All order effective action for charge diffusion from Schwinger-Keldysh holography

Yanyan Bu,Michael Lublinsky,Tuna Demircik

doi:10.1007/jhep05(2021)187

Abstract

An effective action for diffusion of a conserved U(1) charge is derived to all orders in the derivative expansion within a holographic model dual to the Schwinger-Keldysh closed time path. A systematic approach to solution of the 5D Maxwell equations in a doubled Schwarzschild-AdS5 black brane geometry is developed. Constitutive relation for the stochastic charge current is shown to have a term induced by thermal fluctuations (coloured noise). All transport coefficient functions parameterising the effective action and constitutive relations are computed analytically in the hydrodynamic expansion, and then numerically for finite momenta.

Highlights

Hydrodynamics [2,3,4] is an effective long-time long-distance description of many-body systems at nonzero temperature
The derivative expansion is fixed by thermodynamics and symmetries, up to a finite number of transport coefficients (TCs) such as viscosity and diffusion coefficients
When discussing the hydrodynamic expansion, one has to be very careful with the non-commutativity of the hydrodynamic limit vs. the near horizon limit

Summary

Introduction

Hydrodynamics [2,3,4] is an effective long-time long-distance description of many-body systems at nonzero temperature. The entire dynamics of a microscopic theory is reduced to that of conserved macroscopic currents, such as expectation values of energy-momentum tensor or charge current operators computed in a locally near equilibrium thermal state. The derivative expansion is fixed by thermodynamics and symmetries, up to a finite number of transport coefficients (TCs) such as viscosity and diffusion coefficients. The latter are not calculable from hydrodynamics itself, but have to be determined from underlying microscopic theory or extracted from experiments. Causality is restored only after all (infinite) order gradients are resummed, in a way providing a UV completion of the “old” hydrodynamics. A compact way of organising the resummation is by introducing, instead of order by order transport coefficients, momenta-dependent transport coefficient functions (TCFs) [5]

Objectives

Results

Conclusion