Anatomy of the fragmented Hilbert space: eigenvalue tunneling, quantum scars and localization in the perturbed random regular graph

Abstract

We consider the properties of the random regular graph with node degree $d$ perturbed by chemical potentials ${\mu }_k$ for a number of short $k$-cycles. We analyze both numerically and analytically the phase diagram of the model in the $({\mu }_k,d)$ plane. The critical curve separating the homogeneous and clusterized phases is found and it is demonstrated that the clusterized phase itself generically is separated as the function of $d$ into the phase with ideal clusters and phase with coupled ones when the continuous spectrum gets formed. The eigenstate spatial structure of the model is investigated and it is found that there are localized scar-like states in the delocalized part of the spectrum, that are related to the topologically equivalent nodes in the graph. We also reconsider the localization of the states in the non-perturbative band formed by eigenvalue instantons and find the semi-Poisson level spacing distribution. The Anderson transition for the case of combined ($k$-cycle) structural and diagonal (Anderson) disorders is investigated. It is found that the critical diagonal disorder gets reduced sharply at the clusterization phase transition, but does it unevenly in non-perturbative and mid-spectrum bands, due to the scars, present in the latter. The applications of our findings to $2$d quantum gravity are discussed.

27 pages, 9 figures, 87 references + 4 pages, 4 figures in Appendices

References in corpus (23)

• Many body localization and thermalization in quantum statistical mechanics
• Quantum Many-Body Scars and Weak Breaking of Ergodicity
• Quantum Many-Body Scars and Hilbert Space Fragmentation: A Review of Exact Results
• Quantum Many-Body Scars: A Quasiparticle Perspective
• Non-ergodic phases in strongly disordered random regular graphs
• Fractality of wave functions on a Cayley tree: Difference between a tree and a locally tree-like graph without boundary
• Quantum scars from zero modes in an Abelian lattice gauge theory on ladders
• From Anderson localization on Random Regular Graphs to Many-Body localization
• Spectra of random graphs with arbitrary expected degrees
• Delocalized Glassy Dynamics and Many Body Localization
• Dynamical phases in a "multifractal" Rosenzweig-Porter model
• Multifractality of wave functions on a Cayley tree: From root to leaves
• Group theoretic approach to many-body scar states in fermionic lattice models
• Exact many-body scars and their stability in constrained quantum chains
• Robustness of delocalization to the inclusion of soft constraints in long-range random models
• Eigenvalue tunnelling and decay of quenched random networks
• Gravitational orbits, double-twist mirage, and many-body scars
• Self-Assembly of Geometric Space from Random Graphs
• Quantum scars and bulk coherence in a symmetry-protected topological phase
• Interacting Thermofield Doubles and Critical Behavior in Random Regular Graphs
• Mobility Edge in the Anderson model on partially disordered random regular graphs
• A Flow in the Forest
• Quantum Scar States in Coupled Random Graph Models

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