Non-Abelian anomalies in multi-Weyl semimetals

Non-Abelian anomalies in multi-Weyl semimetals

Renato M. A. Dantas rmad@pks.mpg.de Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany    Francisco Peña-Benitez pena@pks.mpg.de Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany    Bitan Roy bitanroy@pks.mpg.de Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany    Piotr Surówka surowka@pks.mpg.de Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
July 2, 2019
Abstract

We construct the effective field theory for time-reversal symmetry breaking multi-Weyl semimetals (mWSMs), composed of a single pair of Weyl nodes of (anti-)monopole charge , with in crystalline environment. From both the continuum and lattice models, we show that a mWSM with can be constructed by placing flavors of linearly dispersing simple Weyl fermions (with ) in a bath of an non-Abelian static background gauge field. Such an field preserves certain crystalline symmetry (four-fold rotational or in our construction), but breaks the Lorentz symmetry, resulting in nonlinear band spectra (namely, , but , for example, where momenta is measured from the Weyl nodes). Consequently, the effective field theory displays non-Abelian anomaly, yielding anomalous Hall effect, its non-Abelian generalization, and various chiral conductivities. The anomalous violation of conservation laws is determined by the monopole charge and a specific algebraic property of the Lie group, which we further substantiate by numerically computing the regular and “isospin” densities from the lattice models of mWSMs. These predictions are also supported from a strongly coupled (holographic) description of mWSMs. Altogether our findings unify the field theoretic descriptions of mWSMs of arbitrary monopole charge (featuring copies of the Fermi arc surface states), predict signatures of non-Abelian anomaly in table-top experiments, and pave the route to explore anomaly structures for multi-fold fermions, transforming under arbitrary half-integer or integer spin representations.

I Introduction

Anomalies are traditionally studied in the realm of relativistic field theories that are pertinent in high-energy physics Bertlmann (1996); Bell and Jackiw (1969); Adler (1969); Nielsen and Ninomiya (1983); Fujikawa and Suzuki (2004); Kharzeev et al. (2016); Bardeen (1969a); Bardeen and Zumino (1984). They show up as the violation of symmetries of the classical action upon quantization of chiral massless fermions. An intrinsic feature of high-energy theories is that they are Lorentz symmetric, stemming from the linear dispersion of chiral fermions. Also in the world of condensed matter systems an emergent relativistic symmetry results from the quasiparticle spectra that are linear in momentum, but at low energies. This is the quintessential feature of Weyl semimetals - a class of materials where quantum anomaly has been studied theoretically Jackiw (2000); Goswami and Tewari (2013); Grushin (2012); Rebhan et al. (2010); Landsteiner et al. (2014); Zyuzin and Burkov (2012); Parameswaran et al. (2014); Goswami et al. (2015) and its signature has possibly been observed in experiments Huang et al. (2015); Wang et al. (2016); Zheng et al. (2016); Wiedmann et al. (2016); Zhang et al. (2016); Arnold et al. (2016); Kikugawa et al. (2016); Li et al. (2016a); Gooth et al. (2017); Schindler et al. (2018); Zhang et al. (2017).

More intriguingly, condensed matter systems offer a unique opportunity to further extend our understanding of anomalies in quantum field theories and its connections with transport. Recent developments have allowed us to go beyond the original paradigm of linearly dispersing chiral fermions, as nowadays gapless chiral systems with finite band curvatures can be realized in various solid state compounds. The main motivation of our work is to pedagogically develop a comprehensive understanding of such systems from their effective low energy field theory and anchor various field theoretic predictions from concrete, but simple lattice models (here defined on a cubic lattice). One representative class of systems where such a theory should be applicable, is so-called the multi-Weyl semimetals. These systems possess linear dispersion only along one component of the momentum, while displaying finite band curvature along the remaining two crystalline directions, see Fig. 1. The power-law dependence of the band dispersion () is set by the charge of the corresponding pairs of (anti-)monopole in the momentum space that act as source and sink of Abelian Berry curvature, and in turn also determines the integer topological invariant of the system. Therefore, the present discussion should allow us to pave the path to connect the notion of quantum anomalies with the topological invariant of gapless chiral systems. Besides the genuine fundamental importance of our quest, it should also be relevant for real materials, as Weyl points with (known as double-Weyl nodes) can be found in HgCrSe Xu et al. (2011); Fang et al. (2012) and SrSi Huang et al. (2016), whereas A(MoX) (with A=Rb, Tl and X=Te) can accommodate Weyl points with (known as triple-Weyl nodes) Liu and Zunger (2017). Even though at a formal level our conclusions hold for any arbitrary integer value of , crystalline environment forbids realization of symmetry protected Weyl nodes with  Yang and Nagaosa (2014).

A direct approach to construct an effective field theory for multi-Weyl semimetals has been discussed in Refs. Huang et al. (2017); Lepori et al. (2018), where a Lagrangian with an anisotropic energy spectrum and the corresponding anomalous violation of chiral symmetry was computed. We emphasize that this approach is cumbersome and may even be problematic for the following reasons. First of all, we stress that departure from the Lagrangian that is linear in (space-time) four-momenta, changes the structure of fermionic operators and all the anomalies need to be calculated from the scratch. Secondly, the Lagrangian corresponding to anisotropic dispersion obscures the underlying symmetry structure (and may even hinder additional features, about which more in a moment). Finally, previous studies on Lorentz violating theories suggest that certain ambiguities may appear in the formulation that cannot be removed within the effective theory description Grushin (2012). To circumvent these pitfalls and address the anomaly structure in multi-Weyl systems in an unambiguous and transparent fashion, we here develop a completely new theoretical approach, highlighted below. For the sake of concreteness, we focus on the minimal model for time-reversal symmetry breaking Weyl semimetals, composed of only a single pair of (anti-)monopole of charge .

Given this motivation, we construct an effective field theory for multi-Weyl systems that is always linear in all momenta, but augmented by a Lorentz violating perturbation, that ultimately leads to the multi-Weyl spectrum in the low energy limit. As we show this formulation has several advantages over the direct approach and improves the analysis in every aspect mentioned above. For instance, from a computational point of view, we do not need to perform additional computations of anomalies, as the theory is always linear. Furthermore, we find that the requisite Lorentz symmetry breaking perturbation yielding the multi-Weyl spectra at low-energy, couples to linearly dispersing chiral fermions as a non-Abelian constant gauge field. As a result the anomaly structure is much richer than the ones inferred from previous studies Huang et al. (2017); Lepori et al. (2018). Namely, in addition to the usual (but generalized) anomalies, we also unveil non-Abelian anomalies for multi-Weyl semimetals when . Due to the extensive nature of our study, it is worth pausing at this point to offer an overview of the main results, before delving into the details.

Figure 1: Energy dispersion () for multi-Weyl fermions in the plane for (a) , (b) and (c) around a Weyl node. Respectively for and the valence and conduction bands display linear, quadratic and cubic touching in this plane. However, the dispersion always scales linearly with irrespective of , as shown in (d). The momenta is measured from the Weyl nodes, placed at .

i.1 Extended Summary

The minimal effective low-energy model for a multi-Weyl semimetal can be described in terms of two-component chiral (left or right) fermions [see Sec. II]. The resulting quasiparticle spectra in the vicinity of each Weyl node scale as and , where [see Fig. 1], since in our construction the Weyl nodes are separated along the direction. Here, is an integer that determines the (anti-)monopole charge of the Weyl nodes and hence the topological invariant of the system. Therefore, when the energy dispersion in the plane displays nontrivial band curvature. In Sec. II, we also show that such a nonlinear dispersion for multi-Weyl semimetal can be achieved at low energies by coupling copies of simple Weyl fermions (with , possessing only linear dispersion, see Fig. 1) with a symmetry preserving perturbation (). In the language of effective field theory, such a perturbation breaks the Lorentz invariance and couples with simple Weyl fermions as an non-Abelian constant gauge field. Consequently, a multi-Weyl semimetal gets immersed in a constant non-Abelian magnetic field . Nonetheless, these two constructions are shown to be equivalent as they yield identical band dispersion at low energies and the topological invariant of the system. But, the later construction allows us to derive an effective field theory for generalized Weyl systems with in terms of simple Weyl fermions, subject to Lorentz symmetry breaking perturbation that enormously simplifies the analysis and keeps the outcomes transparent.

In Sec. III.1, we introduce simple tight-binding models for multi-Weyl semimetals on a cubic lattice. First, we present effective two band models for such systems with and , and argue that discrete four-fold rotational () symmetry protects such higher order touchings of Kramers non-degenerate valence and conduction bands at two Weyl nodes. Therefore, multi-Weyl nodes are symmetry protected. The resulting band structures are shown in Fig. 2. Moreover, we show that multi-Weyl semimetals with and can be constructed by coupling copies of the lattice model for simple Weyl semimetal (with ) by the symmetry preserving perturbation. The resulting band structures of these two systems, shown in Fig. 3, are identical to the ones obtained from their corresponding two-band models [see Fig. 2(b) and (c)], but only at low energies.

The topological equivalence between these two constructions for the multi-Weyl systems is then further substantiated from the bulk-boundary correspondence, encoded through the number of Fermi arc surface states, see Sec III.2. Note that a multi-Weyl semimetal, constituted by (anti-)monopole of charge , supports copies of the Fermi arcs. Indeed we find copies of the Fermi arcs connecting two Weyl nodes of charge from the two-band models, see Fig. 4. Furthermore, we also observe and copies of the arc states for double and triple Weyl semimetals, respectively, when they are constructed by coupling and copies of simple Weyl fermions by a symmetry preserving perturbation, see Fig. 5.

Upon constructing multi-Weyl semimetals by coupling simple Weyl fermions with a static non-Abelian gauge field, we derive the effective field theory of these systems in Sec. IV. The effective field theory for multi-Weyl semimetals displays both as well as non-Abelian (only for ) anomalies. To this end, we compute the Ward identities for both covariant and consistent (related by the Bardeen-Zumino polynomials) Abelian and non-Abelian currents. One of our main results is the generalization of anomalous Hall effect for multi-Weyl semimetals, and its non-Abelian variation, respectively captured by, for example, the regular () and “isospin” () charge densities, given by

(1)

In the above expressions, is the electric charge, is the separation of left and right Weyl nodes, is the external Abelian magnetic field, is the static non-Abelian magnetic field (present only for ), is a coefficient set by the representation of Lie group. Specifically, and for and , respectively. We also find that the chiral magnetic effect vanishes for both vector Abelian and non-Abelian currents.

To test the validity of the field theoretic predictions from Sec. IV, we first compute Abelian or charge density () in the presence of a static external magnetic field from all the lattice models for multi-Weyl systems, introduced in Sec. III. The methodology is discussed in Sec. V and the results are displayed in Fig. 6. We find that the field theoretic predictions [see Eq. (1)] show excellent agreement with the scaling of the Abelian charge density with the external magnetic field flux, at least when it is small (cyclotron frequency being much smaller than lattice momenta), irrespective of the microscopic details. As a penultimate topic, we compute the non-Abelian or isospin density (), capturing the signature of non-Abelian anomalies [see Eq. (1)], for the multi-Weyl semimetals with and , but only from their four and six band lattice models, respectively. The results are shown in Fig. 7, displaying an excellent agreement with the field theoretic predictions.

The topological nature of anomalies in certain cases protects their associated transport, showing universalities even when some symmetries are broken Amado et al. (2014); Copetti et al. (2017). Therefore, the microscopical details of different models become irrelevant, as long as the anomalous structure does not differ between them Son and Surowka (2009); Neiman and Oz (2011). On the other hand, the computation of anomaly induced transport coefficients with standard quantum field theory techniques can be plagued with subtleties and ambiguities Jackiw (2000); Goswami and Tewari (2013); Grushin (2012); Rebhan et al. (2010); Landsteiner et al. (2014); Zyuzin and Burkov (2012); Parameswaran et al. (2014), which have been solved and understood with the help of the holographic techniques Gynther et al. (2011); Landsteiner et al. (2013). Therefore, we address the imprint of the symmetry breaking parameter () in various anomaly induced transports from a simple toy model for a (strongly) interacting multi-Weyl semimetal using the holographic techniques, see Sec. VI. The particular model we study is consistent with the predictions of the effective field theory, and shows a renormalization of the non-Abelian current in the infrared regime, as expected due to the explicit symmetry breaking introduced by . The main outcome from this section is the survival of the non-Abelian transport at low energies, opening a possibility of observing non-Abelian anomaly and the non-renormalization of the Abelian anomaly-induced transport in multi-Weyl semimetals. Therefore, altogether the current discussion presents a comprehensive study of anomalies in Lorentz symmetry violating multi-Weyl semimetals, which in future can be extended to address similar issues for multi-fold fermions Bradlyn et al. (2016, 2017).

i.2 Outline

The rest of the paper is organized as follows. In the next section, we discuss the low-energy models for multi-Weyl semimetal and compute it topological invariant. Section III is devoted to the discussion on the lattice models for these systems on a cubic lattice. In this section we also establish the bulk-boundary correspondence by constructing (numerically) the Fermi arc surface states for multi-Weyl semimetal. The effective field theories, capturing the signature of quantum anomalies, for multi-Weyl semimetals are derived in Sec. IV. The field theoretic predictions from this section are numerically anchored from the representative tight-binding models in Sec. V. The holographic description and transport coefficients are derived from the gauge-gravity duality in Sec. VI. Discussions on our findings and some future directions are highlighted in Sec. VII. Additional technical details are relegated to the appendices.

Ii Multi-Weyl fermions

We begin the discussion by focusing on the effective low-energy models for multi-Weyl systems, constituted by a pair of (anti-)monopole of charge  Huang et al. (2017); Gorbar et al. (2017); Lü et al. (2019); Sinha and Sengupta (2019); Menon and Basu (2019); Xu et al. (2011); Fang et al. (2012); Huang et al. (2016); Liu and Zunger (2017); Yang and Nagaosa (2014); Roy and Sau (2015); Li et al. (2016b); Armitage et al. (2018). The Hamiltonian operator describing such system takes the form

(2)

where and correspond to two valleys, respectively acting as the monopole (source) and antimonopole (sink) of Abelian Berry curvature. Around these two points low-energy excitations are described in terms of left and right chiral fermions, respectively. Momentum is measured from the Weyl node. The set of Pauli matrices operate on the pseudospin indices. The energy spectra in the close vicinity of the Weyl nodes take the form , where respectively correspond to the conduction and valence bands, and

(3)

Respectively, for and the parameter bears the dimension of velocity and inverse mass, while is the Fermi velocity in the direction. The energy dispersions along various high symmetry directions for and are shown in Fig. 1.

The topological invariant of Weyl systems is given by the integer (anti-)monopole charge, which can be computed in the following way. For concreteness, we now focus near one valley, hosting left chiral fermions and introduce the following coordinate system

(4)

where . The Berry curvature of the conduction band then takes the form Dantas et al. (2018)

(5)

where is the unit-norm radial vector and

(6)

The integer monopole charge can then be obtained by integrating the Berry curvature over a unit sphere (), defined by , around the Weyl node, yielding

(7)
Figure 2: Band structure of (a) simple (), (b) double () and (c) triple () Weyl semimetals from the corresponding effective two band models, see Eqs. (15)-(18). We set in the tight-binding model, and the lattice constant . We here take the path , where , , and stand for , , and , respectively. The energy () is measured in units of . Note that Weyl nodes are located at , around which dispersion always scales linearly along the directions, while it respectively display linear, quadratic and cubic dispersion in the plane (see insets). Due to the absence of particle-hole asymmetry, the Weyl nodes are always pinned at zero energy. Realizations of double and triple Weyl systems respectively from four and six band models are shown in Fig. 3.

Even though the low energy model for multi-Weyl semimetals correctly captures the topological invariant of the system, one can construct the Weyl models with by coupling copies (hereafter referred ‘flavor’) of simple Weyl fermions in the following way. This construction follows the spirit of realizing higher order band touchings in multi layer graphenelike systems by introducing interlayer tunneling Castro Neto et al. (2009); Roy (2013); Huang et al. (2016). This construction opens an efficient route to arrive at the effective field theoretic description for Weyl systems with [see Sec. IV]. We focus near the left chiral valley and introduce the following Hamiltonian operator

(8)

where , and operate on the flavor index, while operate on pseudospin index. Note that the first term in corresponds to decoupled flavors of simple Weyl fermions, while the term proportional to introduces nontrivial coupling between them 111Note that the form of such inter-flavor coupling is not unique. One can choose it to be , which leaves all the physical outcomes unchanged.. For any integer , is -dimensional identity matrix, are the generators of the spin- representation of . In particular for , , where are the Pauli matrices, while for

where are the Gell-Mann matrices Gell-Mann (1962). In principle, one can generalize this construction for arbitrary integer value of . But, in crystalline environment only Weyl nodes with are symmetry protected. So, we here focus on Weyl systems with and .

For the energy spectra are composed of four branches, given by , where

(9)

for and , and correspond to the conduction and valence bands, respectively. Note that only the branch displays band touching at , while the branch is fully gapped, for any . Expanding for large and small , we obtain

(10)

which agrees with the expression from Eq. (3) to the order , with , bearing the dimension of inverse mass. Shortly, we show that the pair of split off bands are topologically trivial, while the band touching point within the sector act as a monopole of charge . On the other hand, for , the energy spectra are composed of six branches for , where

Note that only the branch displays band touching at , which acts as monopole of charge , while the remaining four bands are completely gapped and topologically trivial. Expanding for large and small , we obtain

(12)

which agrees with Eq. (3) to the order , with . The above construction of generating multi-Weyl systems by coupling simple-Weyl fermions is, however, not an artifact of low-enegry approximation. In Sec. III, we show that such construction is operative even when we start from the lattice regularized models.

Finally, we compute the Berry curvature for each band for multi-Weyl systems with , obtained by coupling flavors of simple Weyl fermions. For , we perform this exercise analytically, by introducing the nonorthogonal curvilinear coordinate system

(13)

where now , and is displayed in Eq. (9). In this coordinate system, the Berry curvature for the lower conduction band takes a compact form

(14)

where and are the covariant basis vectors. From the above expression for the Berry curvature, we can immediately compute the integer charge assocaited with the band touching point from Eq. (7), yielding for any . The expression for the Berry curvature for the gapped valence and conduction bands are quite lengthy and not very instructive. However, when we integrate the Berry curvature over a closed surface [see Eq. (7)], it yields a trivial answer. Therefore, in the four-band construction for double Weyl fermions, only the two bands touching each other are topologically nontrivial. For triple Weyl fermions four gapped bands are topologically trivial, while the monopole charge of the band touching points, where valence and conduction bands meet is .

Iii Lattice model, Bulk-boundary correspondence and Fermi arcs

In this section, we introduce effective tight-binding models on cubic lattice yielding multi-Weyl semimetals, possessing only two Weyl nodes. Subsequently, we establish the bulk-boundary correspondence for these systems by computing the Fermi arc surface states. These analyses substantiate our discussion from the last section. In addition, we also subscribe to these lattice models to test the predictions from the effective field theory for multi-Weyl semimetals [see Sec. IV], discussed in Sec. V.

Figure 3: Band structure of (a) double and (b) triple Weyl semimetals from four and six band models, respectively [see Eqs. (III.1) and (20)]. For numerical diagonalization we set in the tight-binding models, and the lattice spacing . Note that these models also display quadratic [see panel (a)] and cubic [see panel (b)] dispersion in the () plane around the Weyl nodes located at [compare with subfigures (b) and (c) of Fig. 2]. The dispersion always scales linearly with .
Figure 4: Topologically protected Fermi arc surface states for (a) simple (), (b) double () and (c) triple () Weyl semimetals, obtained from their two band models [see Eqs. (15)-(18)]. For numerical diagonalization we implement a mixed momentum (along and ) and real space or Wannier (along ) representation. The linear dimensionality of the system along is [see Sec. III.2 for details]. Consequently, the Fermi arc surface states are localized on the top and bottom surfaces. We set , and plot the square of the amplitude of the low-energy states within an energy window around zero energy. The number of Fermi arcs is equal to , anchoring the bulk-boundary correspondence for multi-Weyl semimetals. In addition, the Fermi arcs from opposite surfaces (top and bottom) get connected through the bulk Weyl points [acting as defects, namely monopole and antimonopole, in the momentum space], where the band gap vanishes.

iii.1 Lattice models

The lattice model for general Weyl fermions can compactly be written as

(15)

where is a two-component spinor, and is the fermion annihilation operator with momentum and pseudospin projection . The effective two-band theory emerging from the above tight-binding model give rise to left and right chiral Weyl fermions respectively near , if we choose

(16)

For convenience, we set the lattice spacing to be unity. Then simple, double and triple Weyl fermions are realized when we take Roy et al. (2017); Bera et al. (2016); Roy et al. (2018)

(17)

and

(18)

The resulting band structures for and are shown in Fig. 2. Notice that the above tight-binding models produce only a pair of Weyl nodes at , around which the effective low-energy models assume the form announced in Sec. II.

The band touching points in multi-Weyl semimetals are protected by the four-four or rotation about the axis, a bonafide symmetry operation of point group. Under such a rotation . When such rotation in the momentum space is augmented by a rotation by an angle in the pseudo spin space, captured by the unitary operator , the Hamiltonian operator for and remains completely invariant. The situation for is slightly more subtle, as . Nonetheless, monopole and anti-monopole maps onto themselves under the rotations, leaving the triple-Weyl points symmetry protected. On the other hand, if we take for , all the outcomes remain unchanged, but the corresponding Hamiltonian operator remains completely invariant under the rotation. Hence, multi-Weyl points are symmetry protected in a system possessing a symmetry.

The multi-Weyl semimetals with can also be realized by properly coupling copies of simple Weyl semimetals. We discussed this construction from the continuum or low-energy models in Sec. II. We now test the validity of such a construction, starting from the lattice models for Weyl fermions, given by

Following Eq. (8), we construct the lattice model for multi-Weyl semimetals by coupling copies simple Weyl semimetals according to

(20)

The notation is the same as in Sec. II. The resulting band structure for and are shown in Fig. 3. Respectively for and , two and four bands are completely gapped, while the remaining two bands touch each other at . The energy dispersions around these points are respectively quadratic and cubic with the in-plane components of momenta, but always scales linearly with its component.

iii.2 Fermi arcs

Previously in Sec. II, we showed that the four and six band models respectively for double and triple Weyl fermions [see Eq. (8)] and their low-energy description in terms of the two band models [see Eq. (2)] yield identical topological invariant (the monopole charge). The monopole charge determines the integer topological invariant of the system that in turn also dictates the number of topologically protected Fermi arc surface states, connecting two Weyl nodes of opposite chiralities. Therefore, equivalence between four (six) band model for the double (triple) Weyl fermions [see Eq. (20)] and their two band models [see Eqs. (15)-(18)] can be established by comparing the number Fermi arcs for multi-Weyl systems from these two sets of tight-binding models. The results are shown in Figs. 4 and 5.

To compute the Fermi arc surface states we impose periodic boundaries in the and directions, such that and can be treated as good quantum numbers. But, we implement open boundary in the direction, along which the linear dimensionality of the system is denoted by  Slager et al. (2017); Nandy et al. (2018). In such a mixed Bloch-Wannier representation the Fermi arcs are localized on the top and bottom surfaces, as shown in Figs. 4 and 5. Specifically in Fig. 4 we show the topologically protected Fermi arcs for multi-Weyl semimetals, constructed from their two band tight-binding models [see Eqs. (15)-(18)]. We find that a multi-Weyl semimetal, characterized by integer (anti-)monopole charge , supports exactly copies of Fermi arc surface states. This observation establishes the bulk-boundary correspondence for this family of gapless topological semimetals. On the other hand, in Fig. 5 we show the Fermi arcs for double and triple Weyl semimetals, but constructed from the four and six band models [see Eqs. (III.1) and  (20)], respectively. Once again we find that these two systems respectively host two and three copies of the Fermi arcs on the top and bottom surfaces. This outcome besides supporting the bulk-boundary correspondence, also anchors the topological equivalence between the multi and two band representations for the double and triple Weyl semimetals on a lattice. To appreciate some additional salient features of the arc states, next we consider their microscopic origin.

Any general Weyl semimetal hosting (anti-)monopole of charge can be constructed by stacking two-dimensional layers of quantum anomalous Hall insulator, occupying the -plane, in the momentum space along the direction within the range , where in our lattice construction. The first Chern number of each such anomalous Hall insulator is and it supports copies of one-dimensional chiral edge modes, with states at precise zero energy. The collection of such zero-energy states within the range constitutes copies of the Fermi arc surface states, shown in Figs. 4 and 5. Also note that the localization length of each zero-energy mode is inversely proportional to bulk gap of the underlying two-dimensional anomalous Hall insulator for a given . In our lattice models, such a gap is maximal when and it vanishes at . Otherwise, this gap decreases smoothly as we approach from the center of the surface Brillouin zone (). Consequently, the surface localization of each copy of Fermi arcs decreases monotonically as we approach two Weyl points from the center of the arcs. Ultimately, at the arcs are completely delocalized, and at these two points arcs from the top and bottom surfaces get connected via the bulk Weyl nodes. This feature can be seen from Figs. 4 and 5. Next we proceed to derive the effective field theory of these systems.

Figure 5: Fermi arc surface states for (a) double () and (b) triple () Weyl semimetals, obtained respectively from four and six band models [see Eqs. (III.1) and  (20)]. For numerical diagonalization we follow the same approach, mentioned in the caption of Fig. 4, and set , . Note that double and triple Weyl semimetals respectively support two and three copies of Fermi arc surface states, as we obtained from their two band representations [see (b) and (c) of Fig. 4]. This observation establishes the topological equivalence among these models and the bulk-boundary correspondence in these gapless topological systems. The color scale is the same as in Fig. 4.

Iv Effective field theory

All global symmetries present in a classical action do not necessarily survive after quantization Bertlmann (1996). Possibly the best known examples of this phenomenon are the ones related to the chiral anomalies. In particular, the axial anomaly is responsible for the celebrated decay of pion into two photons Bell and Jackiw (1969); Adler (1969). Furthermore, it also leaves signatures on anomaly-induced transports that have attracted ample attention in recent time in the context of Dirac and Weyl semimetals in condensed matter systems, quark-gluon plasma in heavy-ion colliders and magnetized plasmas in cosmology, for example. In this section, we derive the effective field theory for multi-Weyl semimetals, unveil the anomaly structure therein and discuss its imprints on various transports.

After establishing the Hamiltonian description for multi-Weyl semimetals, we seek to formulate the corresponding Lagrangian formalism, which allows us to derive the effective field theory for these systems. Performing a Legendre transformation on the continuum Hamiltonian [see Eq. (8)], we obtain the Lagrangian for left chiral fermions ()

(21)

where . Throughout Einstein’s summation convention over repeated indices is assumed. The above expression allows us to construct a generalized formalism for multi-Weyl semimetals with a non-Abelian flavor symmetry, in presence of a non-Abelian background gauge field according to

(22)

where with are the generators of . In particular, the static background field giving rise to nonlinear dispersion (in the plane) in multi-Weyl systems can be written as

(23)

From now on, we denote the field strength associated to by , the Abelian gauge field by and the gauge field by . Note that the non-Abelian field, giving rise to the multi-Weyl semimetals, picks a preferred direction and reduces the initial symmetry group of decoupled copies of simple Weyl fermions to the diagonal .

Let us assume that we have a theory for left chiral fermions, transforming in some representation of the Lie group . The corresponding generators of the Lie algebra satisfy

(24)

where is the structure factor of the Lie group . For theories with such a flavor symmetry the associated anomalous currents () satisfy the following Ward identities in its covariant form

(25)

where is the covariant derivative containing the gauge and metric connections, is the curved space covariant derivative, is the Rieman curvature tensor, is the stress-energy tensor, and the anomalous coefficients are and Bertlmann (1996); Landsteiner et al. (2013)222For right chiral fermions and .. The currents cannot be obtained by varying an action with respect to the background fields , and normally it is called covariant currents. Nonetheless, there exist the consistent currents related with by the addition of a Chern-Simons polynomial. The consistent currents can be defined as a functional derivative of the action with respect to the background field according to , where and

(27)

Thus, the consistent Ward identity reads

To illustrate the applicability of the general theoretical framework discussed so far, we now focus on a theory with one copy of left and right handed fermions, coupled to Abelian gauge fields. The Ward indentities for the consistent current that couples to the gauge field and the axial current, defined as , respectively read

(29)
(30)

The conservation of electric charge requires that the combination should be conserved. However, in Eq. (29) the vector current is not conserved333Notice that Eq. (29) is an operator equation. Therefore, even though at the fundamental level axial gauge fields do not exist, the three point function spoils the consistency of the theory.. Nonetheless, this issue can be resolved by noting that the theory is not gauge invariant. Hence, one can add a counter-term, known as the ‘Bardeen counter-term’

(31)

to the original action, which reestablishes the gauge invariance. After introducing this local polynomial, the Ward identities for the newly defined consistent currents

(32)

respectively read as

(33)
(34)

Only after such a redefinition, the current associated with the electric charge is conserved, while the axial current remains anomalous. The above construction has a natural generalization to the theories with non-Abelian anomalies, which we discuss below.

iv.1 Theory with flavor symmetries

Now we consider multi-Weyl semimetals, in which the left- and right-handed fermions transform under a representation. In this case, the (covariant) anomalous Ward identities [see Eq. (25)] read as follows

(35)
(36)
(37)
(38)

where is defined via the relation for . For our choices of the generators, and . As discussed in the previous section, the covariant current cannot be obtained by differentiating any functional of the gauge fields. Therefore, they do not couple to the gauge fields. However, Bardeen computed the proper counter-term to construct conserved vector consistent currents Bardeen (1969b)444We impose the conservation of the vector non-abelian current because the continuum version of the lattice symmetry corresponds to the , discussed in the previous section.. Combining the Bardeen counter-terms with the Bardeen-Zumino polynomial [see Eq. (27)] Manes et al. (2018a, b), we can write the Chern-Simons current, relating the covariant and consistent currents according to

(39)

where

(40)
(41)

Having understood the anomaly structure of the effective field theory, we now extract the anomaly-induced transport coefficients for multi-Weyl systems.

Figure 6: The charge density (), measured in units of , in the vicinity of the flux tubes placed at (yielding ) and (yielding ) [see Eq. (65)] for simple Weyl (SW), double Weyl (DW) and triple Weyl (TW) semimetals. Results for simple Weyl semimetal are always obtained from the two band model. By contrast, for the double (triple) Weyl semimetals, we compute from (a) four (six) and (b) two band models. Here, is the magnetic flux, measured in units of the flux quanta (in natural units ). For numerical analysis, we always take and , where is the number of points in the momentum space along the direction. Specifically for (a) we set and , while for (b) , where is the linear dimension of the system in the and directions. The dots correspond to numerically computed values of , whereas the straight lines follow the relation , where is the monopole charge of the Weyl nodes, located at , and . This analysis establishes excellent agreement between the field theoretic predictions and the scaling of the corresponding observable computed from the lattice models.

iv.2 Anomaly induced transport

Chiral fermions exhibit non-dissipative transports at finite temperature () and density (), which are intimately related to the chiral anomalies discussed in the previous sections. In particular, the covariant currents within the linear response approximation were computed in Refs. Son and Surowka (2009); Kharzeev and Warringa (2009); Neiman and Oz (2011); Landsteiner et al. (2011); Vilenkin (1979, 1980); Mueller and Venugopalan (2019); Loganayagam and Surowka (2012); Jensen et al. (2013) and read 555These expressions assume linear response, therefore in the definition of the non-abelian magnetic fields only the linear terms in the gauge fields are considered.

(42)
(43)

where . The magnetic and vorticity fields respectively are defined as

(44)

and is a unit norm timelike vector. The nondissipative currents give rise to (a) chiral magnetic conducitivities ( and ) and (b) chiral vortical conductivities ( and ). In the absence of dynamical gauge fields, these quantities are universal and solely determined by the anomaly. And they are given by

(45)
(46)
(47)

where , with and denoting the regular and flavor chemical potentials, respectively.

For a theory with symmetry (describing a simple Weyl semimetal with ), the vector and axial covariant currents are respectively given by

(48)
(49)

On the other hand, the covariant current arising from the energy-momentum tensor, , is

(50)

In the above expressions, we introduce the following quantities , , and . However, as discussed in Sec. IV the covariant electric current does not couple to the electromagnetic field. The proper conserved current that couples to the photon is the consistent current, obtained after including the contribution from the Bardeen-Zumino polynomial and Bardeen couternterm [see Eq. (40)], leading to the following expressions

(51)
(52)
(53)