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arXiv:2503.08825 · Physical review. E

Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians

Wouter Buijsman, Masudul Haque, Ivan M. Khaymovich

Published Feb 23, 2026• 3 citations
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Abstract

We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.

DOI
10.1103/tvms-89w5

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We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.
Eigenstates of local many-body interacting systems that are far from spectral edges are thought to be ergodic and close to being random states. This is consistent with the eigenstate thermalization hypothesis and volume-law scaling of entanglement. We point out that systematic departures from complete randomness are generically present in mid-spectrum eigenstates, and focus on the departure of the entanglement entropy from the random-state prediction. We show that the departure is (partly) due to spatial correlations and due to orthogonality to the eigenstates at the spectral edge, which imposes structure on the mid-spectrum eigenstates.
Does thermal averaging preserve signatures of eigenstate complexity? We study the entropic complexity C = S_1 - S_2 (Shannon minus second-order Renyi entropy) across the ergodic-to-localized crossover of three disordered models: the Rosenzweig-Porter (RP) ensemble, the power-law banded random matrix (PLBRM) ensemble, and the random-field Heisenberg chain. We compare a wavefunction-level quantity C_eig to the complexity of the thermal (Gibbs) state before and after pointer-basis dephasing, via the trace (C_tr) and diagonal (C_diag) complexities. The answer is mostly no: thermal averaging strongly suppresses, but does not eliminate, eigenstate-complexity signatures. C_eig develops a pronounced mid-phase maximum in RP and, at matching fractal dimension, in the structurally independent PLBRM model; a high-statistics scan resolves a weak (about 10%) but reproducible thermal shadow of this peak in the thermal diagonal complexity. This feature is confined to the two random-matrix models; in the Heisenberg chain the eigenstate complexity instead peaks at the many-body localization transition. A second, genuinely thermal feature, an edge just inside the ergodic phase, has no eigenstate counterpart and, in PLBRM, recedes with system size. Both features are tracked by scale-invariant crossover indicators: the log-ratio of the trace and diagonal crossover temperatures, which vanishes upon localization, and the relative entropy of coherence. Entropic complexity thus cleanly separates thermal-state and eigenstate physics: a sharp wavefunction-level feature, reproducible across unrelated random-matrix constructions, leaves only a faint, structurally distinct imprint on thermal observables.
Eigenstates of local many-body interacting systems that are far from spectral edges are thought to be ergodic and close to being random states. This is consistent with the eigenstate thermalization hypothesis and volume-law scaling of entanglement. We point out that systematic departures from complete randomness are generically present in midspectrum eigenstates, and focus on the departure of the entanglement entropy from the random-state prediction. We show that the departure is (partly) due to spatial correlations and due to orthogonality to the eigenstates at the spectral edge, which imposes structure on the midspectrum eigenstates.
The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entanglement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. In particular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We find deviations in the tails both for the disordered XXZ with total SzS_zSz​ conserved in the half-filled sector as well as in a model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue, which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widom distribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduced density matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared to the Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized participation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme value distribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appropriately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recently developed perturbation theory of weakly coupled chaotic systems.
The eigenstate entanglement entropy is a powerful tool to distinguish integrable from generic quantum-chaotic models. In integrable models, the average eigenstate entanglement entropy (over all Hamiltonian eigenstates) has a volume-law coefficient that generally depends on the subsystem fraction. In contrast, it is maximal (subsystem fraction independent) in quantum-chaotic models. Using random matrix theory for quadratic Hamiltonians, we obtain a closed-form expression for the average eigenstate entanglement entropy as a function of the subsystem fraction. We test it against numerical results for the quadratic Sachdev-Ye-Kitaev model and show that it describes the results for the power-law random banded matrix model (in the delocalized regime). We show that localization in quasimomentum space produces (small) deviations from our analytic predictions.
The entanglement spectrum of the reduced density matrix contains information beyond the von Neumann entropy and provides unique insights into exotic orders or critical behavior of quantum systems. Here, we show that strongly disordered systems in the many-body localized phase have power-law entanglement spectra, arising from the presence of extensively many local integrals of motion. The power-law entanglement spectrum distinguishes many-body localized systems from ergodic systems, as well as from ground states of gapped integrable models or free systems in the vicinity of scale-invariant critical points. We confirm our results using large-scale exact diagonalization. In addition, we develop a matrix-product state algorithm which allows us to access the eigenstates of large systems close to the localization transition, and discuss general implications of our results for variational studies of highly excited eigenstates in many-body localized systems.