Entanglement dynamics of many-body quantum states: sensitivity to system conditions and a hidden universality

TL;DR

The paper presents a framework where many-body Hamiltonians represented by multiparametric Gaussian ensembles induce a state-JPDF for eigenfunctions, whose entanglement statistics are governed by a single complexity parameter $Y$. By deriving diffusion-like evolution equations for the state components and Schmidt eigenvalues, it shows that entanglement measures such as the von Neumann entropy $R_1$ depend on a reduced variable $\Lambda = (Y - Y_0)/\Omega^2$ and that universal paths emerge when data are rescaled by $N\Lambda$, independent of microscopic ensemble details. Numerical studies on the quantum random energy model and random-field Heisenberg model validate the theory, revealing collapses of $\langle R_1\rangle$ and its variance across energy scales and system sizes, and exposing finite-size scaling with a possible critical regime characterized by multifractal eigenfunctions. The approach provides a route to classify eigenstates into universality classes and to engineer states progressively toward Haar randomness under fixed global symmetries, with potential implications for quantum state design and non-equilibrium dynamics. Overall, the work links ensemble-level control to microscopic entanglement properties via a unifying complexity parameter, offering a powerful lens for studying entanglement dynamics in complex quantum systems.

Abstract

We consider physical Hamiltonians that can be represented by the multiparametric Gaussian ensembles, theoretically derive the state ensembles for its eigenstates and analyze the effect of varying system conditions on its bipartite entanglement entropy. Our approach leads to a single parametric based common mathematical formulation for the evolution of the entanglement statistics of different states of a given Hamiltonian or different Hamiltonians subjected to same symmetry constraints. The parameter turns out to be a single functional of the system parameters and thereby reveals a deep web of connection hidden underneath different quantum states.

Figures (8)

• Figure 1: Analysis of the RHS of the eq. \ref{['evoleqr2']}. We compare the $|\text{cov}(R_0, R_1)|$ (blue) and the $\langle \delta{R_1^2} \rangle$ (red) for the eigenstates of the QREM at $E=0$ for $L=14$, and with the x-axis on a log scale. As can be seen, the quantities are qualitatively quite similar, and that in the ergodic regime the former tend to zero. The diverging variance of $R_1$ is therefore a consequence of the diverging covariance $\text{cov}(R_0, R_1)$. (inset) We show the dynamics of $Q - \langle R_1^2 \rangle$ (ref. eq. \ref{['evoleqr2']}), which is also diverging but $\sim 1$ as $\Lambda \to \infty$, and hence dominates the covariance term in that limit. Similar results are obtained for the RFHM.
• Figure 2: Average and variance of von Neumann Entropy in the QREM. (a) The dynamics of the average and (b) the variance of the von Neumann entropy $R_1$, for the system size $L = 14$ over $1000$ disorder realizations, and $200$ eigenstates per realizations at different energy scales ($E$) is shown. The y-axes in (b) has been rescaled by the maximum for a better comparison, as the maximum variance is energy dependent (see Fig. \ref{['fig_var_r1:qrem-rfhm']}). As can be seen, the evolution curves for different energies overlap when plotted with $N \Lambda$, as opposed to with the system parameter $b$ (see inset). The $\Lambda$ for the respective measures is shown in the Table \ref{['tab:chi']}.
• Figure 3: Distribution of von Neumann entropy in the QREM. The distribution of $R_1$ (on a log scale) in the localized, non-ergodic, and ergodic regimes is shown. The different regimes are solely characterized by fixing $N \Lambda$, letting the parameters $b$ and the energy $E$ vary; see Table \ref{['tab:lambda-dist']} for details. The difference between the distributions for eigenstates at the boundary and bulk is due to the different in their variance (Fig. \ref{['fig_var_r1:qrem-rfhm']}) although qualitatively they are the same.
• Figure 4: Variance of von Neumann entropy. The figure displays the evolution of the von Neumann entropy variance ($\langle \delta R_1^2 \rangle$) for (a) the QREM and (b) the RFHM. In contrast to Figs. \ref{['fig_r1:qrem']}(b) and \ref{['fig_r1_delta:rfhm']}(b), the y-axes in this case is not rescaled by their maximum. Clearly, the maximum variance for the higher energies in (a) and for smaller anisotropy parameter $D$ in (b) is different from the other cases. Consequently, the corresponding distributions of $R_1$ in Figs. \ref{['fig_r1_dist:qrem']} and \ref{['fig_r1_dist:rfhm']}, also show deviations; nevertheless, they agree qualitatively.
• Figure 5: Average and variance of von Neumann entropy in the RFHM (varying $L$). (a) The dynamics of the average and (b) the variance of the von Neumann entropy over various disorder realizations, and $100$ ($50$ for $L=12$) eigenstates per realizations for various system sizes, but fixed $D=1.0$, is shown. The y-axes have been rescaled by their maximum value to avoid finite-size effects. As can be seen, the evolution curves for different system sizes overlap when plotted with $N \Lambda$, as opposed to with disorder strength $h$ (see inset).