Exact solution of the fully-connected dissipative transverse Ising model

The dissipative transverse-field Ising model is central in the study of open quantum systems, not just due to its close connection with its well-studied closed-system analogue, but also thanks to its ability to be natively realized in a variety of atomic physics platforms, namely ion traps (long-range limit) and arrays of Rydberg atoms (short-range limit). Here, we present a simple closed-form solution for the steady state of the transverse-field Ising model subject to local spontaneous emission, in the limit that the Ising interaction is infinitely-long ranged. The exact solution exploits a "hidden" symmetry in the dissipative dynamics, which is phrased in terms of the square root of the steady-state density matrix [1]. This symmetry persists even when the external fields are inhomogeneous, and also yields closed-form solutions for all stationary moments, such as local magnetization and spin-spin correlations.

[1] D. Roberts et al., PRX Quantum 2, 020336 (2021)