Axial anomaly in multi-Weyl and triple-point semimetals

Journal of High Energy Physics, Jun 2018

Abstract We derive the expression of the abelian axial anomaly in the so-called multi-Weyl and triple-point crossing semimetals. No simplifying restrictions are assumed on the symmetry of the spectrum. Three different computation methods are considered: the perturbative quantum field theory procedure which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method, and the Atiyah-Singer index argument. It is shown that the functional form of the axial anomaly does not depend on the Lorentz symmetry, but it is determined by the gauge structure group. We discuss the stability of the anomaly — stemming from the quantisation of the anomaly coefficient — under smooth modifications of the lagrangian parameters.

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Axial anomaly in multi-Weyl and triple-point semimetals

Revised: May Axial anomaly in multi-Weyl and triple-point semimetals Luca Lepori 0 1 3 5 7 8 Michele Burrello 0 1 3 6 8 Enore Guadagnini 0 1 2 3 4 8 Theories, Topological States of Matter 0 Largo B. Pontecorvo 3 , 56127 Pisa , Italy 1 Juliane Maries Vej 30 , 2100 Copenhagen , Denmark 2 Dipartimento di Fisica E. Fermi, Universita di Pisa 3 via Vetoio , I-67010 Coppito-L'Aquila , Italy 4 INFN , Sezione di Pisa 5 INFN, Laboratori Nazionali del Gran Sasso 6 Niels Bohr International Academy and Center for Quantum Devices, University of Copenhagen 7 Dipartimento di Scienze Fisiche e Chimiche, Universita dell'Aquila 8 anomalous and @ J We derive the expression of the abelian axial anomaly in the so-called multiWeyl and triple-point crossing semimetals. No simplifying restrictions are assumed on the symmetry of the spectrum. Three di erent computation methods are considered: the perturbative quantum eld theory procedure which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method, and the Atiyah-Singer index argument. It is shown that the functional form of the axial anomaly does not depend on the Lorentz symmetry, but it is determined by the gauge structure group. We discuss the stability of the anomaly | stemming from the quantisation of the anomaly coe cient | under smooth modi cations of the lagrangian parameters. Anomalies in Field and String Theories; Chiral Lagrangians; E ective Field - 1 Introduction 2 3 7 8 9 3.1 3.2 3.3 3.4 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 7.1 7.2 Basic features of topological semimetals Overview of the results Anomalous axial symmetry for Weyl fermions Double-Weyl semimetals Triple-Weyl semimetals Triple-point semimetals 4 Gauge anomalies in perturbative quantum eld theory Perturbative approach Anomaly as a solution of a cohomological problem Axial anomaly 5 Perturbative computation 6 Nielsen-Ninomiya procedure Regularization Gauge variation Computation rules Addition of the contributions The nal result Classical external elds Normalizable zero-energy modes Dirac sea and the axial anomaly Atiyah-Singer index Particles and antiparticles Euclidean zero modes Axial anomaly for triple-Weyl semimetals Triple-point semimetals model 10 Quantization of the anomaly coe cient 11 Chemical potentials and stability 11.1 Multi-Weyl semimetals 11.2 Triple-point semimetals { 1 { 1 Introduction Anomalies are known since long ago in the context of quantum eld theory [1{3]. In the construction of a quantum eld theory based on a classical lagrangian, it may happen that a certain symmetry of the classical action cannot be preserved at the quantum level; when this happens, the symmetry is called anomalous. A well known example of anomaly is found when chiral fermions are minimally coupled with gauge elds; in this case, some of the symmetries acting on the fermion elds may be broken at the quantum level because of the so-called chiral anomalies [4{8]. In elementary particle physics, the experimental consequences of the avour chiral anomalies have been described for instance in [5{7, 9]. In the last decade it has been realized that the study of anomalies in eld theories is also a fundamental tool for the e ective description of topological phases of matter such as topological insulators and superconductors [10, 11]. In this scenario it has been shown that a eld theoretical approach accounting for chiral and gravitational anomalies allows us to characterize the peculiar transport properties of topological materials resulting from the coupling to electromagnetic elds or temperature gradients [12, 13]. In particular, anomalies provide a natural description for phenomena like the surface Hall conductance, which are related to the gapless surface modes of these gapped systems and constitute a useful tool for their classi cation. Since the work by Nielsen and Ninomiya [14] it has been known that also gapless models can enjoy similar topological responses under external electromagnetic elds, thus actualizing the e ects of quantum anomalies. Only recently, however, similar topological gapless phases of matter have gained a considerable attention and have been experimentally realized in solid state materials [15{18]. The main example is provided by Weyl semimetals [19, 20], which constitute a remarkable embodiment of the Dirac theory for massless fermions. In the presence of magnetic elds, they display transport properties which are dictated by the corresponding chiral anomaly [13] and have been studied in recent works [21{25]. These three dimensional systems host pairs of inequivalent and isolated Weyl points (or Weyl nodes) in the Brillouin zone. These are points where two energy bands touch each-other, with a linear dispersion that determines the appearance of a cone. The Weyl nodes appear always in pairs for lattice models [26] and can be separated in momentum space by breaking the space inversion or time reversal canonical symmetries [19, 20, 27]. In their neighborhood, thus for energies close to the band touching points, the fermionic quasiparticles display a linear dispersion law and their dynamics can be e ectively described by a Weyl hamiltonian involving a pair of cones with opposite \chirality". The appearance of Weyl points in pairs has a topological origin [26, 28]; moreover it also implies (if spacerotational symmetry is present close to the nodes) the emergence of an e ective Lorentz covariance, characterizing the low-energy dynamics of the Weyl semimetal, the chemical potential is assumed to coincide with the energy of the Weyl nodes. { 2 { Weyl semimetals present indeed non-trivial topological features, exactly due to the Weyl nodes which constitute a monopole source for the Berry connection of the bands [29]. This topology manifests in the presence of gapless chiral modes, exponentially localized on the surfaces of these systems. At the energy of the Weyl nodes, such surface states identify lines in the momentum space connecting the projections on the surface Brillouin zone of two inequivalent nodes; these surface modes are therefore dubbed Fermi arcs [19]. Weyl semimetals and Fermi arcs have been realized and detected in various compounds, including for instance tantalum arsenide [15{17], niobium arsenide [18], bismuth trisodium [30], and tantalum phosphide [31]. Their implementation, however, was not limited to solid-state materials only, but it also encompasses other platforms, for example in photonic [32{34] and phononic [35] crystals. Quite recently, Weyl semimetals (which we will dub single-Weyl semimetals in the following, to avoid confusion) have been shown to admit notable generalizations in the so-called multi-Weyl semimetals [36], lattice systems displaying isolated band touching points, similar to the Weyl nodes, but where the dispersion law is linear along one space direction only and grows with a higher power of the momenta along the other two directions. Double-Weyl nodes have been predicted in a number of rare-earth compounds [37{42], in ultracold-atoms set-ups [43{45] and in photonic crystals [46]. The dispersion relation of the multi-Weyl points implies a breaking of the e ective (lowenergy) Lorentz covariance, which characterizes instead the single-Weyl cones. Despite the absence of the Lorentz symmetry, it has been argued [47{49] that, for particular values of the lagrangian parameters, the axial anomaly assumes the standard functional form that one derives in Lorentz invariant theories. The anomalous Hall conductivity which characterizes multi-Weyl materials does not depend indeed on the e ective Lorentz covariance of the system, which is always violated when one takes into account the band dispersion. The latter property has been exempli ed by even more exotic topological semimetals, characterized by the simultaneous merging of multiple energy bands, such that they can be interpreted as models with a modi ed spin-statistics relationship [50]. A notable example involves triplepoint semimetals [51{53], recently realized [54] in molybdenum phosphide, where three bands cross in a triply-degenerate point with a multiple topological charge. A summary of some of the main features of these topological semimetals is provided in section 2. The main scope of the present work is the computation of the abelian anomalies related to suitable global transformations of anticommuting elds, which appear in the lagrangian models describing double-Weyl, triple-Weyl and triple-point lattice systems around the nodal points. These anomalies are connected with the so-called axial transformations acting on the fermion elds in the presence of an abelian vector potential A (x) (see [49] and references therein). An overview of our results is presented in section 3. No simplifying restrictions are assumed on the symmetry of the spectrum of the various models. The considered low energy lagrangians are not Lorentz invariant; indeed, the relevant di erential operators acting on the fermionic variables are not necessarily described by homogeneous polynomials of the covariant derivatives, and may contain dimensioned parameters in front of them. Let us recall that the standard results [4{8] concerning the axial anomaly have been obtained in the presence of Lorentz invariance. So the computation { 3 { of the axial anomaly in the case of models which are not Lorentz invariant presents original aspects. Therefore, in order to make our article self-contained, in sections 4{7 we describe in detail our anomaly computations in the case of the double-Weyl semimetal model; the triple-Weyl and triple-point models are examined subsequently. Three di erent methods for the derivation of the anomaly are considered: the perturbative quantum eld theory procedure [4{8] which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method [14], and the Atiyah-Singer index argument [55, 56]. The mutual consistency of these methods is illustrated. The general features of the perturbative approach are described in section 4, where the relationship between the chiral anomaly and the axial anomaly is produced. The perturbative computations of the chiral anomaly for the double-Weyl model are contained in section 5, the Nielsen-Ninomiya procedure is presented in section 6, and the anomaly derivation by means of the AtiyahSinger index argument is contained in section 7. The axial anomaly for the triple-Weyl and the triple-point models are derived in section-8 and section 9 respectively. In these cases, the perturbative approach is rather arduous or a ected by singularities; therefore only the Nielsen-Ninomiya and Atiyah-Singer methods are considered. The quantization of the multiplicative anomaly coe cient is discussed in section 10, which also contains a few comments on the structure of the obtained anomaly expressions, as well as on their stability under smooth perturbations of the lagrangian parameters. The e ects on the axial anomaly of modi cations of the chemical potentials are examined in detail in section 11, where we also illustrate certain peculiar features of the anomaly computation in the case of the triple-point model. Finally, our conclusions are presented in section 12. 2 Basic features of topological semimetals The prediction and discovery of symmetry-protected topological matter is considered one of the crucial achievements of the theory of condensed matter in the last decades. The most prominent example in this set is given by topological insulators [10, 11]. These are gapped materials, non-interacting or weakly interacting, which present gapless modes localized on their surface or edges and responsible for their transport properties. These surface modes, in general, maintain their gapless and localized nature as long as certain discrete symmetries of the system are preserved or a phase transition through a critical point of the bulk occurs. Topological insulators can be e ciently described in terms of non-interacting lattice models and the existence of their gapless edge or surface modes can be deduced, in general, by topological indices, such as Chern or winding numbers, characterizing their energy bands in the Brillouin zone of the lattice. The study of topological features in condensed matter materials extended rapidly to gapless systems (see, for example, the reviews [57, 58]), namely topological metals and semimetals. A semimetal is a material that presents two partially lled energy bands at its chemical potential; this implies that there are at least two energy bands overlapping in energy. The limiting case is provided by energy bands with a discrete set of band touching points in the Brillouin zone and the chemical potential lying exactly at the energy of these points. The most typical example of topological semimetal is the Weyl semimetal that { 4 { (a) logical semimetals considered in this paper. ! is an arbitrary energy scale and a labels the lattice spacing. (a) Dispersion as a function of k1 and k2 of a single-Weyl semimetal corresponding to the model in [27]: it displays pairs of linearly dispersing Weyl cones with opposite chirality in its Brillouin zone. (b) Spectrum of a double-Weyl semimetal from the lattice model in [43]: the energy is plotted as a function of k1 and k2, which are the directions with quadratic dispersion. The value of k3, along which the dispersion is linear, has been xed at the band touching points. Around the double-Weyl points, the system displays a C4 rotational symmetry with axis along the k3 direction. (c) Typical spectrum of a triple-Weyl semimetal as a function of k1 and k3. The dispersion is cubic in k1 (and k2, not shown) and linear in k3. k2 has been chosen at the band touching point. (d) Dispersion of the triple-point semimetal model in [59]. The triple point crossings are band touching points connecting a at band at zero-energy and two linearly dispersing bands. belongs indeed to this case: a Weyl semimetal is a three-dimensional material with two bands touching with a linear dispersion along all the directions in an even number of points in the Brillouin zone (see gure 1(a)). These gapless points are robust against any small translational-invariant perturbation of the system, as long as they lie at di erent momenta of the Brillouin zone. Any pair of these nodes with opposite chiralities can be e ciently described in terms of two decoupled Weyl hamiltonians; therefore, for energies close to the band-touching points (thus small temperatures and chemical potentials close to the Weyl points) these systems are characterized by an emerging Lorentz invariance; increasing or decreasing the chemical potential, instead, the non-trivial dispersion of the bands become relevant to determine the physical properties of the system. The linear-dispersing single-Weyl points correspond to unitary monopoles of the Berry curvature calculated on the two touching energy bands [29]. These points do not exhaust { 5 { the possible topological nodes among di erent bands: in the presence of additional symmetries (e.g. rotational crystal symmetries) it is possible to engineer materials displaying multi-Weyl points [36], which correspond to higher monopoles of the Berry curvature, at the price of giving away the emergent Lorentz invariance. In particular, double-Weyl and triple-Weyl points are stabilized in physical material by the discrete rotational symmetries C4 and C6 and, in general, they do not display a full rotational SO( 2 ) symmetry. DoubleWeyl points are characterized by a quadratic dispersion along two directions (k1 and k2 in gure 1(b)) and by a linear dispersion in the third direction. Triple-Weyl points, instead, display a cubic dispersion along two directions and a linear dispersion along the third one (in gure 1 (c) an example with linear dispersion along k3 and cubic dispersion along k1 is displayed). The triple-Weyl case is the one with the highest power in the dispersion law allowed by point group symmetries [36] in spatial dimensions equal or lower than three; by including multi-component fermions with additional symmetries, however, it is possible to engineer band-touching points with even larger dispersion powers and monopole charges. Finally it is also possible to create systems where three bands connect together in a topological node with double monopole charge [50]. This is the case of the triple-point semimetals with a typical dispersion depicted in gure 1(d). If time-reversal symmetry is preserved, the upper and lower bands display a linear dispersion, whereas the central band is at (it may display as well a quadratic dispersion along all directions, but the gradient of the energy with momentum vanishes in the triple-point crossing). When time-reversal symmetry is broken, also the central band may acquire a dispersion [59]. In the following, our analysis of the axial anomalies which characterize these models is based on lagrangian descriptions of these systems, for energies close to the band-touching points, where the lattice e ect can be considered negligible. 3 Overview of the results Our results are summarized in this section. In order to x the notation, it is useful to add a brief introductory note. 3.1 Anomalous axial symmetry for Weyl fermions The concept of chiral anomaly developed originally in quantum eld theory [4{6]. Let us consider for instance the model of massless electrodynamics with lagrangian (3.2) { 6 { and under global axial U( 1 )A transformations e i 2 U( 1 )A ; 8 > > < R(x) L(x) >>: A (x) ! ! ! A (x) R(x) = ei L(x) = e i R(x) L(x) : (3.3) At the classical level, the Noether axial current J A corresponding to the symmetry (3.3) A = 0. However, at the quantum level the axial current is not conserved because of the so-called axial anomaly. When the vector gauge symmetry (3.2) is preserved, the axial anomaly can be written in the form [4{6] (x) F (x) ; (3.4) (x) = 1 4 F (x). The same result is known to be also valid for the single-Weyl semimetals [13, 14], where the theory (3.1) describes the low-energy dynamics around the nodal points; there the Pauli matrices act on appropriate \chiral indices", generally labelling the sublattices that form the Weyl semimetal. 3.2 Double-Weyl semimetals Double-Weyl semimetals contain two inequivalent band touching points, protected by a symmetry C4, with a linear dispersion relation along the direction connecting them, and a quadratic dispersion relation along the other two directions [36]. In real space-time, the corresponding low energy lagrangian for the fermionic variables associated with these points takes the form (~ = c = 1) L = and denote three nonvanishing real constants (in the following, and should not be confused). The fermionic (anticommuting) elds two components but, di erently form the case of massless electrodynamics, they do not represent Lorentz spinor elds. The variables form of the P T -symmetry, which is required to obtain stable multi-Weyl nodes [19]{[27]. 1 shown in equation (3.2), and under global axial U( 1 )A transformations of equation (3.3). The Noether axial current JA(x) corresponding to the symmetry (3.3) can be written as JA(x) = JR(x) JL(x) ; J R0 = y R R J R1 = i Ry J R2 = i Ry J R3 = y 3 R R 1 D!1 + 1 D!2 + 1 2 1 2 2 D!2 2 D!1 R R JL0 = Ly L JL1 = i Ly JL2 = i Ly JL3 = 1 D!1 + 1 D!2 + 1 2 1 2 2 D!2 2 D!1 L L y 3 L L where where (3.7) (3.8) (3.9) (3.10) in which we have introduced the notation y D!j y (Dj ) Dj y . In this article we show that, for arbitrary nonvanishing values of , and , when the gauge invariance (3.2) is maintained the axial anomaly is given by 1 and , expression (3.9) is twice the chiral anomaly (3.4) that one nds in massless electrodynamics. In section 4 and section 5, we demonstrate equation (3.9) by means of a one-loop calculation based on Feynman diagrams. Then the same result is rederived by means of the Nielsen-Ninomiya procedure in section 6, and by means of the Atiyah-Singer index argument in section 7. The expression for the axial anomaly in equation (3.9) makes its Lorentz independence explicit. The Minkowski metric tensor does not appear in equation (3.9) while the tensor has a \cohomological" origin | as described in section 4 | in view of the fact that the anomaly can be described by a di erential 4-form. 3.3 Triple-Weyl semimetals Triple-Weyl semimetals contain two inequivalent band touching points, protected by a symmetry C6, with a linear dispersion relation along the direction connecting them and a cubic dispersion relation along the two remaining directions [36]. The corresponding low energy lagrangian can be written in real space-time as L = Ry(x) iD0 where the symbol S[P 2; Q] P QP + P P Q + QP P implements the correct symmetrization of the covariant derivatives, and , and denote three nonvanishing real constants. { 8 { In this case also L(x) represent two-components anticommuting elds. Precisely like the case of the double-Weyl semimetals, the lagrangian (3.11) is invariant under local vector U( 1 )V gauge transformations (3.2) and under global axial U( 1 )A transformations (3.3). By means of the Nielsen-Ninomiya procedure and the Atiyah-Singer index argument, in section 8 we shown that, when the gauge invariance (3.2) is maintained, the axial anomaly is given by (3.12) HJEP06(218) The results in equations (3.4), (3.9) and (3.12) have been rst inferred in [60] through a semiclassical calculation based on the kinetic theory of Landau Fermi liquids, and in [48] by means of numerical and analytical approaches. In [48], the authors analyzed numerically the case with parallel electric and magnetic elds, and presented analytical calculations in the spirit of Nielsen and Ninomyia [14] for a system in which both elds are aligned along the direction of linear dispersion and in the presence of a full SO( 2 ) rotational symmetry. Finally, equation (3.9) has also been veri ed in [49] through a eld theoretical approach, based on the Fujikawa's method (see for example [56, 61]), by evaluating the chiral anomaly for a double-Weyl point with SO( 2 ) rotational symmetry. In all these papers, in the expression of the anomaly | which has been derived in these articles | the ( ; ; ) factor (3.10) is missing. But the absolute value of the anomaly, which has been proposed in [48, 49, 60], appears to be correct. The di erence between the two derivations is that, in [48, 49, 60], the left eld has been considered set since the beginning by the condition topological charge N The derivations in [48, 49] do not fully explain the quantization of the anomaly, with ( ; ; ), and, from a more fundamental point of view, do not clarify why the anomaly for multi-Weyl semimetals is proportional to the di erential form F (x) ^ F (x), in spite of the breaking of Lorentz covariance of the corresponding low energy lagrangians. In facts, doubts [62, 63] have been cast upon the use of the regularization scheme of the path-integral measure exploited in the Fujikawa's method [49] in cases different from the standard Weyl theory. 3.4 Triple-point semimetals Triple-point semimetals [50] are characterized by two zero-energy points in which three bands join. The lagrangian of the low energy model takes the form L = Ry(x) i [D0 + Ly(x) i [D0 vM1D1 vM2D2 vM3D3] R(x) vM1D1 vM2D2 + vM3D3] L(x) ; (3.13) { 9 { where the real parameter v is positive and D = @ + iA (x). The fermionic elds and L(x) have three components. The three matrices Mj are given by M1 = M1( ) = BB0 M3 = M3( ) = BB ie i 0 0 0 iei CC ; 0 0 0 iei 0 is a parameter of the model which breaks time-reversal symmetry. The Mj matrices can be interpreted ad deformed generators of the rotation group in the adjoint representation; in our notation these matrices satisfy the commutation relations: Mj ( 1 ); Mk( 2 ) = i jk` M`( 1 The lagrangian (3.13) is invariant under local vector gauge transformations (3.2) and under global axial transformations (3.3). When the vector gauge invariance is preserved, the axial anomaly is found to be This expression is derived in section 9 by means of the Nielsen-Ninomiya and Atiyah-Singer methods. 4 Gauge anomalies in perturbative quantum eld theory Before proceeding with the direct computation of the anomaly, it is useful to discuss the relationship between the so-called left-right and vector-axial possible forms of the anomaly, together with a few general properties of gauge anomalies. 4.1 Perturbative approach In order to simplify the exposition and avoid repetitions, in the following discussion we concentrate directly on the double-Weyl model (3.5); but the results of this section clearly have a general validity. It is convenient to examine rst the lagrangian terms for the massless fermionic elds R(x) and L(x) separately; afterwards, the anomalous behaviours of their corresponding one-loop diagrams will be combined in order to determine the desired axial anomaly. Let us consider the \right-handed" component R(x). In order to simplify the exposition, in the intermediate steps of the computation the gauge eld coupled with R(x) will be denoted by V (x). We shall recover the previous A (x) notation at the end of the present section. The corresponding lagrangian density LR takes the form LR = Ry(x) R(V ) R(x) = (3.14) (3.15) (3.16) where D = @ + iV (x). The function LR is invariant under local U( 1 )R gauge transformations 8 > < > R(x) ! ei R(x) R(x) >>: V (x) ! V (x) @ R(x) : local U( 1 )R : (4.2) The operator R(V ) which enters equation (4.1) can be written as the sum of two terms, R(V ) = R(0) + e R(V ), in which the free part i @3 3 does not depend on V . Therefore the free spinor propagator [1{3] is given by 2 1 2 R(x) Ry(y)j0i = i h0jT Z d4p ( 2 )4 e ip(x y) p0 + (p22 p 2 0 2(p21 and the interaction component of the action takes the form 1 Z Z + 1 2 Z V3 3 R(x) : (4.4) In expression (4.3), the Feynman "-prescription [1, 3] has been introduced in order to guarantee causality and energy positivity. Let i R[V ] denote the sum of the connected oneloop vacuum-to-vacuum diagrams [1{3] of the R(x) eld in the presence of the classical background eld V (x), ei R[V ] = h0j T eiSRI[V ] j0i ; (4.5) where the symbol T denotes the Wick time-ordering. From the de nition (4.5) it follows that, under a gauge transformation V (x) ! V (x) R R[V ] of R[V ] is given by the sum of the connected diagrams (4.6) Z The gauge invariance of the lagrangian LR under the transformations in (4.2) would suggest that R[V ] also is gauge invariant, and consequently R R[V ] = 0. However, because of ultraviolet divergences, the functional R[V ] is not well de ned. Therefore the central question is whether one can de ne or not a renormalized R[V ] which is gauge invariant. If a renormalized gauge invariant R[V ] exists, then the gauge symmetry (4.2) is not R R[V ] 6= 0, and one nds an anomaly (4.7) where, in agreement with the action principle, PR(V ) is a local polynomial of the eld V (x) and of its space-time derivatives. Usually, the construction of a renormalized R[V ] consists of two steps: 1. de nition of a regularized functional rReg[V ], which depends on a cuto , 2. introduction of local conterterms ct[V ], containing in general both divergent and nite parts, which reabsorb the ultraviolet divergences. The renormalized functional R[V ], which is well de ned (free of divergences), corresponds to the sum rReg[V ]+ ct[V ] in the limit in which the cuto is removed. The particular choice of the regularisation is totally irrelevant. In the renormalization procedure, the freedom of adding nite local counterterms completely removes the dependence of the result on the particular choice of the regularization, because two di erent regularizations di er (in the limit of removed cuto ) by the sum of nite local counterterms [1{3]. Thus the expression of the polynomial PR(V ) of V (x), which appears in Z is not uniquely determined, because one can add the gauge variation nite local counterterm Lct[V ] to the integral R RPR. Consequently, if R R Lct[V ] of some RPR can be written as the gauge variation of a local counterterm, then there is no anomaly since, by introducing the appropriate counterterm, one can de ne a renormalized gauge invariant functional R[V ]. The existence of the anomaly means that expression (4.8) cannot be written as the gauge variation of a local term. In this case, even if one can modify the expression of R R[V ] | by means of local counterterms | one cannot eliminate PR(V ). Precisely for this reason, the existence of the anomaly does not depend on the choice of the regularization. Of course, the presence or the absence of the anomaly is determined by the lagrangian (4.1) which speci es how the eld R(x) interacts with the gauge eld. 4.2 Anomaly as a solution of a cohomological problem By developing the consequences of the Wess-Zumino consistency conditions [7], it has been found [8, 64{68] that the search of possible nontrivial solutions to equation (4.8) can actually be reduced to a cohomological problem. Indeed, the gauge variation of any function f [V ] can be represented by the action on f [V ] of a nilpotent BRST [69] operator T , which is de ned (in the abelian case) by the relations T V (x) = the anticommuting variable c(x) takes the place of the gauge parameter R(x). Since the anomaly is determined by the gauge variation of R[V ], the anomaly is T -closed, but it is not T -exact in the set of local counterterms fLct[V ]g. The anomaly then represents a nontrivial solution of the following cohomological problem T Z Z c(x)PR(V ) = 0 ; c(x)PR(V ) 6= T Lct[V ] ; with T 2 = 0 : (4.9) In this general approach, the gauge elds are described by di erential forms, V = V (x)dx ; no Lorentz invariance is assumed and only the properties of the gauge transformations group enter the solutions. In this way, the possible forms of the gauge anomalies can generally be determined without the need of introducing any corresponding eld theory model. More precisely, all the local polynomials of the eld V (x), which are not equal to the gauge variation of a local counterterm, have been produced. The only parameter which is not xed a priori by cohomological arguments is the overall normalization factor of each polynomial. The value of this normalization factor is speci ed by the lagrangian of each particular model. In the case of the abelian gauge symmetry V (x) ! V (x) can always [4{6, 70{73] be written in the form (4.10) where N represents an overall multiplicative factor that must be computed. If N = 0, there is no anomaly. One can easily verify that, when N 6= 0, expression (4.10) cannot be written as the gauge variation of a local counterterm. Therefore the anomaly exists for N 6= 0. In general, the coe cient N takes integer values; this point will be discussed in section 10. For instance, in the case of a relativistic right-handed Weyl spinor minimally coupled with the gauge eld V (x), one nds N = 1. In the case analyzed in this section, the value of N is determined by the speci c form of the lagrangian density (4.1); in particular, the anomaly is speci ed by the structure of the operator our case the eld R(V ). By a direct computation, we will show that N 6= 0. Even if in R(x) does not represent a spinor eld, in agreement with the standard notation, expression (4.10) will be called the chiral anomaly. 4.3 Axial anomaly In order to derive the general form of the U( 1 )A axial anomaly in the double-Weyl model with lagrangian density (3.5), let us now consider the eld L(x) and let us denote by W (x) the gauge eld which is coupled with L(x) according to the lagrangian LL = Ly(x) L(W ) L(x) = Ly(x) iD0 D12 where D = @ + iW (x). LL is invariant under local U( 1 )L gauge transformations local U( 1 )L : 8 > < > L(x) ! ei L(x) L(x) >>: W (x) ! W (x) (4.12) Let i L[W ] be the sum of the connected one-loop vacuum-to-vacuum diagrams of L(x) in the presence of the classical eld W . The in nitesimal variation L L[W ] of L[W ] under the transformation W (x) ! W (x) L(x) is strictly related with R R[V ]. Indeed, the lagrangian for L(x) can be obtained from the lagrangian for R(x) by means of the substitution 3 ! 3 . This means that the expression of the anomaly for L(x) can be obtained from expression (4.10) provided we introduce, in addition to the obvious change @3, and a change of the sign of the third component of the eld W (x), W3(x) ! W3(x). Therefore L L[W ] = N 24 2 1 Z d4x Let us now consider the complete model with lagrangian (3.5) in which both and L(x) are present. The gauge elds V (x) and W (x), that refer to the components of the group U( 1 )R U( 1 )L, can be written as combinations of the vector elds associated with the components of U( 1 )V U( 1 )A: U( 1 )V ; U( 1 )A; V e[A; B] = 1 Z d4x The in nitesimal variation of e[A; B] = R[V (A; B)] + L[W (A; B)] under the vector U( 1 )V transformation A (x) ! A (x) @ V (x) is obtained by combining equations (4.10) and (4.13) while the in nitesimal variation of e[A; B] under the U( 1 )A axial transformation B (x) ! 1 2 1 2 A (x) = (4.15) (4.16) (4.13) A e[A; B] = N 12 2 1 Z d4x Let us introduce the functional and where the nite local counterterm L[A; B] is given by The in nitesimal variations of [A; B] under U( 1 )V U( 1 )A transformations take the form V [A; B] = 0 ; One can easily verify that expression (4.20) is not the gauge variation of a local counterterm. Equation (4.19) shows that the subgroup U( 1 )V is anomaly free; consequently, the vector gauge invariance (3.2) is preserved and the corresponding local gauge theory is consistent. The gauge anomaly only concerns the axial subgroup U( 1 )A. In the model which is described by the lagrangian density (3.5), the eld B (x) is vanishing; therefore expression (4.20) evaluated at B (x) = 0 gives A [A; B] B =0 = N 4 2 1 Z d4x (4.21) So, in the double-Weyl model (3.5), the divergence of the axial current | or the expression of the axial anomaly | takes the form N 4 2 (4.22) which is in agreement with equation (3.9); the value of N remains to be computed. Equation (4.22) shows that, in multi-Weyl semimetals as well as in other generic eld theory models, the axial anomaly | if present | is proportional to the standard axial anomaly of massless electrodynamics. Indeed, on the one hand, the cohomological problem (4.9) admits a universal nontrivial solution and, on the other hand, in the presence of U( 1 )R symmetry the vector U( 1 )V invariance is required to be preserved. Thus no functional modi cation | due to the absence of e ective Lorentz covariance of the classical lagrangian | appears in the axial anomaly. We mention nally that if an explicit gauge symmetry breaking is induced, modi cations from the expression in (4.22) are expected; one example has been considered recently in [74]. 5 Perturbative computation In this section we shall derive the expression (4.10) of the chiral anomaly for the doubleWeyl model by means of perturbation theory. In particular, the value of N | appearing in equation (4.10) | will be determined. This means that, as it has been shown in section 4, the result of this section provides a proof of equation (3.9). 5.1 Regularization As it has been shown in section 4, the origin of the chiral anomaly is represented by the nontriviality of the gauge variation (4.10) of the functional R[V ]. The sum of the connected one-loop diagrams entering the de nition (4.5) is given by [1{3] d4x1 : : : d4xn hx1ji R yR e R(V )jx2i hxnji R yR e R(V )jx1i ; where, in agreement with the Schwinger notations [4], the symbol Tr represents the trace Tr (Q) = Z d4x tr hxj Q jxi ; in which Tr denotes the trace over the indices of the sigma matrices. Since the fermion propagator takes the form shown in equation (4.3), equality (5.2) can be written as i R[V ] = Tr ln 1 + ( ) (5.1) (5.2) (5.3) Indeed the expansion of expression (5.3) in powers of e R(V ) coincides with equation (5.2). The terms of the sum (5.2) which correspond to the divergent Feynman diagrams are not well de ned. So we now introduce a regularisation. Let us recall that, if y is a positive number, one has ln y + constant = lim !0 Z 1 ds s e sy : (is)Tr nes Therefore, according to the Schwinger proper-time regularisation [4], the regularised oneloop functional is de ned as (5.4) (5.6) (5.7) o R : enters the de nition of the propagator (4.3). The sign in the exponent of equation (5.5) is xed by the positivity of the analytic extension of R(V ) in the euclidean region for the momenta. The parameter > 0 represents the cut-o , and the limit of vanishing cut-o is Under a gauge transformation V (x) ! V (x) R(x), the in nitesimal variation of rReg[V ] = i Z 1 ds s Tr h es R(V ) i + constant ; (5.5) where the constant does not depend on V , R(V ) is shown in equation (4.1), and the operator, obtained by taking the ! 0 limit. 5.2 Gauge variation rReg[V ] is given by R rReg[V ] = i Z 1 ds s By means of the relation one obtains R rReg[V ] = Tr du e(1 u) 2 e R[V ]eu 2 Z 1 0 udu u2du Z 1 0 Z 1 0 euv(1 t) 2 dv e(1 u) 2 e R[V ]eu(1 v) 2 e R[V ]euv 2 and e R(V ) is shown in equation (4.4). Note that 2 is symmetric under the exchange The trace (5.9) is computed by moving all the space-time derivatives on the right and the terms which do not contain derivatives on the left so that Z ( 2 )4 Tr fF (x) G(p)g ; HJEP06(218) ! p has been used. Since 2 contains p0 and p3 at power 2, and p1 and p2 at power 4, the integration over the momenta in the euclidean region gives rise to the following powers of Z d4p e 2 ! 0 limit, expression (5.9) is a sum of a large number of nonvanishing contributions. Many of these contributions do not play a part in the anomaly because they are just equal to the variation of local counterterms. So, let us concentrate on the relevant (as far as the anomaly is concerned) terms which are of the type relevant terms (with 6= 6 = 6 = ) (5.13) (5.10) (5.11) (5.12) (5.14) (5.15) integrable contributions of in which there is not a couple of the indices ; ; ; which assume the same value. Let ct[V ] be the sum of the local counterterms whose gauge variation cancels precisely the R rReg[V ] which are not of the type (5.13). With the de nition ct[V ], we shall now consider the gauge variation of R[V ] in the ! 0 There are 4! = 24 contributions of type (5.13), which are contained in the 2 term of the expansion (5.9). In order to obtain a nonvanishing result in the to compensate the powers of the cut-o by powers of the momenta in the integrals (5.11) and (5.12). We will need to extract powers of the momenta also from the exponential factors of the type eq 2 . More precisely, when one exponential factor eq 2 commutes with ! 0 limit, one needs a function f (x), it gives the expression [eq 2 ; f (x)] = +2( 2 where, in agreement with relation (5.12), the rst two terms give rise to contributions of order 1=2, 1=2 ; cn 1 2 2 1 1 1 2 1 2 2 1 2 n in the form dp3 e 2p23 = 1=2 1=2 1 ; Z dp3 e j j 2)p41p22 2)p41p2 p2p2 1 3 p2p2 1 3 p2p2 1 0 p2p2 1 0 p2p2 1 0 p2p2 1 0 whereas the remaining two terms give rise to contributions of order 1=4 o ! 1=4 ; (5.16) and the dots stand for terms which turn out to be irrelevant (they produce vanishing outcomes in the ! 0 limit). 5.4 Addition of the contributions We have found 144 nonvanishing contributions to R R[V ] and their sum can be written where the sum contains 24 addenda and the values of cn, Fn(x) and Gn(p) are shown in e 2 = e p02 e 2p23 e [ 2(p21 p22)2+ 2p21p22] : In agreement with the Feynman "-convention of the propagator, the analytic continuation in the euclidean region is obtained according to p0 ! ip4 with real p4. One gets Z 2 dp4 e p4 = 1=2 1=2 ; Z dp4 e p24 p24 = 2 1 1=2 3=2 ; Z from equations (5.21) and (5.22) one derives HJEP06(218) Z dp1 dp2 e [ 2(p21 p22)2+ 2p21p22] Therefore, the momenta integrals with appear in equation (5.17) take the values Z d4p (2 4) e 2 ( 2p61 + ( 2 Consequently, the sum (5.17) is given by Z Z d4p (2 4) e d4p (2 4) e i i i j j 1 j j X( ; ) ; X( ; ) ; X( ; ) : Even if the fermion operator Dirac operator, and even if various contributions to R(V ) is not Lorentz covariant and di ers from the standard R(V ) contains dimensioned parameters, still the sum of the R R[V ] | quite remarkably | reproduces the standard form of the chiral anomaly (4.10). As it has been mentioned in section 4, this is a consequence of the fact that, in the abelian case, the nontrivial solution of the cohomological problem (4.9) is given precisely by F ^ F . 5.5 The nal result Let us now derive the value of X( ; ). By means of a rescaling of the integration variables pi ! pi= 1=4, equation (5.21) can be written as X( ; ) = dp1 dp2 e [ 2(p21+p22)2+( 2 4 2)p21p22] (p12 + p22) : By introducing two dimensional spherical coordinates, p1 = p cos ', p2 = p sin ', one nds 2(p21 + p22)2 + ( 2 4 2)p21p22 = (4 2 + 2) + (4 2 2) cos(4') ; p 4 8 (5.23) (5.24) (5.25) (5.26) (5.27) (5.28) (5.29) With the de nition of ! given by i@0 (x) = ! (x), we derive: and In the case < 0, instead, the wavefunctions (0i) possess only the second component (see equation (6.16)) and the resulting time dependence is: " " fR(x0) = exp i k3x0 + fL(x0) = exp i k3x0 + E x0 2 !# E x0 2 !# 2 2 ; ; !R = (k3 + Ex0) ; !L = (k3 + Ex0) ; !R = (k3 + Ex0) ; !L = (k3 + Ex0) ; ( if < 0) : ( if ( if > 0) ; < 0) : (6.19) (6.20) (6.21) HJEP06(218) These equations are consistent with a constant acceleration of the particles along x3 given by @0! and de ne the chiral nature of these gapless modes, re ecting the linear dispersion as a function of k3. Vacuum stability requires that the values of !R and !L must be nonnegative. Therefore, in agreement with the Dirac sea interpretation of the fermions ground state, the stable vacuum of the system corresponds to the state in which all the single-particle states with negative frequencies are occupied. All the other Landau levels L(x), which are orthogonal to the modes (6.17), are not chiral, namely they have frequencies with a symmetric dispersion for k3 ! k3 and they never cross zero energy: during the time evolution, the sign of their frequency does not change. This implies that these gapped Landau levels are either totally empty or totally lled, and, when considering the e ect of the acceleration caused by E, they do not contribute to the net rate of change of the right or left particle number [14]. Thus, as far as the axial anomaly is concerned, we only need to discuss the vacuum stability with respect to the modes (6.17). Let us consider the case > 0. The value (6.20) of !R is negative for k3 < Ex0. Therefore all the right-handed one-particle states with k3 < cupied, and the available states for a R particle are only those with k3 > Ex0 must be ocEx0. Similarly, since !L is negative for k3 > Ex0, all the left-handed one-particle states with k3 > must be occupied and the available states for a L particle are only those with k3 < Ex0 Ex0. Let us recall that, if the one-particle states are labelled by the values k of the momentum, the number N of available states for one particle moving inside a cubic box of volume V = L3 is speci ed (in the large V limit) by dN = L3d3k=( 2 )3. Therefore, in our system the number NR of available R states is determined by the product degeneracy is determined by the range of k2 which guarantees the particle localisation inside the box, 0 (x1 k2=B) L, Landau degeneracy = L 0 Z BL dk2 = ( 2 ) BL2 2 : (6.22) (6.23) By means of the same argument, one determines the number NL of available L states NL = 2 L Ex0 dk3 = ( 2 ) 2BV Z ( 2 )2 Ex0 1 dk3 ; (6.24) (6.25) (6.26) (6.27) 2EBV : 2EBV 4 2 : One gets and then and thus Consequently, one nds Z V 4EB 4 2 = V 2 (6.28) which is in agreement with the perturbative computation of the axial anomaly of section 5. It is now easy to verify that, for all the possible nontrivial values of ; and , the axial anomaly computed by means of the Nielsen-Ninomiya method coincides with expression (3.9). This concludes the derivation of the result (3.9) by means of the NielsenNinomiya method. Finally we observe that the vector current is conserved, since Z (6.29) The Nielsen-Ninomiya method suggests that the axial anomaly is stable against perturbations of the chemical potentials around the (zero) energy of the multi-Weyl nodes. We shall discuss this issue in section 11 7 Atiyah-Singer index The axial anomaly can also be interpreted [55, 76{79] as the index of the euclidean analytic extension of the operator which acts on the fermion eld in the expression (3.5) of the lagrangian. The index can be de ned as the number of nontrivial normalizable solutions with zero eigenvalues of this lagrangian operator, having support in R4. Moreover, the null solutions for the right and left parts can be identi ed separately, and the index results from the di erence of their numbers. By using the Atiyah-Singer approach, in this section we shall rederive the result (3.9). 7.1 In the presence of the classical external electric E and magnetic B elds shown in equation (6.1), from the lagrangian (3.5) one can derive the equations of motion for the \righthanded" particle wave functions R(x) = 0 ; and for the \left-handed" particle wave functions L, L(x) = 0 : Let RC and LC represent the wave functions of the \right-handed" and \left-handed" antiparticles respectively, RC = ( R)C = ( i 2) R(x) ; LC = ( L)C = (i 2) L(x) : RC and LC in the given classical electromagnetic back(7.1) (7.2) (7.3) (7.4) (7.5) HJEP06(218) The equations of motion for ground (6.1) take the form and R, 2 2 2 2 1 + RC (x) = 0 ; 1 + LC (x) = 0 : The eld operators R and L describe four kinds of particles: one \right-handed" particle and its antiparticle, and one \left-handed" particle and its antiparticle. The four relations (7.1), (7.2), (7.4) and (7.5) represent precisely a complete set of corresponding equations. 7.2 Euclidean zero modes With xed electromagnetic background, let us consider the analytic extension of the operacorresponding normalizable zero modes in R4. tors entering the equations of motion (6.1), (6.2), (6.4) and (6.6) in the euclidean region [80] (which is obtained by means of the replacement p0 ! ip0). We need to determine [55] the Let us examine the case > 0. One can specify the values of the two spatial components p2 and p3 of the momentum by putting R(x) = eik2x2 eik3x3 ~R(x1; x0) ; RC (x) = e ik2x2 e ik3x3 ~RC (x1; x0) ; L(x) = eik2x2 eik3x3 ~L(x1; x0) LC (x) = e ik2x2 e ik3x3 ~LC (x1; x0) : (7.6) In addition to the ladder operators E 2 E 2 E 2 E 2 and y, r E 2 0 0 2 2 2 2 The complete set of normalizable solutions in R 4 of equations (7.8){(7.11) can easily be determined because these equations depend on separated variable, since and y commute with and y. Equations (7.9) and (7.10) do not admit normalizable solutions. Whereas equation (7.8) admits the normalizable solutions and equation (7.11) admits the normalizable solutions eR(x1; x0) = eLC (x1; x0) = (0i;)"(x1) f0(x0)! (0i;)"(x1) f0(x0) ! ; ; with i = 1; 2 ; with i = 1; 2 ; and y operators de ned in equation (6.4), it is useful to introduce the = p 1 E ; y = p 1 E ; (7.7) satisfying the canonical commutation relations [ ; y] = 1. Let f0(x0) be the normalised ground state wave function satisfying f0(x0) = 0. The euclidean analytic extensions of equations (7.1), (7.2), (7.4) and (7.5) assume the form B( 2 + ( y)2) 1 + i B( 2 ( y)2) 2 ( + y) 3 eR = 0 ; (7.8) (7.9) (7.10) (7.11) (7.12) (7.13) (7.14) where the functions (0i;)" are de ned in equations (6.10) and we are exploiting the mapping between right and left sectors. By exploiting the separability of the Landau level structures de ned by the and operators, we can map the previous wavefunction in a 4D quantum Hall problem [81, 82]. We obtain that inside a hypercube in R4 of hypervolume V4 = L4 the Landau degeneracy of the each of the zero modes (7.12) and (7.13) is given by Landau degeneracy = L Therefore the number R of euclidean zero modes, which are associated with the \righthanded particles", is given by and the number L of euclidean zero modes, which are associated with the \left-handed antiparticles", is found to be Therefore the index of the euclidean extension of the lagrangian operator | acting on the fermion elds | turns out to be R + L = 4 EB 0 B( ) 3 + B( + ) y3! B( + ) 3 + B( 0 Therefore, we search again zero-energy (! = 0) normalizable solutions of the equation for For this purpose, similarly to equation (6.10), we introduce the following ansatz: R = 2 L = 2 EBL4 which is in agreement with the expression (3.9) of the axial anomaly. One can easily verify that this agreement still holds for arbitrary values of ; and . This concludes the rederivation of the result (3.9) by means of the Atiyah-Singer index argument. We note that the approach followed in the present section suggests a direct relation between the lagrangian zero modes and the chiral states derived from the corresponding hamiltonians in section 6, allowing for a parallelism between the two related methods to obtain the chiral anomalies. 8 Axial anomaly for triple-Weyl semimetals In this section, the axial anomaly for the triple-Weyl semimetals model is derived by means of the Nielsen-Ninomiya and the Atiyah-Singer arguments because, in this case, the perturbative quantum eld theory procedure requires considerable e ort. As it has been discussed in section 6, in order to implement the Nielsen-Ninomiya procedure we need to consider the equations of motion which are derived from the lagrangian (3.11) in the presence of the gauge elds background (6.1). For the moment, let us consider the case in which > 0. The analogues of equations (6.6) take the form (k3 + Ex0) B( + ) 3 + B( ) y 3 B( ) 3 + B( + ) y3! (k3 + Ex0) " # ! ; (8.1) ) y 3 " # ! = Pn an hn(x1)! Pn bn hn(x1) ; (7.15) (7.16) (7.17) # = 0 : (8.2) (8.3) where hn(x1) are the wavefunction of the n-th Landau level. By inserting this expansion in eq. (8.2) we obtain, for n 3: ( + ) bn 3 Kn 3;+ + ( ) bn+3 Kn+3; = 0 ( ) an 3 Kn 3;+ + ( + ) an+3 Kn+3; = 0 (8.4) (8.5) where Kn+3; = p(n + 1)(n + 2)(n + 3) and Kn 3;+ = pn(n 1)(n 2). We impose bn 3 = Kn 3;+ = 0 if n < 3 . The two equations (8.4) and (8.5) decouple. Therefore, by following the same argument presented in section 6 after equation (6.13), we conclude that normalizable solutions with ! = 0 exist only if fang = 0 or fbng = 0. By direct inspection, we nd that fang = < 0, while fbng = 0 if > 0. In each case, we obtain three independent normalizable solutions with ! = 0, corresponding to values fa0; a1; a2g or fb0; b1; b2g to be xed. Following the same arguments of the section 6 one obtains: 3EBV 4 2 3EBV = V 6EB which is in agreement with equation (3.12). One can easily verify that a modi cation of the signs of the coe cients , and is taken into account by the ( ; ; ) factor de ned in expression (3.10). The derivation of the axial anomaly by means of the Atiyah-Singer argument is quite simple. Indeed, the counting of the euclidean zero modes is carried out by means of two (8.6) (8.7) steps: reads (by adopting the notations of section 7), the ; y part in the R4 euclidean space gives a factor 1, precisely as the case of the double-Weyl model; whereas the ; y part gives a multiplicative factor 3, because there are three independent normalizable solutions of equation (8.2) with E = 0. Consequently, in agreement with equation (8.6), the axial anomaly of the triple-Weyl model This concludes the proof of equation (3.12). 9 Triple-point semimetals model The lagrangian of the triple-point semimetals model is shown in equation (3.13). In order to compute the axial anomaly, we shall rst use the Nielsen-Ninomiya method, and then the Atiyah-Singer argument, because the standard perturbative approach su ers from difculties due to the existence of singular points in the parameter space. Let us consider the fermionic elds in the presence of the gauge background A0(x) = 0 ; A1(x) = 0 ; A2(x) = Bx1 ; A3(x) = Ex0 ; 1 with E > 0 and B > 0. Since the gauge background does not depend on x2 and x3, one can specify the values k2 and k3 of the components p2 and p3 of the momentum R(x) = eik2x2 eik3x3 (x0; x1) ; L(x) = eik2x2 eik3x3 (x0; x1) ; and the equations of motion following from the lagrangian (3.13) take the form iBx1) + M3(ik3 + iEx0) iBx1) M3(ik3 + iEx0) : Let us recall that we need to determine [14] the crossing rate of the energy eigenvalues of the single particle states through the zero level. Since the (linear) time dependence of the hamiltonian is contained in the covariant derivative D3 exclusively, it is convenient to introduce the two normalised eigenvectors of M3 with nontrivial eigenvalues, A normalizable (in the x1 variable) zero eigenvector u0(x1) of the \reduced hamiltonian" iv (M1D1 + M2D2) must satisfy the equation (9.1) (9.2) (9.3) (M1D1 + M2D2) u0 = BB The normalizable solutions of equation (9.4) are given by u0(x1) = + exp u0(x1) = exp 1 2B 1 2B Bx1 k2 2 ei3 Bx1 k2 2 ei3 M3 = : 0 0 e i (Bx1 k2) 1 CC u0(x1) = 0 : (9.4) when when cos(3 ) > 0 ; cos(3 ) < 0 : (9.5) 0 ei =21 = p 12 @BBie i =2CC ; A 0 0 0 0 ei (Bx1 In the case cos(3 ) = 0, equation (9.4) does not admit normalizable solutions. We point out that this condition corresponds to the condition sin(3 ) = 0 in the notation of [50]. The fermionic modes R(x) = eik2x2 eik3x3 u0(x1)fR(x0) ; L(x) = eik2x2 eik3x3 u0(x1)fL(x0) ; (9.6) satisfy the equation i@0 (x) = ! (x), then fR=L(x0) de ned similarly as in (6.18) and (6.19) , with frequencies !R = v(k3 + Ex0) ; !L = v(k3 + Ex0) ; ( if cos(3 ) > 0 ) ; (9.7) and !R = v(k3 + Ex0) ; !L = v(k3 + Ex0) ; ( if cos(3 ) < 0 ) : (9.8) During the time evolution, the values of these frequencies crosses the zero value; thus the modes (9.6) contribute to the axial anomaly. While the remaining fermionic modes have frequencies with xed signs and can be neglected [14]. Therefore, when cos(3 ) > 0, for particles moving inside a cubic box of volume V = L3, the numbers NR and NL of available one particle states are given by NR = L Hence the vector gauge symmetry is not anomalous whereas Z Z Similarly, when cos(3 ) < 0, one nds V (9.12) As illustrated in section 7, the computation of the axial anomaly by means of the Atiyah-Singer approach is strictly connected with the Nielsen-Ninomiya method. Indeed, in the presence of the gauge background (6.1), the number of euclidean zero modes can be written as the product of the Landau degeneracy (7.14) with the number of the normalised zero modes of the \reduced hamiltonian" of equation (9.4). Therefore, also for the triplepoint model one can easily verify that the Atiyah-Singer argument leads to a result in complete agreement with equations (9.11) and (9.12). To sum up, in the case of the triple-point semimetals model with lagrangian (3.13), when the vector gauge invariance is maintained, the axial anomaly is given by V 2EB (9.13) This concludes the derivation of equation (3.16). Let us recall that the axial anomaly (9.13) has been obtained for vanishing chemical potential; in our notations, this means that all the one-particle states with zero energy are assumed to be occupied. A discussion on the stability of the result (9.13) against perturbations of the chemical potentials is contained in section 11, as well as on its relation with the presence of semi-chiral states with asymptotically vanishing energy [50, 83]. These semi-chiral states could also contribute to the normalization of the vector current associated to the chiral magnetic e ect [13, 84], which is strictly related to the axial anomaly. 10 Quantization of the anomaly coe cient In section 4, it has been mentioned that the overall multiplicative factor N , which appears in equation (4.10), can only assume integer values. According to the interpretations of Nielsen-Ninomiya and Atiyah-Singer of the axial anomaly [14, 55, 56], the quantisation of N naturally emerges. We would like to present here another argument | con rming the quantisation of N | which is based on perturbation theory, and which is of interest because it makes use of the relationship between the abelian and the nonabelian gauge anomalies [71{73, 85]. In the case of non-abelian anomalies, the quantisation of the anomaly normalization factor has been discussed for instance in [86{89]. Let us brie y recall where the emergence of the chiral gauge anomaly is found in perturbation theory. Suppose that the lagrangian for a fermion eld R(x) in the presence of a classical gauge potential V (x) takes the form LR = Ry(x) R(V ) R(x) ; in which R(V ) represents a certain di erential operator which is a function of the covariant derivatives D + iV . As we have seen in the previous sections, the eld may contain several components, and not necessarily it represents a spinor eld. Let us assume that LR is invariant under local gauge transformations R(x) ! ei R(x) R(x), R(x). The renormalized sum of the connected one-loop vacuum-tovacuum diagrams of the R(x) eld | in the presence of the classical background V (x) | is denoted by i R[V ]. Here it is assumed that R[V ] admits a perturbative expansion in powers of the background gauge eld V (x). Despite the gauge invariance of LR, the in nitesimal gauge variation of R[V ] may be nonvanishing and, modulo the variation of local counterterms, it is given by (10.1) R(x) HJEP06(218) d4x (10.2) When N 6= 0, the classical gauge invariance V (x) ! V (x) presence of an anomaly. We have already mentioned the universality of the local function quantisation of N . Let us consider the chiral anomaly in the case of a non-abelian gauge symmetry. The eld theory model de ned by the lagrangian (10.1) will now be modi ed in order to introduce a non-abelian symmetry. Suppose that a certain eld theory model contains N (with N > 2) copies of the fermion eld R(x). Thus the variables of this new model can be described by the elds Rj(x), where the index j, that we call the avour index, takes Rj(x) of values j = 1; 2 : : : ; N ; this set of elds will be denoted by R(x). An internal symmetry group acts on the N components R(x) according to the fundamental representation of SU(N )R. Let now the gauge eld V (x) take values in the Lie algebra of SU(N )R, V (x) = V a(x)T a, where fT ag (with a = 1; 2; : : : ; N 2 Let the lagrangian of the model be 1) are the generators of SU(N )R. LR = yR(x) R(V ) R(x) ; (10.3) where, in the di erential operator R(V ), the abelian covariant derivative D = @ +iV (x) has been replaced by the non-abelian covariant derivative (D )jk = jk@ + iV a(x)Tjak, and a sum over all the avour indices is understood. The lagrangian (10.3) is invariant under SU(N )R gauge transformations. Under an in nitesimal SU(N )R gauge transformation R(x) + i[ R(x); V (x)], the gauge variation of the renormalized oneloop functional 0R[V ] of the nonabelian model is given by d4x h R(x) V (x)@ V (x) + (i=2)V (x)V (x)V (x) : i (10.4) The elds polynomial which must be integrated in expression (10.4) satis es [7] the WessZumino consistency conditions. It is important to note that the multiplicative coe cient N that appears in equation (10.4) is exactly the same coe cient N entering equation (10.2). This equality is well known in the context of the computation [71, 72] of the chiral anomalies by means of perturbation theory. Indeed, in the perturbative computation of the anomaly (10.4), the Feynman diagrams which contribute to the term of expression (10.4) which is quadratic in V (x) precisely coincide with the diagrams which enter the computation of the abelian anomaly (10.2). In the perturbative computation of this term, the non commutativity of the SU(N ) generators is harmless because the eld combination $ @ V . As far as these Feynman diagrams are concerned, the only di erence between the abelian and the nonabelian case is that, in the nonabelian case, in the end of the computations one has to take a sum over the avour indices, or, one needs to introduce a trace over the indices of the fundamental SU(N )R representation. Note that the presence of this trace is explicitly indicated in expression (10.4). Therefore the same Feynman diagrams which produce the coe cient N in equation (10.2) necessarily yield the same coe cient N in equation (10.4). At this point, in order to complete the argument, we need to show that the coe cient N multiplying the nonabelian anomaly (9.4) must take integer values. Instead of displaying a formal proof, let us produce a physical argument. The relationship of the abelian chiral anomaly (4.10) and the corresponding abelian axial anomaly (4.21) has been discussed in section 4; let us consider the non-abelian generalisation of this relationship. In the nonabelian case, suppose that, in addition to the eld R(x), one also has the fermionic eld L(x) made of N components j = 1; 2; : : : ; N and the corresponding lagrangian term is LL = yL(x) L(W ) L(x) : The di erential operator L(W ) is a function of the covariant derivative (D )jk = jk@ + iW a(x)Tjak, where W (x) = W a(x)T a is the connection of the gauge group SU(N )L, which acts on the components Lj(x). It is assumed that the lagrangian (10.5) is invariant under SU(N )L gauge transformations, with L(x) transforming according to the fundamental SU(N )L representation. Let us assume that the chiral SU(N )L anomaly takes the form L 0L[W ] = N 24 2 1 Z d4x h L(x) W (x)@ W (x) + (i=2)W (x)W (x)W (x) ; Lj(x), with (10.5) i (10.6) where the multiplicative factor N is the same factor N appearing in expression (10.4). This is precisely what one nds with fermion multiplets (as quarks and leptons elds) of ordinary spinor elds L(x) minimally coupled with gauge elds. In the composed eld theory model which contains both the N -components eld and the N -components eld L(x) and total lagrangian L = yR(x) R(V ) R(x) + yL(x) L(W ) L(x) ; where V (x) and W (x) are classical background elds, one has an anomalous avour symmetry group SU(N )R SU(N )L. The SU(N )R SU(N )L gauge variation of the one-loop functional 0[V; W ] = 0R[V ] + 0L[W ] is given by the sum of expressions (10.4) and (10.6). Similarly to the abelian case, by adding a suitable nite local Bardeen counterterm LB[V; W ] to the one-loop functional 0[V; W ], one can de ne [71] a new functional 0[V; W ] + LB[V; W ] which is invariant under transformations of the vector subgroup SU(N )V of SU(N )R SU(N )L. Only axial transformations (with in nitesimal parameters given by the di erence R L) are anomalous. The resulting Bardeen avour anomaly [71] of the axial component of the group SU(N )R SU(N )L is proportional to N . When this avour anomaly is integrated [7] by employing for instance the so-called \Goldstone bosons" eld [90, 91] U(x) 2 SU(N ), the corresponding Wess-Zumino term is proportional to N . The Wess-Zumino term is well de ned [7, 92, 93] | and the ambiguities which are originated by the obstruction given by the non triviality of 5(SU(N )) are harmless | only when N 2 Z. On the other hand, the Wess-Zumino term must be well de ned because, in the case of the low energy e ective lagrangian [90, 91] of the hadrons physics in which N = 3, for instance, it describes part of the hadronic interactions of the light pseudo scalar mesons of the octet | and part of the interactions between these mesons and the avour gauge elds of the Standard Model | which can be observed in laboratory. A few consequences of the quantisation of N in particles physics can also be found, for instance, in references [86{89, 94{102]. Therefore the value of the multiplicative factor N , entering expressions (10.6), (10.4) and (10.2) must be an integer. The quantization of the anomaly multiplicative factor N has nontrivial consequences in the eld theory models in which the fermion lagrangian terms contain free parameters, as in the case of the parameters f ; ; g in the multi-Weyl models (3.5) and (3.11). Indeed, since any smooth variation of a quantised coe cient must be vanishing, in these models the expression of the axial anomaly must be invariant under \smooth" variations of the parameters, i.e. \smooth" modi cation of the operator acting on the fermion elds in the lagrangian. And in facts, this is precisely the outcome of the explicit computations of the axial anomalies (3.9) and (3.12), which depend on f ; ; g through the variable R(x) (10.7) HJEP06(218) ( ; ; ) = j j : The function ( ; ; ) is locally constant. All the modi cations of the value of ( ; ; ) are found when one of the parameters f ; ; g changes its sign; that is, when the value of one of the parameters crosses the zero point. Note that the zero value of one of these parameters represents a critical point for the lagrangian operators R(V ) or L(W ) entering the lagrangian. Indeed, when = 0, for instance, there are no more normalizable solutions of equations (6.7) and (8.2). Consequently, a modi cation of the sign of one of the parameters f ; ; g does not corresponds to a \smooth" modi cation of the operators L(W ). Similarly, in the case of the triple-point semimetals model (3.13) the dependence of the axial anomaly (3.16) on the parameter is given by the multiplicative factor which is locally constant. The change of sign of this factor occurs for 3 = =2, which correspond to critical points for the operators appearing in the lagrangian (3.13). Indeed, as it has been shown in section 9, when 3 = =2 the zero eigenvectors of the reduced hamiltonian (9.4) are not normalizable. A consequence of the quantization of the anomaly is that, if a term q kx Ry(k) R(k) Ly(k) L(k) (terms proportional to / q ky or / q kz are possible as well) is added to the Weyl lagrangian (3.1), no variation of the anomaly from the form (3.4) is obtained, until q is strong enough to induce a transition to a type-II Weyl-semimetal [103]. In the latter condition, the Fermi surface becomes extended and important deviations in the anomaly are expected. We predict the same situation for type-II generalizations of doubleand triple-Weyl semimetals, driven by terms as E(k) / q kxl Ry(k) R(k) Ly(k) L(k) , E(k) / with l = 2; 3 respectively. 11 Chemical potentials and stability The computations of the axial anomaly that have been presented in the previous sections refer to the case in which all the single-particle states with negative energy are occupied. This corresponds to the situation in which the chemical potential coincides with the energy of the band-touching nodes of the semimetals. Let us now consider the stability of our results under modi cations of the chemical potentials. 11.1 Multi-Weyl semimetals In the perturbative approach, a discussion on the stability of the chiral gauge anomalies for a single Weyl model can also be found, for instance, in the paper [71] by Bardeen, in which the most general bilinear couplings of the fermions with external sources have been considered. Actually, the stability of the axial anomaly has a general validity that we shall now examine. As it is shown in equations (3.5), (3.11) and (3.13), the lagrangian density L of the various models that we have considered in the present article has the common structure L = y(x) fiD0 H(D) g (x) = H(D) g (x) ; (11.1) where = ( R; L) and H(D) represents a di erential operator constructed with the spatial components of the covariant derivative Dj = @j + iAj (x), with j = 1; 2; 3. The modi cation of the chemical potentials can be described by the introduction of the additional lagrangian term L = R Ry(x) R(x) + L Ly(x) L(x) ; where R and L are constant parameters. For su ciently small values of R and L, does the addition of L 6= 0 modify the expression of the axial anomaly? In other words, is the axial anomaly, computed for the model which is described by the lagrangian L + L , equal to the axial anomaly which is found for the model with lagrangian L? When R = L = , the modi ed lagrangian density L + L takes the form (11.2) (11.3) L + L = )] H(D) g ; which is equal to expression (11.1) with the only replacement A0(x) ! A0(x) . The axial anomaly expression A0(x) . Therefore the introduction of the chemical potential 6= 0 does not alter the expression of the axial anomaly. The stability of the axial anomaly if R = L is suggested directly also by the Nielsen-Ninomiya method. Indeed, the addition of a term y(x) (x) modi es the zero-point of the energy spectrum but, in the presence of electric and magnetic elds, it does not modify the crossing rate of the energy values through the zero level. Therefore, the introduction of would simply amount to a constant energy shift ! ! ! + which does not a ect the values of the rates @0NR and @0NL of equations (6.25) and (6.27). When L = R = 5, the lagrangian density L + L takes the form L + L = H(D) (11.4) where 5 = diag(1; 1). This expression describes the lagrangian of a model in which the fermion variable (x) is coupled with the gauge connection A of the group U( 1 )V in the usual way, and (x) is also coupled with the gauge connection B (x) of the group U( 1 )A in which B0(x) = 5 and B(x) = 0. As it has been demonstrated in section 4, in this case, when the U( 1 )V gauge invariance is preserved, the axial anomaly is given by expression (4.20) d4x This equation shows that the contribution of the B eld to the axial anomaly is vanishing when B0(x) = 0 and B(x) = 0, and it is vanishing also when B0(x) = 5 and B(x) = 0. Therefore, the introduction of the chemical potential 5 6= 0 also does not modify the expression of the axial anomaly. Note that the presence of 5 6= 0 gives rise to the chiral magnetic e ect [13, 84], which is connected to the axial anomaly and is described by the vector current e 2 2 2 0 (11.6) The computation of the axial anomaly and the stability of the corresponding result in the case of the triple-point semimetal deserves a particular discussion. Let us recall that the lagrangians of the type shown in equations (3.5), (3.11) and (3.13) represent phenomenological approximations of more complicated theories which describe the dynamics of the relevant fermionic degrees of freedom in the various materials. Usually, the validity of these simpli ed expressions is limited to a neighbourhood of the locations of the touching nodes in the Brillouin zone; in our notations, these neighbourhoods correspond to the low momenta regions. The study of the e ective models, which are de ned by the phenomenological lagrangians, may by useful because, in certain cases, one can easily deduce interesting features which are common to both these low-energy models and to the true physical systems. The computation of the axial anomaly is precisely one example of this strategy, in which it is assumed that the axial anomaly of the e ective theories coincides with the axial anomaly of the corresponding real systems. However, it turns out that the low-energy model associated with the triple-point semimetal, which is de ned by the lagrangian (3.13), presents certain unrealistic peculiarities that must be taken into account in order to determine the axial anomaly. In the case of vanishing gauge potential, A (x) = 0, the energy spectrum of the oneparticle states, which is de ned by the lagrangian (3.13) when = 0, for instance, contains a at band with vanishing energy, such that !(k) = 0, for all values of k. As a consequence, standard perturbation theory | in which one makes an expansion of the Feynman diagrams in powers of the classical elds A (x) | cannot be de ned for the theory (3.13), because the Feynman propagator does not exist. The existence of a at band of this type, with arbitrarily large momentum and zero energy, is usually unstable under realistic perturbations of the Hamiltonian and can be considered as an unphysical artifact of the model. k3 ! 1, whereas ! e and vanishing energy as k3 ! 1. Although the Feynman diagrams approach cannot be utilised, in order to determine the axial anomaly, one can still use the Nielsen-Ninomiya method (or, equivalently, the Atiyah-Singer approach), as it has been illustrated in section 9. But also in this case one nds certain unphysical features that must be taken into account. Indeed, as it has been shown in [50, 83], in the presence of a magnetic eld directed, for instance, along the x3 direction, in addition to the chiral states with wave functions (9.6), other two Landau levels of normalizable states (here quoted \semi-chiral") emerge, which have zero energy only asymptotically (k3 ! 1); for xed chirality, in appropriate simpli ed notations, the corresponding dispersion relations have the hyperbolic form !e The branch !e(+)(k3) describes particle states with decreasing and vanishing energy as ( )(k3) corresponds to antiparticle (or hole) states with decreasing ( )(k3) = k3 pk32 + B2. The asymptotic behaviour of !e( )(k3) in the large jk3j limit appears to be rather unreliable. Indeed, for realistic lattice (tight-binding) models, it is likely that the semichiral states display a modi ed dispersion at momenta su ciently far from the nodes, and a nite separation in energy from the zero level, due at least to the nite extension of the Brillouin zone. Moreover, these semi-chiral states could be subject to a strong mixing 0 !1 k3 HJEP06(218) theory of equation (3.13). e ect due to the magnetic eld. Their peculiar dispersion which approaches zero energy at large momenta k3 can indeed favor the coupling of the hyperbolic branches belonging to triple-point crossings with opposite chiralities and we reckon the consequent mixing e ect to be stronger than the one for Weyl semimetals [104]. In this scenario, the semi-chiral modes would develop a nite energy gap, separating them from the at band. Let us now concentrate on the two chiral Landau levels that appear for positive energies and are relevant for the computation of the axial anomaly. The rst has a dispersion relation !1(k3) linear in k3 (in appropriate notations one can put !1(k3) = 2k3), the wave behaviours of !1(k3) and !2(k3) are sketched in gure 2. functions of the corresponding states are shown in equation (9.6). The second has instead a dispersion relation of the hyperbolic form !2(k3) = !e(+)(k3) = k3 + pk32 + B2. The For vanishing chemical potential, only the branch !1(k3) intersects the zero energy level and, according to the Nielsen-Ninomiya procedure, it gives a unitary contribution to the integer coe cient N appearing in the axial anomaly. However, as it is shown in gure 2, both energy branches !1(k3) and !2(k3) intersect the energy level determined by a nonvanishing value > 0 of the chemical potential. So, one could claim that, in this case, the multiplicative integer coe cient of the axial anomaly must be doubled, jN j = 2 (with a negative value of the chemical potential, one needs to consider the branch !e of !(+)(k3), getting the same conclusion). However, the value of k3 corresponding to the ( )(k3) instead intersection point of !2(k3) with the energy level tends to 1 as approaches to zero. Thus, for su ciently small , the intersection point is outside the validity range of the lowenergy e ective theory (3.13), and the lattice corrections mentioned above should no more be neglected. This is why, in the computation of the axial anomaly for the triple-point semimetals presented in section 9, we have chosen to take into account of the dispersion relation !1(k3) exclusively, thus modeling the behavior for approaching zero. Summing up, we expect that our result jN j = 1 for the axial anomaly coe cient of the triple-point semimetals is stable under su ciently small modi cations of the chemical potentials from zero energy. If the mixing of the semi-chiral modes due to the magnetic eld is not strong enough to develop a gap, by varying the chemical potential of the system, we possibly expect to nd a transition from a phase displaying jN j = 1 for small , to a phase with jN j = 2 for larger values of it. 12 In this article we have derived the expression of the abelian axial anomaly for the doubleWeyl, triple-Weyl and triple-point semimetal models. Three di erent computation methods have been considered: the perturbative quantum eld theory procedure which is based on the evaluation of the one-loop Feynman diagrams, the Nielsen-Ninomiya method, and the Atiyah-Singer index argument. The consistency of these methods, which have been shown to be closely related, has been illustrated in detail in the case of the double-Weyl model. For the triple-Weyl and the triple-point models the perturbative approach is rather burdensome or a ected by singularities; therefore only the Nielsen-Ninomiya and Atiyah-Singer methods have been discussed. It has been shown that the dependence of the anomaly on the vector gauge eld A (x) is not contingent on the Lorentz symmetry, but is determined by the gauge symmetry structure. In facts, the axial anomaly takes the general form F ^ F ; @ A ) dx ^dx denotes the curvature 2-form, and the value of the multiplicative factor N is determined by the lagrangian of each model. General arguments, still based on gauge invariance, suggest that the factor N must be quantized and must match the topological charge of the corresponding band touching points. Indeed, this is precisely the outcome of the explicit anomaly computations. We have found that jN j = 2 for the double-Weyl model, jN j = 3 for the triple-Weyl model and jN j = 1 for the triple-point model. The last result has been obtained by neglecting the hyperbolic Landau levels with asymptotical vanishing energy in the limit of large momenta. Indeed, we have presented arguments supporting this choice. For this reason, our result is not in contradiction with the usual counting of the chiral states in the triple-point semimetals presented in [50, 83]). We have further discussed the stability of the anomaly under smooth modi cations of the lagrangian parameters, showing that the value of N is invariant under these deformations. The modi cation of the sign of N in the considered models has been examined. We have veri ed that, in the parameter space, the points in which the value of N undergoes a change of sign indeed correspond to critical points. Finally we have shown that, in agreement with the case of a single-Weyl model, a modi cation of the chemical potentials does not change the expression of the axial anomaly. 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Luca Lepori, Michele Burrello, Enore Guadagnini. Axial anomaly in multi-Weyl and triple-point semimetals, Journal of High Energy Physics, 2018, 110, DOI: 10.1007/JHEP06(2018)110