Thermodynamics of polarized relativistic matter

Journal of High Energy Physics, Jul 2016

We give the free energy of equilibrium relativistic matter subject to external gravitational and electromagnetic fields, to one-derivative order in the gradients of the external fields. The free energy allows for a straightforward derivation of bound currents and bound momenta in equilibrium. At leading order, the energy-momentum tensor admits a simple expression in terms of the polarization tensor. Beyond the leading order, electric and magnetic polarization vectors are intrinsically ambiguous. The physical effects of polarization, such as the correlation between the magneto-vortically induced surface charge and the electro-vortically induced surface current, are not ambiguous.

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Thermodynamics of polarized relativistic matter

HJE Thermodynamics of polarized relativistic matter Pavel Kovtun 0 1 2 0 PO Box 1700 STN CSC, Victoria BC, V8W 2Y2 , Canada 1 Department of Physics and Astronomy, University of Victoria 2 @ T =T is the acceleration We give the free energy of equilibrium relativistic matter subject to external gravitational and electromagnetic elds, to one-derivative order in the gradients of the external elds. The free energy allows for a straightforward derivation of bound currents and bound momenta in equilibrium. At leading order, the energy-momentum tensor admits a simple expression in terms of the polarization tensor. Beyond the leading order, electric and magnetic polarization vectors are intrinsically ambiguous. The physical e ects of polarization, such as the correlation between the magneto-vortically induced surface charge and the electro-vortically induced surface current, are not ambiguous. Holography and quark-gluon plasmas; Thermal Field Theory 1 Introduction 2 Free energy 3 Leading order in the derivative expansion Next order in the derivative expansion Weak electromagnetic elds Strong electromagnetic elds: 1+1 dimensions Strong electromagnetic elds: 2+1 dimensions Strong electromagnetic elds: 3+1 dimensions 5 Summary 1 Introduction We would like to understand collective macroscopic behaviour of matter subject to external elds. In the high-temperature limit this is often captured by classical hydrodynamics. The ingredients that go into writing down the hydrodynamic equations are: the identi cation of relevant variables (conserved densities, order parameters), the derivative expansion (small gradients near equilibrium), and symmetry constraints. The hydrodynamic equations are modi ed when the system is subject to external electric and magnetic elds. The latter will induce polarization (electric, magnetic, or both) in a uid, and as a result the transport properties of the uid will change. Our focus here will be on isotropic relativistic matter because a) electromagnetic elds are intrinsically relativistic, b) relativistic uids have more symmetry than non-relativistic uids, and c) relativistic uids have been a subject of much recent attention in the literature due to their appearance in heavy-ion physics [1, 2], in gravitational physics, through the holographic duality [3], and even in condensed matter cedure for obtaining the energy-momentum tensor and the current density for stationary equilibrium polarized matter subject to external gravitational and electromagnetic elds. We will nd simple expressions for \bound" currents, including equilibrium surface currents and surface momenta. Let us start with the standard description of equilibrium thermodynamics without external elds. In the grand canonical ensemble at temperature T0 = 1= 0 and chemical potential 0, extensivity in the large-volume limit dictates that the logarithm of the grandcanonical partition function Z[T0; 0] is proportional to the d-dimensional spatial volume, 0, see e.g. [7]. In the path integral action, the fundamental elds of the microscopic theory can then be coupled to time-independent external sources: the (Euclidean) metric gE and the (Euclidean) gauge eld AE. The gauge eld couples to the conserved current, whose time component is the charge density corresponding to the chemical potential. See ref. [8] for a convenient parametrization of the Euclidean sources gE and AE. The Euclidean path integral gives rise to the partition function Z = Z[T0; 0; gE; AE], where T0 = 1= 0 is the coordinate periodicity of the Euclidean time. We assume that the coupling to time-independent external sources leaves the system in equilibrium, so that no entropy is produced. The temperature and the chemical potential will be altered by the external sources and are not uniform any more. For example, the equilibrium temperature becomes T (x) = T0=pg0E0(x) [6]. Similarly, the chemical potential will be shifted by the time component of the external gauge eld. We can write W = i ln Z as W [T0; 0; gE; AE] = i 0 Z ddx pgE F (T0; 0; gE; AE) ; (1.1) where pgE is the square root of the determinant of gE , and F is the negative of the grand canonical free energy density. In at space and without external gauge elds, F reduces to the pressure P , and in general F is a complicated function of the spatially varying external sources. In a slight abuse of terminology, we will refer to F as the free energy density, and to W as the free energy. Varying W with respect to a time-independent source gives rise to a zero-frequency insertion in the Euclidean path integral of the operator coupled to the source. The relevant operators are the energy-momentum tensor (coupled to the metric), and the conserved current (coupled to the gauge eld). Thus W is the generating functional for zero-frequency correlation functions of the energy-momentum tensor and the current in equilibrium. The Euclidean external sources gE and AE may be \un-Wick-rotated" to Minkowski time to obtain the (...truncated)


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Kovtun, Pavel. Thermodynamics of polarized relativistic matter, Journal of High Energy Physics, 2016, pp. 1-25, Volume 2016, Issue 7, DOI: 10.1007/JHEP07(2016)028