions, here we perform a series of \nnumerical experiments, based on the GPE, of a superfluid being spun-up \nin a bucket whose walls are microscopically rough. \nThese numerical experiments allows us to build a physical picture \nof how vorticity enters the superfluid and forms a vortex lattice, \nand of the role of remanent vortices, \nsharp intrusions, rotation rate, and dimensionality. \n\nThe plan of the paper is the following.\nIn Section~\\ref{section2} we introduce our model and details our of numerical\nsimulations. In Section~\\ref{section3} we present our main results for the\nspin up of a quiescent superfluid. Section~\\ref{section4} explores\nthe possibility that a single strong imperfection in the shape of\na protuberance, remanent vortex lines or dimensionality may affect the main results described\nin Section~\\ref{section3}. Finally, in Section~\\ref{section5} \nwe discuss and conclude our findings.\n\n\n\n\n\n\n\n\\section{Model and method}\n\\label{section2}\n\n\\subsection{Gross-Pitaevskii equation}\n\n\nWe model the superfluid dynamics using the Gross-Pitaevskii equation. Within this model, the superfluid is parametrised by a mean-field complex\nwavefunction $\\Psi({\\bf r},t)=\\vert \\Psi({\\bf r},t)\\vert e^{iS(\\bf r,t)}$. \nThe particle density follows as $n({\\bf r},t)=|\\Psi({\\bf r},t)|^2$ and \nthe fluid velocity as ${\\bf v}({\\bf r},t)=(\\hbar\/m)\\nabla S({\\bf r},t)$, \nwhere $\\hbar=h\/(2 \\pi)$ and $S({\\bf r},t)$ is the phase distribution \nof $\\Psi$. The dynamics of $\\Psi({\\bf r},t)$ follows the \nGPE \\cite{Pitaevskii,Barenghi2016},\n\n\\begin{equation}\ni \\hbar \\frac{\\partial\\Psi }{\\partial t}= \\hat{\\mathcal{H}} \\Psi, \n\\label{eq:GPE}\n\\end{equation}\n\n\\noindent\nwith Hamiltonian operator,\n\n\\begin{equation}\n\\hat{\\mathcal{H}} = \n-\\frac{\\hbar^2}{2m}\\nabla^2\n+V+g\\left|\\Psi \\right|^{2}.\n\\end{equation}\n\n\n\\noindent\nHere $m$ is the particle mass, $g$ $(>0)$ is a nonlinear coefficient \ndescribing the inter-particle interactions,\n and $V({\\bf r},t)$ is the external potential acting on the fluid. \nStationary solutions of the GPE satisfy \n$\\hat{\\mathcal{H}}\\Psi= \\mu_0 \\Psi$, \nwhere $\\mu_0$ is the chemical potential of the fluid. \n\nWe make two physically-motivated modifications to the basic GPE above. \nFirstly, since the GPE conserves energy, we follow other works \n\\cite{Choi1998,Tsubota2001} in introducing a phenomenological dissipation \nterm into the GPE to model, at least in a qualitative way, \nthe damping of excitations of the superfluid (for example, by their \ninteraction with the normal fluid).\nThis is achieved by replacing the left-side of Eq. (\\ref{eq:GPE}) \nwith $(i-\\gamma) \\hbar \\, \\partial \\Psi\/\\partial t$, where $\\gamma$ \nspecifies the strength of the dissipation. \nAlthough not as accurate the friction included within the\n VFM, this phenomenological dissipation\nwill help damp out the oscillations of the vortex lines (Kelvin waves),\nwhich is the main effect of the friction which concerns us here.\n Secondly, given our rotating scenario, we work in the reference\nframe rotating at constant angular frequency $\\Omega$ about the $z$ axis; this is achieved by modifying the GPE Hamiltonian to $\\hat{\\mathcal{H}}-i \\Omega L_z$, where $L_z$ is the angular momentum operator about $z$. In Cartesian coordinates $L_z=i \\hbar (y \\partial_x-x \\partial_y)$.\n\n\n\\subsection{Bucket set-up}\n\nWe consider the fluid to be confined within \na cylindrical bucket of radius $R$ and height $H$. The axis of the cylinder\nis the z-axis of rotation. \nThe bucket is modelled through the potential $V({\\bf r})$: in the interior \nof the bucket we set $V=0$ while at the boundary and beyond we set \n$V \\gg \\mu_0$. In the ground state, the fluid density has the bulk value \n$n_0$ in the centre of the bucket, while close to the bucket wall \nit heals to zero density over a length scale characterised by the healing \nlength $\\xi=\\hbar\/\\sqrt{m n_0 g}$. The healing length also characterises \nthe size of the cores of vortices in the fluid.\nNote that the chemical potential in the bulk is $\\mu_0=n_0 g$. \nThe speed of sound in the uniform systems is\n$c=\\sqrt{n_0 g \/m}$.\n\nIt is clearly computationally impossible\nto simulate the range of length scales which are realistic for a typical\nexperiment with liquid helium in the context of the GPE model.\nThe dimensions (radius and height) of typical buckets used \nin the experiments are of the order of the\ncentimetre, which is around eight orders of magnitude larger than the \nvortex core size in helium-4, $a_0=10^{-10}\\rm m$ (in helium-3 the vortex\ncore is about 100 times larger).\nInstead, in our numerical experiments we employ buckets whose scale \nis around 2 orders of magnitude larger than the vortex core size.\nWhile this is clearly a vast scale reduction compared to real systems, \nthe separation of scales between the vortices and the bucket size \nis sufficient to give us a qualitative insight into the dynamics\nof the vortex lines.\n\n\\subsection{Surface roughness}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig1.pdf}\n\\caption{(a) An example of the 2D fractional Brownian motion function $f$, \nnormalised to the range [0,1], shown as a surface plot and a heatmap.\n(b) The rough cylindrical boundary of our bucket\nis formed by using the surface in (a) to modulate\nthe radius of the bucket boundary with an amplitude $a$. Here $a=0.1$. }\n\\label{fig1}\n\\end{figure}\n\n\n\nTo mimic the experimentally unavoidable surface roughness, \nwe modify the azimuthal face of the bucket away from a perfect \ncylindrical shape using a noisy two-dimensional (2D) function. \nThis function is numerically generated through a two-dimensional \nfractal Brownian motion \\cite{MandelbrotVanNess}\nwith Hurst index of $0.3$, a parameter which describes the fractal dimension\nof the surface \\cite{Mandelbrot1985}.\nThe choice to model the roughness in this way is motivated by the well established fractal properties of real surfaces, including machined surfaces (of relevance to helium experiments), and the success of fractal brownian motion in modelling a wide variety of real rough surfaces \\cite{Majumdar1990}. \nThe function is normalised between $0$ and $1$, and is mirrored \nabout its edge and recombined with itself in order to create periodicity \nacross one dimension; a single realisation of the function is \ndepicted in Fig.~\\ref{fig1}(a). The function is mapped onto \nthe space of axial coordinate $z$ and azimuthal angle $\\theta$, \nand used to modify the radius of the bucket according to the form,\n\n\\begin{equation}\nr(z,\\phi)=R(1-a f(z,\\theta)),\n\\label{eq:rbucket}\n\\end{equation}\n\n\\noindent\nwhere $R$ is the smooth bucket radius and $a$ is the \n(dimensionless) roughness parameter.\nThis numerical procedure\ngenerates all of our rough 3D bucket shapes. \nBy computing the local curvature of the surface roughness,\nwe find that the values of the average radius of curvature corresponding to\nvalues $a=0.05$, $0.1$, $0.2$ and $0.3$ of the roughness parameter are\n$10.4 \\xi$, $5.2 \\xi$, $2.6 \\xi$ and $1.7\\xi$ respectively\n(small values of $a$ correspond to large radius of curvature, i.e. smoother\nsurface).\nFor simplicity, the top and bottom surfaces of the bucket are left\nsmooth. The reason is that, by providing the vortex lines with\npinning sites, any roughness on these surfaces \nwill act essentially as an extra friction (an effect which already\nqualitatively account for via the dissipation parameter $\\gamma$) \nslowing down the final stage of cristallization of the vortex lattice.\n\n\n\n\n\n\\subsection{Simulation set-up}\n\nThe initial condition for $\\Psi$ in all of our simulations is the \nnon-rotating ground state solution, found by the method of imaginary \ntime propagation of the GPE, supplemented with low-amplitude \nwhite noise to $\\Psi$ (amplitude $0.001$) to break any symmetries \nartificially presented in the initial condition. \nWe then impose a constant rotation on the system for $t>0$, \nwith fixed rotation frequency $\\Omega$. Note that $\\Omega$ \nfar exceeds the critical rotation frequency to support vortices \n$\\Omega_{\\rm c}$, such that the lowest energy state of the fluid \nis a vortex lattice.\n\n\n\nThe non-dimensionalisation of the GPE is based on the natural units \nof the homogeneous fluid \\cite{Barenghi2016}:\nthe unit of length is the healing length $\\xi$, the unit of speed\nis $c$, the unit of time is \n$\\tau=\\xi\/c=\\hbar\/\\mu_0$, the unit of energy is $\\mu_0$, \nand the unit of density is $n_0$. Both our 3D and 2D numerical simulations \nare performed using XMDS2 \\cite{xmds2}, an open-source partial and \nordinary differential equation solver. \nThe time evolution of the dimensionless GPE is computed\nvia an adaptive fourth-fifth order Runge-Kutta integration scheme \nwith typical time step $dt=0.01 \\tau$ and grid spacing $dx = 0.4 \\xi$; \nthese discretization numbers are sufficiently small to resolve \nthe smallest spatial features (vortices and the fluid boundary layer, \nwhich are of the order of few healing lengths) and the shortest \ntimescales in the fluid. We typically conduct our 3D simulations \non a cubic grid of size $256^3$. Threaded parallel \nprocessing is employed using the OpenMP standard across typically $44$ \nthreads to improve processing speeds on computationally intensive simulations. \n\n\n\\section{Results}\n\\label{section3}\n\n\\subsection{Typical spin-up dynamics} \n\n\\begin{figure*}\n(a) \\hspace{3.8cm} (b) \\hspace{4cm} (c) \\hspace{5cm} \\\\\n\\includegraphics[width = 0.24\\textwidth]{fig2a.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2b.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2c.pdf}\n\\\\(d) \\hspace{4cm} (e) \\hspace{3.8cm} (f) \\hspace{3.8cm} \\\\\n\\includegraphics[width = 0.24\\textwidth]{fig2d.pdf}\n\\includegraphics[width = 0.24\\textwidth]{fig2e.pdf}\t\n\\includegraphics[width = 0.24\\textwidth]{fig2f.pdf}\t\t\n\\caption{\nThree-dimensional snapshots at times \n$t\/\\tau=100$ (a), $200$ (b), $500$ (c), $1000$ (d), $1500$ (e) and $3000$ (f)\nduring the spin-up of the initially quiescent fluid. \nThe vortex cores are identified by density isosurfaces;\nvortices with positive and negative circulation (as determined by \ntheir pseudo-vorticity \\cite{Villois2016} in the $z$ direction)\nare visualised in red and blue respectively. The faint yellow\nisosurface represents\nthe confining bucket. A false-color shadow is projected\nonto the bottom surface to enhance the visualisation of the 3D vortex lines.\n}\n\\label{fig2}\n\\end{figure*} \n\nWe now demonstrate the typical spin up of an initially \nquiescent fluid. Unless otherwise indicated, we present results\nfor the following choice of parameters: \nbucket radius $R=50\\xi$, bucket height $H=100 \\xi$, \nrotation frequency $\\Omega = 0.02~ \\tau^{-1}$,\ndissipation parameter $\\gamma=0.05$, and\nroughness parameter $a=0.1$ (meaning that the irregular surface\nof the bucket extends radially from $45\\rm \\xi$ to $50~\\xi$, corresponding to\nirregular `surface bumps' of height up to 5 healing lengths).\n\n\nThe evolution of the fluid is illustrated by the\nsnapshots shown in Fig.~\\ref{fig2} in which the vortex lines are tracked\nin 3D space using a precise method introduced in Ref. \\cite{Villois2016}; movies of the evolution\nare available in Supplementary Material \\cite{movies}. From the initial quiescent and vortex-free \nfluid, first we see the nucleation of vortex lines at the \ncylindrical boundary of the fluid [Fig.~\\ref{fig2}(a)]. These vortices are all singly-quantized; we do not detect the presence of multiply-charged vortices in any of our simulations, which is consistent with the energetic instability of multiply-charged vortices and the favourability of singly-charged vortices \\cite{Barenghi2016}.\nThe nucleation takes place at the sharpest features on the surface, as seen \nin a previous calculation over a flat rough surface \\cite{Stagg2017}: at \nthese features the local (potential) flow velocity is raised by the \ncurvature of the boundary, and exceeds the critical velocity of \nvortex nucleation, which, according to Landau's criterion,\nin a Bose gas is $v_c \\approx c$.\nSince the local flow speed around a moving obstacle always exceeds \nthe translational speed of the obstacle, Landau's criterion can be\nsatisfied by a translational speed less than $c$.\nFor example, a cylindrical obstacle moving at speed approximately\nequal to $0.4c$ will nucleate vortices \\cite{Frisch1992,Stagg2014}. \nIn our case ($\\Omega =0.02\/\\tau$ and $R=50 \\xi$), the translational \nspeed of the prominences on the rough boundary\nis approximately $\\Omega R \\approx c$, which is sufficient to exceed \nLandau's criterion and nucleate vortices.\nFigure~\\ref{fig2} (a) and (b) show that the\nvortex lines which nucleate at the rough boundary have the shape of\nsmall half-loops or handles; similar vortex shapes have been reported \nin trapped Bose-Einstein condensates\n\\cite{Aftalion2003} and turbulent superfluid helium-4 near\na heated cylinder \\cite{Rickinson2020}, and have been called respectively\n``U-vortices\" and ``handles\". \n\nWe next consider the angular momentum of the fluid, exploring its evolution and distribution. We define the density of the $z$-component of the angular momentum of the fluid as,\n\\begin{equation}\nL_z(x,y,z) = \\psi^* i\\hbar\\left(y\\frac{\\partial}{\\partial x}-x\\frac{\\partial}{\\partial y} \\right) \\psi.\n\\end{equation}\nFigure \\ref{fig_Lz}(a) shows the angular momentum density as a function of the radial coordinate, averaged vertically and azimuthally, $\\langle L_z \\rangle$. At $t=0$, this is zero throughout the fluid. At time evolves, angular momentum builds at the edge of the bucket and drifts inwards, corresponding to the nucleation and inward drift of the vortex lines. At steady-state, the angular momentum forms a stepped curve, with each step corresponding to a concentric ring of vortex lines in the final lattice. Note how the final distribution of the angular momentum density approximately follows the result of solid-body rotation with constant mass density, $m n_0 \\Omega r^2$. The total $z$-component of the angular momentum $\\mathcal{L}=\\iiint L_z\\, {\\rm d}x\\,{\\rm d}y\\,{\\rm d}z$ (inset in Fig. \\ref{fig_Lz}(a)) grows in time, saturating at a final value by around $t\\sim 2000~\\tau$, which is when the vortex line have settled into the lattice configuration. In gaseous superfluids confined within smooth potentials, recent results of merging superfluids \\cite{Kanai2018,Kanai2020} suggest that the rate of angular momentum transfer between a static and rotating state is constant; however, here the growth of the angular momentum follows a sigmoidal curve, rather than a linear one. \n\n\nWe also consider the distribution of the vortex length projected in the $z$-direction, \n$\\Lambda_z$, as a function of radius, shown in Figure \\ref{fig_Lz}(b). At early times, vortex length exists only near the bucket edge, spreading progressively into the bulk. Later, the vortex length converges towards falling at discrete peaks at $r=0$, $r\\approx 13 \\xi$ and $r\\approx 26 \\xi$, corresponding to the concentric arrangement of vortices in the lattice configuration.\n\n\n\n\\begin{figure}\n\\centering\n\\hspace{0.5cm}(a)\\\\\n\\includegraphics[width = 0.7\\columnwidth]{fig3a.pdf}\\\\\n\\hspace{0.5cm}(b)\\\\\n\\includegraphics[width = 0.7\\columnwidth]{fig3b.pdf}\n\\caption{(a) Angular momentum density as a function of radius at various times during the spin-up of the rough bucket (same parameters as in Fig. \\ref{fig2}). Plotted is the angular momentum density averaged over the $z$-dimension, $\\langle L_z \\rangle$. The lines correspond to times indicated by circular markers on the inset The distribution of the angular momentum of a solid-body of uniform density is shown by the red dashed line. The inset shows the evolution of the total angular momentum of the fluid $\\mathcal{L}=\\iiint L_z\\, {\\rm d}x\\,{\\rm d}y\\,{\\rm d}z$. (b) The vortex length projected in the $z$-direction \n$\\Lambda_z$ plotted as a function of radius, at the same times as in (a). The data is binned in radial intervals of $3\\xi$.\n}\n\\label{fig_Lz}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\hspace{0.5cm}(a)\\\\\n\\includegraphics[width = 0.6\\columnwidth]{fig4a.pdf}\\\\\n\\hspace{0.5cm}(b)\\\\\n\\includegraphics[width = 0.6\\columnwidth]{fig4b.pdf}\n\\caption{Azimuthal velocity $v_{\\theta}$ of the fluid as a function of radius $r$, for\nrotation frequencies of $\\Omega=0.02~\\tau^{-1}$ (a) and\n$\\Omega=0.06~\\tau^{-1}$ (b). We use roughness parameter $a=0.1$. The solid red lines \nrepresent solid body rotation $v_{\\theta}=\\Omega r$; the blue lines\nare values of $v_{\\theta}(r)$ averaged in the $\\theta$ direction;\nthe pale blue rectangles are histograms with bin size $\\Delta r=5 \\xi$\n(therefore the outer bins contain more data points). It is apparent that\nthe more rapid rotation (b) creates a vortex lattice in better agreement\nwith the solid body rotation, and that there is a vortex-free region\nnear the boundary.\n}\n\\label{fig3}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig5.pdf}\n\\caption{\nEvolution of the total vortex length, $\\Lambda$ (solid lines), as well\nas the vortex length in the $z$-direction, $\\Lambda_z$ (dashed lines) and\nthe vortex length in the $xy$-plane, $\\Lambda_{xy}$ (dot-dashed lines), \nplotted versus time $t$ for different angular velocity of rotation\n$\\Omega=0.02~\\tau^{_1}$ (black), $0.04~\\tau^{-1}$ (blue) and\n$0.06 ~\\tau^{-1}$ (red) \nachieving final values of the vortex\nlength $\\Lambda_{\\infty}=2184 \\xi$, $6007 \\xi$ and $5568 \\xi$ respectively.\nAll curves refer to roughness parameter $a=0.1$. \n}\n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.9\\columnwidth]{fig6.pdf}\n\\caption{\nEvolution of the total vortex length, $\\Lambda$ (solid lines), as well\nas the vortex length in the $z$-direction, $\\Lambda_z$ (dashed lines) and\nthe vortex length in the $xy$-plane, $\\Lambda_{xy}$ (dot-dashed lines), \nplotted versus time $t$ for different boundary roughness\n$a=0.05$ (red), $0.1$ (black), $0.2$ (blue), and $0.3~\\rm \\xi$ (magenta),\nachieving final values $\\Lambda_{\\infty}=1899~\\xi$, $2184~\\xi$,\n$2161~\\xi$ and $1643~\\xi$. All curves refer to the same\nangular velocity $\\Omega=0.02~\\tau^{-1}$. \n}\n\\label{fig5}\n\\end{figure}\n\nThe collection of U-vortices nucleated at the boundary\nis the superfluid's analog of a boundary layer, the region separating the\nrotating boundary from the still quiescent bulk of the fluid.\nThe U-vortices tend to be aligned along the $z$-direction, creating\na superflow in the same direction of the rotating boundary.\nThe vortex nucleation is therefore short-lived, since the nucleated\nU-vortices reduce the relative motion between the fluid and the\nboundary, suppressing further nucleations.\nIn time, the U-vortices grow in size and extend further into the \nfluid [Fig. \\ref{fig2}(b,c)], ultimately filling the bulk \n[Fig. \\ref{fig2}(d)]. During this stage of the evolution,\nthe U-vortices also grow in vertical extent in the $z$-direction, \noccasionally connecting and merging with each other, thus increasing\ntheir vertical extent. When the length in the $z$-direction becomes of the order \nof the bucket's height $H$, one or both vortex endpoints start sliding along\nthe smooth top and\/or bottom of the bucket.\nOnce most of the vortex lines are fully extended from the top to the bottom\nof the bucket, they quickly drift into the \nbulk of the fluid. Although the vortex lines are aligned along the direction of\nrotation, they remain highly excited and undergo reconnection events when\nthey collide with each other. Over time they relax towards a regular \nconfiguration of straight vortices. A small proportion of U-vortices\nremain attached to the side of the bucket for a longer period of time \n[Fig.~\\ref{fig2}(d)]; over a longer time they detach,\nand relax to the final lattice configuration. Some of the vortex lines\nend up diagonally across the rest of the vortex lattice\n[Fig.~\\ref{fig2}(e)]: eventually they also relax to the final\nlattice configuration \n[Fig. \\ref{fig2}(f)]. The vortex lattice is stationary in \nthe rotating frame, representing the lowest energy state of \nthe rotating superfluid. In this final state, the coarse-grained \nfluid velocity approximates the solid-body result \n${\\bf v} = v_{\\theta}\\, {\\bf e_\\theta}= \\Omega \\, r \\, {\\bf e_\\theta}$, \nwhere ${\\bf e_\\phi}$ is the azimuthal unit vector, as shown in Fig.~\\ref{fig3};\nas expected, the agreement improves with increasing $\\Omega$, and there is\na vortex-free region near the boundary. \n\nOur 3D results are presented for a fixed bucket size due to computational constraints of simulating a larger system. For a larger bucket we would expect qualitatively similar dynamics; indeed our 2D results in a larger bucket presented in Section IV C support this. The most significant change under a larger bucket is more vortices in the final state (at a fixed rotation frequency) and as a result a better approximation to solid-body rotation.\n\n\n\n\\subsection{Role of angular velocity and roughness} \n\nTo analyse the vortex dynamics further it is useful to distinguish the total vortex length, \n$\\Lambda$, from the vortex length projected in the $z$-direction, \n$\\Lambda_z$, and the vortex length projected in the $xy$-plane, \n$\\Lambda_{xy}$. In the final vortex lattice all vortex lines \nare aligned along $z$, hence we expect that, after a sufficiently long time, $\\Lambda_{xy}\\approx 0$ and\n$\\Lambda_z \\approx \\Lambda$, with $\\Lambda \\rightarrow N_v H$, where $N_v$ is the final number of straight vortex lines. \nFigure~\\ref{fig4} displays $\\Lambda$ (solid lines), $\\Lambda_z$ (dashed lines)\nand $\\Lambda_{xy}$ (dot-dashed lines) as a function of time for different\nangular velocities of rotation, $\\Omega=0.02$, $0.04$ and $0.06$\nat the same roughness parameter $a=0.1$. \n It is apparent that in the initial stage, \na great amount of vorticity is in the $xy$-plane, before realignment \nof the vortex lines along the $z$-axis of rotation takes place. The effect\nis particularly noticeable at the largest angular velocities, for which,\nduring the initial transient, the vortex length is considerably larger\nthan the value $\\Lambda_{\\infty}$ achieved in the final vortex lattice\nconfiguration. Moreover, we see that the final vortex line length increases with $\\Omega$ due to the increasing number of vortices in the final state. \n\nFigure~\\ref{fig5} shows $\\Lambda$, $\\Lambda_z$ and $\\Lambda_{xy}$ \nplotted versus time at the same angular velocity $\\Omega=0.02~\\tau^{-1}$ \nfor different values of \nroughness parameter $a$. The largest values of the final vortex\nlength $\\Lambda_{\\infty}$ are achieved with $a=0.1 \\xi$ and $a=0.2\\xi$. \nSmoother ($a=0.05\\xi$) and rougher ($a=0.3\\xi$) boundaries generate \nless vortex length. These variations in the final line length arise to the final number of vortex lines varying by a few vortices across these cases.\nIt is not surprising that the final vortex lattice depends\non the roughness which has nucleated the initial \nvorticity. Feynman's rule [Eq.~\\eqref{eq:Feynman}] only refers to \nan idealised homogeneous system. Boundaries are known to have effects \n(e.g. missing vortex lines near the boundary) and \nit has been observed that the formation of the vortex lattice may be \nhistory-dependent and involve\nmetastability \\cite{CampbellZiff,Wood2019} and hysteresis \\cite{Mathieu}. \n\nFigure~\\ref{fig_Lz_total} compares the growth of angular momentum between the default case (blue line), the case where the rotation frequency is doubled (red line), and the case where the roughness amplitude is doubled (yellow line). The growth behaviour is qualitatively similar in all cases. Doubling the rotation frequency leads to a much faster rate of injection of angular momentum, and a higher final value, consistent with the faster injection rate of vortex lines from the boundary and the higher density of vortex lines in the final lattice state. Doubling the surface roughness has little effect on the growth of the angular momentum, just slightly increasing the rate of angular momentum injection, which can be attributed to the greater injection rate of vortices from the rougher surface.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.8\\columnwidth]{fig7.pdf}\\\\\n\\caption{\nEvolution of the total angular momentum for the default parameters used in Fig. \\ref{fig2} (blue line), double the rotation frequency (red line) and double the roughness amplitude (yellow line). \n}\n\\label{fig_Lz_total}\n\\end{figure}\n\n\\begin{figure*}\n(a) \\hspace{3.8cm} (b) \\hspace{4cm} (c) \\hspace{7cm} \\\\\n\t\\includegraphics[width = 0.24\\textwidth]{fig8a.pdf}\n \\includegraphics[width = 0.24\\textwidth]{fig8b.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8c.pdf}\n\\\\(d) \\hspace{4cm} (e) \\hspace{3.8cm} (f) \\hspace{3.8cm} \\\\\n\t\\includegraphics[width = 0.24\\textwidth]{fig8d.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8e.pdf}\n\t\\includegraphics[width = 0.24\\textwidth]{fig8f.pdf}\t\n\\caption{\nDensity profile in the $xy$-plane at the half-height of the bucket, showing the final vortex configurations for the following cases:\n(a): $\\Omega=0.02~\\tau^{-1}$ and $a=0.1$ (default choice);\n(b): $\\Omega=0.04~\\tau^{-1}$ and $a=0.1$ (double rotation);\n(c): $\\Omega=0.06~\\tau^{-1}$ and $a=0.1$ (triple rotation);\n(d): $\\Omega=0.02~\\tau^{-1}$ and $a=0.05$ (half roughness);\n(e): $\\Omega=0.02~\\tau^{-1}$ and $a=0.2$ (double roughness);\n(f): $\\Omega=0.02~\\tau^{-1}$ and $a=0.1$ (single strong protuberance added,\nsee Section~\\ref{section4}). \n}\n\\label{fig6}\n\\end{figure*} \n\n\nFigure~\\ref{fig6} illustrates some of the final vortex patterns which we have\ncomputed by plotting the superfluid density,\n$\\vert \\psi(x,y) \\vert^2$ in the $xy$-plane at\nhalf-height of the bucket. In these pictures\nthe vortices appear as small holes; to clarify the lengthscales,\nwe recall that on the vortex axis the density is zero and that\nat distance $r=2 \\xi$ from the axis, the density recovers about $80 \\%$ of the\nbulk value at infinity.\nIt is interesting\nto compare the different final vortex configurations for halved\/doubled rotation velocity and the roughness parameter with respect\nto our default choice ($\\Omega=0.02~\\tau^{-1}$ and $a=0.1$). While the ideal\n2D vortex lattice has a vortex at the centre, surrounded by a first row of 6 vortices,\na second row of 12 vortices, etc, the vortex configurations shown\nin Fig.~\\ref{fig6} contain slightly different vortex numbers; in\nparticular some configurations contain vortex lines which seem \nmisplaced [Fig.~\\ref{fig6}(c)]\nor lack the vortex at the centre [Fig.~\\ref{fig6}(e)];\nthese configurations are metastable states corresponding to local \nminima of the free energy in the rotating frame \\cite{CampbellZiff}. \nMoreover, at slow rotations [Fig.~\\ref{fig6}(a,d)] the predicted vortex-free region near the boundary \n\\cite{NorthbyDonnelly1970,ShenkMehl1971,StaufferFetter1968} is clearly\nvisible; this phenomenon affects the coarse-grained azimuthal velocity near the\nboundary shown previously in Fig.~\\ref{fig3}(a). The depletion of the background fluid density in the centre of the bucket - particularly evident in Fig. \\ref{fig6}(b) and (c) - is due to coarse-grained centrifugal effects, analogous to the classical rotating case \\cite{Barenghi2016}. \n\n\n\n\n\n\n\n\n\\section{Other effects}\n\\label{section4}\n\nIn this section we\nrepeat the simulation of Section~\\ref{section3} with several \nsignificantly modifications: the presence of a single strong\nprotuberance, the presence of remanent vortex lines, and the 2D case.\nThe aim is to determine whether these effects change qualitatively the dynamics \ndescribed in Section~\\ref{section3}.\n\n\\subsection{Effect of a strong protubance}\n\n\\begin{figure*}\n\\hspace{-0.2cm} (a) \\hspace{2.9cm} (b) \\hspace{2.9cm}(c) \\hspace{2.9cm} (d) \\hspace{2.9cm} (e) \\hspace{1cm}\\\\\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9a.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9b.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9c.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9d.pdf}\n\t\t\\includegraphics[width = 0.19\\textwidth]{fig9e.pdf}\t\n\\caption{Early-time dynamics during the spin-up of the fluid in the \npresence of a single strong protuberance added to the rough\ncylindrical boundary. The snapshots, taken at \n$t\/\\tau=26$, $50$, $100$, $200$ and $500$, are presented in the same way \nas Fig.~\\ref{fig2}. }\n\\label{fig7}\n\\end{figure*}\n\n\nFirst we consider the effect of a single strong imperfection in the\nform of a protuberance on the cylindrical wall. The question is\nwhether, by enhancing vortex nucleation, the protuberance \ncan induce a turbulent boundary layer.\nThe protuberance is numerically created by adding a Gaussian-shaped \npotential to the existing (small-scale) roughness potential. \nEquation~\\eqref{eq:rbucket} is replaced by\n\n\n\\begin{equation}\nr(z,\\phi)=R[1-a (f(z,\\theta) + G f_G(z,\\theta))],\n\\label{eq:rbucket2}\n\\end{equation}\n\n\\noindent\nwhere $G=2$ and $f_G(z,\\theta)$ is a Gaussian-shape function taking values\nfrom $0$ to $1$ and rms width $4 \\xi$. The approximate height of the \nstrong protuberance \nin the simulation which we present is $10 \\xi$, as also visible\nin Fig.~\\ref{fig6}(f).\n\n\\noindent\nSnapshots taken during the time evolution for $\\Omega=0.02~\\tau^{-1}$\nand $a=0.1$ are shown in Fig.~\\ref{fig7}; a movie can be viewed \nin Supplementary Material \\cite{movie_gaussian}.\nThe protuberance catalyses the local nucleation of vortices \nat early times: large vortex loops \n(of the same size order as the protuberance) are rapidly generated\n[Fig.~\\ref{fig7}(a)], leading to a downstream trail of \nloops [Fig.~\\ref{fig7}(b, c)], in addition to the slower \nnucleation of U-vortices from the rough bucket wall. \nThe vortex configuration becomes\nclearly anisotropic near the bucket edge [Fig.~\\ref{fig7}(d)]. \nHowever, once the vortices fill the bulk [Fig.~\\ref{fig7}(e)],\nmemory of this effect is lost, and the subsequent evolution is\nvery similar to the evolution without the strong protuberance.\nIn fact, the final vortex lattice is not significantly different from the\nlattices considered in Section~\\ref{section3}, as shown in Fig.~\\ref{fig6}(f). Figure~\\ref{fig10} shows the time evolution of $\\Lambda$,\n$\\Lambda_z$ and $\\Lambda_{xy}$ in the presence of the protuberance (magenta lines) and its absence (black lines). This confirms that the protuberance accelerates the generation of vortex line length at early times, but that its effect becomes washed out at later times.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width = 0.45\\textwidth]{fig10.pdf}\n\\caption{\nTime evolution of $\\Lambda$ (solid line), $\\Lambda_z$ (dashed line), \n$\\Lambda_{xy}$ (dot-dashed line). The colours correspond to the \nsimulations with default parameters $\\Omega=0.02 \\tau^{-1}$,\n$a=0.1 \\xi$ (black),\nthe added surface protuberance (magenta),\nthe added remanent negative vortex (blue), and\nthe added remanent positive vortex (red), respectively.\n}\n\\label{fig10}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Effect of remanent vortices}\n\nSecondly, we consider the effect of remanent vortex lines.\nIn experiments with liquid helium, it is believed that so-called \n`remanent vortices' may be present in the fluid, created via the \nKibble-Zurek mechanism when cooling the helium sample through the \nsuperfluid transition to the final experimental temperature. \nThe presence of remanent vortices may modify the vortex nucleation and the\nformation of the vortex lattice when the sample is rotated.\nTo explore this idea, we have repeated the simulations imposing\na suitable phase profile to add a vortex to the initial state during the \nimaginary-time propagation. \nFor simplicity we position the remanent vortex along the $z$-axis \nof rotation. \n\nThe evolution of the superfluid with the standard rough cylindrical\nwall and a ``positive\" remanent vortex, that is, one whose circulation is oriented in the same direction of the\nbucket's rotation is shown through Fig.~\\ref{fig8} \nand the movie in the Supplementary Material \\cite{movie_pos_remnant}. \nCompared to Section~\\ref{section3},\nthe only significant modification is a dampening of the initial injection \nof U-vortices; the effect is visible by eye when comparing like-time \nsnapshots [Fig. \\ref{fig2}(b) and Fig.~\\ref{fig8}(a)]. \nThe remanent vortex acts in the same direction as the rotating container: \nit reduces the relative speed between the bucket's wall and the superfluid,\nand remains largely undisturbed at early times \n[Fig.~\\ref{fig8}(a)] until the \nU-vortices that are nucleated fill the bulk and interact \nwith it [Fig.~\\ref{fig8}(b)]; at this point\nthe remanent vortex becomes subsumed within the other like-signed \nvortices [Fig.~\\ref{fig8}(c)], and the \nsubsequent relaxation of the vortex configuration into a vortex\nlattice largely proceeds as if there was not any remanent vortex initially. Confirming this, we see that in Figure~\\ref{fig10} that the presence of the positive vortex (red lines) depletes the generation of vortex line length at early times, but this recovers at later times such that the system reaches the same line length as in the absence of any remanent vortices (black lines).\n\n \n\\begin{figure*}\n\\hspace{-2.2cm} (a) \\hspace{3.7cm} (b) \\hspace{3.9cm} (c)\\\\\n\\includegraphics[width = 0.22\\textwidth]{fig11a.pdf}\n\\includegraphics[width = 0.22\\textwidth]{fig11b.pdf}\n\\includegraphics[width = 0.22\\textwidth]{fig11c.pdf}\n\\caption{Spin-up of the superfluid in the presence of a positively-charged \nremanent vortex. Snapshots are taken at $t\/\\tau=200$, $400$ and $500$, \nand are presented in the same way as Fig.~\\ref{fig2}.}\n\\label{fig8}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\hspace{0.2cm} (a) \\hspace{3.7cm} (b) \\hspace{3.2cm} (c)\n\\hspace{0.2cm} (d) \\hspace{3.7cm} (e) \\hspace{3.2cm} \n\\includegraphics[width = 0.19\\textwidth]{fig12a.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12b.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12c.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12d.pdf}\n\\includegraphics[width = 0.19\\textwidth]{fig12e.pdf}\n\\caption{Spin-up of the superfluid in the presence of a negatively-charged \nremanent vortex. \nSnapshots are taken at $t\/\\tau=200$ (a), $360$ (b), $386$ (c), $420$ (d)\nand $500$ (e). The images are presented in the same way as Fig.~\\ref{fig2}. \nVortices with negative circulation are coloured blue.}\n\\label{fig9}\n\\end{figure*}\n\n\n\n\n\nIf the remanent vortex is oriented in the direction opposite to the\nrotation of the bucket, i.e. a negative vortex, the evolution proceeds differently, as seen in Fig.~\\ref{fig9} and the movie in Supplementary Material\n\\cite{movie_neg_remnant}. \nThe remanent vortex enhances the nucleation of U-vortices\nfrom the boundary, as evident from comparing \nFig.~\\ref{fig2}(b) and Fig.~\\ref{fig9}(a). \nThis effect is caused by the counter-flow induced by the remanent\nvortex, which increases the relative speed of the fluid over the rough \nboundary. \nOnce the other vortices drift close to the remanent vortex, the \nremanent vortex becomes excited by their interaction\n[Fig.~\\ref{fig9}(b)]. \nA series of vortex reconnections break up the remanent vortex, \nforming progressively smaller vortex loops \n[Fig.~\\ref{fig9}(c,d)]. This leads to the rapid \nremoval of vorticity of the `wrong' sign from the fluid\n [Fig.~\\ref{fig9}(e)]. \nHereafter the fluid evolves in a similar manner to when the remanent vortex in absent [Section~\\ref{section3}], albeit with a slightly higher final vortex line length [Fig.~\\ref{fig10}].\n\n\n\\subsection{2D case}\n\nFinally, we have also performed the corresponding 2D simulations of the spin-up of a 2D superfluid within a rough circular boundary; the boundary is taken from the central slice of the 3D rough bucket. A movie showing the typical dynamics\nis available in the Supplementary Material \\cite{movie_2d_large_bucket}.\nThe 2D geometry\nallows calculations of much larger buckets, up to $R=200 \\xi$ with\na $1024^2$ numerical grid. We observe the same qualitative behaviour as in\n3D in smaller buckets, albeit with many more vortices and without 3D effects\nsuch as vortex reconnections. Collisions of vortices of the opposite\ncirculation result in the annihilation of the vortices and the emission of\nsound pulses \\cite{Kwon2014,Stagg2015,Groszek2016}.\nIn general, we find that, in 2D, the timescales of injection, diffusion and \nlattice crystallisation are faster than in 3D. \nA particular feature that we see in the early-time dynamics of\nthe 2D simulations is the nucleation of vortices with both positive \nand negative circulation (i.e. with circulation which is inconsistent\nwith the imposed rotation). We notice that some negative vortices \noriginate from localised rarefaction pulses generated \nfrom the rough boundary when the bucket is set into rotation. \nWe associate these pulses with Jones-Roberts solitons \n\\cite{Jones1982,Tsuchiya2008}, which are low energy\/momentum \nsolutions of the 2D GPE. \nAt higher energy\/momentum, these solutions become pairs of vortices \nof opposite sign (also called vortex dipoles in the literature).\nThe conversion of Jones-Roberts solitons into vortex dipoles occurs\nif the pulse gains energy from the large positive vortex cluster\nwhich starts forming in the centre of the bucket. Occasionally,\nthe vortices which are parts of a dipole separate\nand mix with the rest of the vortices. Over time, the vortices \nof negative circulation are lost from the system, either \ncolliding (hence annihilating) with positive vortices within the bulk,\nor by exiting the fluid at the bucket's boundary (effectively\nannihilating with their images). \n\\vspace{0.3cm}\n\n\\section{Conclusions}\n\\label{section5}\n\nIn conclusion, we have employed simulations of the Gross-Pitaevskii equation to study the spin-up of a superfluid in a rotating bucket featuring microscopically rough walls. Within this model, we see several key stages of the dynamics. Firstly, vortices are nucleated at the boundary by the flow over the rough features, typically in the form of small U-shaped vortex lines. Secondly, these U-shaped vortices interact strongly and reconnect, creating\na transient turbulent state. This becomes increasingly polarised by the imposed rotation until the vortex configuration consists of vortices of the correct orientation extending from the top to the bottom of\nthe bucket. Finally, the vortex lines slowly straighten and arrange themselves in the expected final vortex lattice configuration. Our results highlight the importance of vortex \nreconnections \\cite{Galantucci2019}: it is\ngenerally assumed that vortex reconnections are important in turbulence,\nbut here we have seen that reconnections are essential to create, starting\nfrom potential flow, something as simple as solid body rotation\n(the vortex lattice). \nThe addition\nof a single large protuberance or one additional remanent vortex line does not\nchange the dynamics significantly, only speeding up or slowing down the injection of vorticity. Moreover, analogous dynamics arise in the 2D limit. \n\nWe reiterate that the GPE is not a quantitatvely accurate model of superfluid helium and these results should be interpreted qualitatively only. For example, the role of friction is introduced into the GPE through a widely-used phenomenological dissipation term; however, a more accurate physical model of this stage of the dynamics would be provided by the VFM. Also, a distinctive physical property of superfluid helium is its strong non-local interactions. This, for instance, supports a roton minimum in its excitation spectrum. While this is absent from the GPE model we have employed, it can be introduced through an additional non-local term \\cite{Berloff2014,Reneuve2018}. It would be interesting to see if this causes any significant departures from the dynamics we have reported. \n\n\n\n\n\\section*{Acknowledgements}\n\nN.P., L. G. and C.F.B. acknowledge support by the Engineering and Physical \nSciences Research Council (Grant No. EP\/R005192\/1).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Backpropagation through time}\n\nThere are many approaches to training recurrent networks~\\cite{A1987,P1987},\nand the most successful one by far is backpropagation\nthrough time \\cite{W1990}.\nBackpropagation through time (BPTT) is an approach to compute\nthe gradient of a recurrent network when consuming a sequence.\nA computation graph is created based on the decoding approaches,\nand the gradients are propagated back through the computation\nvertices. It is equivalent to unrolling the recurrent networks\nfor several time steps depending on the decoding approaches;\nhence the name backpropagation through time.\n\nThere are many variants of BPTT with the most popular one\nbeing truncated BPTT \\cite{WP1990}. In the original\ndefinition \\cite{WP1990}, instead of propagating\ngradients all the way to the start of the unrolled chain,\nthe accumulation stops after a fixed number of time steps.\nTraining recurrent networks with truncated BPTT can be justified\nif the truncated chains are enough to learn\nthe target recursive functions.\nIn the modern setting~\\cite{SSEP2014},\ntruncated BPTT is treated as regular BPTT with batch decoding,\nand the number of unrolled steps before truncation is\nthe number of context frames in batch decoding.\nHenceforth, we will use the term BPTT with batch decoding\nto avoid confusion.\n\nBPTT with batch decoding is seldom used in practice\ndue to the high computation cost,\nwith the only exception being~\\cite{SSEP2014}\nwhere the context is set to six frames.\nA more common approach used in conjunction with BPTT and batch decoding\nis to predict a batch of frames rather than a single one \\cite{SSB2014, CYH2015}.\nFormally, for a single chain of recurrent network to\npredict $p$ consecutive frames at time $t$, we have\n$\\hat{y}_{t+i-1} = o(h_{t+\\ell+i-2})$ for $i = 1, \\dots, p$\nwhere $\\ell$ is the number of lookahead.\nNote that this decoding approach is only used for training\nand is never actually used at test time.\nFigure~\\ref{fig:online-batch} summarizes\nthe decoding approaches and the hyperparameters, including\nlookahead, context frames, and consecutive prediction.\n\nIn practice, recurrent networks are trained with a combination\nof BPTT, batch decoding, lookahead, and consecutive prediction \\cite{SSB2014, CYH2015}.\nTo speed up training, no frames in an utterance are predicted more than once,\nand many sequences are processed in batches to better\nutilize parallel computation on GPUs.\nIn addition, the hidden vectors are sometimes cached \\cite{SSEP2014}.\nApplying these heuristics, however, creates a mismatch\nbetween training and testing.\nPrevious work has not addressed this issue,\nand the distinction between online and batch decoding\nunder BPTT has only been lightly explored in~\\cite{SSEP2014, KTLK2015}.\nWe will examine these in detail in our experiments.\n\n\\begin{figure*}\n\\begin{center}\n\\begin{tikzpicture}\n\\node (x1) at (0, 0) {$x_1$} ;\n\\node (x2) at (1, 0) {$x_2$} ;\n\\node (xdots) at (2, 0) {$\\cdots$} ;\n\\node (xT) at (3, 0) {$x_T$} ;\n\n\\node (h1) at (0, 1) {$h_1$} ;\n\\node (h2) at (1, 1) {$h_2$} ;\n\\node (hdots) at (2, 1) {$\\cdots$} ;\n\\node (hT) at (3, 1) {$h_T$} ;\n\n\\node (y1) at (0, 2) {$y_1$} ;\n\\node (y2) at (1, 2) {$y_2$} ;\n\\node (ydots) at (2, 2) {$\\cdots$} ;\n\\node (yT) at (3, 2) {$y_T$} ;\n\n\\draw[->] (x1) -- (h1);\n\\draw[->] (x2) -- (h2);\n\\draw[->] (xT) -- (hT);\n\n\\draw[->] (h1) -- (h2);\n\\draw[->] (h2) -- (hdots);\n\\draw[->] (hdots) -- (hT);\n\n\\draw[->] (h1) -- (y1);\n\\draw[->] (h2) -- (y2);\n\\draw[->] (hT) -- (yT);\n\n\\node (x2T3) at (6, 0) {$x_{t-3}$} ;\n\\node (x2T2) at (7.5, 0) {$x_{t-2}$} ;\n\\node (x2T1) at (9, 0) {$x_{t-1}$} ;\n\\node (x2T) at (10.5, 0) {$x_t$} ;\n\\node (x2Tp1) at (12, 0) {$x_{t+1}$} ;\n\\node (x2Tp2) at (13.5, 0) {$x_{t+2}$} ;\n\\node (x2Tp3) at (15, 0) {$x_{t+3}$} ;\n\n\\node (h2T3) at (6, 1) {$h_{t-3}$} ;\n\\node (h2T2) at (7.5, 1) {$h_{t-2}$} ;\n\\node (h2T1) at (9, 1) {$h_{t-1}$} ;\n\\node (h2T) at (10.5, 1) {$h_t$} ;\n\\node (h2Tp1) at (12, 1) {$h_{t+1}$} ;\n\\node (h2Tp2) at (13.5, 1) {$h_{t+2}$} ;\n\\node (h2Tp3) at (15, 1) {$h_{t+3}$} ;\n\n\\node (y2T) at (10.5, 2) {$y_t$} ;\n\\node (y2Tp1) at (12, 2) {$y_{t+1}$} ;\n\\node (y2Tp2) at (13.5, 2) {$y_{t+2}$} ;\n\n\\draw[->] (h2T2) -- (h2T1);\n\\draw[->] (h2T1) -- (h2T);\n\\draw[->] (h2T) -- (h2Tp1);\n\\draw[->] (h2Tp1) -- (h2Tp2);\n\\draw[->] (h2Tp2) -- (h2Tp3);\n\n\\draw[->] (x2T2) -- (h2T2);\n\\draw[->] (x2T1) -- (h2T1);\n\\draw[->] (x2T) -- (h2T);\n\\draw[->] (x2Tp1) -- (h2Tp1);\n\\draw[->] (x2Tp2) -- (h2Tp2);\n\\draw[->] (x2Tp3) -- (h2Tp3);\n\n\\draw[->] (h2Tp1) -- (y2T);\n\\draw[->] (h2Tp2) -- (y2Tp1);\n\\draw[->] (h2Tp3) -- (y2Tp2);\n\n\\end{tikzpicture}\n\\caption{\\emph{Left}: Online decoding for $(x_1, \\dots, x_T)$.\n \\emph{Right}: One chain of batch decoding\n at time $t$ with 6 context frames, a lookahead of 2 frames, and consecutive prediction\n of 3 frames. The chains are repeated for $t = 1, \\dots, T$.}\n\\label{fig:online-batch}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\\section{Conclusion}\n\nWe study unidirectional recurrent networks, LSTMs in particular,\ntrained with two decoding approaches, online and batch\ndecoding, and explore various hyperparameters,\nsuch as lookahead, context frames, and consecutive prediction.\nOnline decoding can be as broad as recursive functions,\nwhile batch decoding can only include Markov functions.\nTraining LSTMs with the matching decoding approaches\nperforms best. LSTMs trained with online decoding\ncan still have decent performance with batch decoding,\nbut LSTMs trained with batch decoding tend\nnot to perform well with online decoding.\nThe amount of lookahead is also critical\nfor LSTMs with online decoding to get better\ntraining errors. The number of context frames\nstrictly limits the history that can be used\nfor prediction, while increasing the amount of consecutive\nprediction gears LSTMs closer to recursive functions.\nThe results depend on the long-term dependency in the data,\nand we confirm this by exploring subphonetic states, phonemes, and words.\nFinally, we show that the vanishing gradient phenomenon\nis a necessary condition for Markov functions\nwith respect to variables beyond the necessary history.\nIt is important to note that the same LSTM architecture\nand the same number of model parameters can\nhave drastically different results and behaviors.\nWe hope the results improve the understanding\nof recurrent networks, what they learn from\nthe data, and ultimately the understanding\nof speech data itself.\n\n\\section{Experiments}\n\nTo study how well recurrent networks model Markov\nand recursive functions, we conduct experiments on\nframe classification tasks with three different\nlinguistic units, i.e., subphonetic states, phonemes,\nand words. We expect that subphonetic states and phonemes are\nrelatively local in time, while words require\nlonger contexts to predict. By varying the target units, we control\nthe amount of long-term dependency in the data.\n\nExperiments are conducted on the Wall Street Journal data set\n(\\textfw{WSJ0} and \\textfw{WSJ1}), consisting of about 80 hours of\nread speech and about 20,000 unique words,\nsuitable for studying rich long-term dependency.\nWe use 90\\% of \\textfw{si284}\nfor training, 10\\% of \\textfw{si284} for development,\nand evaluate the models on \\textfw{dev93}.\nThe set \\textfw{dev93} is chosen because\nit is the only set where frame error rates are\nreported for deep networks \\cite{GJM2013}.\nThe time-aligned frame labels, including\nsubphonetic states, phonemes, and words,\nare obtained from speaker-adaptive hidden Markov models\nfollowing the Kaldi recipe~\\cite{P+2011}.\n\nIn the following experiments, we use, as input to the frame classifiers,\n80-dimensional log Mel features without appending i-vectors.\nFor recurrent networks, we use 3-layer LSTMs with 512 hidden units\nin each layer. In addition, for the baseline we use\na 7-layer time-delay neural network (TDNN)\nwith 512 hidden units in each layer and an architecture\ndescribed in~\\cite{THGG2018,PWPK2018}.\nThe network parameters are initialized based on~\\cite{HZRS2015}.\nThe networks are trained with vanilla stochastic gradient descent\nwith step size 0.05 for 20 epochs.\nThe batch size is one utterance and the gradients are clipped\nto norm 5. The best performing model is chosen based on\nthe frame error rates (FER) on the development set.\nUtterances are padded with zeros when lookahead or\ncontext frames outside the boundaries is queried.\n\n\\subsection{Long-term dependency in subphonetic states}\n\nWe first experiment with recurrent networks on subphonetic states.\nWe expect LSTMs to perform best when trained and\ntested with online decoding, so we first explore\nthe effect of lookahead.\nResults are shown in Table~\\ref{tbl:state-lookahead}.\nLookahead has a significant effect on the frame error rates.\nThe error rate improves as we increase the amount of lookahead\nand plateaus after 10 frames.\nThe improvement is due to better training error (not regularization or other factors),\nas shown in Figure~\\ref{fig:train}.\nIn addition, the fact that increasing the amount of lookahead does\nnot hurt performance suggests that LSTMs are able\nto retain information for at least 20 frames.\nCompared to prior work, our 3-layer LSTM is unidirectional\nand uses fewer layers than the ones used in~\\cite{GJM2013},\nbut the results are on par with theirs.\nThe best LSTM achieves 11.7\\% word error rate on \\textfw{dev93},\nsimilar to the results in~\\cite{GJM2013}.\\footnote{\nWe are aware of the state of the art on this data set \\cite{CL2015}.\nAs the results are in the ballpark,\nwe do not optimize them further.}\nThe TDNN serves as an instance of a Markov function.\nWe present its result, but more investigations are needed\nto conclude anything from it.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different LSTM lookaheads\n trained with online decoding, on subphonetic states.\n The FER of a 7-layer TDNN is provided as a reference.}\n\\label{tbl:state-lookahead}\n\\begin{center}\n\\begin{tabular}{lll}\nlookahead & dev & \\textfw{dev93} \\\\\n\\hline \n1 & 45.46 & 43.45 \\\\\n5 & 33.64 & 32.43 \\\\\n10 & 30.43 & 29.83 \\\\\n15 & 29.89 & 28.69 \\\\\n20 & 29.36 & 28.59 \\\\\n\\hline\n\\hline\nTDNN 512x7\n & 33.56 & 34.19\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=5cm]{lookahead}\n\\caption{Running average of training losses\n for comparing different amount of lookahead for LSTMs\n with online decoding trained on subphonetic states.}\n\\label{fig:train}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\nTo see how decoding approaches affect performance,\nwe examine the error rates with batch decoding \non the best LSTM in Table~\\ref{tbl:state-lookahead},\ni.e., the one trained with online decoding\nand a lookahead of 20 frames. Results are shown in\nTable~\\ref{tbl:state-batch}. The performance\ndeteriorates as we reduce the number of context frames.\nThis suggests that LSTMs do utilize information\n40 frames away. On the other hand, the degradation\nis not severe, suggesting that much of\nthe long-term dependency is Markov.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different context frames\n under batch decoding\n with the best LSTM trained with online decoding,\n and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-batch}\n\\begin{center}\n\\begin{tabular}{lll}\ncontext & dev & \\textfw{dev93} \\\\\n\\hline\n40 & 37.65 & 36.45 \\\\\n35 & 41.73 & 40.12 \\\\\n30 & 48.79 & 46.70 \\\\\n\\hline\n\\hline\nonline & 29.36 & 28.59\n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\nWe then examine whether LSTMs with\nbatch decoding can learn recursive functions.\nThe same LSTMs are trained with batch decoding\nof different context frames.\nConsecutive prediction is not used in this setting.\nIn other words, for an utterance of $T$ frames,\nwe create $T$ chains of LSTMs,\neach of which predicts the label of a frame.\nResults are shown in Table~\\ref{tbl:state-tbptt}.\nWith a context of 40 frames, the LSTM trained\nwith batch decoding is only slightly behind the LSTM trained\nwith online decoding. This again suggests\nthat the class of Markov functions can perform reasonably well,\nand much of the long-term dependency is likely Markov.\nHowever, the error rate degrades significantly\nwhen we switch from batch decoding to online decoding.\nThis strongly suggests that these LSTMs do not\nbehave like recursive functions.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing online and batch decoding\n for LSTMs trained with batch decoding,\n and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-tbptt}\n\\begin{center}\n\\begin{tabular}{lll}\ncontext & batch & online \\\\\n\\hline\n40 & 31.30 & 80.29 \\\\\n35 & 30.99 & 84.78 \\\\\n30 & 32.80 & 85.74 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\nTo understand what makes LSTMs\nbehave like recursive functions,\nwe train LSTMs with batch decoding\nand increase the amount of consecutive\nprediction. To simplify the setting,\nwe allow each frame to be predicted multiple\ntimes. In fact, if a network predicts\nconsecutively for $p$ frames,\nthen each frame gets predicted $p$ times.\nResults are shown in Table~\\ref{tbl:state-pred}.\nAs we increase the number of consecutively predicted frames,\nthe error rate for batch decoding\nstays about the same, while the one for online\ndecoding improves. This suggests that\nit is the amount of consecutive prediction that gears\nthe behavior of LSTMs towards recursive functions.\nIn addition, as seen in Table~\\ref{tbl:state-batch},\nLSTMs can achieve reasonable performance\nwith both online and batch decoding.\nPerhaps the data is a complex mix of long-term\ndependency, or perhaps the LSTM learns to\nforget the history. More analyses are needed\nto tease the factors apart.\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing different\n numbers of consecutive prediction\n for LSTMs trained with batch decoding,\n and a lookahead of 20 frames on subphonetic states.}\n\\label{tbl:state-pred}\n\\begin{center}\n\\begin{tabular}{llll}\n\\# of prediction & batch & online \\\\\n\\hline \n1 & 31.30 & 80.29 \\\\\n5 & 31.29 & 80.21 \\\\\n10 & 31.65 & 46.82 \\\\\n15 & 32.27 & 33.03 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\subsection{Long-term dependency in phonemes and words}\n\nWe repeat the same experiments with phonemes and words.\nNote that the number of classes in the case of words\nis the number of unique words in the training set.\nWe do not handle out-of-vocabulary words (OOV).\n\nFor the lookahead experiments in Table~\\ref{tbl:phone-word-lookahead},\nthe general trend is similar to that in Table~\\ref{tbl:state-lookahead},\nexcept that the frame error rate plateaus at around 15 frames\nfor words, longer than the 10 frames in subphonetic states\nand phonemes.\n\nThe fact that the TDNN achieves\na low frame error rate for phonemes in Table~\\ref{tbl:phone-word-lookahead}\nsuggests that much phonetic information\nis concentrated within a 30-frame window.\nHowever, for words we expect a larger context\nwindow to predict words, so we evaluate a 10-layer\nTDNN with an effective context window of 48 frames.\nThe 10-layer TDNN performs better than the 7-layer\none, but is still behind LSTMs.\n\nThe conclusion in Table~\\ref{tbl:phone-word-batch}\nis the same as in Table~\\ref{tbl:state-batch},\ni.e., for LSTMs trained with online decoding,\nreducing the amount of context during batch decoding\nhurts the accuracy. However, the conclusion\nin Table~\\ref{tbl:phone-word-tbptt} is different\nfrom that in Table~\\ref{tbl:state-tbptt}.\nThe LSTMs with batch decoding are\nable to perform well with online decoding\non phonemes but not on words.\nWe suspect that phonemes are more local than subphonetic states,\nand in general the results are affected by the choice of\nlinguistic units.\n\nIn Table~\\ref{tbl:phone-word-pred}, we observe a similar trend to\nthat in Table~\\ref{tbl:state-pred}, adding consecutive prediction\nimproves the LSTMs' performance with online decoding.\nIn fact, the performance with online decoding,\ncompared to that in Table~\\ref{tbl:phone-word-lookahead},\nis fully recovered when using 15 frames of consecutive prediction.\n\n\n\\begin{table}[t]\n\\caption{FERs (\\%) comparing different lookaheads\n for LSTMs trained with online decoding, on phonemes and words.\n The FERs of a 7-layer TDNN and a 10-layer TDNN are provided as reference.}\n\\label{tbl:phone-word-lookahead}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\nlookahead & dev & \\textfw{dev93} & dev & \\textfw{dev93} \\\\\n\\hline \n1 & 18.31 & 16.34 & 32.78 & 45.60 \\\\\n5 & 13.57 & 12.01 & 28.39 & 40.36 \\\\\n10 & 12.33 & 11.01 & 24.68 & 36.62 \\\\\n15 & 11.90 & 10.67 & 21.57 & 32.82 \\\\\n20 & 11.86 & 10.76 & 20.85 & 31.45 \\\\\n25 & & & 21.02 & 30.43 \\\\\n\\hline\n\\hline\nTDNN 512x7\n & 14.22 & 12.46 & 39.27 & 38.93 \\\\\nTDNN 512x10\n & & & 28.97 & 32.24 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) for batch decoding with different context frames\n with the best LSTM trained with online decoding,\n and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-batch}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\ncontext & dev & \\textfw{dev93} & dev & \\textfw{dev93} \\\\\n\\hline \n40 & 16.74 & 14.99 & 48.39 & 44.84 \\\\\n35 & 19.28 & 17.10 & 53.33 & 48.55 \\\\\n30 & 23.26 & 20.59 & 58.78 & 53.00 \\\\\n\\hline \n\\hline \nonline & 11.86 & 10.76 & 20.85 & 31.45 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing online and batch decoding\n for LSTMs trained with batch decoding,\n and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-tbptt}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\ncontext & batch & online & batch & online \\\\\n\\hline \n40 & 11.50 & 11.24 & 31.65 & 56.96 \\\\\n35 & 12.12 & 15.11 & 34.21 & 58.26 \\\\\n30 & 12.89 & 21.97 & 37.83 & 64.74 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{FERs (\\%) on \\textfw{dev93} comparing different\n numbers of consecutive prediction\n for LSTMs trained with batch decoding,\n and a lookahead of 20 frames on phonemes and words.}\n\\label{tbl:phone-word-pred}\n\\begin{center}\n\\begin{tabular}{l|ll|ll}\n & \\multicolumn{2}{c}{phonemes} & \\multicolumn{2}{|c}{words} \\\\\n\\# of prediction & batch & online & batch & online \\\\\n\\hline \n1 & 11.50 & 11.24 & 31.65 & 56.96 \\\\\n5 & 11.58 & 11.68 & 31.79 & 45.88 \\\\\n10 & 11.49 & 10.89 & 32.31 & 36.40 \\\\\n15 & 11.48 & 10.72 & 32.34 & 30.93 \n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table}\n\n\n\\section{Introduction}\n\nRecurrent neural networks (RNNs) have been extensively used for speech\nand language processing, and are particularly successful at\nsequence prediction tasks,\nsuch as language modeling \\cite{MKS2018-1,MKS2018-2,KHQJ2018}\nand automatic speech recognition \\cite{SSEP2014,SSB2014,PWPK2018}.\nTheir success is often attributed to their ability to learn long-term dependencies\nin the data or, more generally, their ability to remember.\nHowever, exactly how far back recurrent networks can remember \\cite{WLS2004, KHQJ2018},\nwhether there is a limit \\cite{CSS2017}, and how the limit is characterized\nby the network architecture~\\cite{Z+2016} or by the optimization method ~\\cite{PMB2013},\nare still open questions.\nIn this paper, we examine how training and decoding approaches affect\nrecurrent networks' ability to learn long-term dependencies.\nIn particular, \nwe study the behavior of recurrent networks in the context of speech recognition\nwhen their ability to remember is constrained.\n\nThe ability to remember in recurrent networks\nis affected by many factors, such as the amount of long-term dependency\nin the data, as well as the training and decoding approaches.\nWe differentiate between two types of decoding, the online approach\nand the batch approach. In the online case,\nthere is a single chain of recurrent network that predicts\nat all time steps, while in the batch case,\nmultiple chains of recurrent networks that start\nand end at different time points are used.\nIn the online case, a recurrent network resembles a recursive function\nwhere the prediction of the current time step depends on\nthe memory that encodes the entire history. In the batch\ncase, a recurrent network resembles a fixed order Markov process,\nbecause the prediction at the current time step strictly depends\non a fixed number of time steps in the past.\nBy changing the decoding approach, we restrict\nthe ability of recurrent networks to model certain classes of functions.\n\nThe speech signal is particularly rich with long-term dependencies.\nDue to the causal nature of time, speech is assumed to be Markovian\n(though with a potentially high order).\nThe order of the Markov property depends on the time scale\nand the linguistic units, such as subphonetic states, phonemes, and words.\nThe Markov assumption has a strong influence in many design choices,\nsuch as the model family or the training approaches.\nPartly due to the Markov assumption and partly due to\ncomputational efficiency \\cite{KTLK2015,CYH2015}, recurrent networks\nin speech recognition \\cite{SSEP2014,SSB2014}\nand in language modeling \\cite{MKS2018-1, MKS2018-2, KHQJ2018} are commonly\ntrained with truncated backpropagation through time (BPTT) \\cite{W1990, WP1990},\nwhere a recurrent network is unfolded for a fixed number of time steps\nfor the gradients to propagate.\nThe hypothesis is that even with truncation,\nrecurrent networks are still able to learn recursive\nfunctions that are richer than fixed order Markov processes.\nIn fact, recurrent networks at test time are typically applied to sequences much\nlonger than the ones they are trained on~\\cite{SSB2014,PWPK2018}.\nUnder certain conditions, it has be shown that\ntruncation does not affect the performance of RNNs \\cite{MH2018}.\nBy our definition, these recurrent networks are trained in batch mode,\nand are used in an online fashion during testing.\nWe will examine how the number of BPTT steps affects\nthe training loss and how the decoding approaches\nat test time affect the test loss.\n\nAnother factor that impacts the ability of a recurrent network \nto remember is optimization \\cite{BSF1994, PMB2013}.\nVanilla recurrent networks are known to be difficult\nto train \\cite{HBFS2001,PMB2013}.\nPrevious work has attributed this difficulty to the\nvanishing gradient problem \\cite{H1998}.\nLong short-term memory networks (LSTMs)\nhave been proposed to alleviate the vanishing gradient problem and are\nassumed to be able to learn longer dependencies in the data \\cite{HS1997}.\nWhether these statements are true is still debatable, but we will draw a connection between the\nMarkov assumption, Lipschitz continuity, and vanishing\ngradients in recurrent networks. The vanishing gradient phenomenon\nis in fact a necessary condition for Markov processes.\n\nThe contribution of the paper is a comprehensive set of\nexperiments comparing two decoding approaches, various numbers of BPTT steps during\ntraining, and their respective results on three types of\ntarget units, namely, subphonetic states, phonemes, and words.\nThe results have implications for studying long-term dependency\nin speech and how well they are learned by recurrent networks.\n\n\n\\section{Markov Assumption, Lipschitz Continuity, and Vanishing Gradients}\n\nFrom the experiments, it is difficult to conclude\nwhether the LSTMs really learn Markov or recursive functions.\nIn this section, we derive a necessary condition\nthat we can check empirically for Markov functions.\nA function $f$ is $G$-Lipschitz with respect to $i$-th coordinate and norm $\\|\\cdot\\|$ if\n\\begin{align}\n& \\bigg|f(a_1, \\dots, a_{i-1}, a, a_{i+1}, \\dots, a_t) \\notag \\\\\n& \\quad - f(a_1, \\dots, a_{i-1}, b, a_{i+1}, \\dots, a_t)\\bigg| \\leq G\\|a - b\\|\n\\end{align}\nfor any $a_1, \\dots, a_t$ and any $a$, $b$.\nIn words, the function does not change too much when we perturb\nthe $i$-th coordinate.\nBy definition, a Markov function does not change at all\nwhen we perturb the coordinate beyond the necessary history,\nor more formally, a $\\kappa$-th order Markov function is 0-Lipschitz with\nrespect to $x_1, \\dots, x_{t-\\kappa}$.\nThe Lipschitz property relates to the gradient through\nthe perturbation interpretation.\nA gradient of a function can be regarded as having\nan infinitesimal perturbation of the coordinates,\nso the $i$-th coordinate of a gradient has a value at most $G$\nfor a $G$-Lipschitz function with respect to the $i$-th coordinate.\nA $\\kappa$-th order Markov function should have zero gradient\nwith respect to $x_1, \\dots, x_{t-\\kappa}$,\nbecause it is 0-Lipschitz with respect to those coordinates.\nIn other words, the gradients to the input, and thus to the parameters,\nmust vanish after $\\kappa$ steps for a $\\kappa$-th order Markov function.\n\nIn practice, partly because of noise and partly\nbecause we do not know if the data\nis really Markov, it is better not to enforce vanishing\ngradients in the model.\nRegardless, this gives a necessary condition we can check empirically.\nWe take the best online and batch LSTMs trained on subphonetic states\nfrom Tables~\\ref{tbl:state-lookahead} and \\ref{tbl:state-tbptt}.\nThe norms of gradients to the input at time $t-20$\nwhen predicting at time $t$\nis shown in Figure~\\ref{fig:grad-hist}.\nThe norms of the LSTM trained with batch decoding concentrate near\nzero compared to the ones with online decoding.\nThis confirms that LSTMs trained with batch decoding behave\nlike Markov functions.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=5cm]{grad-0.eps}\n\\caption{The norms of gradients to the input $x_{t-20}$ when predicting $y_t$\n on \\textfw{dev93}\n comparing LSTMs trained with online and batch decoding with 40 context frames.\n Both LSTMs are trained on subphonetic states with a lookahead of 20 frames.\n Note that the counts are in log scale.}\n\\label{fig:grad-hist}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{figure}\n\n\n\\section{Problem Setting}\n\\label{sec:prob}\n\nWe first review the definition of recurrent networks\nand discuss the types of functions they aim to model.\n\nLet $X$ be the set of input elements, $L$ be the label set,\nand $Y = \\mathbb{R}^{|L|}$ for representing\n(log-)probabilities or one-hot vectors of elements in $L$.\nFor example, in the case of speech recognition, the set $X = \\mathbb{R}^{40}$\nif we use 40-dimensional acoustic features, and $L$ can be the set of\nsubphonetic states, phonemes, or words.\nLet $\\mathcal{X}, \\mathcal{Y}$ be the sets of sequences whose elements are in\n$X$, $Y$, respectively.\nWe are interested in finding a function that maps a sequence in $\\mathcal{X}$\nto a sequence in $\\mathcal{Y}$ of the same length.\nIn other words, the goal is to map $(x_1, \\dots, x_T) \\in \\mathcal{X}$,\nwhere each $x_i \\in X$, to $(y_1, \\dots, y_T) \\in \\mathcal{Y}$,\nwhere each $y_i \\in Y$, for $i = 1, \\dots, T$.\nWe assume we have access to a loss function\n$\\ell: Y \\times Y \\to \\mathbb{R}^+$.\nFor example, the loss function for classification at each time step $t$\ncan be the cross entropy $\\ell(y_t, \\hat{y}_t) = - y_t^\\top \\log \\hat{y}_t$,\nwhere $y_t$ is the one-hot vector\nfor the ground truth label at time $t$, and $\\log \\hat{y}_t$\nis a vector of log probabilities produced by the model.\n\nA general recurrent network is a function $s: H \\times X \\to H$\nwhere $H$ is the space of hidden vectors.\nFor an input sequence $(x_1, \\dots, x_T)$, we repeatedly apply\n\\begin{align}\nh_t = s(h_{t-1}, x_t)\n\\end{align}\nfor $t = 1, \\dots, T$\nto obtain a sequence of hidden vectors $(h_1, \\dots, h_T)$\nwhere $h_0 = 0$.\nThe hidden vectors are then used for prediction.\nNote that we are interested in the setting where the length of the\ninput sequence matches the length of the output sequence.\nSpecifically, we have a function $o: H \\to Y$ and apply\n\\begin{align}\n\\hat{y}_t = o(h_t)\n\\end{align}\nfor $t = 1, \\dots, T$ to obtain the label sequence $(\\hat{y}_1, \\dots, \\hat{y}_T)$.\nFor our purposes, it suffices to use $o(h_t) = \\text{softmax}(W h_t)$\nfor some trainable parameter matrix $W$.\n\nA vanilla recurrent neural network implements the above with\n\\begin{align}\ns(h_{t-1}, x_t) & = \\sigma(U x_t + V h_{t-1}),\n\\end{align}\nwhile a long short-term memory network (LSTM) \\cite{HS1997} uses\n\\begin{align}\n\\begin{bmatrix}\ng_t \\\\\ni_t \\\\\nf_t \\\\\no_t\n\\end{bmatrix}\n& =\n\\begin{pmatrix}\n\\tanh \\\\\n\\sigma \\\\\n\\sigma \\\\\n\\sigma\n\\end{pmatrix}\n(U x_t + V h_{t-1}) \\\\\nc_t &= i_t \\odot g_t + f_t \\odot c_{t-1} \\\\\ns(h_{t-1}, x_t) &= o_t \\odot \\tanh(c_t)\n\\end{align}\nto implement the recurrent network, where\n$\\odot$ is the Hadamard product,\n$\\sigma$ is the logistic function, $\\tanh$\nis the hyperbolic tangent, and the matrices\n$U$ and $V$ are trainable parameters.\n\nRecurrent networks can benefit from stacking on top of each other~\\cite{GMH2013}.\nWe abstract away stacking in the discussion, but will use\nstacked recurrent networks in the experiments.\n\n\\subsection{Markov and recursive functions}\n\\label{sec:markov}\n\nRecurrent networks can be seen as recursive functions\nwith a constant size memory.\nTo be precise, consider $y_t$ as a function of $x_1, \\dots, x_t$ for any $t$.\nWe say that $y_t$ is a recursive function if it satisfies\n$y_t = f(m_t)$ where $m_i = g(m_{i-1}, x_i)$\nfor $i = 1, \\dots, t$ and any function $f$ and $g$.\nThe memory is of size constant if the size of $m_t$\nis independent of $t$.\nIn contrast, we say that $y_t$ is a $\\kappa$-th order Markov function if \n$y_t$ is a function of $x_{t-\\kappa+1}, \\dots, x_t$.\nBy this definition, the set of recursive functions\nincludes all Markov functions, and the inclusion is proper.\nFor example, the sum $y_t = \\sum_{i=1}^t x_i$\nis not Markov of a fixed order, but it can be realized with\na recursive function $y_t = m_t$ where $m_i = m_{i-1} + x_i$\nfor $i = 1, \\dots, t$.\nIn the language of signal processing,\na Markov function resembles a finite-impulse response filter,\nand a recursive function resembles an infinite-impulse response\nfilter.\n\n\\subsection{Online and batch decoding}\n\nBased on the two function classes,\nwe define two decoding approaches, online and batch\ndecoding, respectively. In online decoding, a sequence of\npredictions is made one after another,\nwhile in batch decoding, the predictions\nat different time steps are made independently\nfrom each other.\n\nUsing the notation in the previous section,\nto predict $\\hat{y}_t$ at time $t$, we define online decoding as\n$\\hat{y}_t = f(m_t)$ where $m_i = g(m_{i-1}, x_i)$ for $i = 1, \\dots, t$\nand any function $f$ and $g$.\nDuring decoding, only the vector $m_i$ at time $i$\nneeds to be stored in memory, and no history is actually maintained;\nhence the name online decoding.\nTo implement online decoding with recurrent networks,\nwe simply let $f = o$, $g = s$, and $m_i = h_i$ for all $i$.\n\nWe define batch decoding as $\\hat{y}_t = f(x_{t-\\kappa+1}, \\dots, x_t)$\nfor some function $f$ and context size $\\kappa$.\nBy this definition, batch decoding is not limited to recurrent\nnetworks, and can be used with other neural networks.\nTo implement batch decoding with recurrent networks,\nwe let $f$ compute $\\hat{y}_t = o(h_t)$, $h_t = z^t_t$, and\n$z^t_i = s(z^t_{i-1}, x_i)$ where $z^t_{t - \\kappa} = 0$,\nfor $i = t - \\kappa + 1, \\dots, t$.\n\nIn terms of computation graphs, the graph for online\ndecoding with recurrent networks is a single chain, while the graph\nfor batch decoding consists of multiple parallel chains.\nNote that in batch decoding the hidden vector of each chain\nstarts from the zero vector. In other words, $h_i$ is not a function of $h_{i-1}$\nfor any $i$, so the hidden vectors cannot be reused when\npredicting at different time points.\nThough batch decoding requires more computation and space,\nit can be parallelized. Online decoding has to be computed\nsequentially.\n\nBy the above definition, batch decoding with $\\kappa$ context frames aims to\nrealize a $\\kappa$-th order Markov function.\nOnline decoding, however, aims to realize a recursive function.\nIt has been shown that the number of unrolled steps in\nvanilla recurrent networks controls the capacity of the model\nin terms of the Vapnik-Chervonenkis dimension \\cite{KS1996}.\nIt is unclear whether recurrent networks truly learn this class of functions.\nHowever, by changing decoding approaches, we explicitly restrict\nthe class of functions that recurrent networks can realize.\n\nFinally, one distinct property in speech recognition\nthat is absent in language modeling is the option to\nlook beyond the current time frame. \nThe task of predicting the next word becomes meaningless\nwhen the next word is observed.\nHowever, it is useful to look a few frames\nahead to predict the word while the word is being spoken.\nFormally, we say that a recurrent\nnetwork decodes with a lookahead $\\ell$ if $\\hat{y}_t = o(h_{t + \\ell - 1})$.\nLookahead can be applied to both online and batch decoding,\nand it falls back to regular decoding when the lookahead is one frame.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}