Exact solution of the boundary-dissipated transverse field Ising model: Structure of Liouvillian spectrum and dynamical duality

TL;DR

This paper provides an exact analytical solution for the Liouvillian spectrum of a boundary-dissipated transverse field Ising model, revealing spectral structures, phase boundaries, and a dynamical duality in relaxation behavior.

Contribution

It introduces an exact solution method for the Liouvillian spectrum of the model by mapping it to an SSH model with imaginary boundary potentials, uncovering spectral structures and duality relations.

Findings

Liouvillian spectrum has four distinct structures.

Phase boundaries are analytically determined.

A dynamical duality in relaxation dynamics is demonstrated.

Abstract

We study the boundary-dissipated transverse field Ising model described by a Lindblad Master equation and exactly solve its Liouvillian spectrum in the whole parameter space. By mapping the Liouvillian into a Su-Schrieffer-Heeger model with imaginary boundary potentials under a parity constraint, we solve the rapidity spectrum analytically and thus construct the Liouvillian spectrum strictly with a parity constraint condition. Our results demonstrate that the Liouvillian spectrum displays four different structures, which are characterized by different numbers of segments. By analyzing the properties of rapidity spectrum, we can determine the phase boundaries between different spectrum structures analytically and prove the Liouvillian gap fulfilling a duality relation in the weak and strong dissipation region. Furthermore, we unveil the existence of a dynamical duality, i.e., the long-time relaxation dynamics exhibits almost the same dynamical behavior in the weak and strong dissipation region as long as the duality relation holds true.