Equilibrium Fluctuations in Maximally Noisy Extended Quantum Systems. (arXiv:1811.09427v3 [cond-mat.stat-mech] UPDATED)

We introduce and study a class of models of free fermions hopping between
neighbouring sites with random Brownian amplitudes. These simple models
describe stochastic, diffusive, quantum, unitary dynamics. We focus on periodic
boundary conditions and derive the complete stationary distribution of the
system. It is proven that the generating function of the latter is provided by
the Harish-Chandra-Itzykson-Zuber integral which allows us to access all
fluctuations of the system state. The steady state is characterized by non
trivial correlations which have a topological nature. Diagrammatic tools
appropriate for the study of these correlations are presented. In the
thermodynamic large system size limit, the system approaches a non random
equilibrium state plus occupancy and coherence fluctuations of magnitude
scaling proportionally with the inverse of the square root of the volume. The
large deviation function for those fluctuations is determined. Although
decoherence is effective on the mean steady state, we observe that sub-leading
fluctuating coherences are dynamically produced from the inhomogeneities of the
initial occupancy profile.

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