Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomaly

The study of quantum corrections in the QED fermionic sector of the Standard Model Extension (SME) Kostelecky0 ; Kostelecky1 due to the electromagnetic interaction was sparkled in Ref. Kostelecky1 which posed the problem of obtaining corrections to the Chern-Simons interaction $\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}{\Delta k}_{\mu}A_{\nu}F_{\alpha\beta}$ as arising from the one loop vacuum polarization tensor of fermions with the additional coupling ${\bar{\Psi}}\gamma^{5}\gamma^{\mu}b_{\mu}\Psi$. Since then, a large number of authors have carried the calculation of the CPT odd contribution to the vacuum polarization obtaining a result that can be presented as $\Delta k^{\mu}=\zeta\,(e^{2}/\pi^{2})\,b^{\mu}$, where $\zeta$ is always a finite coefficient adopting different values, including $\zeta=0$, according to the regularization prescription chosen to deal with the superficially linear divergence of the vacuum polarization tensor $\Pi^{\mu\nu}$ CG ; JK ; PV ; CHUNG1 ; CHUNG2 ; CHEN ; BONNEAU ; CHAICHIAN ; PV1 ; BATTISTEL ; ANDRIANOV ; BALT1 ; BALT2 ; ALFARO . A survey of the principal approaches regarding this issue can be found in Refs. PV1 ; ALFARO ; JACKIW2 ; BALT3 . The search of a physical criterion to select one specific value of $\zeta$ has been unsuccessful and the current understanding is that $\zeta$ would be fixed either by an experimental condition or by a more fundamental theory. In order to briefly summarize some of the previous results we assume that $b_{\mu}$ is timelike. The case $\zeta=0$ is obtained by demanding gauge invariance of the effective Lagrangian density CG , and has the drawback of eliminating the Chern-Simons contribution from the outset since this term changes by a total derivative under gauge transformations. Using the full non-perturbative expression in $b_{\mu}$ for the fermion propagator and different regulators consistent with symmetric integration in four dimensions, the authors of Refs. JK ; PV ; CHUNG1 ; CHUNG2 find $|\zeta|={3}/{16}$. In an attempt to elucidate a further significance of the non-perturbative approach, which demands the use of a unique regulator for all the contributions of $\Pi^{\mu\nu}$, the CPT even contribution to $\Pi^{\mu\nu}$ (second order in $b_{\mu}$) was calculated in the $m=0$ case, finding that gauge invariance is violated BALT1 . The use of Pauli-Villars regularization was subsequently introduced in the non-perturbative calculation of the massive case, yielding gauge invariance to second order but two possibilities for the CPT odd contribution: either the expected value $\zeta=0$ or $|\zeta|=3/8$ BALT2 . Again, there is no unique way of fixing the $\zeta$ contribution by demanding gauge invariance of $\Pi^{\mu\nu}$. An alternative regularization prescription based upon the maximal residual symmetry group of the vector $b_{\mu}$, proposed in Ref. ANDRIANOV , provides a definition on how to perform the loop integration $d^{4}k$ in $\Pi^{\mu\nu}$. For example, in the timelike case, the maximal residual symmetry group is $SO(3)$ which calls for a spherically symmetric integration in the tri-momentum ${\mathbf{k}}$ instead of on the four-momentum $k^{\mu}$. This prescription yields $|\zeta|=1/4$.

In this work we extend previous calculations of effective electromagnetic action induced by radiative correction in the SME by including additional terms of the fermionic sector in the minimal QED extension of the SME TABLE . Our starting point is the action

which is coupled to the electromagnetic field $A^{\mu}=(A^{0},A^{i})=(A^{0},{\mbox{\boldmath$A$}})$, with the notation ${\mbox{\boldmath$A$}}=(A_{x},\,A_{y},\,A_{z})$ in terms of the Cartesian components. The metric is $\eta_{\mu\nu}=\mbox{diag}(+,-,-,-)$ and we use $\hbar=c=1$ units henceforth. We set $m=m_{5}=H_{\mu\nu}=e_{\lambda}=f_{\lambda}=g_{\lambda\kappa\nu}=0$, in the notation of Table XVI of Ref. TABLE . Thus we restrict ourselves to

with $\gamma^{\mu}$ being the standard gamma matrices. The term $c^{\mu}{}_{\nu}$ already includes the $\delta^{\mu}_{\nu}$ contribution corresponding to the free Dirac action. The terms ${\bar{\Psi}}\,\Gamma^{\mu}i\partial_{\mu}\Psi$ are CPT even, while those in ${\bar{\Psi}}M\Psi$ are CPT odd. Still, since $\gamma^{5}$ is PT odd, each of them separately contains a mixture of PT even and odd terms.

Let us emphasize that the methods and techniques employed in the calculation of these extended effective actions in high energy physics, besides being valuable in their own, can be of relevance in some areas of condensed matter physics. Indeed, the identification of fermionic quasiparticles of Dirac and/or Weyl type in the linearized approximation of Hamiltonians in topological phases of matter provides the opportunity of studying them under the perspective of the SME. Such approach has been particularly fruitful in the case of Weyl semimetals (WSMs) whose electronic Hamiltonians naturally include some of the LIV terms considered in the fermion sector of the SME. Nevertheless, in this case, the LIV parameters need not be highly suppressed, since they are determined by the electronic structure of the material and are subjected to experimental determination GRUSHIN ; Burkov ; Goswami ; TI ; REVLANDS ; Kostelecky3 . Our choice of the LIV parameters in Eq. (2) are those relevant for the description of a general WSM. The simplest example arises in the Hamiltonian of a Weyl semimetal with no tilting and with an isotropic Fermi velocity, which can be embedded in the fermionic action

where ${\tilde{b}}^{\mu}=({\tilde{b}}^{0},{\mbox{\boldmath${\tilde{b}}$}})$. Starting from the action (3) coupled to an electromagnetic field, the chiral rotation method of Ref. Burkov , which provides an alternative quite different from the standard vacuum polarization calculation, yields

which determines $|\zeta|=1/4$. Another outstanding example relevant in condensed matter is the case of the superfluid ³He-A, where the value $\zeta=1/2$ is reported VOLOVIK .

The action (4) captures the electromagnetic response of WSMs. While the axion coupling $\Theta(x)$ arises from the nontrivial topology of the band structure of the material, the appearance of the abelian Pontryagin density (APD) $\epsilon^{\alpha\beta\mu\nu}F_{\alpha\beta}F_{\mu\nu}$ is related to the axial anomaly

From Eq. (4) we obtain the effective sources FRANZ

where $E$ and $B$ are the electromagnetic fields. In our index notation the above current is $J_{i}=\delta S_{\rm eff}/\delta A^{i}$. In particular, for ${\tilde{b}}_{0}=0$, this yields $J_{x}=\sigma_{xy}E_{y}$ with the transverse anomalous Hall conductivity $\sigma_{xy}=-\frac{e^{2}}{2\pi^{2}}{\tilde{b}}_{z}$, a distinguishing feature of Weyl semimetals BURKOVBALENTS . As it will become clear in Section III.1, the knowledge of this quantity allows to unambiguously fix the otherwise arbitrary coefficient $\zeta$ in this case. This additional information is a general consequence of the underlying microscopic theory describing the material, as opposed to the lack of a more fundamental theory containing the SME.

Recently, the appearance of the APD in effective actions arising from (1) has promoted the use of anomaly calculations to obtain them. For example, in the path integral approach, $S_{\rm eff}$ was obtained by introducing the electromagnetic coupling in Eq. (3) and subsequently eliminating the fermionic term proportional to ${\tilde{b}}_{\mu}$ through a chiral rotation. Nevertheless, this produces an electromagnetic contribution to the action arising from the nonzero Jacobian of the chiral rotation which is proportional to the Pontryagin density Burkov . Following this idea, the Fujikawa prescription to obtain the chiral anomalies Fujikawa ; Bertlement has also been used to calculate the effective electromagnetic action of different materials in Refs. Goswami ; TI ; REVLANDS as well as to give an alternative explanation of the indeterminacy of the coefficient $\zeta$ related to some freedom in the definition of the fermionic axial vector current CHUNG3 .

Nevertheless, as pointed out in Ref. LUAGA , the anomaly does not directly yield the effective action. Also, the method of eliminating the modifying fermionic contributions via a chiral rotation cannot be easily extended to deal with the more complicated configurations envisaged in the action (1).

The reasons indicated above suggest the convenience of applying and extending the quantum field theory methods developed in high energy physics to obtain the required effective electromagnetic actions corresponding to fermionic systems described by the action (1) which are of interest in condensed matter physics. In particular this procedure should clarify how the LIV corrections enter in the effective action, while the chiral anomaly remains insensitive to them LUAGA ; FIDEL ; SALVIO ; Scarpelli . Motivated by the inclusion of temperature effects in the odd contribution of the vacuum polarization tensor arising from the $b_{\mu}$ coupling in the action (3), reported in Refs. EBERT ; TEMP1 ; TEMP2 ; TEMP3 ; TEMP4 ; TEMP5 , we provide the first steps to incorporate thermal field theory in the case of the more general LIV couplings defined in Eqs. (2). In this way, we incorporate non-zero chemical potential effects, but still remain in the zero temperature limit. We follow the conventions of Ref. PESKIN .

Integrating the fermions in Eq. (1) defines the effective action

which we write as

to second order in the electromagnetic potential. This introduces the vacuum polarization tensor $\Pi^{\mu\nu}(p)$

where $S(k)=i/(\Gamma^{\mu}k_{\mu}-M)$ is the exact fermion propagator in momentum space including LIV modifications. Since $S_{\rm eff}^{(2)}$ is real we must have $\Pi^{*}_{\mu\nu}(p)=\Pi_{\nu\mu}(p)$. As we will show, the new effective action (8) will keep the form of Eq. (4) with all the modifications entering through a new vector ${\cal B}_{\lambda}$ to be determined. The link between both expressions is accomplished with the identifications

In the following we calculate the CPT odd contribution of the effective action (8). In the massless case ($m=0$), $\Gamma^{\mu}k_{\mu}-M$ is linear in $\gamma^{\mu}$ and $\gamma^{5}\gamma^{\mu}$. The appearance of the matrix $\gamma^{5}$ suggests the convenience of using left and right chiral projectors in order to evaluate $\gamma^{5}=\pm 1$ BALT1 ; SALVIO ; TEMP1 . Therefore, we can work in the chiral basis, where the decomposition of the Dirac spinor into the right and left Weyl spinors is manifest. These latter are eigenspinors of $\gamma^{5}$ with the eigenvalues $\pm 1$. We now define the projection operators

which project onto right- and left-handed spinors, respectively. Note that $\gamma^{\mu}P_{L}=P_{R}\gamma^{\mu}$. The projectors (11) allow us to define the matrices $\Gamma_{R}^{\mu}$ such that

which identifies $\Gamma^{\mu}_{R}=r^{\mu}{}_{\nu}\gamma^{\nu}$ with $r^{\mu}{}_{\nu}=c^{\mu}{}_{\nu}-d^{\mu}{}_{\nu}$. In an analogous way we define the left-handed part of the matrices $\Gamma^{\mu}$ as

Following the same idea, we can also split the fermion propagator $S(k)$ into its right- and left-handed parts, i.e.

Note that the propagators $S_{L}$ and $S_{R}$, having the generic form $i/(Z_{\nu}\gamma^{\nu})$, can be readily rationalized as $i(Z_{\nu}\gamma^{\nu})/Z^{2}$.

To proceed forward with the calculation we now split the combination $T^{\mu\nu}(k,p)=S(k-p)\Gamma^{\mu}S(k)\Gamma^{\nu}$ under the trace in $\Pi^{\mu\nu}(p)$ into its left- and right-handed parts, i.e.

which implies that the vacuum polarization can be written as the sum $\Pi^{\mu\nu}(p)=\Pi_{L}^{\mu\nu}(p)+\Pi_{R}^{\mu\nu}(p)$, where

is the vacuum polarization for a left-(right-)handed massless fermion. We now concentrate in the calculation of $\Pi_{L(R)}^{\mu\nu}(p)$. Using Eqs. (12)-(14) and the cyclic property of the trace, the left-handed part $\Pi_{L}^{\mu\nu}$ can be written as

A similar procedure yields the right-handed part $\Pi^{\mu\nu}_{R}(p)$ by means of the replacements $L\to R$, $l^{\mu}{}_{\nu}\to r^{\mu}{}_{\nu}$, $b_{\mu}\to-b_{\mu}$, $P_{L}\to P_{R}$ in Eq. (17). In the following we restrict ourselves to the axial contributions $\Pi^{\mu\nu}_{A,L}$ and $\Pi^{\mu\nu}_{A,R}$ of the left- and right-handed terms, respectively, which are obtained by isolating the terms $P_{R}\to+\gamma^{5}/2$ and $P_{L}\to-\gamma^{5}/2$ in the corresponding expressions for the vacuum polarization. Both expressions can be summarized in the general form

with the following assignments

Clearly, the full axial contribution to the vacuum polarization is the sum of the $L$ and $R$ parts, i.e. $\Pi_{A}^{\mu\nu}=\Pi_{A,L}^{\mu\nu}+\Pi_{A,R}^{\mu\nu}$. For the sake of simplicity, we do not introduce an additional subindex $\chi$ either in the matrix $m^{\alpha}{}_{\beta}$ or in the vector $C_{\rho}$, which are to be restored at the end of the calculations according to Eq. (19). The calculation in Eq. (18) proceeds as follows: to simplify the notation we introduce the primed vectors $q_{\nu}^{\prime}=q_{\mu}m^{\mu}{}_{\nu}$ maintaining the original measure $d^{4}k$. Also we rationalize the denominators in the propagators and take the trace $\mbox{tr}(\gamma^{\rho}\gamma^{\beta}\gamma^{\sigma}\gamma^{\alpha}\gamma^{5})=-4i\epsilon^{\rho\beta\sigma\alpha}$, with $\epsilon^{0123}=+1$. We use the antisymmetry of the Levi-Civita symbol to eliminate the term $\epsilon^{\alpha\beta\rho\sigma}(k^{\prime}-C)_{\rho}(k^{\prime}-C)_{\sigma}$ in the numerator. Since we are interested in the contribution to the effective action including the product of two electromagnetic tensors without additional derivatives, we calculate the integral Eq. (18) only to first order in $p_{\alpha}$. Finally, and assuming that $m^{\mu}{}_{\nu}$ is invertible, the identity $m^{\mu}{}_{\beta}\,m^{\nu}{}_{\alpha}\,m^{\kappa}{}_{\sigma}\epsilon^{\beta\alpha\rho\sigma}=(\det m)\,(m^{-1})^{\rho}{}_{\lambda}\epsilon^{\mu\nu\lambda\kappa}$ yields the further simplification

where we have defined the integral

and we recall that ${k^{\prime}}_{\mu}={k}_{\alpha}m^{\alpha}{}_{\mu}$. Previous to regularization, the above expression is our final result for the vacuum polarization tensor in Minkowski spacetime, which can be evaluated for the left- and right-handed fermions according to Eq. (20) with the assignments in Eq. (19). The result (20) holds for arbitrary LIV terms $c^{\mu}{}_{\nu},\,d^{\mu}{}_{\nu},\,a_{\mu}$ and $b_{\mu}$ as long as these produce invertible matrices $l^{\mu}{}_{\nu}$ and $r^{\mu}{}_{\nu}$.

As a first step in the inclusion of thermal field theory methods in the description of the radiative corrections to the fermionic action (1) under the new conditions (3), here we shall consider the case of a non-zero chemical potential $\mu$ but still remain in the zero-temperature limit. The inclusion of both will be published elsewhere. To this end we adopt the finite temperature approach in the imaginary time formulation KAPUSTA , which we recover after the substitution BERNARD ; DITTRICH ; DAS

with $k_{0}\to k_{0}=i\omega_{n}+\Lambda$.

In order to have a specific system which determines the choice of the finite but undetermined parameters, together with a better physical understanding of the subsequent steps, we introduce the Hamiltonian

borrowed from condensed matter, which describes a Weyl semimetal with two linear 3D band crossings of chirality $\chi=\pm 1$ close to $\pm{\mbox{\boldmath$\tilde{b}$}}$ in momentum space and $\pm{\tilde{b}}_{0}$ in energy. Here $v_{F}$ is the isotropic Fermi velocity at each band crossing, ${\mbox{\boldmath$\sigma$}}=(\sigma_{x},\sigma_{y},\sigma_{z})$ is the triplet of spin-$1/2$ Pauli matrices, ${\mbox{\boldmath$v$}}_{\chi}$ is the tilting parameter and $p$ is the momentum. In the Weyl basis for the gamma matrices

both $2\times 2$ Hamiltonians in Eq. (23) can be embedded in the action (1) with the choices $c^{0}{}_{0}=1,c^{0}{}_{i}=d^{0}{}_{i}=d^{0}{}_{0}=0$, i.e. $\Gamma^{0}=\gamma^{0}$ together with $\Gamma^{i}$ and $M$ being determined by the parameters in (23). For simplicity, the anisotropy in the Fermi velocities is not included in (23) but enters naturally in our final result through the coefficients $c^{i}{}_{j}$ and $d^{i}{}_{j}$. The Hamiltonian (23) is realized in terms of chiral fermions whose dispersion relation behaves linearly, with band crossing points localized both in momentum and energy, as schematically indicated in the Fig. 1(a). As usual, the position of the chemical potential determines most of the transport properties of a given material. In a metal for instance, it has to be measured from the gap closing, since it represents the filling of either the conduction or the valence bands. This suggests that in WSMs, the transport properties will depend on the band filling, i.e. on the chemical potential as measured from the node of the cones. Therefore, in order to apply the finite temperature approach (22) to WSMs, the parameter $\Lambda$ in Eq. (22) has to be understood as the chemical potential measured from the band-crossing points, as shown in Fig. 1(a). To be precise, $\Lambda=\mu-E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$, where ${\mbox{\boldmath$p$}}_{\chi}$ is the location in momentum of the node with chirality $\chi$, and $E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$ the corresponding energy. Both ${\mbox{\boldmath$p$}}_{\chi}$ and $E_{\chi}({\mbox{\boldmath$p$}}_{\chi})$ will be determined later for the problem at hand. In the simplified model (3), it is clear that ${\mbox{\boldmath$p$}}_{\chi}=\chi{\mbox{\boldmath$\tilde{b}$}}$ and $E_{\chi}({\mbox{\boldmath$p$}}_{\chi})=-\chi{\tilde{b}}_{0}$.

The sum in Eq. (22) is over the Matsubara frequencies $\omega_{n}=(2n+1)\pi T$ required to produce anti-periodic boundary conditions for the fermions KAPUSTA . Next we focus in Eq. (21) and we make use of the additional relation KAPUSTA

where the contour $\Omega$ is shown in the Fig. 1(b).

We extend the vacuum polarization method of high energy physics, used in the calculation of the electromagnetic response of fermionic systems, to a large class of fermionic couplings included in the Standard Model Extension (SME) describing Lorentz symmetry violations Kostelecky0 . Emphasis is made in the CPT odd contribution to the effective action which exhibits remnants of the abelian chiral anomaly due to the appearance of the Pontryagin density $\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$. Our approach does not rest in a direct manipulation of chiral transformations which induce the chiral anomaly via the corresponding Jacobian Fujikawa . Rather, it is based in the standard perturbative expansion in powers of the electromagnetic potential of the determinant resulting from the integration of the fermions in the functional integral corresponding to the action (1) plus the specific choices (2) . This amounts to the calculation of the vacuum polarization tensor to second order in $A_{\mu}$ starting from the exact modified fermion propagators obtained from the SME. A distinguishing feature of the method is the inclusion of corrections depending upon the chemical potential $\mu$, which enlarge the range of possible phenomenological applications. This is possible through a systematic application of thermal field theory methods KAPUSTA , which is exemplified here in the case of the zero temperature limit. The fermionic systems we consider exhibit the axial anomaly and the resulting electromagnetic actions are fully described by axion electrodynamics in the form of Eq. (4). All corrections induced by the different parameters in the SME enter only in the coupling $\Theta(x)=2{\cal B}_{\lambda}x^{\lambda}$ to the Pontryagin density, as shown in our general expressions (27) and (47). This is consistent with the property that the axial anomaly is insensitive to the Lorentz invariance violating parameters of the SME LUAGA ; FIDEL ; SALVIO ; Scarpelli . Along the text we have insisted in the advantage of extending these high energy physics methods to condensed matter physics. In fact we envisage interesting applications in the realm of topological quantum matter, where a rich phenomenology in the electromagnetic response can be explored. Our general results can be applied to type-I Weyl semimetals (WSM) with arbitrary tilting and anisotropies. A simpler system results from the restriction to a WSM with arbitrary tilting but equal isotropic Fermi velocity at each node, described by the Hamiltonian (23). In this setting, our result (51) for the vector contribution ${\cal B}^{i}$ to the $\Theta$ coupling has been validated by a direct calculation of the conductivity using the Boltzmann semiclassical approach, as shown in Eq. (53). The calculation and proper regularization of the corrections for the case $|\mbox{\boldmath$\mathcal{V}$}|\geq 1$ (relevant for type-II WSM) is left unsettled for future research. Pending also is the introduction of a chiral chemical potential which is allowed by the chiral structure of the Hamiltonian describing a WSM with two band-crossings, i.e. each node can be held at different chemical potential. We find appealing also to pursue the comparison between the quantum field theory approach and the semiclassical Boltzmann formalism. In particular, it would be interesting to elucidate the role of the Berry phase in the first strategy ENPREP . Finally, the recent calculation of the one-loop Heisenberg-Euler effective action for zero chemical potential in two of the most studied minimal Lorentz-violating extensions of QED PETROV , motivates the challenge of extending the thermal field theory approach to the calculation of the CPT even contributions to the effective electromagnetic action arising from the fermionic sector of the SME considered in Eq. (2).