Introduction to the nonequilibrium functional renormalization group

TL;DR

This paper introduces the nonequilibrium functional renormalization group framework, emphasizing real-time analysis for quantum systems out of equilibrium, and explores fixed points and turbulence phenomena.

Contribution

It develops a nonequilibrium FRG approach using the Schwinger-Keldysh formalism, extending the equilibrium methods to out-of-equilibrium quantum systems.

Findings

Derived hierarchy of fixed point solutions from vacuum to nonequilibrium.

Identified scale-invariance at fixed points independent of initial conditions.

Connected nonequilibrium fixed points to turbulence in quantum field theory.

Abstract

In these lectures we introduce the functional renormalization group out of equilibrium. While in thermal equilibrium typically a Euclidean formulation is adequate, nonequilibrium properties require real-time descriptions. For quantum systems specified by a given density matrix at initial time, a generating functional for real-time correlation functions can be written down using the Schwinger-Keldysh closed time path. This can be used to construct a nonequilibrium functional renormalization group along similar lines as for Euclidean field theories in thermal equilibrium. Important differences include the absence of a fluctuation-dissipation relation for general out-of-equilibrium situations. The nonequilibrium renormalization group takes on a particularly simple form at a fixed point, where the corresponding scale-invariant system becomes independent of the details of the initial density matrix. We discuss some basic examples, for which we derive a hierarchy of fixed point solutions with increasing complexity from vacuum and thermal equilibrium to nonequilibrium. The latter solutions are then associated to the phenomenon of turbulence in quantum field theory.