Exact solution of Markovian master equations for quadratic fermi systems: thermal baths, open XY spin chains, and non-equilibrium phase transition

TL;DR

This paper develops an exact method to solve Markovian master equations for quadratic fermionic systems, enabling detailed analysis of non-equilibrium phenomena, phase transitions, and heat transport in open quantum systems.

Contribution

It introduces a unified exact approach using third quantization for solving Redfield and Lindblad equations in quadratic fermionic systems, including time-dependent problems.

Findings

Non-equilibrium quantum phase transition persists under thermal driving.

Long-range magnetic correlations are hypersensitive to external parameters.

Negative thermal conductance occurs at strong thermal driving.

Abstract

We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of 4n x 4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how to compute all physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time dependent problems. The general formalism is then applied to study a thermally driven open XY spin 1/2 chain. We find that recently proposed non-equilibrium quantum phase transition in the open XY chain survives the thermal driving within the Redfield model. In particular, the phase of long-range magnetic correlations can be characterized by hypersensitivity of the non-equilibrium-steady state to external (bath or bulk) parameters. Studying the heat transport we find negative thermal conductance for sufficiently strong thermal driving, as well as non-monotonic dependence of the heat current on the strength of the bath coupling.