The Schwinger-Keldysh Coset Construction

TL;DR

This paper extends the coset construction to the Schwinger-Keldysh formalism, enabling systematic derivation of effective actions for Nambu-Goldstone modes in mixed states and thermal environments, with explicit examples in magnetic systems.

Contribution

It introduces a Schwinger-Keldysh coset construction for Nambu-Goldstone modes, applicable to mixed states and thermal systems, without requiring additional diffusive symmetries.

Findings

Constructed Schwinger-Keldysh effective actions for Nambu-Goldstone modes.

Demonstrated the approach with finite-temperature correlators in magnetic systems.

Clarified the implementation of the dynamical KMS symmetry in different magnetic phases.

Abstract

The coset construction is a tool for systematically building low energy effective actions for Nambu-Goldstone modes. This technique is typically used to compute time-ordered correlators appropriate for $S$-matrix computations for systems in their ground state. In this paper, we extend this technique to the Schwinger-Keldysh formalism, which enables one to calculate a wider variety of correlators and applies also to systems in a mixed state. We focus our attention on internal symmetries and demonstrate that, after identifying the appropriate symmetry breaking pattern, Schwinger-Keldysh effective actions for Nambu-Goldstone modes can be constructed using the standard rules of the coset construction. Particular emphasis is placed on the thermal state and ensuring that correlators satisfy the KMS relation. We also discuss explicitly the power counting scheme underlying our effective actions. We comment on the similarities and differences between our approach and others that have previously appeared in the literature. In particular, our prescription does not require the introduction of additional ``diffusive'' symmetries and retains the full non-linear structure generated by the coset construction. We conclude with a series of explicit examples, including a computation of the finite-temperature two-point functions of conserved spin currents in non-relativistic paramagnets, antiferromagnets, and ferromagnets. Along the way, we also clarify the discrete symmetries that set antiferromagnets apart from ferromagnets, and point out that the dynamical KMS symmetry must be implemented in different ways in these two systems.