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Physical Review A

Universal classification scheme for the spatial-localization properties of one-particle states in finite,d-dimensional systems

Jānos Pipek, Imre Varga

Published Sep 1, 1992• 116 citations
Publisher

Abstract

The spatial-localization properties of general lattice distributions (e.g., charge distribution of one-particle eigenstates) are studied. We have introduced a localization quantity, the structural entropy characteristic of the decay form (shape) of the distribution. We have found that there exists a nontrivial relation between the structural entropy and the well-known delocalization index (participation ratio) where the latter parameter measures the spatial extension of the distribution. This relation is universal in the sense that it is independent of the geometrical arrangement of the atomic network, of the lattice constant, and of the size of the system, and is fully determined by the decay form of the eigenstates and the dimension d of the lattice. Based on this relation we have developed a classification scheme for the characterization of the shape and extension of the one-particle states, which makes possible the identification of various decay forms (Gaussian, exponential, power law, etc.) in numerical calculations even on finite systems. The implementation of our method in already existing program systems is particularly easy. Besides presenting the theoretical background we demonstrate its applicability on some numerical calculations.

DOI
10.1103/physreva.46.3148

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The spatial-localization properties of general lattice distributions (e.g., charge distribution of one-particle eigenstates) are studied. We have introduced a localization quantity, the structural entropy characteristic of the decay form (shape) of the distribution. We have found that there exists a nontrivial relation between the structural entropy and the well-known delocalization index (participation ratio) where the latter parameter measures the spatial extension of the distribution. This relation is universal in the sense that it is independent of the geometrical arrangement of the atomic network, of the lattice constant, and of the size of the system, and is fully determined by the decay form of the eigenstates and the dimension d of the lattice. Based on this relation we have developed a classification scheme for the characterization of the shape and extension of the one-particle states, which makes possible the identification of various decay forms (Gaussian, exponential, power law, etc.) in numerical calculations even on finite systems. The implementation of our method in already existing program systems is particularly easy. Besides presenting the theoretical background we demonstrate its applicability on some numerical calculations.
We investigate a one-dimensional tight-binding model in which onsite potentials [Formula: see text] exhibit power-law spatial correlations (with exponent α) and the hopping amplitudes decay as [Formula: see text]. This two-parameter family interpolates continuously between short-range Anderson-like disorder, correlated disorder with conventional hopping, and long-range hopping models with nontrivial delocalization tendencies. Using large-scale exact diagonalization, we construct a comprehensive phase map in the [Formula: see text] plane by combining spectral statistics, density-of-states analysis, and energy-resolved localization indicators such as the participation ratio, single-particle entanglement entropy, level-spacing ratio r, and the ratio of the geometric to arithmetic density of states. From these observables we define phase-indicator functions that compactly quantify localization behavior across the spectrum. Our analysis reveals robust mobility edges and multiple regimes of spectral coexistence between localized, extended, resonant, and critical states. Finite-size scaling, implemented via an explicit smoothness-based cost function, enables extraction of critical exponents and delineation of transition lines across the [Formula: see text] parameter space. To validate and complement these physics-based diagnostics, we employ a supervised autoencoder that learns high-level representations of eigenstate structure directly from raw features and reliably reproduces the phase classification defined by the indicator functions. Together, these approaches provide a coherent and internally consistent picture of the spectral transitions driven by correlated disorder and long-range hopping, establishing a unified framework for characterizing mobility edges in long-range one-dimensional systems.
In the spectrum of many-body quantum systems appearing in condensed matter physics, the low-energy eigenstates were the traditional focus of research. The interest in the statistical properties of the full eigenspectrum has grown more recently, in particular in the context of non-equilibrium questions. Wave functions of interacting lattice quantum systems can be characterized either by local observables or by global properties such as the participation ratio (PR) in a many-body basis or the entanglement between various partitions. We present a study of the PR and of the entanglement entropy (EE) between two roughly equal spatial partitions of the system, in all the eigenfunctions of local Hamiltonians. Motivated by the similarity of the PR and EE—both are generically larger in the bulk and smaller near the edges of the spectrum—we quantitatively analyze the correlation between them. We elucidate the effect of (proximity to) integrability, showing how low-entanglement and low-PR states appear also in the middle of the spectrum as one approaches integrable points. We also determine the precise scaling behaviour of the eigenstate-to-eigenstate fluctuations of the PR and EE with respect to system size and characterize the statistical distribution of these quantities near the middle of the spectrum.
Using the density matrix renormalization group algorithm, we investigate the lattice model for spinless fermions in one dimension in the presence of a strong interaction and disorder. The phase sensitivity of the ground state energy is determined with high accuracy for systems up to a size of 60 lattice constants. This quantity is found to be log normally distributed. The fluctuations grow algebraically with system size with a universal exponent of $\ensuremath{\approx}2/3$ in the localized region of the phase diagram. Surprisingly we find, for an attractive interaction, a delocalized phase of finite extension. The boundary of this delocalized phase is determined.
We study a partially disordered one-dimensional system with interacting particles. Concretely, we impose a disorder potential to only every other site, followed by a clean site. Our numerical analysis of eigenstate properties is based on the entanglement entropy and density distributions. Most importantly, at large disorder, there exist eigenstates with large entanglement entropies and significant correlations between the clean sites. These states have volume-law scaling, embedded into a sea of area-law states, reminiscent of inverted quantum-scar states. These eigenstate features leave fingerprints in the nonequilibrium dynamics even in the large-disorder regime, with a strong initial-state dependence. We demonstrate that certain types of initial charge-density-wave states decay significantly, while others preserve their initial inhomogeneity, the latter being the typical behavior for many-body localized systems. This initial-condition-dependent dynamics may give extra control over the delocalization dynamics at large disorder strength and should be experimentally feasible with ultracold atoms in optical lattices.
We show that the one-particle density matrix ρ can be used to characterize the interaction-driven many-body localization transition in closed fermionic systems. The natural orbitals (the eigenstates of ρ) are localized in the many-body localized phase and spread out when one enters the delocalized phase, while the occupation spectrum (the set of eigenvalues of ρ) reveals the distinctive Fock-space structure of the many-body eigenstates, exhibiting a steplike discontinuity in the localized phase. The associated one-particle occupation entropy is small in the localized phase and large in the delocalized phase, with diverging fluctuations at the transition. We analyze the inverse participation ratio of the natural orbitals and find that it is independent of system size in the localized phase.
Developing a local description of quantum information is a central problem in quantum many-body physics: the goal is to identify how correlations are organized across positions and scales without choosing specific observables. We introduce a higher-dimensional information lattice, an inclusion-exclusion construction that decomposes the von Neumann information of reduced density matrices into contributions labeled by subsystem position and scale. Above one dimension, overlapping subsystems can form loops, allowing information associated with one subsystem to be inferred nonlocally from the density matrices of others. The sign of the inclusion-exclusion information resolves such overlap redundancies in position and scale: positive values mark positions and scales with net information gain, while negative values occur where redundant contributions dominate new ones. We implement the construction in two dimensions and apply it to many-body ground states with distinct entanglement structures. For localized states, such as disordered-localized states or gapped ground states of local Hamiltonians, the local information decays exponentially along different directions, defining direction-dependent information length scales. In critical states with a Fermi surface, decay becomes algebraic, with dominant directions tied to the velocity of gapless modes. In the toric code, topological order manifests as a negative information value at the largest lattice scale, interpreted as global information redundancy between the smallest loop-enclosing overlapping subsystems. We also isolate the universal information carried by edge states and locate the signatures of non-Abelian fusion channels in position and scale. Our results establish the higher-dimensional information lattice as a framework for resolving universal correlation structures through the spatial and scale distribution of quantum information.