Statistics of eigenstates near the localization transition on random regular graphs

Abstract

Dynamical and spatial correlations of eigenfunctions as well as energy level correlations in the Anderson model on random regular graphs (RRG) are studied. We consider the critical point of the Anderson transition and the delocalized phase. In the delocalized phase near the transition point, the observables show a broad critical regime for system sizes $N$ below the correlation volume $N_{\xi }$ and then cross over to the ergodic behavior. Eigenstate correlations allow us to visualize the correlation length $\xi \sim \log N_{\xi }$ that controls the finite-size scaling near the transition. The critical-to-ergodic crossover is very peculiar, since the critical point is similar to the localized phase, whereas the ergodic regime is characterized by very fast "diffusion" which is similar to the ballistic transport. In particular, the return probability crosses over from a logarithmically slow variation with time in the critical regime to an exponentially fast decay in the ergodic regime. Spectral correlations in the delocalized phase near the transition are characterized by level number variance ${\Sigma }_2(\omega )$ crossing over, with increasing freqyency $\omega$, from ergodic behavior ${\Sigma }_2=\left(2/{\pi }^2\right)\ln \omega /\Delta$ to ${\Sigma }_2\propto {\omega }^2$ at ${\omega }_c\sim (N{N}_{\xi }{)}^{-1/2}$ and finally to Poissonian behavior ${\Sigma }_2=\omega /\Delta$ at ${\omega }_{\xi }\sim N_{\xi }^{-1}$. We find a perfect agreement between results of exact diagonalization and those resulting from the solution of the self-consistency equation obtained within the saddle-point analysis of the effective supersymmetric action. We show that the RRG model can be viewed as an intricate $d\to \infty$ limit of the Anderson model in $d$ spatial dimensions.

References in corpus (26)

• Many body localization and thermalization in quantum statistical mechanics
• Anderson Transitions
• Localization of interacting fermions at high temperature
• Many-body localization edge in the random-field Heisenberg chain
• Spectral signatures of many-body localization with interacting photons
• Anomalous diffusion and Griffiths effects near the many-body localization transition
• Absence of diffusion in an interacting system of spinless fermions on a one-dimensional disordered lattice
• Signatures of Many-Body Localization in a Controlled Open Quantum System
• Many-body localization and delocalization in large quantum chains
• Non-ergodic phases in strongly disordered random regular graphs
• Depolarization dynamics in a strongly interacting solid-state spin ensemble
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• Many-Body Delocalization in Strongly Disordered System with Long-Range Interactions: Finite Size Scaling
• Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph
• Return probability for the Anderson model on the random regular graph
• Many-body localization transition with power-law interactions: Statistics of eigenstates
• Quantum ergodicity in the SYK model
• Delocalized Glassy Dynamics and Many Body Localization
• Critical properties of the Anderson localization transition and the high dimensional limit
• Multifractality of wave functions on a Cayley tree: From root to leaves
• A mean-field theory of nearly many-body localized metals
• Dimensional dependence of the metal-insulator transition
• Finite-size scaling and multifractality at the Anderson transition for the three Wigner-Dyson symmetry classes in three dimensions
• Wave function multifractality and dephasing at metal-insulator and quantum Hall transitions
• Suppression of inelastic electron-electron scattering in Anderson Insulators
• Thermalization Processes in Interacting Anderson Insulators

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• Anderson Localization on Husimi Trees and its implications for Many-Body localization
• Critical Dynamics of the Anderson Transition on Small-World Graphs
• Large deviations in the many-body localization transition: The case of the random-field XXZ chain
• Localization properties of the sparse Barrat-Mézard trap model
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