Eigenstate condensation in quantum systems with finite-dimensional Hilbert spaces

TL;DR

This paper investigates eigenstate condensation in finite-dimensional quantum systems, revealing three distinct phases and demonstrating how phase transitions become exponentially sharp with increasing system size.

Contribution

It characterizes the phase diagram of eigenstate condensation in finite systems, including the nature of phase transitions and finite-size effects.

Findings

Identifies three phases: ground-state, high-temperature, and anti-ground-state.

Shows phase transition sharpness increases exponentially with system size.

Describes the behavior of local spin systems across different phases.

Abstract

Random quantum states drawn from the Haar ensemble with a constraint on the energy expectation value $E_{\mathrm{av}} = \langle \psi | H | \psi\rangle$ display \textit{eigenstate condensation}: for $E_{\mathrm{av}}$ 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 $1/[\text{system size}]$, 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.