A variational approach for many-body systems at finite temperature

TL;DR

This paper presents a variational method based on a non-linear flow equation for density matrices, capable of analyzing equilibrium states in many-body systems, including complex fermionic and bosonic states, with applications to superconductivity and polaron physics.

Contribution

It introduces a novel variational approach using a differential flow equation for density matrices applicable to a broad class of many-body states.

Findings

Successfully benchmarks on BCS lattice Hamiltonian.

Reproduces phase transition in the Holstein model.

Identifies phase separation between superconducting and charge-density wave phases.

Abstract

We introduce a non-linear differential flow equation for density matrices that provides a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal state. We use this equation to build a variational approach for analyzing equilibrium states of many-body systems and demonstrate that it can be applied to a broad class of states, including all bosonic and fermionic Gaussian states, as well as their generalizations obtained by unitary transformations, such as polaron transformations, in electron-phonon systems. We benchmark this method with a BCS lattice Hamiltonian and apply it to the Holstein model in two dimensions. For the latter, our approach reproduces the transition between the BCS pairing regime at weak interactions and the polaronic regime at stronger interactions, displaying phase separation between superconducting and charge-density wave phases.