# Emergent Supersymmetry in Local Equilibrium Systems

## TL;DR

This paper demonstrates that classical systems in local thermal equilibrium inherently exhibit emergent supersymmetry at low energies, linking non-equilibrium effective field theory, BRST symmetry, and dynamical KMS symmetry.

## Contribution

It establishes a theoretical connection between local equilibrium conditions and emergent supersymmetry, providing a new framework for analyzing non-equilibrium systems.

## Key findings

- Supersymmetry emerges in classical local equilibrium systems.
- A theorem links the special dynamical KMS condition to supersymmetrizability.
- Explicit examples include dynamical critical phenomena and hydrodynamics.

## Abstract

Many physical processes we observe in nature involve variations of macroscopic quantities over spatial and temporal scales much larger than microscopic molecular collision scales and can be considered as in local thermal equilibrium. In this paper we show that any classical statistical system in local thermal equilibrium has an emergent supersymmetry at low energies. We use the framework of non-equilibrium effective field theory for quantum many-body systems defined on a closed time path contour and consider its classical limit. Unitarity of time evolution requires introducing anti-commuting degrees of freedom and BRST symmetry which survive in the classical limit. The local equilibrium is realized through a $Z_2$ dynamical KMS symmetry. We show that supersymmetry is equivalent to the combination of BRST and a specific consequence of the dynamical KMS symmetry, to which we refer as the special dynamical KMS condition. In particular, we prove a theorem stating that a system satisfying the special dynamical KMS condition is always supersymmetrizable. We discuss a number of examples explicitly, including model A for dynamical critical phenomena, a hydrodynamic theory of nonlinear diffusion, and fluctuating hydrodynamics for relativistic charged fluids.