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# Non-Hermitian chiral anomalies in interacting systems

Sharareh Sayyad<EMAIL_ADDRESS>Max Planck Institute for the
Science of Light, Staudtstraße 2, 91058 Erlangen, Germany

(August 30, 2024)

###### Abstract

The emergence of chiral anomaly entails various fascinating phenomena such as
anomalous quantum Hall effect and chiral magnetic effect in different branches
of (non-)Hermitian physics. While in the single-particle picture, anomalous
currents merely appear due to the coupling of massless particles with
background fields, many-body interactions can also be responsible for
anomalous transport in interacting systems. In this Letter, we study anomalous
chiral currents in systems where interacting massless fermions with complex
Fermi velocities are coupled to complex gauge fields. Our results reveal that
incorporating non-Hermiticity and many-body interactions gives rise to
additional terms in anomalous relations beyond their Hermitian counterparts.
We further present that many-body corrections in the subsequent non-Hermitian
chiral magnetic field or anomalous Hall effect are nonvanishing in
nonequilibrium or inhomogeneous systems. Our results advance efforts in
understanding the anomalous transport in interacting non-Hermitian systems.

##  Introduction.

The chiral anomaly emerges due to the violation of classical chiral symmetry
by quantum fluctuations in odd spatial dimensions. This quantum anomaly has
given rise to a plethora of exotic phenomena, including anomalous decay of
neutral pion in high-energy physics Bell and Jackiw (1969); Adler (1969);
Adler and Bardeen (1969); Bilal (2008); Zinn-Justin (2019), anomaly-induced
charges in baryons in quantum chromodynamics Eto  _et al._ (2012), anomalous
transport in condensed matter physics Zyuzin  _et al._ (2012); Hosur and Qi
(2013); Landsteiner (2016); Fröhlich (2018); Araki (2020); Arouca  _et al._
(2022); Froehlich (2023), and the magnetic helicity transfer in the early
universe due to chiral asymmetry in cosmology Boyarsky  _et al._ (2012);
Brandenburg  _et al._ (2017). Aside from deepening our understanding and their
experimental realizations Huang  _et al._ (2015); Hirschberger  _et al._
(2016); Zhang  _et al._ (2016); Li  _et al._ (2016); Ong and Liang (2021),
chiral anomalies are proposed to be used in advancing quantum computing, e.g.,
as chiral qubits Kharzeev and Li (2019); Babaev  _et al._ (2023).

While chiral anomaly in condensed matter physics usually describes
noninteracting massless fermions under electromagnetic fields, generalizations
of this theory allow incorporating Weyl node-mixing terms Raines and Galitski
(2017) and treating short-range interactions between fermions Kikuchi (1992);
Rylands  _et al._ (2021); Parhizkar  _et al._ (2023). The latter unveils novel
contributions to the anomaly equations through its nonperturbative
formulation. This differs from the traditional convention of perturbative
treatment of interactions in high-energy physics to explore the chiral anomaly
Adler and Bardeen (1969). These perturbative studies reported the cancellation
of higher-order corrections upon respecting Lorentz and chiral symmetries
Giuliani  _et al._ (2021) and concluded the universality of chiral anomaly. As
Lorentz and chiral symmetries are usually broken in interacting condensed
matter systems Raines and Galitski (2017); Giuliani  _et al._ (2021); Rylands
_et al._ (2021), violating the chiral symmetry by many-body interactions in
the absence of background fields opens new directions to explore chiral
anomaly in interacting lattice models 111We note that all reported additional
terms in Ref. Kikuchi (1992); Rylands  _et al._ (2021); Parhizkar  _et al._
(2023) originated from many-body corrections in the anomaly equation can be
written as total derivatives of physical quantities..

Condensed matter systems with different constituent particles, e.g., electrons
and phonons, can be studied as closed or open quantum systems. While in the
framework of closed systems, all degrees of freedom are treated self-
consistently, the formulation of open quantum systems takes advantage of
tracing out some degrees of freedom with the expense of losing unitarity. It
has been shown that the path-integral formulations for open or closed systems
can provide an effective non-Hermitian description for these systems Kamenev
(2005); van Leeuwen  _et al._ (2005); Sieberer  _et al._ (2016); Michishita
and Peters (2020); Zirnstein  _et al._ (2021); Gomez-Leon  _et al._ (2022);
Gal  _et al._ (2023). Here non-Hermiticity originates from the dissipative
nature of open quantum systems or the imaginary parts of self-energies
accounting for interactions between different subsystems in closed quantum
systems.

Upon constructing non-Hermitian models, the underlying physics of open/closed
systems can be unraveled using methods in non-Hermitian physics and their
unique properties with no counterparts in Hermitian physics Zyuzin and Simon
(2019); Ashida  _et al._ (2020); Wang  _et al._ (2021); Bergholtz  _et al._
(2021); Okuma and Sato (2023). The emergence of defective Sayyad  _et al._
(2021); Ding  _et al._ (2022); Sayyad and Kunst (2022) and non-defective Zhang
_et al._ (2021); Sayyad (2022); Sayyad  _et al._ (2022a) degeneracies and the
occurrence of exotic boundaries modes Zhang  _et al._ (2022) exemplify the
fascinating features of non-Hermitian models. While most studies focus on
exploring noninteracting systems, investigating non-Hermitian many-body
physics and their dynamics have gained momentum in recent years Fukui and
Kawakami (1998); Buča  _et al._ (2020); Zhang and Song (2021); Nakagawa  _et
al._ (2021); Hyart and Lado (2022); Yoshida and Katsura (2023); Sayyad and
Lado (2023); Mak  _et al._ (2023). Despite these studies, the rich transport
properties of interacting non-Hermitian models are mainly unexplored.
Addressing the anomalous chiral response of interacting Dirac fermions in non-
Hermitian models is the primary goal of this work.

In this letter, we explore the chiral anomaly in $(1+1)$ and $(3+1)$
dimensions for non-Hermitian Dirac fermions with complex Fermi velocities
coupled to complex background gauge fields. By introducing a unified notation,
we bring the non-Hermitian model and its symmetrized version under the same
umbrella, enabling us to identify purely non-Hermitian contributions in
anomalous currents. We further present the physical consequences of the non-
Hermitian chiral anomaly in non-Hermitian systems and discuss plausible
platforms to realize them.

##  Non-Hermitian chiral anomaly in many-body systems.

We consider a non-Hermitian model of interacting massless fermions $\Psi$,
with complex Fermi velocities, in the presence of non-Hermitian gauge fields
$(V,W)$ in even $d$ dimensions. To facilitate later comparison of our non-
Hermitian results with previous Hermitian calculations, we employ a notation,
shown by a tilde, which unifies non-Hermitian ($\rm nh$) and its symmetrized
form, a.k.a., Hermitionized ($\rm h$), models Sayyad  _et al._ (2022b).

In this notation, the partition function ($\cal Z$), the Euclidean-space
action ($\cal S$) and the Dirac operator ${\mathscr{D}}$ for our model in the
units where $c=e=\hbar=1$ read

$\displaystyle\tilde{\cal Z}\propto\int{\cal D}\Psi{\cal
D}\overline{\Psi}e^{\tilde{\cal S}},\quad\text{with }\tilde{\cal
S}=\tilde{\cal S}_{0}+\tilde{\cal S}_{\rm int},$ (1) $\displaystyle\tilde{\cal
S}_{0}=\mathop{\mathrm{i}}\int
d^{d}x\overline{\Psi}\gamma^{\mu}\tilde{\mathscr{D}}_{\mu}\Psi,$ (2)
$\displaystyle\tilde{\cal S}_{\rm int}=\int
d^{d}x\left(-\frac{\lambda_{\mu\nu}^{2}}{2}j^{\mu}j^{\nu}-\frac{\lambda_{5,\mu\nu}^{2}}{2}j^{5,\mu}j^{5,\nu}\right),$
(3)
$\displaystyle\tilde{\not{\mathscr{D}}}=\gamma^{\mu}\tilde{\mathscr{D}}_{\mu}=\gamma^{\mu}\tilde{d}_{\mu}-\mathop{\mathrm{i}}\gamma^{\mu}\tilde{V}_{\mu}-\mathop{\mathrm{i}}\gamma^{\mu}\gamma^{5}\tilde{W}_{\mu},$
(4)

with $\tilde{d}_{\mu}=f_{\mu}^{\nu}\partial_{\nu}$. The mapping between
elements of the unified notation and their counterparts in the Hermitionized
and non-Hermitian models is introduced in Table 1. The gamma matrices
$\gamma^{\mu}$ satisfy the Clifford algebra
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$ with the Euclidean metric being
$g^{\mu\nu}=-\delta^{\mu\nu}$ and Greek indices run from 1 to $d$. The
Hermitian fifth gamma matrix reads $\gamma^{5}=-\prod_{\mu}\gamma^{\mu}$ and
$\gamma^{0}=\mathop{\mathrm{i}}\gamma^{d}$ used in obtaining the Dirac adjoint
$\overline{\Psi}=\Psi^{\dagger}\gamma^{0}$. The Fermi velocities are elements
of the rank $d$ diagonal matrix $M={\rm diag}[v_{1},\ldots,v_{d}]$ with
$v_{d}=1$ and $v_{i\neq d}$ are in general complex-valued Fermi velocities.

$\tilde{\mathcal{S}}$ | $f_{\mu}^{\nu}$ | $\tilde{A}_{\mu}$ | $\tilde{\cal F}_{2}$ | $\tilde{\cal F}_{4}$
---|---|---|---|---
$\mathcal{S}_{\rm{h}}$ | $\,\mathop{\mathrm{Re}}[M_{\mu}^{\nu}]\,$ | $\,\quad\mathop{\mathrm{Re}}[M_{\mu}^{\nu}A_{\nu}]\,$ | $\,4\pi|\mathop{\mathrm{Re}}[v_{\rm f}]|\,$ | $\,32\pi^{2}|\det[\mathop{\mathrm{Re}}[M]]|$
$\mathcal{S}_{\rm{nh}}$ | $\,M_{\mu}^{\nu}\,$ | $\,M_{\mu}^{\nu}A_{\nu}\,$ | $\,4\pi\sqrt{\det[B]}\,$ | $\,32\pi^{2}\sqrt{\det[B]}$
Table 1:  Mapping $f$, $\tilde{A}$ and $\tilde{\cal F}$ from the unified
notation into Hermitianized and non-Hermitian notation. $A$ stands for gauge
fields $V$ and $W$. $\tilde{\cal F}$ is presented for $2$ and $4$ dimensions.
The matrix $B$ for non-Hermitian models is given in Eq. (11). The elements of
the diagonal matrix $M$ are complex-valued Fermi velocities.

The interacting action $\tilde{\cal S}_{\rm int}$ consists of short-range
four-fermion interactions between currents
($j^{\mu}=\overline{\Psi}\gamma^{\mu}\Psi$) and chiral currents
($j^{5,\mu}=\overline{\Psi}\gamma^{\mu}\gamma^{5}\Psi$) with real-valued
interaction strengths
$\lambda_{\mu\nu}^{2}=\lambda_{\mu\alpha}\lambda_{\nu}^{\alpha}$ and
$\lambda_{5,\mu\nu}^{2}=\lambda_{5,\mu\alpha}\lambda_{5,\nu}^{\alpha}$,
respectively 222We emphasize that merely real-valued interaction strengths can
be handled in this work as the key Hubbard-Stratonovich transformation breaks
down for complex coefficients.. The current-current interaction with
interaction strengths $\lambda_{\mu\nu}^{2}=\lambda^{2}g_{\mu d}g_{\nu d}$
describes a density-density interaction. Similarly, the interaction between
chiral currents with $\lambda_{5,\mu\nu}^{2}=\lambda^{2}_{5}g_{\mu d}g_{\nu
d}$ in $d=4$ dimensions embed the spin-spin interaction in its spatial part
Parhizkar  _et al._ (2023).

The action $\tilde{S}$ respects $U_{A}(1)\times U_{V}(1)$ symmetry classically
Rylands  _et al._ (2021); Parhizkar  _et al._ (2023), where $U_{A(V)}(1)$
denotes the chiral (vector) symmetry. However, this classical symmetry does
not hold in the presence of quantum fluctuations resulting in the emergence of
the chiral anomaly. In the following, we present the covariant form of this
anomaly using Fujikawa’s path integral approach Fujikawa (1979, 1980, 1981);
Bertlmann (1996); Fujikawa and Suzuki (2004).

We start with introducing two auxiliary field $a$ and $s$ which brings the
interaction action $\tilde{S}$ in Eq. (1) into a free-fermion action
$\tilde{\cal S}_{a,s}$ through a Hubbard-Stratonovich transformation such that

$\displaystyle\int{\cal D}\Psi{\cal D}\overline{\Psi}e^{\tilde{\cal
S}}=\int{\cal D}\Psi{\cal D}\overline{\Psi}{\cal D}a{\cal D}s\,e^{\tilde{\cal
S}_{a,s}}\equiv\tilde{\cal Z}_{a,s},$ (5) $\displaystyle\tilde{\cal
S}_{a,s}=\int
d^{d}x\left[\mathop{\mathrm{i}}\overline{\Psi}\gamma^{\mu}\tilde{\mathscr{D}}_{a,s,\mu}\Psi+\frac{1}{2}a_{\mu}a^{\mu}+\frac{1}{2}s_{\mu}s^{\mu}\right],$
(6)
$\displaystyle\tilde{\mathscr{D}}_{a,s,\mu}=\tilde{d}_{\mu}-\mathop{\mathrm{i}}\tilde{V}_{\mu}-\mathop{\mathrm{i}}\gamma^{5}\tilde{W}_{\mu}-\mathop{\mathrm{i}}{\lambda}_{\mu\nu}a^{\nu}-\mathop{\mathrm{i}}{\lambda}_{5\mu\nu}\gamma^{5}s^{\nu}.$
(7)

Integration over $a$ and $s$ fields in the above equations reproduces
$\tilde{\cal S}$ in Eq. (1). Performing an infinitesimal chiral transformation
wth angle $\beta$ on the spinor $\Psi_{\rm
rot}=\exp[-\mathop{\mathrm{i}}\gamma^{5}\beta(x)]\Psi$, keeps the action
invariant but gives rise to an anomalous term due to a change in the Jacobian
of the path integral measure such that

$\displaystyle{\cal D}\Psi_{\rm rot}{\cal D}\overline{\Psi}_{\rm
rot}=e^{\mathop{\mathrm{i}}\int d^{d}x\beta(x)\tilde{{\cal A}}_{a,s}^{5}}{\cal
D}\Psi{\cal D}\overline{\Psi}.$ (8)

To evaluate this anomalous contribution, we express $\Psi$ and
$\overline{\Psi}$ in terms of eigenbasis of the Hermitian Laplacian operators
$\tilde{\not{\mathscr{D}}}_{a,s}\tilde{\not{\mathscr{D}}}^{\dagger}_{a,s}$ and
$\tilde{\not{\mathscr{D}}}^{\dagger}_{a,s}\tilde{\not{\mathscr{D}}}_{a,s}$
Kikuchi (1992); Rylands  _et al._ (2021) and employ the Heat-kernel method
Nakahara (1990); Bertlmann (1996); Fujikawa and Suzuki (2004) to regularize
the divergent sum in the exponent of the Jacobian; see Ref. Sayyad  _et al._
(2022b) for further details on a similar approach. Introducing
$\overline{V}_{\mu}=\tilde{V}_{\mu}+\lambda_{\mu\nu}a^{\nu}$ and
$\overline{W}_{\mu}=\tilde{W}_{\mu}+\lambda_{5\mu\nu}s^{\nu}$, $\tilde{\cal
A}_{a,s}^{5}$ in $d=2$ dimensions and up to the first order in fields casts

$\tilde{{\cal A}}_{a,s}^{5}=\frac{-\varepsilon^{\mu\nu}}{\tilde{\cal
F}_{2}}\left[\mathop{\mathrm{i}}(\tilde{F}_{\mu\nu}[\overline{V}^{\dagger}]-\tilde{F}^{\dagger}_{\mu\nu}[\overline{V}])\right].$
(9)

In $d=4$ dimensions and up to the second-order in the fields, $\tilde{\cal
A}_{a,s}^{5}$ reads

$\displaystyle\tilde{\cal A}_{a,s}^{5}=$
$\displaystyle\frac{\varepsilon^{\mu\nu\eta\zeta}}{\tilde{\cal
F}_{4}}\left[\tilde{F}_{\mu\nu}[\overline{V}^{\dagger}]\tilde{F}_{\eta\zeta}[\overline{V}^{\dagger}]+\tilde{F}^{\dagger}_{\mu\nu}[\overline{V}]\tilde{F}^{\dagger}_{\eta\zeta}[\overline{V}]\right.$
$\displaystyle\left.+\tilde{F}^{\dagger}_{\mu\nu}[\overline{W}]\tilde{F}^{\dagger}_{\eta\zeta}[\overline{W}]+\tilde{F}_{\mu\nu}[\overline{W}^{\dagger}]\tilde{F}_{\eta\zeta}[\overline{W}^{\dagger}]\right].$
(10)

Here,
$\tilde{F}_{\mu\nu}[\overline{A}]=\tilde{d}_{\mu}\overline{A}_{\nu}-\tilde{d}_{\nu}\overline{A}_{\mu}$,
and
$\tilde{F}^{\dagger}_{\mu\nu}[\overline{A}]=\tilde{d}^{\dagger}_{\mu}\overline{A}_{\nu}-\tilde{d}^{\dagger}_{\nu}\overline{A}_{\mu}$
with $\tilde{d}_{\mu}^{\dagger}=-f^{*\nu}_{\mu}\partial_{\nu}$. $\tilde{{\cal
F}}_{d}$ for Hermitionized and non-Hermitian models in $d=2,4$ dimensions are
presented in Table. 1, where it is written in terms of the determinant of a
matrix $B$ given in Euclidean space, with matrix elements

$B^{\alpha\beta}=\delta^{\mu\nu}f_{\mu}^{*\alpha}f_{\nu}^{\beta}-\frac{1}{2}[\gamma^{\mu},\gamma^{\nu}]\frac{f_{\mu}^{*\alpha}f_{\nu}^{\beta}-f_{\mu}^{\alpha}f_{\nu}^{*\beta}}{2}.$
(11)

Carrying out the same steps for an infinitesimal vector transformation
$\Psi_{\rm rot}=\exp[-\mathop{\mathrm{i}}\kappa(x)]\Psi$ with ${\cal
D}\Psi_{\rm rot}{\cal D}\bar{\Psi}_{\rm
rot}=\exp\left[{\mathop{\mathrm{i}}\int{\rm d}^{d}x\kappa(x)\tilde{\cal
A}_{a,s}}\right]{\cal D}\Psi{\cal D}\bar{\Psi}$ in $d=2$ dimensions and up to
the first order in the fields, results in

$\displaystyle\tilde{\cal A}_{a,s}=$
$\displaystyle\frac{-\varepsilon^{\mu\nu}}{\tilde{\cal
F}_{2}}\left[\mathop{\mathrm{i}}\big{(}\tilde{F}^{\dagger}_{\mu\nu}[\overline{W}]-\tilde{F}_{\mu\nu}[\overline{W}^{\dagger}]\big{)}\right],$
(12)

and in $d=4$ dimensions and up to the second order in fields, $\tilde{\cal
A}_{a,s}$ reads

$\displaystyle\tilde{\cal A}_{a,s}$
$\displaystyle=\frac{-\varepsilon^{\mu\nu\eta\zeta}}{\tilde{\cal
F}_{4}}\left[\tilde{F}_{\mu\nu}[\overline{V}^{\dagger}]\tilde{F}_{\eta\zeta}[\overline{W}^{\dagger}]+\tilde{F}^{\dagger}_{\mu\nu}[\overline{V}]\tilde{F}^{\dagger}_{\eta\zeta}[\overline{W}]\right].$
(13)

When $\lambda_{\mu\nu}$ and $\lambda_{5\mu\nu}$ are zero, $\tilde{\cal
A}_{a,s}$ and $\tilde{\cal A}_{a,s}^{5}$ reproduce the results of Ref. Sayyad
_et al._ (2022b).

Combining all results, the rotated generalized action in Eq. (6) under the
vector and chiral transformation casts

$\displaystyle\tilde{\cal S}_{a,s}^{{\rm rot}}-\tilde{\cal S}_{a,s}=$
$\displaystyle-\int{\rm
d}^{2}x\Big{[}-\beta(x)\tilde{d}_{\mu}j_{a,s}^{5,\mu}-\kappa(x)\tilde{d}_{\mu}j_{a,s}^{\mu}\Big{]}.$
(14)

Enforcing the invariance of the partition function $\tilde{\cal Z}_{a,s}$ in
Eq. (5) under the vector and chiral transformations, results in satisfying
$\tilde{\cal A}_{a,s}^{5}=-\mathop{\mathrm{i}}\tilde{d}_{\mu}j_{a,s}^{5,\mu}$
and $\tilde{\cal A}_{a,s}=-\mathop{\mathrm{i}}\tilde{d}_{\mu}j_{a,s}^{\mu}$.
To obtain the anomalous relations for the interacting model, we should shift
the auxiliary fields by their on-shell values as $a_{\mu}\rightarrow
a_{\mu}-\lambda_{\mu\alpha}j^{\alpha}$ and $s_{\mu}\rightarrow
s_{\mu}-\lambda_{5\mu\alpha}j^{5,\alpha}$ and integrate over Hubbard-
Stratonovich fields $a$ and $s$ Parhizkar  _et al._ (2023). The subsequent
anomalous equation in the Euclidean space in $d=2$ dimensions is

$\tilde{d}_{\mu}j^{5,\mu}=\frac{\varepsilon^{\mu\nu}}{\tilde{\cal
F}_{2}}\left(\tilde{F}_{\mu\nu}[\tilde{V}^{\dagger}]-\tilde{F}^{\dagger}_{\mu\nu}[\tilde{V}]-4\mathop{\mathrm{Re}}[f_{\mu}^{\eta}]\partial_{\eta}[\lambda_{\nu\alpha}^{2}j^{\alpha}]\right),$
(15)

and in $d=4$ dimensions reads

$\displaystyle\tilde{d}_{\mu}j^{5,\mu}$
$\displaystyle=\frac{\varepsilon^{\mu\nu\eta\zeta}}{\tilde{\cal
F}_{4}}\left(\tilde{F}_{\mu\nu}[\tilde{V}^{\dagger}]\tilde{F}_{\eta\zeta}[\tilde{V}^{\dagger}]+\tilde{F}^{\dagger}_{\mu\nu}[\tilde{V}]\tilde{F}^{\dagger}_{\eta\zeta}[\tilde{V}]\right.$
$\displaystyle+\tilde{F}^{\dagger}_{\mu\nu}[\tilde{W}]\tilde{F}^{\dagger}_{\eta\zeta}[\tilde{W}]+\tilde{F}_{\mu\nu}[\tilde{W}^{\dagger}]\tilde{F}_{\eta\zeta}[\tilde{W}^{\dagger}]$
$\displaystyle-4\tilde{d}_{\eta}[\lambda^{2}_{\zeta\xi}j^{\xi}]\,\tilde{F}_{\mu\nu}[\tilde{V}^{\dagger}]-4\tilde{d}_{\eta}^{\dagger}[\lambda^{2}_{\zeta\xi}j^{\xi}]\,\tilde{F}^{\dagger}_{\mu\nu}[\tilde{V}]$
$\displaystyle-4\tilde{d}_{\eta}[\lambda^{2}_{5,\zeta\xi}j^{5,\xi}]\,\tilde{F}_{\mu\nu}[\tilde{W}^{\dagger}]-4\tilde{d}_{\eta}^{\dagger}[\lambda^{2}_{5,\zeta\xi}j^{5,\xi}]\,\tilde{F}^{\dagger}_{\mu\nu}[\tilde{W}]$
$\displaystyle+8\mathop{\mathrm{Re}}[f_{\mu}^{\kappa}f_{\eta}^{\iota}]\partial_{\kappa}[\lambda^{2}_{\nu\alpha}j^{\alpha}]\partial_{\iota}[\lambda^{2}_{\zeta\alpha}j^{\alpha}]$
$\displaystyle\left.+8\mathop{\mathrm{Re}}[f_{\mu}^{\kappa}f_{\eta}^{\iota}]\partial_{\kappa}[\lambda^{2}_{5,\nu\alpha}j^{5,\alpha}]\partial_{\iota}[\lambda^{2}_{5,\zeta\alpha}j^{5,\alpha}]\right).$
(16)

When the gauge field $V$ is real, $W$ is absent and $M=\mathbbm{1}_{d\times
d}$, the above $\tilde{d}_{\mu}j^{5,\mu}$ are in agreement with the Hermitian
results Rylands  _et al._ (2021); Parhizkar  _et al._ (2023). We note that in
the absence of interactions, the above relations reproduce results of Ref.
Sayyad  _et al._ (2022b).

In the Minkowski space, the divergence of chiral currents in Eqs. (15) and
(16) read

$\displaystyle\tilde{d}_{\mu}j^{5,\mu}=$ $\displaystyle\frac{2}{\tilde{\cal
F}_{2}}\Big{(}\tilde{E}_{1}^{\dagger}+\tilde{E}_{1}$
$\displaystyle-4\mathop{\mathrm{Re}}[v_{\mu}]\delta^{\mu\eta}\partial_{\eta}[\lambda_{\nu\alpha}^{2}j^{\alpha}]\Big{)}\,\,\text{
in $d=1+1$},$ (17) $\displaystyle\tilde{d}_{\mu}j^{5,\mu}=$
$\displaystyle\frac{8}{\tilde{\cal
F}_{4}}\left(\overline{\bm{E}}^{\dagger}\cdot\overline{\bm{B}}^{\dagger}+\overline{\bm{E}}^{5\dagger}\cdot\overline{\bm{B}}^{5\dagger}\right.$
$\displaystyle\qquad\left.+\overline{\bm{E}}\cdot\overline{\bm{B}}+\overline{\bm{E}}^{5}\cdot\overline{\bm{B}}^{5}\right)\,\,\,\text{
in $d=3+1$}.$ (18)

Here, the generalized electric and magnetic fields cast

$\displaystyle\overline{E}_{\mu}$
$\displaystyle=v_{\mu}^{*}\delta^{\mu\iota}\tilde{E}_{\iota}-\big{(}\partial_{0}{\lambda}^{2}_{\mu\alpha}-v_{\mu}^{*}\delta^{\mu\iota}\partial_{\iota}{\lambda}^{2}_{0\alpha}\big{)}j^{\alpha},$
(19) $\displaystyle\overline{E}^{5}_{\mu}$
$\displaystyle=v_{\mu}^{*}\delta^{\mu\iota}\tilde{E}^{5}_{\iota}-\big{(}\partial_{0}{\lambda}^{2}_{5,\mu\alpha}-v_{\mu}^{*}\delta^{\mu\iota}\partial_{\iota}{\lambda}^{2}_{5,0\alpha}\big{)}j^{5,\alpha},$
(20) $\displaystyle\overline{B}_{\xi\zeta}=$ $\displaystyle
v_{\xi}^{*}v_{\zeta}^{*}\delta^{\xi\iota}\delta^{\zeta\kappa}\tilde{B}_{\iota\kappa}$
$\displaystyle-\frac{1}{2}\big{(}v_{\xi}^{*}\delta^{\xi\iota}\partial_{\iota}{\lambda}^{2}_{\zeta\alpha}-v_{\zeta}^{*}\delta^{\zeta\kappa}\partial_{\kappa}{\lambda}^{2}_{\xi\alpha}\big{)}j^{\alpha},$
(21) $\displaystyle\overline{B}^{5}_{\xi\zeta}=$ $\displaystyle
v_{\xi}^{*}v_{\zeta}^{*}\delta^{\xi\iota}\delta^{\zeta\kappa}\tilde{B}^{5}_{\iota\kappa}$
$\displaystyle-\frac{1}{2}\big{(}v_{\xi}^{*}\delta^{\xi\iota}\partial_{\iota}{\lambda}^{2}_{5,\zeta\alpha}-v_{\zeta}^{*}\delta^{\zeta\kappa}\partial_{\kappa}{\lambda}^{2}_{5,\xi\alpha}\big{)}j^{5,\alpha}.$
(22)

The complex electric fields are also given by
$\tilde{E}_{j}=(\exp[2\mathop{\mathrm{i}}\phi_{j}]\partial_{t}V_{j}-\partial_{j}V_{0})$,
$\tilde{E}^{5}_{j}=(\exp[2\mathop{\mathrm{i}}\phi_{j}]\partial_{t}W_{j}-\partial_{j}W_{0})$
with $i,j,k\neq t$. Similarly, the complex magnetic fields cast
$\tilde{B}^{i}=\varepsilon^{ijk}\tilde{B}_{jk}$ and
$\tilde{B}^{5,i}=\varepsilon^{ijk}\tilde{B}_{jk}^{5}$ with
$\tilde{B}_{jk}=\exp[2\mathop{\mathrm{i}}\phi_{k}]\partial_{j}V_{k}-\exp[2\mathop{\mathrm{i}}\phi_{j}]\partial_{k}V_{j}$
and
$\tilde{B}_{jk}^{5}=\exp[2\mathop{\mathrm{i}}\phi_{k}]\partial_{j}W_{k}-\exp[2\mathop{\mathrm{i}}\phi_{j}]\partial_{k}W_{j}$.
The phase $\phi_{j}$ with $j$ being a spatial index satisfies
$\exp[\mathop{\mathrm{i}}\phi_{j}]=v_{j}/|v_{j}|$.

Our main results in Eqs. (17) and (18) retain the general forms of chiral
anomaly, namely $\tilde{d}_{\mu}j^{5,\mu}\propto E$ in $(1+1)$ dimensions and
$\tilde{d}_{\mu}j^{5,\mu}\propto\bm{E}.\bm{B}$ in $(3+1)$ dimensions, in
Hermitian systems Fujikawa (1979); Bertlmann (1996); Rylands  _et al._ (2021);
Parhizkar  _et al._ (2023). We emphasize that anomalous relations carry
additional terms originating from complex Fermi velocities and complex gauge
fields not present in Hermitionized anomalous equations. Note also that the
terms proportional to interaction strengths can be written as total
derivatives of physical quantities. The prefactors of $\tilde{\cal F}_{d}$ in
non-Hermitian systems with complex Fermi velocities also differ from their
counterparts in Hermitionized models.

The complex generalized fields in Eqs. (19, 20, 21, 22) can be viewed as
complex background fields screened by interactions between currents and
densities of spinors. These screened fields are then responsible for breaking
the chiral symmetry and giving rise to the chiral anomaly. In other words,
anomalous relations in Eqs. (17) and (18) account for violating the chiral
symmetry by the gauge fields $(V,W)$ and also by the induced contributions
from interactions between different constituent of our systems.

Considering $(1+1)$-dimensional systems with real Fermi velocities and
$\lambda^{\mu\nu}=\lambda^{2}\delta^{\mu\nu}$ with $\lambda$ being a constant,
we rewrite Eq. (17) as

$\displaystyle\tilde{d}_{\mu}j^{5,\mu}=$
$\displaystyle\frac{1}{1+4\lambda^{2}/\tilde{\cal F}_{2}}\frac{2}{\tilde{\cal
F}_{2}}\left(\tilde{E}_{1}^{\dagger}+\tilde{E}_{1}\right),$ (23)

where we use the relation between the chiral and vector currents
$j^{5,\mu}=\epsilon^{\mu\nu}j_{\nu}$ to obtain the above relation. In the
absence of the axial $W$ field in $(3+1)$ dimensions, keeping
$M\in\mathbb{R}$, $\lambda^{\mu\nu}=\lambda^{2}\delta^{\mu\nu}$ and setting
$\bm{E}=E_{z}\hat{z}$ and $\bm{B}=B_{z}\hat{z}$ results in

$\displaystyle\tilde{d}_{\mu}j^{5,\mu}=$
$\displaystyle\frac{1}{1+\frac{8\lambda^{2}(\tilde{B}_{z}+\tilde{B}^{\dagger}_{z})}{\tilde{\cal
F}_{4}}}\frac{8}{\tilde{\cal
F}_{4}}\left(\tilde{E}_{z}^{\dagger}\tilde{B}_{z}^{\dagger}+\tilde{E}_{z}\tilde{B}_{z}\right),$
(24)

with the relation $\varepsilon^{12\mu\nu}j_{\nu}=j^{5,\mu}$. We note that the
current-current contributions, e.g., the last two terms in Eq. (16), do not
appear in the above relation. This is due to the translational and rotational
symmetry on the $x-y$ plane, with fields along the $z$ directions. We can
interpret our results in Eqs. (23) and (24) as renormalization of
$(\bm{E},\bm{B})$ fields by interactions. These equations coincide with the
Hermitian results in $(1+1)$ Shei (1972); Parhizkar  _et al._ (2023) and
$(3+1)$ Rylands  _et al._ (2021); Parhizkar  _et al._ (2023) dimensions upon
imposing $(E,B)$ to be real fields.

## Physical consequences in non-Hermitian Weyl semimetals.

To obtain physical phenomena stemming from the non-Hermitian chiral anomaly in
interacting systems, we proceed with presenting the Chern-Simons description
of our $(3+1)$ dimensional model in the absence of the axial field $W$ and
without interactions between chiral currents. In this case, the change of
action $\tilde{S}$ under an infinitesimal rotation $\beta$ reads

$\displaystyle\tilde{\cal S}_{\rm rot}-\tilde{\cal S}=\int
dtd^{3}x\beta(t,x)[\tilde{d}_{\mu}j^{5,\mu}-\tilde{A}_{5}],$ (25)

where the first term in the r.h.s. of the above relation is due to the
classical shift of the action and the anomalous second term ($\tilde{S}^{5}$)
accounts for the change of measure and should satisfy
$\tilde{A}_{5}=\tilde{d}_{\mu}j^{5,\mu}$ as we discussed in the previous
section. The anomaly-induced action after performing integration by parts and
neglecting a total derivative term can be rewritten as the Chern-Simons action
in the Minkowski spacetime as

$\displaystyle\tilde{S}^{5}[\beta]=$ $\displaystyle-\int
dtd^{3}x\frac{8\varepsilon^{\mu\nu\eta\zeta}}{\tilde{F}_{4}}\left[\tilde{d}_{\eta}\tilde{j}_{\zeta}\tilde{d}_{\mu}\beta\tilde{V}^{\dagger}_{\nu}+\tilde{d}^{\dagger}_{\eta}\tilde{j}_{\zeta}\tilde{d}^{\dagger}_{\mu}\beta\tilde{V}_{\nu}\right]$
$\displaystyle+\int
dtd^{3}x\frac{8\varepsilon^{\mu\nu\eta\zeta}}{\tilde{F}_{4}}\mathop{\mathrm{Re}}[f_{\mu}^{\kappa}f_{\eta}^{\iota}]\partial_{\kappa}\beta\tilde{j}_{\nu}\partial_{\iota}\tilde{j}_{\zeta}$
$\displaystyle+\int
dtd^{3}x\frac{4\varepsilon^{\mu\nu\eta\zeta}}{\tilde{F}_{4}}\tilde{d}_{\mu}\beta\tilde{V}^{\dagger}_{\nu}\tilde{d}_{\eta}\tilde{V}^{\dagger}_{\zeta}$
$\displaystyle+\int
dtd^{3}x\frac{4\varepsilon^{\mu\nu\eta\zeta}}{\tilde{F}_{4}}\tilde{d}^{\dagger}_{\mu}\beta\tilde{V}_{\nu}\tilde{d}^{\dagger}_{\eta}\tilde{V}_{\zeta},$
(26)

where $\tilde{j}_{\mu}=\lambda^{2}_{\mu\alpha}j^{\alpha}$. The associated
currents for the above action are then evaluated by summing the functional
derivatives of $\tilde{S}^{5}$ with respect to $V$ and $V^{\dagger}$. These
currents are given by

$\displaystyle M_{\nu}^{\alpha}j^{\nu}=$
$\displaystyle\frac{8\varepsilon^{\mu\nu\eta\zeta}\partial_{\delta}\beta}{\tilde{F}_{4}}\mathop{\mathrm{Re}}\left[M_{\mu}^{\delta}M_{\eta}^{\iota}M_{\zeta}^{\rho}M_{\nu}^{*\alpha}\partial_{\iota}V_{\rho}\right]$
$\displaystyle-\frac{16\varepsilon^{\mu\nu\eta\zeta}\partial_{\delta}\beta}{\tilde{F}_{4}}\mathop{\mathrm{Re}}\left[M_{\eta}^{\iota}M_{\mu}^{\delta}M_{\nu}^{*\alpha}\right]\partial_{\iota}\tilde{j}_{\zeta}.$
(27)

Imposing $V$ field to be real and $M=\mathbbm{1}_{4\times 4}$ in Eq. (27)
recovers the Hermitian chiral magnetic effect when $\delta=0$ and the
Hermitian anomalous Hall effect with $\delta$ being a spatial index Zyuzin
_et al._ (2012). We note that the interaction-induced terms, second line in
Eq. (27), are merely present in nonequilibrium systems
($\partial_{\iota}j_{\nu}\neq 0$ with $\nu\neq\iota=0$) or in inhomogeneous
systems with $\partial_{\iota}j_{0}\neq 0$ with $\iota\neq 0$. Hence, in
nonequilibrium or inhomogeneous systems, Eq. (27) describes the interacting
non-Hermitian chiral magnetic effect when the temporal component of $\beta$
($\delta=0$) is nonzero and when a spatial component of $\beta$ with
$\delta\in\\{1,2,3\\}$ is nonvanishing, Eq. (27) results in the interacting
non-Hermitian anomalous Hall effect.

$\partial_{0}\beta$ and $\partial_{\delta}\beta$ are related to the complex-
valued energy and spatial separation of the Weyl nodes in non-Hermitian Weyl
semimetals Sayyad  _et al._ (2022b). We emphasize that these Weyl points
should be non-defective degeneracies. This is because the coalescence of
eigenvectors in defective degeneracies prevents introducing a well-defined
basis to express the Laplacian operators for Fujikawa’s path integral method.
As all non-defective degeneracies are symmetry-protected Sayyad (2022),
respecting their underlying symmetry maintains nodal intersections and offers
platforms to realize non-Hermitian chiral anomalies. One approach for
constructing non-Hermitian Weyl semimetals is the effective non-Hermitian
description of open quantum systems. Starting from a Hermitian model for Weyl
semimetals and allowing the coupling of this system with external
environments, e.g., by a Boltzmann factor (see Ref. Nuske  _et al._ (2020) for
an example), results in an effective Hamiltonian whose imaginary part of the
spectrum is nonpositive.

## Conclusion and outlook.

In conclusion, we have presented how non-Hermitian chiral anomalies for
massless fermions with complex Fermi velocities coupled to complex background
fields are modified in the presence of four-particle interactions. We have
shown that anomalous relations cast the same form as the chiral anomaly in
Hermitian (non)interacting systems. Despite the similar structure, the
embedded terms in the presented non-Hermitian chiral anomaly exceed those in
the Hermitionized model. Our results show that nonperturbative interaction
corrections to chiral anomalies are nonvanishing in nonequilibrium or
inhomogeneous systems. This can be seen in the presented non-Hermitian chiral
magnetic effect and non-Hermitian anomalous Hall effect.

An experimental study on light-driven anomalous Hall effect in graphene
reported that the Hall conductance is unquantized, despite the theoretical
expectation of quantized conductivity in Hermitian systems McIver  _et al._
(2020). Theoretical efforts to explain this observation revealed the essential
roles played by out-of-equilibrium and dissipative properties of the
experimental system Sato  _et al._ (2019); Nuske  _et al._ (2020). As these
two factors are well incorporated within our theory and measuring additional,
terms proportional to currents, in Eqs. (15, 16) are experimentally feasible,
we expect to find signatures of our findings in light-driven Weyl semimetals
with a similar experimental setup as in Ref. McIver  _et al._ (2020). In
addition, combining circuits to explore real-time chiral dynamics Kharzeev and
Kikuchi (2020) with algorithms to simulate open quantum systems Kamakari  _et
al._ (2022) may also pave the way to realize our findings digitally.

Finally, extending these results to understand the parity-like anomaly Burkov
(2018); Matsushita  _et al._ (2020), the axial-torsional anomaly Ferreiros
_et al._ (2019) and axial–gravitational anomaly Gooth  _et al._ (2017);
Chernodub  _et al._ (2022) in non-Hermitian interacting systems is also of
interest which we leave for future studies.

## Acknowledgement.

I thank A. Parhizkar for the clarifications on reproducing some of the results
in Ref. Parhizkar  _et al._ (2023). Fruitful discussions with K. Landsteiner
and helpful comments on the manuscript from A. G. Grushin and P. Surwoka are
gratefully acknowledged.

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