Eigenstate condensation in quantum systems with finite-dimensional Hilbert spaces

Quantum states drawn at random with constrained energy expectation value $E_\av = \braket{\psi | H | \psi}$ display eigenstate condensation: for $E_{av}$ below a critical value $E_c$ they develop O(1) overlap with the ground state. We use analytical calculations and numerical methods drawn from the Bayesian statistics community to study eigenstate condensation in systems with finite-dimensional Hilbert spaces. These systems display three phases: a ground-state phase, in which energy-constrained random states have $O(1)$ overlap with the ground state; a high-temperature phase, in which they have exponentially small overlap with each energy eigenstate; and an anti-ground-state phase, in which they have $O(1)$ overlap with the most highly excited state. In local spin systems the ground- and anti-ground-state phases approach mid-spectrum as 1/[system size], but the crossover becomes exponentially sharp in system size and the high-temperature phase is best understood as an extended phase. We also comment on prospects for realizing these states in physical systems: although they are not easily realizable via Lindblad dynamics, because they are far from thermal, they may be realizable in mixed quantum-classical or -semiclassical systems.