Linear response and exact hydrodynamic projections in Lindblad equations with decoupled Bogoliubov hierarchies

TL;DR

This paper analyzes a class of spinless-fermion Lindblad equations with decoupled hierarchies, deriving exact hydrodynamic projections and linear response functions, revealing diffusive late-time behavior and connections to integrability.

Contribution

It introduces a method to obtain exact hydrodynamic projections and linear response functions for Lindblad equations with decoupled BBGKY hierarchies, including integrable models.

Findings

Late-time behavior is characterized by diffusive dynamics leading to infinite temperature steady states.

Exact hydrodynamic projections of quadratic fermionic operators are derived.

Linear response functions in non-equilibrium Lindbladian dynamics are explicitly computed.

Abstract

We consider a class of spinless-fermion Lindblad equations that exhibit decoupled BBGKY hierarchies. In the cases where particle number is conserved, their late time behaviour is characterized by diffusive dynamics, leading to an infinite temperature steady state. Some of these models are Yang-Baxter integrable, others are not. The simple structure of the BBGKY hierarchy makes it possible to map the dynamics of Heisenberg-picture operators on few-body imaginary-time Schr\"odinger equations with non-Hermitian Hamiltonians. We use this formulation to obtain exact hydrodynamic projections of operators quadratic in fermions, and to determine linear response functions in Lindbladian non-equilibrium dynamics.