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

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.