Eigenstate condensation in quantum systems with finite-dimensional Hilbert spaces

Random quantum states drawn from the Haar ensemble with a constraint on the energy expectation value E_av = langle ψ| H | ψrangle display eigenstate condensation: for E_av below a critical value E_c, they develop macroscopic overlap with the ground state. We 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 macroscopic 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 macroscopic overlap with the most highly excited state. In local spin systems the ground-state and anti-ground-state phases approach the middle of the spectrum as 1/\[system size\], but - because the condensation phase transitions have exponential, rather than polynomial, finite-size scaling - the crossover becomes exponentially sharp in system size and the high-temperature phase is best understood as an extended phase.