Rapid mixing in long-range Lindbladians and static properties of their fixed points

The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information(CMI), with the latter encapsulated in the notion of Markov length. In this talk, we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generator: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove i)that local Lindbladians satisfying
(global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperature.