Anomalous transport in second order hydrodynamics

We study the non-dissipative transport effects appearing at second order in the hydrodynamic expansion for a non-interacting gas of chiral fermions by using the partition function formalism. We discuss some features of the corresponding constitutive relations, derive the explicit expressions for the conductivities and compare with existing results in the literature.


Introduction
Hydrodynamics is an effective description of out-of-equilibrium systems in which it is assumed local thermodynamical equilibrium. The hydrodynamical systems should obey the conservation laws of the energy-momentum tensor and spin one currents, and these quantities are written in terms of fluid variables in the so-called constitutive relations. Some of the transport phenomena are related to dissipative effects, as they lead to entropy production: examples are the shear viscosity η and bulk viscosity ζ [1]. However new phenomena on the hydrodynamics induced by quantum anomalies have recently received much attention and interest. In presence of anomalies the currents are no longer conserved, and this has important effects in the constitutive relations. Two relevant phenomena appear at first order in the hydrodynamic expansion as a consequence of chiral anomalies: the chiral magnetic effect, which is responsible for the generation of an electric current induced by a magnetic field [2], and the chiral vortical effect, in which the electric current is induced by a vortex [3]. It is believed that these phenomena can produce observable effects in heavy ion physics [4], as well as in condensed matter systems [5]. These effects are non-dissipative, and the associated conductivities are almost completely fixed by imposing the requirement of zero entropy production. At second order a plethora of dissipative and non-dissipative conductivities have been studied, see e.g. [6] and references therein.
Some methods to compute the transport coefficients from a microscopic theory, either dissipative or non-dissipative, include kinetic theory [7,8], Kubo formulae [9], diagrammatic methods [10] and fluid/gravity correspondence [11]. Recently it has been proposed a new formalism to obtain the non-dissipative part of the anomalous constitutive relations, and it is based on the existence of an equilibrium partition function in a stationary background. It has been observed that the equations of hydrodynamics are significantly constrained by the requirement of consistency with the partition function [12,13], and these constraints seem to overlap with the ones obtained from the existence of an entropy current with non-negative divergence [14]. In this work we study, within this formalism, the non-dissipative constitutive relations up to second order in the hydrodynamic expansion for an ideal gas of chiral fermions. We will be able to get explicit results for some of the transport coefficients.

Hydrodynamics of relativistic fluids
Hydrodynamics is based on the assumption that the scales of variation of the observables are much longer than any microphysical scale in the system, and the result can be organized in a gradient expansion, also called hydrodynamic expansion [1]. The constitutive relations for the energy-momentum tensor and charged currents write generally in the form where ε is the energy density, P the pressure, ρ the charge density and u µ the local fluid velocity. In addition to the equilibrium contributions, there are extra terms which lead to dissipative and anomalous effects. Within the Landau frame 1 and in presence of external electromagnetic fields, these terms write up to first order in derivatives as 2 where P µν = G µν +u µ u ν , and the electric, magnetic fields and vorticity are defined as E µ = F µν u ν , B µ = 1 2 ǫ µνρλ u ν F ρλ and ω µ = ǫ µνρλ u ν ∇ ρ u λ respectively. The coefficients appearing in Eqs. (3)-(4) are the shear η and bulk η viscosities, the electric σ, chiral magnetic σ B and chiral vortical σ V conductivities respectively. At this point it is worth analyzing the parity P and time reversal T properties of these coefficients. The spatial component of the charged current, J i , is P-odd and T -odd, while the magnetic field and vorticity are P-even and T -odd. Then one concludes from Eq. (4) that σ B and σ V are P-odd and T -even. On the other hand, the second law of thermodynamics states the increase of entropy with time, i.e. ∂ ∂t and this means that only T -odd contributions can lead to entropy production of the system. This is not the case of the chiral conductivities, so that these terms should be related to non-dissipative transport. A similar analysis for the shear, bulk and electric conductivities implies that these transport coefficients are P-even and T -odd, and they are associated with dissipative transport phenomena. In Sections 3 and 4 we present the partition function formalism, which is suitable to compute T -even conductivities. We use this formalism to obtain the constitutive relations up to second order in the hydrodynamic expansion in Section 5.

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Field Theory with a time independent U(1) gauge connection on the manifold The fields σ, a i , g i j , A 0 and A i are smooth functions of the spatial coordinates x. As usual, the partition function of the system writes where H is the Hamiltonian of the theory, Q is the charge associated to the gauge connection, while T 0 and µ 0 are the temperature and chemical potential at equilibrium. An obvious question is: what is the dependence of Z on the fields σ, a i , g i j , A 0 and A i ? This has important implications for the hydrodynamics of the system, as we will see later. To answer this, the standard procedure is to build the most general partition function of the system consistent with the allowed symmetries, in and iii) U(1) time-independent gauge invariance (up to an anomaly). It is convenient to introduce the combination A i ≡ A i − A 0 a i , which is invariant under the Kaluza-Klein transformation.
From the partition function we can compute the energy-momentum tensor and U(1) charged current by performing the appropriate t-independent variations, i.e.
where g 3 = det(g i j ). In particular, for a general partition function of the form log Z = W(e σ , A 0 , a i , A i , g i j , T 0 , µ 0 ), one gets and a similar expression for T i j . This illustrates the fact that W plays the role of a generating functional for the hydrodynamic constitutive relations, and one expects that its form matches order by other with the derivative expansion in hydrodynamics. In the rest of this section we present the most important properties of this partition function up to second order in derivatives.

Equilibrium partition function at zeroth order
We study first the zeroth order in derivatives. The most general partition function at this order, which is consistent with the symmetries mentioned above, reads [12] where P is an arbitrary function of its arguments. From Eq. (12) and after applying the variational formulae (10)-(11), one gets T 00 = e 2σ (P − a∂ a P − b∂ b P) , EPJ Web of Conferences where for convenience we have defined the variables a ≡ e −σ T 0 and b ≡ e −σ A 0 . This result should correspond to the equilibrium contribution of the hydrodynamic constitutive relations. Finally, after a comparison with Eqs. (1)-(2) one gets Note that P is identified with the pressure of the system, and the thermodynamic variables ε, P and ρ are determined in terms of this single master function. Thermodynamical consistency requires to identify the local value of the temperature, T , and chemical potential, µ, with a and b respectively.

Equilibrium partition function at higher orders
Let us discuss now the properties of the partition function at higher derivative orders. The most general partition function at first order in the derivative expansion was presented in [12], and it reads where while Eq. (16) contains only P-odd contributions, as one cannot build P-even contributions to the partition function at first order in derivatives. Following the method of Section 4, a computation of the partition function for an ideal gas of Weyl fermions leads to the following result [17] α 1 (T, ν) = − C 6 ν , α 2 (T, ν) = − 1 2 where are constants related to the axial anomaly [3,18] and gauge-gravitational anomaly [19] respectively. After a suitable computation of the constitutive relations as it will be explained in Section 5, one gets the well known expressions for the chiral conductivities in the Landau frame 3 where we have used that ν = µ/T . These results have been obtained in a wide variety of methods, see e.g. [2,3,9,[18][19][20][21][22][23][24][25].
Finally, the most general partition function at second order in derivatives is built from seven scalar and two pseudo-scalar quantities as follows [14,16] where R is the Ricci scalar in 3 dim, with M i = M i (T, ν) and N i = N i (T, ν). We keep for the moment both P-odd and P-even contributions.
We present in this section the procedure to compute the second order partition function of Eq. (21) for a theory of free massless Dirac fermions. We refer to Ref. [17] for full details in the computation.

Free theory of Dirac fermions
We will derive the partition function by using Pauli-Villars (PV) regularization, and this demands the consideration of the massive theory for the vacuum contribution. The action of the theory is The space-time dependent Dirac matrices satisfy {γ µ (x), γ ν (x)} = 2G µν (x), and they are related to the Minkowski matrices by γ µ (x) = e µ a (x)γ a , where e µ a (x) is the vierbein, {γ a , γ b } = 2η ab and η ab = diag(−1, 1, 1, 1). The U(1) current and energy-momentum tensor write where it has been assumed that the spinor field Ψ = ψ L ψ R satisfies the Dirac equation. Using the explicit form of the background Eqs. (6)-(7) one has where σ i are the Pauli matrices, and ψ is the two-component Weyl fermion ψ L . The same expressions are obtained for ψ R , but with opposite sign in J i .

Thermal Green's function
The expectation values of J µ and T µν at equilibrium may be computed from the thermal Green's function, defined as where T denotes time ordering. The explicit calculation of the Green's function is the most computationally demanding step in the derivation. After rotating to imaginary time t → −iτ, the Green's function satisfies where H is the Hamiltonian with ω ab µ the spin connection and γ ab = 1 2 [γ a , γ b ]. Then by expanding Eq. (27) in derivatives, the Green's function can be computed recursively order by order in a derivative expansion, i.e. G = G 0 + G 1 + G 2 + · · · , where G ℓ is the contribution at order ℓ. Note that to get G ℓ one needs to know the Green's function at all the orders lower than ℓ. The complete expressions for G up to second order are very lengthy and will not be presented here. Finally, from Eqs. (24)- (26) one obtains the precise form of the current and energy-momentum tensor up to ℓ-th order in the derivative expansion Note that T 00 ℓ receives a contribution ∝ rot a · J ℓ−1 .

Charge density at second order
To get W 2 it is enough to compute J 0 2 and T 00 2 including only bilinear terms ∼ ∂ i X∂ j Y. The evaluation of Eq. (29) produces the following renormalized expression where Q(ν) is the analytic continuation of the series Q(ν) = −2 ∞ n=1 (−1) n cosh(nν) log(n 2 ). A similar expression is obtained for T 00 2 . To derive this result we have regularized the vacuum contribution in a gauge invariant way by using the PV regularization procedure. We have defined the rescaled PV mass asM = 2 −3/2 e γ E M. The distinction between vacuum and finite temperature and chemical potential contributions in the expectation values can be obtained, for instance, by considering the Poisson summation formula in the Matsubara sums.