Parity-Violating Hydrodynamics in 2+1 Dimensions
We study relativistic hydrodynamics of normal fluids in two spatial dimensions. When the microscopic theory breaks parity, extra transport coefficients appear in the hydrodynamic regime, including the Hall viscosity, and the anomalous Hall conductivity. In this work we classify all the transport coefficients in first order hydrodynamics. We then use properties of response functions and the positivity of entropy production to restrict the possible coefficients in the constitutive relations. All the parity-breaking transport coefficients are dissipationless, and some of them are related to the thermodynamic response to an external magnetic field and to vorticity. In addition, we give a holographic example of a strongly interacting relativistic fluid where the parity-violating transport coefficients are computable.
Hydrodynamics is an effective long-distance description of many classical or quantum many-body systems at non-zero temperature. The form of the hydrodynamic equations is dictated by the symmetries of the microscopic Hamiltonian, and is not sensitive to the precise nature of the short-distance degrees of freedom. When the microscopic description exhibits Lorentz invariance, the collective flow is described by the relativistic analogue of the Navier-Stokes equations. For normal fluids with an unbroken global symmetry (such as baryon number), the hydrodynamic equations take the form [1, 2],
Here is the energy-momentum tensor of the fluid, is the symmetry current, and we have allowed for the possibility of coupling the fluid to an external non-dynamical gauge field (with field strength ) and metric (with covariant derivative ). The gauge field couples to the conserved current . The relativistic analogue of the Navier-Stokes equations have a wide range of applications. For example, their study in dimensions has led to significant progress in understanding the quark-gluon plasma . In dimensions the equations were proposed as an effective description of thermo-magnetic transport in cuprates  and in graphene .
A complete hydrodynamic description must, of course, supplement
equations (1) with constitutive relations
which express and in terms of macroscopic parameters
such as the local fluid velocity , local temperature and local chemical potential . A conventional description of the constitutive relations can be found, for example, in the classic textbook by Landau and Lifshitz .
In this paper we take a closer look at the equations of relativistic hydrodynamics in dimensions,
and argue that the canonical constitutive relations need to be modified when the microscopic theory does not respect
parity P. By parity we mean invariance under reflection of one of the spatial coordinates.
Our expressions for the constitutive relations may be written as follows:
The tensor quantities appearing in the constitutive relations (2) are
The thermodynamic parameters , and are the values of the pressure, energy density and charge density respectively in an equilibrium configuration in which , where is the rest-frame magnetic field and the vorticity. They satisfy
where is the entropy density. The velocity field is denoted and is normalized so that . In this paper we study hydrodynamics to first order in derivatives. For counting purposes, the derivatives of and are of the same order as derivatives of the hydrodynamic variables. As a result, we take the magnetic field and vorticity as first order in derivatives, and work to linear order in and .
The remaining parameters in (2) characterize the transport properties of the fluid, or its thermodynamic response. The shear viscosity , bulk viscosity , and charge conductivity are the canonical dissipative transport coefficients and must satisfy
as a consequence of either positivity of the divergence of the entropy current, or positivity of the spectral functions in the corresponding Kubo formulas.
The Hall viscosity and a new parameter
, which contributes to the Hall effect in the absence of external magnetic fields, are both dissipationless. Our analysis does not constrain the values of and ,
The remaining parameters , , and are not independent, and are specified in terms of three thermodynamic functions, , and , such that
where we have defined . All derivatives in (8) are evaluated at constant or except for and which are evaluated at constant and respectively. The Kubo formulas for the parameters appearing in the constitutive relations (2) are
where is an antisymmetric tensor with . Here denotes the retarded Green’s functions,
in the thermal equilibrium state at and , defined by varying the one-point functions with respect to the appropriate sources. One important difference between the Kubo formulas (9) and (10) is that the former are given in terms of zero-momentum response functions, while the latter are given in terms of zero-frequency response functions. As emphasized in , the zero-frequency response functions are inherently Euclidean, and therefore only contain thermodynamic information. For this reason the parameters , , , are not transport coefficients, but should be thought of as thermodynamic quantities, consistent with (8). We will refer to , , and as transport coefficients and to , , and as thermodynamic response parameters.
Our parametrization of the constitutive relations (2) was not general in the sense that we have chosen a particular out-of-equilibrium definition of energy density, charge density, and fluid velocity. Such a choice is referred to as a “frame”. The choice (2) is usually referred to as the Landau frame. We will find it convenient to use an alternative frame that is naturally suited to fluids whose thermodynamics depends on and . Indeed, a static magnetic field does not lead to an increase in (fluid) energy and therefore may be non-zero in equilibrium. Similarly, on a compact manifold, one may have non-zero vorticity and still remain in thermal equilibrium, e.g., a system which executes rigid rotation. For such equilibrium states the pressure is , so that
Here and in the rest of this paper, all thermodynamic derivatives with respect to and are evaluated at and . The constitutive relations in a ‘magnetovortical’ frame which is adapted to the thermodynamic relation (11) are given by
where and are
The constitutive relations (2) (or (13)) together with the subsequent relations for the transport coefficients and thermodynamic response parameters are the main results of this paper. The relations (6) and the Kubo formulas for , , and are well known, while Kubo formulas for were discussed recently in . Our strategy for obtaining the relations (6), (7), (8), and (10) involved the imposition of several physical constraints on the constitutive relations. These constraints amount to requiring that the response functions of a hydrodynamic theory must: (i) obey positivity constraints, (ii) have their zero-frequency limits coincide with the corresponding thermodynamic susceptibilities, and (iii) transform covariantly under time-reversal, T. In addition we ensure that (iv) a local version of the second law of thermodynamics holds. Some of these constraints are more familiar than others. Relations (i) and (iv) have often been used in the literature , and (iii) is the basis for Onsager’s reciprocity relations [11, 12].
Parity-violating systems in dimensions have been considered in the condensed matter literature. The simplest such example is a theory of free massive Dirac fermions at zero temperature and in 2+1 dimensions. Parity breaking leads to a remarkable transport property at zero temperature: the Hall conductivity is quantized although no magnetic field is present . This is an example of the anomalous Hall effect . The transverse response to a thermal gradient (the thermal Hall conductivity) was recently discussed in several classes of topological insulators in 2+1 (and 3+1) dimensions, and related to anomalies in various dimensions . In 2+1 dimensions the parity-odd analogue of the shear viscosity, which is called the Hall viscosity, has been studied from the condensed matter physics perspective in [16, 17, 6, 18, 19], using an effective field theory [7, 20], and also using the AdS/CFT correspondence [10, 21, 22, 23].
Parity-violating transport effects were also studied in 3+1 dimensions with QCD applications in mind: using field theory techniques [24, 25, 26, 27, 28], in the context of hydrodynamics [29, 30, 31, 32], and in parallel using the AdS/CFT correspondence along with hydrodynamics [33, 34, 35, 36, 37, 38, 39, 40, 41]. A relation linking parity-odd transport in 3+1 dimensions with the chiral anomaly was first found in [42, 43], and later effects of a gravitational anomaly were considered in [44, 45]. Recently, parity-odd transport in superfluids was discussed in [9, 46, 47]. An effective field theory for non-dissipative transport in 1+1 dimensions was suggested in . For considerations of hydrodynamics in arbitrary dimensions see [32, 49].
The rest of this paper is organized as follows. In Section 2 we construct the most general constitutive relations allowed for a relativistic P-violating system in 2+1 dimensions. In Section 3 we construct an entropy current with positive divergence, and determine the ensuing constraints on the constitutive relations. We independently derive restrictions on these constitutive relations in Section 4 using linearized hydrodynamics, by computing the retarded Green’s functions and imposing the conditions (i)-(iii) described above. We discuss an alternative hydrodynamic frame (13) in Section 5, which provides a more transparent picture of -dimensional thermodynamics in the presence of non-zero and . We check our results against an AdS/CFT computation in Section 6, and conclude our analysis with a discussion of the results in Section 7.
2 The hydrodynamic expansion
In hydrodynamics, the chemical potential, temperature and velocity field are allowed to vary slowly in space and time. The four equations of motion which determine the values of the hydrodynamic variables are energy-momentum conservation and charge conservation, while the explicit relations between the energy-momentum tensor (and the current) and the hydrodynamic variables are called constitutive relations. Given a time-like vector (satisfying ), the energy-momentum tensor and the current can be decomposed into pieces which are transverse and longitudinal with respect to ,
where, as before, projects onto the space orthogonal to the velocity field. In this decomposition, , , and are Lorentz scalars, , , and are transverse, , , , and is symmetric and traceless. Some readers may be familiar with a decomposition of the energy-momentum tensor and current into an ideal and dissipative part. For example, one may write
where , and where defined below (3). We point out that the decomposition in (16) is of a different nature—it is a decomposition into scalar, transverse vector and transverse tensor modes which can be carried out for any tensor and vector. In other words, , and do not necessarily take on their values in equilibrium. As a consequence, the scalars, transverse vectors and transverse tensors in (16a) and (16b) depend, a priori, on the hydrodynamic variables , and and on quantities built from their derivatives. Needless to say, one may easily go from (17) to (16) by comparing appropriate terms in the current or energy-momentum tensor. For example: .
Out of equilibrium one can redefine the fields , , and in a way that simplifies the decomposition (16). This four-parameter field redefinition is referred to as a choice of frame  (for a detailed recent discussion see ). In what follows we will choose a conventional Landau frame, in which the four-parameter ambiguity is fixed by requiring that and that and retain their values in an equilibrium configuration with zero magnetic field and zero vorticity, i.e. and . This choice of frame gives
where the transverse current and transverse traceless tensor vanish in the equilibrium state, and similarly
We now need to specify the constitutive relations which express the energy-momentum tensor and the current in terms of the hydrodynamic variables , , and , their derivatives and possible electromagnetic and gravitational sources. Since we consider small deviations from thermal equilibrium, we may expand , , and to first order in derivatives of hydrodynamic variables. As is generally the case in effective field theories, we must allow all possible one derivative contributions to , , and consistent with the symmetries of the system, but rule out those expressions that are forbidden by physical constraints such as thermodynamic laws, unitarity and time reversal symmetry. In the remainder of this section, we will classify all possible independent contributions to , , and . By independent we mean expressions which are inequivalent under the equations of motion (1) to first order in derivatives. The additional constraints which need to be implemented in order for the theory to satisfy all physical requirements will be described in Section 3 and Section 4.
In formulating the first-order constitutive relations, we take the external fields to be small, with the field strengths and the connection coefficients of the same order as gradients of the hydrodynamic variables. This is the scaling required to study the response of the fluid to sources to first order in a derivative expansion. Magnetohydrodynamics or fluid dynamics with large values of vorticity would require a separate treatment.
To carry out a classification of the scalars, vectors and tensors it is convenient to supplement the transverse projector , satisfying , with a transverse antisymmetric tensor which satisfies , and . A generic vector or pseudovector can be projected into orthogonal components in the plane transverse to via and . This allows us to straightforwardly write down all possible structures contributing to the constitutive relations.
At first order in derivatives there are three scalars, , , , and two pseudoscalars, , that one could construct out of the hydrodynamic variables. However, since there are two scalar equations of motion: and , only one of the three scalars is independent. We take this scalar to be . Thus, there is one scalar and two pseudoscalars which may contribute to to first order in derivatives,
|where is the local thermodynamic pressure, and is the bulk viscosity. The second and third terms in (20a) are forbidden in parity-invariant systems, but are allowed once parity is broken.|
|Next we consider the tensors. Since and have no transverse projections, it is sufficient to focus on projections of . The two structures and with circular brackets denoting a symmetric combination are, in fact, an exhaustive set of first order tensors. Using the properties of and listed above one can show that any other symmetric transverse projection can be represented as a linear combination of these structures and . Forming the trace-subtracted combinations we have, on writing out the tensors more explicitly,|
|where is the shear viscosity. The parameter is a P-violating transport coefficient referred to as the Hall viscosity. It is only allowed once parity is broken, and has been discussed previously for non-relativistic  and relativistic  fluids. If we denote small fluctuations of the spatial component of the velocity field by , then unlike the normal shear viscosity which in flat space gives a response of the stress tensor to , the Hall viscosity gives a response of to .|
|There are four transverse vectors and four transverse pseudovectors which we can construct at first order in derivatives. These vectors and pseudovectors can be formed by projecting , , or with either or . Since we have one transverse vector equation of motion, and one transverse pseudovector equation of motion, , only two vectors and two pseudovectors are independent. We choose to drop the projections of . The constitutive relation for then takes the form,|
where , and is the electric field in the fluid rest frame.
We have written the constitutive relations in the Landau frame. However, a frame-invariant definition of the transport coefficients does exist and will be discussed in the next section.
3 Positivity of entropy production
In the Landau frame the constitutive relations take the form given in (18) and (20). In this section we will study how the second law of thermodynamics restricts the coefficients in the constitutive relations leading to the results (6), (7), and (8) described in Section 1. At intermediate stages of the computation we will find frame invariant expressions for the constitutive relations.
The assumption that the flux of entropy entering any compact spacelike region cannot exceed the amount of entropy produced in that region amounts to the existence of a current whose divergence is positive semi-definite,
where is the entropy density given in (5). When there is no dissipation is conserved. The most general form of the entropy current in a -dimensional relativistic theory must take the form
where we will refer to
Our analysis closely follows [43, 9]. We have described all possible first order transverse vectors, tensors and scalars in Section 2. A list of independent transverse traceless symmetric tensors, transverse vectors, scalars and pseudoscalars is reproduced in Table 1 for convenience.
Transverse pseudovectors and pseudotensors can be obtained from the above transverse vectors and tensors via
We will refer to the first order terms which are independent under the equations of motion of ideal hydrodynamics as first derivative data. In this section we will find it convenient to use two alternative bases for the first order pseudovectors and pseudoscalars. The first basis is defined in (25) and Table 1. In addition to the basis of first-order pseudovectors spanned by , , and the ’s we will also find it convenient to use a different basis of first-order pseudovectors, given in Table 2. In these bases the most general expression for the entropy current takes the form
where the ’s and the ’s are (as yet) undetermined functions of and . By including the pseudovectors and we have parametrized the longitudinal pseudovector contributions and , respectively, to . The entropy current (26) contains all possible parity-even and parity-odd vector contributions both longitudinal and transverse to the velocity field.
We note in passing that
is a divergenceless vector for arbitrary . If we add such a term to the entropy current (26), it will not contribute to entropy production but it will shift , and such that , and . The two combinations of ’s which are invariant under this shift are and . Combined with and , this means that there are only four combinations of the ’s that a priori can participate in entropy production.
The expression for the divergence of the entropy current in first order hydrodynamics can be written as a sum of products of first order data and a sum of genuine second order scalars,
By second order scalar data we mean expressions which are second order in a gradient expansion and cannot be decomposed into a product of first order terms. Equation (21) states that the divergence of the entropy current should be positive semi-definite for any flow which solves the equations of motion and with any background fields. Thus, all second order data in (28) should vanish and all first order data should arrange themselves into complete squares.
In other words, all the expressions in the divergence of the canonical part of the entropy current involve products of first order data.
All second order data in the divergence of the entropy current must vanish. Therefore the coefficients, , and must be tuned so that no second order data appears in the divergence of the non-canonical part of . Using (29) it is not difficult to show that
where is the Ricci tensor. Following an analysis similar to the one carried out in  one can show that all explicit expressions on the right hand side of (30) are genuine second order data. Since the genuine second order terms on the right hand side of (30) should vanish, we find that
There are no additional constraints that arise from demanding positivity on a curved background, so we will work in flat space from now on.
We now require that the remaining data which contributes to the divergence of the entropy current appears quadratically. We allow the undetermined variables to depend on and . To first order in the derivative expansion,
Evaluating the right hand side of (32) we find
where we have used the identities
In this expression, is a scalar, is a transverse vector, and is a transverse traceless symmetric tensor. Therefore, in (35), the first expression in the parentheses can be expanded in the basis of scalars, the second expression in the parentheses can be expanded in the basis of transverse vectors, and can be expanded in the basis of transverse traceless symmetric tensors listed in Table 1: