Dissipative quantum Ising chain as a non-Hermitian Ashkin-Teller model

TL;DR

This paper explores a quantum Ising chain with dissipation, mapping it to a non-Hermitian Ashkin-Teller model and analyzing its eigenstates, steady states, and phase transition behavior through exact formulas for decay modes.

Contribution

It introduces a novel mapping of a dissipative quantum Ising chain to a non-Hermitian Ashkin-Teller model and derives exact decay mode formulas, revealing phase transition features.

Findings

Steady states are completely mixed in each parity sector.

Exact formula for Liouvillian gap g in the thermodynamic limit.

Identification of a cusp in g indicating a phase transition.

Abstract

We study a quantum Ising chain with tailored bulk dissipation, which can be mapped onto a non-Hermitian Ashkin-Teller model. By exploiting the Kohmoto-den Nijs-Kadanoff transformation, we further map it to a staggered XXZ spin chain with pure-imaginary anisotropy parameters. This allows us to study the eigenstates of the original Liouvillian in great detail. We show that the steady state in each parity sector is a completely mixed state. The uniqueness of each is proved rigorously. We then study the decay modes on the self-dual line corresponding to the uniform XXZ chain and obtain an exact formula for the Liouvillian gap $ g $, the inverse relaxation time, in the thermodynamic limit. The gap $ g $ as a function of dissipation strength $ \Delta $ has a cusp, implying a kind of phase transition for the first decay mode.