Exact dynamics of quantum dissipative $XX$ models: Wannier-Stark localization in the fragmented operator space

TL;DR

This paper investigates the dissipative dynamics of a 1D quantum spin chain, revealing operator space fragmentation and Wannier-Stark localization, with exact solutions demonstrating complex decay behaviors and phase transitions.

Contribution

It introduces a class of dissipative terms that fragment the operator space and provides exact solutions showing Wannier-Stark localization and exceptional points in the dissipative quantum spin chain.

Findings

Operator space fragments into disjoint subspaces under dissipation.

Exact solutions exhibit Wannier-Stark localization in operator subspaces.

Identification of an exceptional point at a critical dissipation strength.

Abstract

We address dissipative dynamics of the one-dimensional nearest-neighbour $XX$ spin-$1/2$ chain governed by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the absence of dissipation the model is integrable. We identify a broad class of dissipative terms that generically destroy integrability but leave the operator space of the model fragmented into an extensive number of dynamically disjoint subspaces of varying dimensions. In sufficiently small subspaces the GKSL equation in the Heisenberg representation can be easily solved, sometimes in a closed analytical form. We provide an example of such an exact solution for a specific choice of dissipative terms. It is found that observables experience the Wannier-Stark localization in the corresponding operator subspace. As a result, the expectation values of the observables are linear combinations of essentially a few discrete decay modes, the long time dynamics being governed by the slowest mode. We examine the complex Liouvillian eigenvalue corresponding to this latter mode as a function of the dissipation strength. We find an exceptional point at a critical dissipation strength that separates oscillating and non-oscillating decay. We also describe a different type of dissipation that leads to a single decay mode in the whole operator subspace. Finally, we point out that our exact solutions of the GKSL equation entail exact solutions of the Schr\"odinger equation describing the quench dynamics in closed spin ladders dual to the dissipative spin chains.