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}=〈ψ|H|ψ〉 display eigenstate condensation: For E_{av} below a critical value E_{c-} or above E_{c+}, they develop macroscopic overlap with the ground state or anti-ground state. We use analytical calculations and state-of-the-art numerical methods to investigate the eigenstate condensation phase transition. We give compact expressions for the critical energies, derive an analytical scaling form for the order parameter in systems with an extensive free energy, show that in those systems the phase transition itself has exponential rather than power-law finite-size scaling, and test these results with large-scale numerical sampling in random matrices and a range of local spin Hamiltonians. In local spin systems the two critical energies approach the middle of the spectrum as 1/V with V the number of spins. Our scaling form, however, demonstrates that the condensation phase transitions have exponential, rather than polynomial, finite-size scaling, which justifies treating the high-temperature phase as an extended phase.