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A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems

Paolo Glorioso, Michael Crossley, Hong Liu

TL;DR

The paper introduces a holographic prescription for computing real-time Schwinger-Keldysh correlators in non-equilibrium states via a simple analytic continuation around a dynamical black-hole horizon. It demonstrates that, for slowly varying sources, one can perform a derivative expansion to obtain the generating functional and, from gravity, derive a non-equilibrium diffusion effective action consistent with the CGL framework and dynamical KMS symmetry. Through a scalar two-point function analysis and an explicit gravity-based EFT of diffusion, the authors show both practical simplifications and conceptual clarity in isolating hydrodynamic modes. The results pave the way for higher-point real-time calculations and a full dissipative hydrodynamics holographic derivation in non-equilibrium settings.

Abstract

We develop a prescription for computing real-time correlation functions defined on a Schwinger-Keldysh contour for non-equilibrium systems using gravity. The prescription involves a new analytic continuation procedure in a black hole geometry which can be dynamical. For a system with a slowly varying horizon, the continuation enables computation of the Schwinger-Keldysh generating functional using derivative expansion, drastically simplifying calculations. We illustrate the prescription with two-point functions for a scalar operator. We then use it to derive from gravity the recently proposed non-equilibrium effective action for diffusion.

A prescription for holographic Schwinger-Keldysh contour in non-equilibrium systems

TL;DR

The paper introduces a holographic prescription for computing real-time Schwinger-Keldysh correlators in non-equilibrium states via a simple analytic continuation around a dynamical black-hole horizon. It demonstrates that, for slowly varying sources, one can perform a derivative expansion to obtain the generating functional and, from gravity, derive a non-equilibrium diffusion effective action consistent with the CGL framework and dynamical KMS symmetry. Through a scalar two-point function analysis and an explicit gravity-based EFT of diffusion, the authors show both practical simplifications and conceptual clarity in isolating hydrodynamic modes. The results pave the way for higher-point real-time calculations and a full dissipative hydrodynamics holographic derivation in non-equilibrium settings.

Abstract

We develop a prescription for computing real-time correlation functions defined on a Schwinger-Keldysh contour for non-equilibrium systems using gravity. The prescription involves a new analytic continuation procedure in a black hole geometry which can be dynamical. For a system with a slowly varying horizon, the continuation enables computation of the Schwinger-Keldysh generating functional using derivative expansion, drastically simplifying calculations. We illustrate the prescription with two-point functions for a scalar operator. We then use it to derive from gravity the recently proposed non-equilibrium effective action for diffusion.

Paper Structure

This paper contains 13 sections, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A new analytic continuation procedure
  3. A scalar example
  4. Equilibrium
  5. Slowly-varying horizon
  6. Effective field theory of diffusion from gravity
  7. Review of EFT for diffusion
  8. Obtaining generating functional $W[A_1, A_2]$
  9. Prescription for finding $I_{\rm EFT} [B_1, B_2]$
  10. Explicit bulk solution
  11. Near-horizon behavior to all orders
  12. The effective action
  13. Discussion and conclusions

Figures (3)

  • Figure 1: A Schwinger-Keldysh contour, or often referred to as a closed time path. Operators inserted on the upper and lower segments are labeled respectively by indices $1$ and $2$.
  • Figure 2: The complexified gravity spacetime corresponding to the Schwinger-Keldysh contour in Fig. \ref{['fig:sk']}. The two segments of gravity spacetime are identified with the two segments of the Schwinger-Keldysh contour. It should be understood that in the above plot the circle around $r_h$ is infinitesimal and the two segments along the real $r$-axis have infinitesimal imaginary parts. The two boundaries are denoted respectively as $\infty_{1,2}$ with possible sources $\phi_{1,2}$.
  • Figure 3: A boundary condition \ref{['hbd0']} is imposed at the "stretched horizon" $r=r_c$. As will be discussed below, such a boundary condition implies presence of a nonzero surface charge $\rho_c$ at $r=r_c$, with a discontinuity in the radial component of the electric field at $r_c$.