Chiral Anomaly in Euler Fluid and Beltrami Flow

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 .

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

and use it in the augmented Hamiltonian (3)

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

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

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

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.

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

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

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)

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

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

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.

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

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

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).

We now proceed with the Euler equation. We compute

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

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

and the canonical vorticity 2-form

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

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.

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)

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

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).

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

[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

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

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

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

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

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.

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

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

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

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).

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.