aainstitutetext: Kadanoff Center for Theoretical Physics, University of
Chicago, 5640 South Ellis Ave, Chicago, IL 60637, USAbbinstitutetext:
Department of Physics and Astronomy and Simons Center for Geometry and
Physics, Stony Brook University, Stony Brook, NY 11794, USA

# Chiral Anomaly in Euler Fluid and Beltrami Flow

P.B. Wiegmann b and A.G. Abanov<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We show that the chiral anomaly of quantum field theories with Dirac fermions
subject to an axial background field is an inherent property of kinematics of
a perfect classical fluid. Celebrated Beltrami flows (stationary solutions of
Euler equations with extensive helicity) exhibit the chiral anomaly equivalent
to that known for Dirac fermions. A prominent effect of the chiral anomaly is
the transport electric current at equilibrium. We show that it is also a
property of Beltrami flows.

##  1 Introduction

Interacting quantum fermionic systems can behave as perfect fluids. Prominent
examples are superconductors, the superfluid ${}^{3}\hbox{He}$, the fractional
quantum Hall effect, and one-dimensional electronic systems, e.g., Luttinger
liquid. Recently a possibility that a quark-gluon plasma arising in collisions
of heavy ions may turn into a fluid came to light. The fundamental particles
in quark-gluon plasma, ${}^{3}\hbox{He}$ and some other examples are Dirac
fermions. A characteristic feature of quantum field theories (QFT) with Dirac
fermions is quantum anomalies that are mainly insensitive to interactions.
Therefore, if the Dirac fermions turn into a fluid, anomalies must be an
inherent part of hydrodynamics as well. This argument set off a search for
relativistic hydrodynamics models consistent with the anomalies (see, Refs.
son2009hydrodynamics ; haehl2014effective , and references therein). In a
recent paper, abanov2022axial , the effect of the axial-current anomaly had
been identified in the foundational description of fluid dynamics – the
conventional classical Euler equations (relativistic and non-relativistic
alike).

If the Dirac fermions retain chiral imbalance, a fermionic system exhibits a
physically significant anomaly-driven effect: an electric current runs across
the system at the thermodynamic equilibrium. This work shows that this and
other anomaly-based properties are readily apparent in Beltrami flows of the
perfect classical fluid. We recall that the Beltrami flows are stationary
flows with extensive helicity (i.e., proportional to the volume of the
system). Superfluid ${}^{3}\hbox{He-A}$ may serve as a platform for the
experimental realization of the effects discussed in this work. There is an
experimental evidence of both the axial current anomaly bevan1997momentum ,
and of realization of the Beltrami flow salomaa1987quantized .

#### Chiral anomaly in Dirac fermions

We briefly review the chiral anomaly in the fermionic quantum field theories
before turning to hydrodynamics. Dirac fermions, whose Hamiltonian density is

$\displaystyle H=\psi_{\scriptscriptstyle
L}^{\dagger}\bm{\sigma}(i\bm{\nabla}+\bm{A})\psi_{\scriptscriptstyle
L}-\psi_{\scriptscriptstyle
R}^{\dagger}\bm{\sigma}(i\bm{\nabla}+\bm{A})\psi_{\scriptscriptstyle R}$ (1)

in a time-independent magnetic field $\bm{B}=\bm{\nabla}\\!\times\\!\bm{A}$
possess two conserved charges. They are the electric charge $Q$ and the axial
charge referred to as the chirality $Q_{\scriptscriptstyle\rm A}$

$Q=\int\rho d\bm{x}\bm{=}N_{\scriptscriptstyle L}+N_{\scriptscriptstyle
R},\,\quad Q_{\scriptscriptstyle\rm A}=\int\rho_{\scriptscriptstyle\rm
A}d\bm{x}=N_{\scriptscriptstyle L}-N_{\scriptscriptstyle R}\,.$ (2)

The densities are the sum $\rho=\psi^{\dagger}_{\scriptscriptstyle
L}\psi_{\scriptscriptstyle L}+\psi^{\dagger}_{\scriptscriptstyle
R}\psi_{\scriptscriptstyle R}$ and the difference $\rho_{\scriptscriptstyle\rm
A}=\psi^{\dagger}_{\scriptscriptstyle L}\psi_{\scriptscriptstyle
L}-\psi^{\dagger}_{\scriptscriptstyle R}\psi_{\scriptscriptstyle R}$ of the
left and the right occupation numbers of chiral (Weyl) components of the Dirac
multiplet $(\psi_{\scriptscriptstyle L},\,\psi_{\scriptscriptstyle R})$. They
are temporal components of the 4-vector current $j=(j^{0},\bm{j})$ and 4-axial
current $j_{\scriptscriptstyle\rm A}=(j_{\scriptscriptstyle\rm
A}^{0},{\bm{j}}_{\scriptscriptstyle\rm A})$.

As a system with two conserved charges, Dirac fermions could be brought in
contact with a reservoir which maintains the chemical potential $\mu$ coupled
to the particle density $\rho$ and the chiral chemical potential
$\mu^{\scriptscriptstyle\rm A}$ coupled to the chirality density
$\rho_{\scriptscriptstyle\rm A}$. Such a system may reside in thermodynamic
equilibrium with the reservoir. Then microstates are described by a grand
canonical ensemble with the Gibbs probability $e^{-\mathscr{H}/T}$ with

$\displaystyle\mathscr{H}=H-\\!\int(\mu\rho+\mu^{\scriptscriptstyle\rm
A}\rho_{\scriptscriptstyle\rm A})\,d\bm{x}\,,$ (3)

where $H$ is the energy of the isolated system. The chemical potential $\mu$
may be treated as a temporal component of the external electromagnetic
4-vector potential $A=(\mu,\bm{A})$ and the chiral chemical potential
$\mu^{\scriptscriptstyle\rm A}$ as a temporal component of the external axial
4-vector potential $A^{\scriptscriptstyle\rm A}=(\mu^{\scriptscriptstyle\rm
A},\bm{A}^{\scriptscriptstyle\rm A})$ commonly considered in the Dirac QFT. In
the main part of the paper we focus on the case $\bm{A}^{\scriptscriptstyle\rm
A}=0$. The general case is briefly considered in Sec. 9. We consider a case of
zero temperature, where the equilibrium state is the ground state of
$\mathscr{H}$. There $\mu$ and $\mu^{\scriptscriptstyle\rm A}$ determine the
charges $Q$ and $Q_{\scriptscriptstyle\rm A}$.

In the familiar case, when the chemical potential is coupled to the particle
number, the equilibrium state assumes no flow. The situation is fundamentally
different when the reservoir maintains the chiral chemical potential
$\mu^{\scriptscriptstyle\rm A}$. In this case, the equilibrium state carries
an electric transport current proportional to and directed along the magnetic
field

$\displaystyle\bm{j}_{\rm trans}=k\mu^{\scriptscriptstyle\rm A}\bm{B}\,.$ (4)

This result had a tangled history and met interpretational difficulties.
Formally one could extract it from early works on anomalies
bardeen1969anomalous ; bardeen1984consistent  where fermions are considered in
the background of the axial vector potential. Physics applications started
perhaps from the work of Vilenkin vilenkin1980equilibrium . Recently (4) had
been explored in heavy-ion collisions kharzeev2008effects  among other fields.
A clear interpretation of Eq. (4) as a ballistic transport at equilibrium is
presented in Ref. alekseev1998universality  (see, also frohlich2000new . In
one spatial dimension the analog of Eq. (4) is the Landauer formula
$\bm{j}_{\rm trans}=k\mu^{\scriptscriptstyle\rm A}$ landauer1970electrical ;
buttiker1986four  for the experimentally observed Landauer-Sharvin effect of
the resistance of ballistic transport. There $\mu^{\scriptscriptstyle\rm A}$
is a voltage drop across the bulk and the constant $k$ is the conductance
quantized in fundamental units $\text{}e^{2}/h$. The effect of the transport
current at equilibrium also discussed in connection with non-centrosymmetric
semiconductors and cosmology. For some references and applications of the
effects of the chiral reservoir in various domains of physics, see Refs.
chernodub2021thermal ; rogachevskiilaminar ; Avdoshkin2016Onconsistency .

The origin of this result is the axial-current anomaly obtained by Adler
adler1969axial  and Bell and Jackiw bell1969pcac  in 1969 (see also Ref.
treiman2014current ). That is, under a physics requirement that the vector
current is conserved in an isolated system, the divergence of the axial
current is necessarily given by the formula

$\displaystyle\partial\\!\cdot\\!j_{\scriptscriptstyle\rm
A}=k\bm{E}\\!\cdot\\!\bm{B},\quad k=2\,.$ (5)

The coefficient $k$ is the value of the triangle diagram with one axial and
two vector vertices. In units of the magnetic flux quantum $\Phi_{0}=hc/e$,
the coefficient $k$ is equal $2$, the number of Weyl components in the Dirac
multiplet. Then, if the system is coupled to reservoirs, the same triangular
diagram which entered (5), determines that a gradient of the chiral chemical
potential causes the vector current to diverge

$\displaystyle{\partial\\!\cdot\\!j=k\bm{E}^{\scriptscriptstyle\rm
A}\\!\cdot\\!\bm{B}\,,\quad\bm{E}^{\scriptscriptstyle\rm
A}:=\bm{\nabla}\mu^{\scriptscriptstyle\rm A}\,.}$ (6)

Eq. (4) is a particular stationary solution of the chiral anomaly equation
(6). We comment that Eq. (4) is valid for $\mu^{\scriptscriptstyle\rm A}={\rm
const}$ even though the r.h.s. of (6) vanishes.111Throughout the paper we use
a dot to denote the contraction of adjacent indices, i.e., $\partial\cdot\\!j$
denotes $\partial_{\alpha}j^{\alpha}=\dot{\rho}+\bm{\nabla}\\!\cdot\bm{j}$.

### Helicity and Beltrami flow of a perfect fluid

Now we turn to the electrically charged barotropic perfect (Euler’s) fluid
governed by the Hamiltonian

$\displaystyle
H=\int\left(\tfrac{1}{2}m\rho\bm{v}^{2}+\varepsilon[\rho]\right)d\bm{x}\,.$
(7)

Here $\rho,\bm{v}$ are the fluid density and velocity, $m$ is the mass of
fluid particles and the energy density $\varepsilon$ is a function of $\rho$.

Like Dirac fermions, Euler flows in three dimensions possess two conserved
charges. One is the particle number $Q=\int\rho d\bm{x}$. The second conserved
charge is the fluid helicity

$\displaystyle\mathcal{H}=h^{-2}\int
m\bm{v}\\!\cdot\\!\bm{\omega}\,d\bm{x}\,.$ (8)

Here, $\bm{\omega}=\bm{\nabla}\\!\times\\!m\bm{v}$ is the fluid vorticity, and
$h$ is a normalization constant of the dimension of $\text{\it
energy}\times\text{\it time}$ which makes helicity dimensionless. The root of
the helicity conservation can be traced to its topological nature: helicity as
a linkage of vortex lines moffatt1969degree . Another related root is the
degeneracy of the Poisson structure of the Euler hydrodynamics. The latter
gives helicity the meaning of the Casimir invariant (see, e.g.,
zakharov1997hamiltonian ). It is conserved regardless of the choice of the
Hamiltonian.

For the reasons described above, the fluid could be coupled with two
reservoirs as in (3). One maintains the particle density (the electric
charge), the second maintains the helicity

$\displaystyle\mathscr{H}=\int\left[\tfrac{1}{2}m\rho\bm{v}^{2}+\varepsilon[\rho]-\mu\rho-\mu^{\scriptscriptstyle\rm
A}m\bm{v}\\!\cdot\\!\bm{\omega}\right]d\bm{x}\,.$ (9)

In the next section we explain how to add the electromagnetic field to the
chiral coupling. Then we will show that the formulas for anomalies (5,6,4)
hold to the Euler fluid. Furthermore, if the normalization constant $h$ in (8)
is identified with the Planck constant, the formulas become identical. With
this in mind, we will use the units of $h$ and $\Phi_{0}$, setting
$h=\Phi_{0}=1$ in (9) and below.

We focus on the electric transport current present in the equilibrium state
(4). From the fluid perspectives the equilibrium state is a stationary (or
steady) flow. We will show that the chiral reservoir yields a stationary flow
broadly known as a Beltrami flow arnold2008topological . We recall that the
(generalized) Beltrami flow is defined by the condition that velocity and
vorticity are collinear at every point

${\rm curl}\,\bm{v}=\kappa\bm{v}\,.$ (10)

Beltrami flows carry an extensive helicity, which, as we will show, itself is
determined by $\mu^{\scriptscriptstyle\rm A}$. Then we show that the Beltrami
flow conducts an electric current given by the formulas identical to the
formulas (4,6) of the fermionic QFT.

### Axial current anomaly in fluid dynamics and chiral coupling

In magnetic field the definition of helicity amounts to replacing the momentum
$m\bm{v}$ in (8) by the canonical momentum

$m\bm{v}\to\bm{\pi}:=m\bm{v}+\bm{A}\,,$ (11)

where $\bm{A}$ is the vector potential so that
$\bm{B}=\bm{\nabla}\\!\times\\!\bm{A}$. Under this definition the helicity

$\mathcal{H}=\int\bm{\pi}\cdot\bm{\nabla}\\!\times\\!\bm{\pi}\,d\bm{x}$ (12)

remains conserved bekenstein1987helicity ; khesin1989invariants  (see also the
section VI.2.C of arnold2008topological ). However, the helicity $\mathcal{H}$
is no longer an additive quantity, as its density
$\bm{\pi}\cdot\bm{\nabla}\\!\times\\!\bm{\pi}$ is not a local Eulerian field.
This creates the difficulty of coupling helicity to the reservoir. The problem
had been addressed in abanov2022axial . There we argued that in magnetic field
the analog of the axial charge is the fluid chirality
$Q_{\scriptscriptstyle\rm A}=\int\rho_{\scriptscriptstyle\rm A}d\bm{x}$, not
the helicity. The chirality density $\rho_{\scriptscriptstyle\rm A}$ is a
local Eulerian field

$\displaystyle\rho_{\scriptscriptstyle\rm
A}=m\bm{v}\\!\cdot\\!\left(\bm{\omega}+2\bm{B}\right)\,,$ (13)

and, therefore is an additive quantity which at $\bm{B}=0$ is equal to the
helicity density.222Heuristically, one could arrive at (13) in the following
way. In a local frame rotating with a fluid parcel, the magnetic field
modifies the angular velocity $\bm{\omega}/(2m)$ of a charged parcel by the
Larmor frequency $\bm{B}/m$. This effectively changes vorticity as
$\bm{\omega}\to\bm{\omega}+2\bm{B}$. It is not difficult to find the relation
between chirality and helicity. Their difference depends only on external
fields and is equal to the twice magnetic helicity

$\displaystyle Q_{\scriptscriptstyle\rm
A}=\mathcal{H}-2\int\bm{A}\\!\cdot\\!\bm{B}\,d\bm{x}\,.$ (14)

Based on this arguments we conclude that the chirality density
$\rho_{\scriptscriptstyle\rm A}$ given by (13), not the helicity density (12)
which enters to the augmented energy (3). However, there is a caveat. There
are certain space-time configurations of the electromagnetic field which may
change the magnetic helicity, and therefore change the fluid chirality (14).
It follows from (14) that

$\tfrac{d}{dt}Q_{\scriptscriptstyle\rm
A}=2\int\bm{E}\\!\cdot\\!\bm{B}\,d\bm{x}\,.$ (15)

The local form of this equation

$\displaystyle\partial_{t}{\rho_{\scriptscriptstyle\rm
A}}+\bm{\nabla}\\!\cdot\bm{j}_{\scriptscriptstyle\rm
A}=2{\bm{E}}\\!\cdot\\!\bm{B}\,$ (16)

is the axial-current anomaly (5) [see (43) below for the explicit formula for
the axial flux $\bm{j_{\scriptscriptstyle\rm A}}$]. Like in the case of
fermions the chiral reservoir is coupled to a charge whose conservation is
subject to the anomaly. This is the source of the effects we study.

Perhaps the most interesting and relevant case for applications occurs at
$\mu,\mu^{\scriptscriptstyle\rm A}={\rm const}$. In this case
$\mathscr{H}=H-\mu Q-\mu^{\scriptscriptstyle\rm A}Q_{\scriptscriptstyle\rm A}$
and one may think about the reservoir as contacts at the system boundaries.
Still, we assume that the chemical potentials may vary in space as that
clarifies computational and structural aspects of hydrodynamics and the
anomalies. Furthermore, anomaly-based properties are insensitive to the space-
time structure, Galilean or Minkowski. We choose to describe a non-
relativistic fluid. In the end of the text we list the extension of our
results for a general axial potential $A^{\scriptscriptstyle\rm
A}=(\mu^{\scriptscriptstyle\rm A},\,A^{\scriptscriptstyle\rm A})$ and present
the formulas in a covariant form readily applicable to relativistic fluid
dynamics. Similarities between anomalies in QFT and the fluid dynamics also
hold in neutral rotating systems. In this case, the angular velocity is
identified with the Larmor frequency $\bm{B}/m$ and the electric field with
the gradient of an electrochemical potential or external forces.

From the perspective of fluid dynamics, fluid motion subject to the chiral
reservoir is an interesting problem on its own. To the best of our knowledge,
the hydrodynamics in such a setting has not been addressed so far. We develop
it below within the Hamiltonian description of the fluid.

##  2 The Hamiltonian and the Poisson algebra

First, we express the fluid chirality density (13) in terms of the canonical
momentum $\bm{\pi}$ (11)

$\displaystyle\rho_{\scriptscriptstyle\rm A}=$
$\displaystyle\left(\bm{\pi}-\bm{A})\\!\cdot\\!(\bm{\nabla}\times\bm{\pi}+\bm{B}\right)\,$
(17)

and use it in the augmented Hamiltonian (3)

$\displaystyle\mathscr{H}[\rho,\bm{\pi}]=$
$\displaystyle\int\left[\tfrac{\rho}{2m}(\bm{\pi}\\!-\\!\bm{A})^{2}\\!+\epsilon[\rho]-\mu\rho-\mu^{\scriptscriptstyle\rm
A}\rho_{\scriptscriptstyle\rm A}\right]d\bm{x}\,.$ (18)

The canonical momentum and the fluid density generate the Lie-Poisson algebra

$\displaystyle\\{\rho(\bm{x}),\,\rho(\bm{x}^{\prime})\\}=0\,,$
$\displaystyle\\{\rho(\bm{x}),\,\bm{\pi}(\bm{x}^{\prime})\\}=\bm{\nabla}_{\bm{x}^{\prime}}\delta(\bm{x}\\!-\\!\bm{x}^{\prime})\,,$
(19)
$\displaystyle\\{\bm{\pi}(\bm{x})\times\bm{\pi}(\bm{x}^{\prime})\\}=(2/\rho)(\bm{\nabla}\\!\times\\!\bm{\pi})\,\delta(\bm{x}\\!-\\!\bm{x}^{\prime})\,.$

Throughout the paper we routinely drop the argument $\bm{x}$ when it does not
compromise the meaning. The brackets (19) yield the bracket for the chirality
density

$\displaystyle\\{\rho_{\scriptscriptstyle\rm
A}(\bm{x}),\rho(\bm{x}^{\prime})\\}=(\bm{\nabla}\\!\times\\!\bm{\pi}+\bm{B})\cdot\bm{\nabla}_{\bm{x}^{\prime}}\delta(\bm{x}\\!-\\!\bm{x}^{\prime})\,.$
(20)

For completeness we write down the remaining brackets also following from (19)

$\displaystyle\\{\bm{\pi}(\bm{x}),\rho_{\scriptscriptstyle\rm
A}(\bm{x}^{\prime})\\}\\!=-\rho^{-1}\\!(\bm{\nabla}\\!\times\\!\bm{\pi})\times\\!(m\bm{v}\\!\times\\!\bm{\nabla}_{\bm{x}^{\prime}})\delta(\bm{x}\\!-\\!\bm{x}^{\prime})\,,$
(21) $\displaystyle\\{\rho_{\scriptscriptstyle\rm
A}(\bm{x}),\rho_{\scriptscriptstyle\rm
A}(\bm{x}^{\prime})\\}=-\Big{[}\bm{\nabla}\\!\times\\!\Big{(}{\rho}^{-1}(\bm{\nabla}\\!\times\\!\bm{\pi})\\!\cdot\\!m\bm{v}\Big{)}m\bm{v}\Big{]}\\!\cdot\\!\bm{\nabla}_{\bm{x}^{\prime}}\delta(\bm{x}\\!-\\!\bm{x}^{\prime})\,.$
(22)

The form of the brackets (20,21,22) reflects that the fluid chirality
$Q_{\scriptscriptstyle\rm A}=\int\rho_{\scriptscriptstyle\rm A}d\bm{x}$ is the
Casimir function of the Poisson algebra, i.e., Poisson-commute with all
hydrodynamic fields.

#### Magnetic field as a central extension

We notice that the magnetic field prevents the r.h.s. of (20) from being
expressed solely in terms of the canonical momentum $\bm{\pi}$. The magnetic
field acts as a central extension of the Poisson algebra. In Sec.7 we show
that the central extension in the Poisson structure of hydrodynamics is
essentially equivalent to the formulation of the effect of the anomaly in
terms of the divergence of the currents (5,6). Here we comment on how the
axial anomaly in the form of (15) follows from the bracket (20). First we
write the Hamiltonian equation $\dot{Q}_{\scriptscriptstyle\rm
A}=\\{Q_{\scriptscriptstyle\rm A},\mathscr{H}\\}$ and notice that the only
contribution comes from the term $-\int\mu\rho d\bm{x}$ in (18). Other terms
are divergence and are eliminated by the integration. Then with the help of
(20) we compute

$\displaystyle\dot{Q}_{\scriptscriptstyle\rm A}=-\\{Q_{\scriptscriptstyle\rm
A},\,\int\mu\rho d\bm{x}\\}=2\int\bm{\nabla}\mu\\!\cdot\\!\bm{B}\,d\bm{x}\,.$
(23)

This yields (15) after identifying $\bm{\nabla}\mu$ with the electric field
$\bm{E}$.

We remark that if magnetic field possess monopoles with a charge $q_{\rm
mon}=-\int\bm{\nabla}\\!\cdot\\!\bm{B}\,d\bm{x}$, then (23) yields the central
extension of the Poisson algebra of the charges

$\displaystyle\\{Q,\,Q_{\scriptscriptstyle\rm A}\\}=2q_{\rm mon}\,.$ (24)

The same formula holds for Dirac fermions (see, e.g., Refs. Grossman1983Dyon ;
treiman2014current ). It follows from the current algebra of Faddeev
faddeev1984operator  (see, also alekseev1998universality ), where the magnetic
field also yields the central extension of the algebra of fermionic currents.

### Conjugate fields

We will use the fields conjugate to the density, the momentum, and to the
external electromagnetic vector potential. The first one is defined as

$\displaystyle\pi_{0}:=-\delta\mathscr{H}/\delta\rho|_{\bm{\pi}}=\Phi+\mu\,,$
(25)

where we denoted the Bernoulli function $\Phi$ and the enthalpy $w$ as

$\displaystyle\Phi=-\tfrac{1}{2}m\bm{v}^{2}-w[\rho]\,,\quad
w=d\varepsilon/d\rho\,.$ (26)

The field conjugate to the momentum is the flow vector field

$\displaystyle\bm{\mathcal{J}}={\delta\mathscr{H}}/{\delta\bm{\pi}}|_{\rho}\,,$
(27)

whose meaning we will discuss shortly.

Our primary interest is the electric (or vector) current which we define as a
linear response to the electromagnetic vector potential $\bm{A}$ (at a fixed
fluid momentum $\bm{\pi}$) evaluated at zero electromagnetic field333We
comment that the electric current in alekseev1998universality  is defined as
$\bm{\mathcal{J}}$.

$\displaystyle\bm{j}$
$\displaystyle=-(\delta\mathscr{H}/\delta\bm{A}|_{\bm{\pi}})|_{\bm{A}=0}\,.$
(28)

We further clarify the definition of the electric current in Sec. 6 where we
discuss the Lorentz force.

We remark that in an isolated Galilean invariant fluid, the fluid momentum per
particle, the electric current and the vector flow field $\bm{v}$ are
proportional to the fluid’s velocity $\bm{v}$. The chiral reservoir makes
these three vector fields (11,28,27) different.

##  3 Flow vector field and the continuity equation

Frozen-in fields are fluid substances whose Lie derivatives along a flow
vector field vanish. We denote the flow vector field as $\bm{\mathcal{U}}$.
The physical meaning of $\bm{\mathcal{U}}$ is the true fluid velocity, which
is not necessarily equals to $\bm{v}$ serving as a notation for the fluid
momentum $\bm{p}=m\bm{v}$. Similarly, the advection flux
$\bm{\mathcal{J}}=\rho\,\bm{\mathcal{U}}$, defined by Eq.(27) is the true mass
flux. We demonstrate it obtaining the continuity equation as the Hamiltonian
equation

$\dot{\rho}(\bm{x})=\\{\rho(\bm{x}),\,\mathscr{H}\\}=\int\\{\rho(\bm{x}),\bm{\pi}\\}\frac{\delta\mathscr{H}}{\delta\bm{\pi}}\,,$
(29)

where the integration goes over the omitted argument. Then with the help of
Poisson algebra and (27) we obtain that the 4-advection flux
$\mathcal{J}=(\rho,\bm{\mathcal{J}})$ is divergence-free

$\displaystyle\partial\cdot\\!\mathcal{J}\equiv\dot{\rho}+\bm{\nabla}\\!\cdot\bm{\mathcal{\\!J}}=0\,.$
(30)

We see that the flow along the vector field $\bm{\mathcal{U}}$ advects the
fluid mass. Later we will see that vorticity is also advected along
$\bm{\mathcal{U}}$.

The explicit form of the flow field elementary follows from Eqs. (3,7,11) and
its definition (27)

$\displaystyle\bm{\mathcal{J}}$
$\displaystyle=\rho\bm{v}-2\mu^{\scriptscriptstyle\rm
A}(\bm{\omega}+\bm{B})-{\bm{E}}_{\scriptscriptstyle\rm A}\times m\bm{v}$ (31)
$\displaystyle=\rho\bm{v}-\mu^{\scriptscriptstyle\rm
A}(\bm{\omega}+2\bm{B})-\bm{\nabla}\\!\times\\!(\mu^{\scriptscriptstyle\rm
A}m\bm{v})\,,$ (32)

where we denote $\bm{E}^{\scriptscriptstyle\rm
A}=\bm{\nabla}\mu^{\scriptscriptstyle\rm A}$ as an analog of the axial
“electric” field. In the important case, when $\mu^{\scriptscriptstyle\rm A}$
is constant across the fluid, the term $\int\mu^{\scriptscriptstyle\rm
A}\rho_{\scriptscriptstyle\rm A}d\bm{x}=\mu^{\scriptscriptstyle\rm
A}Q_{\scriptscriptstyle\rm A}$ in (3) is the Casimir invariant that does not
affect the equations of motion. In this case

$\displaystyle\bm{\mathcal{J}}_{\mu^{\scriptscriptstyle\rm A}=\rm const}$
$\displaystyle=\rho\bm{v}-2\mu^{\scriptscriptstyle\rm
A}(\bm{\omega}+\bm{B})\,,$ (33)

so that $\bm{\mathcal{J}}$ differs from $\rho\bm{v}$ by a divergence-free
term.

##  4 Vector current and its anomaly

Applying the definition (28) to the Hamiltonian (18) we obtain the explicit
expression of the vector current in terms of the Eulerian fields $\rho,\bm{v}$
444We assume that the definition (34) remains intact even if the system is in
the magnetic field. In more general situations, the electric current may
contain additional divergence-free micro-currents (aka nonminimal terms)
triggered by the magnetic field.

$\displaystyle\bm{j}=\rho\bm{v}+\bm{E}^{\scriptscriptstyle\rm
A}\\!\times\\!m\bm{v}\,.$ (34)

If $\mu^{\scriptscriptstyle\rm A}$ is uniform $\bm{E}^{\scriptscriptstyle\rm
A}=0$ and the vector current does not deviate from $\rho\bm{v}$. Otherwise,
the vector current (34) receives a component normal to velocity, a feature
reminiscent of the Hall effect.

From (34) and (31) we obtain the relation between the flow current and the
vector current

$\displaystyle\bm{\mathcal{J}}=\bm{j}-2\bm{\nabla}\\!\times\\!(\mu^{\scriptscriptstyle\rm
A}m\bm{v})-2\mu^{\scriptscriptstyle\rm A}\bm{B}\,.$ (35)

Then, the continuity equation (30), written in terms of the vector current
reads yields the chiral anomaly (6)

$\displaystyle\dot{\rho}+\bm{\nabla}\cdot\bm{j}=2\bm{\nabla}\mu^{\scriptscriptstyle\rm
A}\cdot\bm{B}\,.$ (36)

Hence, from the perspective of the fluid dynamics, the chiral anomaly (6) is
yet another form of the continuity equation emphasizing the difference between
the vector current $\bm{j}$ and the advection flux $\bm{\mathcal{J}}$ given by
(35).

##  5 Euler equation

We now proceed with the Euler equation. We compute

$\displaystyle\dot{\bm{\pi}}(\bm{x})\\!$
$\displaystyle=\\!\\{\bm{\pi}(\bm{x}),\mathscr{H}\\}\\!=\int\left[\\{\bm{\pi}(\bm{x}),\rho\\}\frac{\delta\mathscr{H}}{\delta\rho}+\\{\bm{\pi}(\bm{x}),\bm{\pi}\\}\frac{\delta\mathscr{H}}{\delta\bm{\pi}}\right]\,,$
(37)

where the integration goes over the omitted argument. With the help of the
Poisson brackets (19) and the definitions (25,27) we obtain the Euler equation

$\displaystyle(\dot{\bm{\pi}}-\bm{\nabla}\pi_{0})=\bm{\mathcal{U}}\times(\bm{\nabla}\times\bm{\pi})\,.$
(38)

If there is no chiral coupling $\mu_{\scriptscriptstyle\rm A}=0$, the flow
vector field $\bm{\mathcal{U}}$ equals $\bm{v}$ and (38) becomes the familiar
Euler equation for the perfect charged fluid
$m\dot{\bm{v}}-\bm{v}\\!\times\\!\bm{\omega}+\bm{\nabla}(\tfrac{m}{2}\bm{v}^{2}+w)=\bm{E}+\bm{v}\\!\times\\!\bm{B}$.

The form (38) of the Euler equation is the basis of the geometric
interpretation of the Euler hydrodynamics. It emphasizes the Helmholtz law of
advection of vorticity along the flow vector field $\bm{\mathcal{U}}$. It is
instructive to write the Euler equation in the space-time covariant form. Let
us introduce a space-time 4-momentum

$\pi_{\alpha}=(\pi_{0},\bm{\pi})$ (39)

and the canonical vorticity 2-form

$\Omega_{\alpha\beta}=\partial_{\alpha}\pi_{\beta}-\partial_{\beta}\pi_{\alpha}\,.$
(40)

The vector field $\bm{\mathcal{U}}$ relates the components of the vorticity so
that (38) can be written as $\Omega_{0i}={\mathcal{U}}^{j}\Omega_{ji}$. This
means that the flow advects the vorticity along the vector field
$\bm{\mathcal{U}}$ (the Helmholtz law). In the space-time notations
$\mathcal{J}=(\rho,\bm{\mathcal{J})}$, the Euler equation receives the form
known in relativistic hydrodynamics as the Lichnerowitz equation
rieutord2006introduction

$\displaystyle\mathcal{J}^{\alpha}\Omega_{\alpha\beta}=0\,.$ (41)

In particular, this form of the Euler equation suggests that $\pi$ is a space-
time covector, and the advection 4-current $\mathcal{J}$ is the space-time
vector.

##  6  Momentum-energy equations and the Lorentz force

An insight about forces the reservoir exerts on the fluid comes from the
equations for the fluid momentum $\rho{\bm{v}}$ and the energy density
$\mathcal{E}=\tfrac{1}{2}m\rho v^{2}+\varepsilon$ of an isolated fluid. We
obtain them by transforming the Euler equation (38) with the help of (30,31)

$\displaystyle\begin{cases}&\partial_{t}(\rho{\bm{v}})+\partial_{i}(v_{i}\rho\bm{v})+\bm{\nabla}P=\rho\bm{E}+\bm{j}\times\bm{B}+\rho_{\scriptscriptstyle\rm
A}{\bm{E}}_{\scriptscriptstyle\rm A}\,,\\\
&\partial_{t}\mathcal{E}+\bm{\nabla}\\!\cdot\\!\bm{j}_{\mathcal{E}}=\bm{E}\cdot\bm{j}+{\bm{E}}^{\scriptscriptstyle\rm
A}\cdot\bm{\bm{j}}_{\scriptscriptstyle\rm A}\,,\end{cases}$ (42)

where $P=\rho w-\varepsilon$, is the pressure and
$\bm{j}_{\mathcal{E}}=(w+mv^{2}/2)\rho\bm{v}$ is the energy flux of the
isolated fluid and

$\displaystyle\bm{j}_{\scriptscriptstyle\rm
A}=-\Phi(\bm{\omega}+2\bm{B})\,-m\bm{v}\\!\times\\!\Big{(}(\bm{\omega}+\bm{B})\\!\times\bm{\mathcal{U}}\Big{)}+\bm{E}\times
m\bm{v}\,,$ (43)

In the next section, we show that (43) is the axial current of (16). The
l.h.s. of these equations are the space-time divergence of the momentum and
the energy flux of the isolated fluid, the r.h.s. are forces exerted by the
reservoirs and electromagnetic field.

The highlight of these equations is that the vector current $\bm{j}$ defined
by (28) and given explicitly by (34) enters the Lorentz force in the right
hand side of (42). Furthermore, Eq. (42) suggests that (i)
$\bm{E}^{\scriptscriptstyle\rm A}$ acts on the fluid axial current
$(\rho_{\scriptscriptstyle\rm A},\bm{j}_{\scriptscriptstyle\rm A})$ similarly
to the action of the electric field $\bm{E}$ on the vector current
$(\rho,\bm{j})$, and that (ii) the vector current $\bm{j}$ (34) defining the
Lorentz force is, indeed, the electric current defined by (28).

##  7 Axial-current and its anomaly

In a similar manner we derive the axial anomaly equation (16). With the help
of the brackets (20,21) we compute $\partial_{t}\rho_{\scriptscriptstyle\rm
A}=\\{\rho_{\scriptscriptstyle\rm A},\,\mathscr{H}\\}$. Proceeding similarly
to (37) and using the notations (25,27) we obtain

$\displaystyle\partial_{t}\rho_{\scriptscriptstyle\rm
A}(\bm{x})=\int\Big{(}\left\\{\rho_{\scriptscriptstyle\rm
A}(\bm{x}),\bm{\pi}\right\\}\cdot\bm{\mathcal{J}}-\left\\{\rho_{\scriptscriptstyle\rm
A}(\bm{x}),\rho\right\\}\pi_{0}\Big{)}\,,$ (44)

[the integration goes over the omitted argument]. Then with the help of (20)
and (21), we obtain the axial anomaly equation (16) and the hydrodynamic
expression for the axial current. The latter is given by (43) and coincides
with the current ${\bm{j}}_{\scriptscriptstyle\rm A}$ entering the evolution
equation (42) for the energy density.

Finally, with the help of the Euler equation (38) and using the notation

$\displaystyle
p_{\alpha}:=\pi_{\alpha}-A_{\alpha}=\left(-w[\rho]-\tfrac{1}{2}m\bm{v}^{2},m\bm{v}\right)\,,$
(45)

we write the chirality 4-current $j_{\scriptscriptstyle\rm
A}=(\rho_{\scriptscriptstyle\rm A},\bm{j}_{\scriptscriptstyle\rm A})$ in a
space-time covariant form previously obtained in abanov2022axial

$\displaystyle j_{\scriptscriptstyle\rm
A}^{\alpha}=\epsilon^{\alpha\beta\gamma\delta}p_{\beta}(\partial_{\gamma}p_{\delta}+F_{\gamma\delta})\,,$
(46)

where $F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$
is the electromagnetic field tensor.

##  8 Beltrami flow

Now we are equipped to discuss stationary flows. We assume the external fields
are either time-independent, or depend on time adiabatically. The stationary
(or steady) flows give extrema to $\mathscr{H}$. The variation of
$\mathscr{H}$ with respect to the fluid density is the Bernoulli function
(25). The variation with respect to the momentum is the flow vector field
(27). In the stationary flow, whose density and velocity we denote by
$\bar{\rho},\bm{\bar{v}}$, both these variations should be zero which gives
the conditions

$\displaystyle\begin{cases}\pi_{0}=0:\quad&w[\bar{\rho}]+\tfrac{1}{2}m\bm{\bar{v}}^{2}=\mu,\\\
\bm{\mathcal{J}}=0:\quad&\rho\bm{\bar{v}}=2\mu^{\scriptscriptstyle\rm
A}(\bm{\bar{\omega}}+\bm{B})+\bm{E}^{\scriptscriptstyle\rm A}\times
m\bm{\bar{v}}\,.\end{cases}$ (47)

The first line is the Bernoulli condition. The second defines the generalized
Beltrami flow in the magnetic field. Together they determine
$\rho,\rho_{\scriptscriptstyle\rm A}$ as functions of
$\mu,\mu^{\scriptscriptstyle\rm A}$. For example, at $\bm{B}=0$ we have the
relation $\mu^{\scriptscriptstyle\rm A}{\bar{\rho}}_{\scriptscriptstyle\rm
A}=(\mu-w[\bar{\rho}])\bar{\rho}$.

We recall that in the absence of magnetic field, the velocity of the
generalized Beltrami flow is an eigenfunction of the $\rm curl$ operator

${\rm curl}\,\bm{\bar{v}}=\kappa\bm{\bar{v}}\,.$ (48)

Extension by the magnetic field amounts to the shift by the Larmor frequency
${\rm curl}\,\bm{v}\to{\rm curl}\,\bm{v}+\bm{B}/m$. In our case ${\rm\
curl}\,\bm{v}=\tfrac{1}{\sqrt{\mu^{\scriptscriptstyle\rm
A}}}\bm{\nabla}(\sqrt{\mu^{\scriptscriptstyle\rm A}}\times\bm{v})$ and
$\kappa=\rho/2m\mu^{\scriptscriptstyle\rm A}$. Locally Beltrami flow describes
a helix along a vortex line. The pitch of the helix of the order $1/\kappa$
(see, e.g., dritschel1991generalized ).

A particularly interesting case is the Beltrami flow proper, when
$\kappa=\text{const}$. In this case, the Beltrami flow is incompressible as
${\rm div}\,\bar{\bm{v}}=0$, where ${\rm
div}\,\bm{v}=\tfrac{1}{\sqrt{\mu^{\scriptscriptstyle\rm
A}}}\bm{\nabla}\cdot(\sqrt{\mu^{\scriptscriptstyle\rm A}}\,\bm{v})$. An
important feature of the streamlines of the incompressible Beltrami flow is
that they are not integral, and some are known to be chaotic
arnold2008topological . A similar situation occurs in some compressible
(generalized) Beltrami flows morgulis1995compressible  (see also
enciso2016beltrami  for conditions when compressible solutions exist). We do
not elaborate on this aspect further but remark that a possibility of chaotic
advection of particles (Lagrangian turbulence) in the context of fermionic QFT
seems to be an unexplored aspect of anomalies.

### Transport current of the Beltrami flow

Let us evaluate the vector and the axial current on the Beltrami flow. This
way we obtain the equilibrium value of the currents. Combining (34,43,47) we
obtain

$\displaystyle\bar{\bm{j}}=2\mu^{\scriptscriptstyle\rm
A}\bm{B}+2\bm{\nabla}\\!\times\\!(\mu^{\scriptscriptstyle\rm
A}m\bm{\bar{v}})\,,\quad\bar{\bm{j}}_{\scriptscriptstyle\rm
A}=2\mu\bm{B}+\bm{\nabla}\\!\times\\!(\mu m\bm{\bar{v}})\,.$ (49)

We notice that both currents (49) consist of the transport part

$\bm{j}_{\rm trans}=2\mu^{\scriptscriptstyle\rm
A}\bm{B}\,,\quad\bm{j}_{\scriptscriptstyle\rm A\,\rm trans}=2\mu\bm{B}\,,$
(50)

and a microscopic current, the curl

$\bm{j}_{\rm micro}=2\bm{\nabla}\\!\times\\!(\mu^{\scriptscriptstyle\rm
A}m\bm{\bar{v}})\,,\quad\bm{j}_{\scriptscriptstyle\rm A\,\rm
micro}=\bm{\nabla}\\!\times\\!(\mu m\bm{\bar{v}})\,.$ (51)

If the chemical potentials are constant, the vector and axial currents
currents are555 Formulas similar to (52) appear in various hydrodynamic models
(other than Euler hydrodynamics) aimed to be consistent with the chiral
anomaly. There the term proportional to the magnetic field and the term
proportional to the vorticity in the vector current is dubbed by the ‘chiral
magnetic effect’ and the ’chiral vortical effect’, respectively (see, e.g.,
kharzeev2011testing  and references therein). Similar terms in the axial
current are sometimes referred to as the ‘chiral separation effect’ and the
’chiral vortical separation effect’, respectively chernodub2021thermal . The
coefficients vary across the literature.

$\displaystyle\bar{\bm{j}}=$ $\displaystyle 2\mu^{\scriptscriptstyle\rm
A}(\bm{B}+\bm{\omega})\,,\quad{\bar{\bm{j}}_{\scriptscriptstyle\rm
A}}=2\mu\left(\bm{B}+\tfrac{1}{2}\bm{\omega}\right)\,,\quad\mu,\mu^{\scriptscriptstyle\rm
A}=\rm const\,.\ $ (52)

The micro-currents describe the stationary movement of electric and axial
charges localized on vortex loops. In contrast, the transport currents
transfer hydrodynamics substances along magnetic lines propagating across the
system.

As we remarked in the Sec. 7 one can trace the origin of the transport current
to the central extension of the Poisson algebra (20). The expressions (49-52)
are the main results of this work. We have shown that an equilibrium state of
the fluid coupled to the chiral reservoir (i) is a Beltrami flow and that (ii)
the Beltrami flow transports both electric and the chirality charges in a way
identical to the anomaly-driven flows of Dirac fermions.

##  9 Flows in an external axial potential

We conclude the paper by extending the coupling to the chiral reservoir from
$\mu^{\scriptscriptstyle\rm A}$ to a full axial 4-potential
$\mu^{\scriptscriptstyle\rm A}\to A_{\alpha}^{\scriptscriptstyle\rm
A}=(\mu^{\scriptscriptstyle\rm A},\bm{A}^{\scriptscriptstyle\rm A})$. The
extended coupling brings various currents to space-time-covariant forms and
further clarifies the parallels between the fluid dynamics and Dirac fermions
in the axial field. An economical way to obtain the results is to use almost
obvious covariant extensions of the formulas of the main text. We present the
main expressions here and publish a detailed derivation separately.

The covariant extension is based on the main property of the Euler
hydrodynamics: the fluid momentum $p_{\alpha}=\pi_{\alpha}-A_{\alpha}$, and
the advection current $\mathcal{J}^{\alpha}$, are $\mathbb{R}^{4}$ covector
and vector, respectively, regardless of the space-time structure. For that
reason the formulas below are equally valid in the relativistic case.

We start from the currents. The axial current (46) does not depend on the
axial potential and has already appeared in the space-time covariant form. In
turn, the vector current does not depend on the vector potential. The
advection 4-current is the covariant extension of (35). Together the currents
read

$\displaystyle\begin{cases}j^{\alpha}=\rho
u^{\alpha}+^{\star}\\!\\!F_{\scriptscriptstyle\rm
A}^{\alpha\beta}p_{\beta},\quad&\\\ j_{\scriptscriptstyle\rm
A}^{\alpha}=\epsilon^{\alpha\beta\gamma\delta}p_{\beta}(\partial_{\gamma}p_{\delta}\,+F_{\gamma\delta}),\quad&\\\
\mathcal{J}^{\alpha}=j^{\alpha}+2\,{}^{\star}\\!F^{\alpha\beta}A^{\scriptscriptstyle\rm
A}_{\beta}+2\epsilon^{\alpha\beta\gamma\delta}\partial_{\gamma}(A^{\scriptscriptstyle\rm
A}_{\beta}p_{\delta}).\end{cases}$ (53)

In these formulas we denote the 4-velocity $u^{\alpha}=(1,\bm{v})$. Also,
${}^{\star}\\!F^{\alpha\beta}=\tfrac{1}{2}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}$
is the dual field tensor,
$F_{\alpha\beta}=\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}$ and we
use similar notations for the axial field tensors.

As we mentioned in the Sec. 5, the covariant form of the Euler equation
appears as the Lichnerowitz equation (41). We reproduce it here

$\displaystyle\begin{cases}\mathcal{J}^{\alpha}(\partial_{\alpha}\pi_{\beta}-\partial_{\beta}\pi_{\alpha})=0,\\\
\partial_{\alpha}\mathcal{J}^{\alpha}=0\,.\end{cases}$

The covariant form of the divergence of the stress tensor extends the equation
(42) and the divergence of the vector current extends the formula (36).
Together they are given by

$\displaystyle\begin{cases}\partial_{\beta}T_{\alpha}^{\beta}=F_{\alpha\beta}\,j^{\beta}+F^{\scriptscriptstyle\rm
A}_{\alpha\beta}\,j^{\beta}_{\scriptscriptstyle\rm A}\,,\\\
\partial\\!\cdot\\!j=F\\!\cdot\\!{}^{\star}\\!F^{\scriptscriptstyle\rm
A}=2(\bm{E}^{\scriptscriptstyle\rm
A}\\!\cdot\\!\bm{B}+\bm{B}^{\scriptscriptstyle\rm A}\\!\cdot\\!\bm{E})\,,\\\
\partial\\!\cdot\\!j_{\scriptscriptstyle\rm
A}=\tfrac{1}{2}F\\!\cdot\\!^{\star}\\!F=2\bm{E}\\!\cdot\\!\bm{B}\,.\end{cases}$
(54)

The equations (53) and (54) retain their form for the relativistic fluid. For
the non-relativistic fluid the components of the momentum-stress-energy tensor
in (54) are: the energy density
$-T_{0}^{0}=\tfrac{1}{2}m\rho\bm{v}^{2}+\varepsilon[\rho]$, the energy flux
$-T^{i}_{0}=\rho v^{i}(\tfrac{1}{2}mv^{2}+w)$, the momentum density
$T^{0}_{i}=m\rho v_{i}$, and the momentum flux $T_{i}^{j}=m\rho
v_{i}v^{j}+P\delta_{i}^{j}$. For a relativistic fluid
$T_{\alpha}^{\beta}=\rho\,p_{\alpha}u^{\beta}+P\delta_{\alpha}^{\beta}$, and
the entries in (53) are the relativistic 4-velocity
$u^{\alpha}=(1,\bm{v}/c)/\sqrt{1-\bm{v}^{2}/c^{2}}$ and the 4-momentum
$p_{\alpha}=(mc+c^{-1}w)u_{\alpha}$.

We comment that starting from the Eqs.(54) and given momentum-stress-energy
tensor, one could derive the expressions for the currents (53) as a
consistency of (54). This argument had been explored in son2009hydrodynamics
for the relativistic fluid.666 We comment on the formal relation between Eqs.
(53,54) and the formulas of Refs. son2009hydrodynamics ;
monteiro2015hydrodynamics . Equations
$\partial_{\beta}T_{\alpha}^{\beta}=F_{\alpha\beta}\,j_{[1]}^{\beta}$ and
$\partial_{\mu}j_{[1]}^{\mu}=C\bm{E}\cdot\bm{B}$ of son2009hydrodynamics ;
monteiro2015hydrodynamics  follow from (54) in a particular background when
the vector and the axial potentials are set proportional
$A^{\scriptscriptstyle\rm A}=CA/6$. Under this condition $j_{[1]}$ of Ref.
son2009hydrodynamics  is a particular linear combination
$j+Cj_{\scriptscriptstyle\rm A}/6$ of the currents given by (53).

##  10 Summary

Summing up, we have shown that anomaly-based properties of QFT with Dirac
fermions coupled to the axial vector potential are identical to the kinematic
properties of the Euler fluid coupled to the reservoir, which maintains the
fluid helicity. The result is well captured by Eqs. (53,54). From the
perspective of fluid dynamics, the equation for the stress tensor (the first
equation in (54)) is the complete equation of motion. Once the stress tensor
and the currents are expressed through the fluid density and the velocity as
in (53), the divergence of the currents (the last two equations in (54))
follow. Now let us look on (54) from the perspective of Dirac fermions. The
first equation in (54) is valid for the classical Dirac equation, or quantum
alike. However, the divergences of the currents (the last two equations) are
of a quantum nature. They require a short-distance regularization which
preserves the gauge invariance. That regularization yields the anomaly in the
divergence of the currents. Once the divergence is determined, the equations
must be completed by constitutive relations, for example, expressing the
stress tensor and the axial current in terms of the vector current. We have
shown that the Euler fluid provides such constitutive relation in the form of
Eqs. (53). Another result of our work is a close parallel between the Beltrami
flow in fluid dynamics and the ground state of the Dirac fermions. The
Beltrami flow stands out as a fascinating property of fluid dynamics. In turn,
the quantum anomalies might be understood from the properties of the ground
state of the Dirac fermions (perturbed by the external fields). A close
parallel between these objects of seemingly unrelated fields seems noteworthy.
Interpretation and consequences of the chaotic nature of the Beltrami flow in
the quantum field theory are especially interesting.

Equations of motions (54) and the hydrodynamic expressions for the vector and
the axial currents (53) are gauge invariant with respect to both the vector
$A\to A+\partial\varphi$ and the axial $A^{\scriptscriptstyle\rm A}\to
A^{\scriptscriptstyle\rm A}+\partial\varphi^{\scriptscriptstyle\rm A}$ gauge
transformations. However, the advection current $\mathcal{J}$ given by the
third line of (53) and the Beltrami flow defined by the condition
$\bm{\mathcal{J}}=0$ are not invariant under the axial gauge transformation.
As a result, the quantities evaluated on the stationary flow (47) do not
generally possess the axial gauge symmetry. Despite the axial gauge symmetry
of equations of motion, the symmetry does not hold at the equilibrium state
(or the ground state in QFT). This is the essence of the anomaly: the (axial)
symmetry of equations of motions is not a symmetry of their solutions. This
property gives rise to the main effect we discussed in the paper. The
equilibrium state (the Beltrami flow) possesses a non-vanishing mass flux
described in Sec. 38. Even if the axial vector potential is a pure gauge, so
that the axial field is a gradient $A_{\beta}^{\scriptscriptstyle\rm
A}=(2\pi)^{-1}\partial_{\beta}\Theta$, hence drops out of the equations of
motions (54), the advection current $\mathcal{J}$, hence the state of
equilibrium $\bm{\mathcal{J}}=0$ depends on $\Theta$. As a result, the
equilibrium state transports the mass/charge with an adiabatic change of the
theta-angle as $\bm{j}_{\rm trans}=(1/\pi)\bm{B}\dot{\Theta}$. The amount of
charge crossing the surface element $d\sigma$ when $\Theta$ changes by $2\pi$
is $2Bd\sigma$, i.e., twice the magnetic flux threading the surface.

###### Acknowledgements.

We thank A. Alekseev, A. Cappelli, G. Monteiro and V. P. Nair for their
interests in this study and useful discussions. We thank G. Volovik and B.
Khesin for comments and bringing the Refs. bevan1997momentum ;
salomaa1987quantized ; khesin1989invariants  to our attention. The work of
P.B.W. was supported by the NSF under the Grant NSF DMR-1949963. The work of
A.G.A. was supported by NSF DMR-2116767.

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