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

The Lindblad equation is a well-known quantum master equation that describes the evolution of open quantum systems. In general, the Lindblad equation is more challenging to analyze than the Schrödinger equation for closed systems, as one has to deal with the space of operators rather than the Hilbert space. One approach to this difficulty is to construct exactly solvable models. However, very few exact results are available for open many-body systems, although many such examples are known in closed many-body systems.

In this talk, we construct a solvable example of a quantum Ising model with bulk dissipation [1]. By vectorizing the density matrix, its Liouvillian is mapped to a non-Hermitian analog of the Ashkin-Teller model, further reducing to the staggered XXZ chain with pure-imaginary anisotropy parameters. Using this mapping, we show that the steady states are doubly degenerate and these two are unique. Moreover, we obtain the exact formula for the Liouvillian gap, the inverse relaxation time, for a particular set of parameters corresponding to a uniform XXZ chain. The result implies a kind of phase transition for the first decay mode.

[1] N. Shibata and H. Katsura, Phys. Rev. B 99, 224432 (2019)