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arXiv:2610.21631

Finite-size geometric crossover toward fragmentation of the potential energy shell in the quantum random energy model

Ishita Modak, Vatsana Tiwari, Soumya Bera

cond-mat.dis-nncond-mat.str-elquant-phSubmitted Oct 15, 2026• Announced Oct 16, 2026• 0 citations
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Abstract

The quantum random energy model (QREM) is a paradigmatic model for studying delocalization-localization transitions at finite energy density in Fock space. In the computational basis, its diagonal energies are independent Gaussian random variables, in contrast to the correlated energies of local interacting models such as the disordered XXZ chain. In this work, we construct a potential energy shell associated with a basis state, defined through its energy variance, and investigate its connectivity on the Fock-space hypercube. As the transverse field is varied, this energy-selected subgraph undergoes a finite-size geometric crossover between fully connected and fragmented regimes. Furthermore, we observe a correlation between the shell connectivity and the finite-size fractal dimension of the eigenstates, as extracted from the participation entropy. For the QREM, the statistical independence of the diagonal energies allows the shell connectivity to be analyzed as an effective independent site percolation problem on the hypercube, with its crossover scale estimated within a one-hop approximation. At finite energy density, this estimate gives a nonmonotonic dependence of the crossover scale on system size $L$: increasing $L$ initially favors connectivity, while beyond an energy-dependent scale $L^*(\varepsilon)$ the trend reverses towards fragmentation.

Comments
10 pages, 6 figures

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The quantum random energy model (QREM) is a paradigmatic model for studying delocalization-localization transitions at finite energy density in Fock space. In the computational basis, its diagonal energies are independent Gaussian random variables, in contrast to the correlated energies of local interacting models such as the disordered XXZ chain. In this work, we construct a potential energy shell associated with a basis state, defined through its energy variance, and investigate its connectivity on the Fock-space hypercube. As the transverse field is varied, this energy-selected subgraph undergoes a finite-size geometric crossover between fully connected and fragmented regimes. Furthermore, we observe a correlation between the shell connectivity and the finite-size fractal dimension of the eigenstates, as extracted from the participation entropy. For the QREM, the statistical independence of the diagonal energies allows the shell connectivity to be analyzed as an effective independent site percolation problem on the hypercube, with its crossover scale estimated within a one-hop approximation. At finite energy density, this estimate gives a nonmonotonic dependence of the crossover scale on system size LLL: increasing LLL initially favors connectivity, while beyond an energy-dependent scale L∗(ε)L^*(\varepsilon)L∗(ε) the trend reverses towards fragmentation.
Building on recent progress in the study of Anderson and many-body localization via the renormalization group (RG), we examine the scaling theory of localization in the quantum Random Energy Model (QREM). The QREM is known to undergo a localization-delocalization transition at finite energy density, while remaining fully ergodic at the center of the spectrum. At zero energy density, we show that RG trajectories consistently flow toward the ergodic phase, and are characterized by an unconventional scaling of the fractal dimension near the ergodic fixed point. When the disorder amplitude is rescaled, as suggested by the forward scattering approximation approach, a localization transition emerges also at the center of the spectrum, with properties analogous to the Anderson transition on expander graphs. At finite energy density, a localization transition takes place without disorder rescaling, and yet it exhibits a scaling behavior analogous to the one observed on expander graphs. The universality class of the model remains unchanged under the rescaling of the disorder, reflecting the independence of the RG from microscopic details. Our findings demonstrate the robustness of the scaling behavior of random graphs and offer new insights into the many-body localization transition.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}Nc​ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
We investigate numerically the nature of energy eigenstates in one-dimensional bond-disordered Anderson models with hopping amplitudes decreasing as ${H}_{\mathrm{ij}}\ensuremath{\propto}1/|i\ensuremath{-}j{|}^{\ensuremath{\alpha}}.$ The eigenstates become delocalized whenever the hopping amplitudes decay slower than 1/r.1/r.1/r. By performing an exact diagonalization scheme on finite chains, we compute the participation ratio of all energy eigenstates. Employing a finite-size scaling analysis, we report on the relevant scaling exponents characterizing this delocalization transition as well as the level-spacing distribution at the critical point $\ensuremath{\alpha}=1.$ The random hopping amplitudes are taken from both uniform and random sign distributions. We show that these models display similar critical behavior in the vicinity of $\ensuremath{\alpha}=1.$ However, the random sign model exhibits an asymptotic delocalization in the limit of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}\ensuremath{\infty}$ and the universal scaling behavior in this regime is also reported.
We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions DqD_qDq​ lie strictly between the limits of extended and localized states, exhibiting a distinct qqq-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.
The random energy model (REM) provides a solvable mean-field description of the equilibrium spin-glass transition. Its quantum sibling (the QREM), obtained by adding a transverse field to the REM, has similar properties and shows a spin-glass phase for sufficiently small transverse field and temperature. In a recent work, some of us have shown that the QREM further exhibits a many-body localization-delocalization (MBLD) transition when viewed as a closed quantum system, evolving according to the quantum dynamics. This phase encloses the familiar equilibrium spin-glass phase. In this paper, we study in detail the MBLD transition within the forward-scattering approximation and replica techniques. The predictions for the transition line are in good agreement with the exact diagonalization numerics. We also observe that the structure of the eigenstates at the MBLD critical point changes continuously with the energy density, raising the possibility of a family of critical theories for the MBLD transition.
Quantum phase transitions are usually observed in ground states of correlated systems. Remarkably, eigenstate phase transitions can also occur at finite energy density in disordered, isolated quantum systems. Such transitions fall outside the framework of statistical mechanics as they involve the breakdown of ergodicity. Here, we consider what general constraints can be imposed on the nature of eigenstate transitions due to the presence of disorder. We derive Harris-type bounds on the finite-size scaling exponents of the mean entanglement entropy and level statistics at the many-body localization phase transition using several different arguments. Our results are at odds with recent small-size numerics, for which we estimate the crossover scales beyond which the Harris bound must hold.