Eigenstate condensation in quantum systems with finite-dimensional Hilbert spaces

TL;DR

The paper studies eigenstate condensation for Haar-random quantum states constrained by an energy expectation value $E_{\mathrm{av}}$, across various finite-dimensional Hamiltonians. It identifies three phases—ground-state, high-temperature, and anti-ground-state—separated by critical densities $E_c$ (or $\varepsilon_c = E_c / V$ for extensive systems) at which macroscopic weight shifts to the ground or anti-ground state. Analytically, the average weight on eigenstates satisfies $p_\alpha \propto \left(1+\beta E_\alpha\right)^{-1}$ with $\beta$ fixed by the constraint, so crossing $E_c$ yields a non-analytic buildup of ground-state occupation. Numerically, GOE/GUE ensembles yield constant $\varepsilon_{c-}$ and $\varepsilon_{c+}$ in the large-size limit, while local spin models drive these toward the spectrum center as $\sim 1/V$, producing exponential finite-size scaling and an effectively extended high-temperature phase; the study relies on nested sampling with Galilean Monte Carlo to access up to $V \approx 14$ spins.

Abstract

Random quantum states drawn from the Haar ensemble with a constraint on the energy expectation value $E_{\mathrm{av}} = \langle ψ| H | ψ\rangle$ 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.

Figures (8)

• Figure 1: Eigenstate condensation in five models: GOE and GUE random matrix Hamiltonians; the paramagnetic and critical transverse field Ising model (TFIM) in 1D, and the paramagnetic TFIM in 2D. Left column: (heuristic) density of states. Center: combined ground state and anti-ground-state probability $p_\text{g.s.} + p_\text{anti-g.s.} = |\braket{\text{g.s.} | \psi}|^2 + |\braket{\text{anti-g.s.}|\psi}|^2$ across system size; in each case we see $O(1)$ weight on the ground state for energy densities $\varepsilon_\text{av} < \varepsilon_{c-}$ and on the anti-ground-state for $\varepsilon_\text{av} > \varepsilon_{c+}$. Right: (heuristic) phase diagrams. Top row: random matrices have a semicircle density of states; consequently the critical energy densities approach $\varepsilon_{c-} = 1/2, \varepsilon_{c+} = 3/2$ for large systems. Bottom row: local spin systems have Gaussian densities of states with energy-density variance shrinking with system size $V$ like $1/V$. Consequently the ground-state and anti-ground-state phases approach the middle of the spectrum.
• Figure 2: Average weight on energy eigenstates for $400$ random states with energy $E_{\text{av}} \approx {33}$ in the $V = {8}$ paramagnetic TFIM. The weight is well-described by $p_k \propto [1 + \beta E_k]^{-1}$, but not by the Boltzmann probability $p_k \propto e^{-\beta' E_k}$ (best fit $\beta, \beta'$ shown in blue).
• Figure 3: Critical energy density$\varepsilon_{c-}$ from \ref{['eq:Ec-sum']} for paramagnetic and critical 1D Ising models and a paramagnetic 2D Ising model, together with the large-system limit for GOE and GUE matrices (Eq. \ref{['eq:GbE-Ec']}). For the spin models a moment expansion (Eq. \ref{['eq:Ec-large-L']}; dotted lines) predicts that the critical energy density approaches the spectrum center $\varepsilon_\infty$ as $\varepsilon_\infty - \varepsilon_{c-} \propto 1/V$. For the Gaussian random matrix ensembles the Wigner semicircle predicts $\varepsilon_{c-} = 1/2$ independent of system size, though at finite size the random matrix $\varepsilon_{c\pm}$ exhibit considerable fluctuations (not shown).
• Figure 4: Scaling collapse of $p_\text{g.s.}$ for the paramagnetic 1D TFIM with system sizes $V = 10, 11, \dots, 79, 80$. $p_\text{g.s.}$ and $E$, and $E_c$ are computed semianalytically via \ref{['eq:pk-Ek']} and \ref{['eq:Ec-sum']}. The scaling collapse agrees well with the moment-expansion prediction \ref{['eq:scaling']}. Inset: unrescaled data.
• Figure 5: Ground and first excited state probabilities for the ferromagnetic 1D TFIM $J = 1$, $h = {0.5}$ at $L = 11$. At $\varepsilon_{c1} \approx 0.84$ the system undergoes a transition into a regime with $p_\text{g.s.} = p_1 = 0.5(1 - \varepsilon / \varepsilon_{c1})$, i.e. $O(1)$ weight on the nearly-degenerate ground space. At $\varepsilon_{c0} \approx 0.12$ it undergoes a crossover to a regime with $p_\text{g.s.} \approx 1 - \varepsilon/\varepsilon_{c0}$ and $p_1 \approx \varepsilon / \varepsilon_{c0}$. $p_\text{g.s.}$ and $E$ are calculated semianalytically via Eq. \ref{['eq:pk-Ek']}.