Thermal states of random quantum many-body systems

We study a distribution of thermal states given by random Hamiltonians with a local structure. We show that the ensemble of thermal states monotonically approaches the unitarily invariant ensemble with decreasing temperature if all particles interact according to a single random interaction and achieves a state $t$-design at temperature $O(1/\log(t))$. For the system where the random interactions are local, we show that the ensemble achieves a state $1$-design. We then provide numerical evidence indicating that the ensemble undergoes a phase transition at finite temperature.