[Paper Review] Many-body localization and new critical phenomena in regular random graphs and constrained Erdős-Renyi networks

В. А. Аветисов, A. Gorsky+2 more|arXiv (Cornell University)|Nov 25, 2016

Complex Network Analysis Techniques1 references3 citations

TL;DR

This paper proposes that structural disorder induced by 3-cycles in regular random graphs (RRG) and constrained Erdős-Rényi networks (CERN) drives a many-body localization transition in Fock space, realizing 'localization without disorder.' Above a critical chemical potential μ, the spectrum splits into delocalized and localized zones separated by a mobility edge, with the latter corresponding to clusters of resonant triples, indicating non-ergodic delocalized states via memory-dependent statistics.

ABSTRACT

We consider from the localization perspective the new critical phenomena discovered recently for perturbed random regular graphs (RRG) and constrained Erdős-Rényi networks (CERN) \cite{crit2}. At some critical value of the chemical potential of 3-cycles, $μ$, the network decays into the maximally possible number of almost full subgraphs, and the spectrum of the Laplacian matrix acquires the two-zonal structure with a large gap. We find that the Laplacian eigenvalue statistics corresponds to delocalized states in one zone, and to the localized states in the second one. We interpret this behavior in terms of the many-body localization problem where the structure of the Fock space of some interacting many-body system is approximated by the RGG and/or by the CERN. We associate 3-cycles in RRGs and CERNs as resonant triples in the Fock space. We show that the scenario of the "localization without disorder", discussed previously in physical space, can be realized in the Fock space as well. We argue that it is natural to identify clusters in a RRG with particles in a many-body system above the phase transition. We discuss the controversial issue of an additional phase transition between ergodic and non-ergodic regimes in the delocalized phase in the Fock space and find a strong "memory dependence" of the states in the delocalized phase, thus advocating existence of non-ergodic delocalized states.

Motivation & Objective

• To investigate how structural features like 3-cycles in RRG and CERN networks induce localization in the Fock space of many-body systems.
• To test whether 'localization without disorder' can emerge in translationally invariant systems via spontaneous structural organization.
• To determine whether a mobility edge separates delocalized and localized phases in the spectrum, and whether delocalized states are ergodic or non-ergodic.
• To explore the role of resonant triples (3-cycles) as analogs of interacting many-body states in Fock space.
• To assess the validity of the non-ergodic delocalized phase conjecture using level spacing statistics and memory dependence.

Proposed method

• The authors model the Fock space of interacting many-body systems using RRG and CERN networks, treating vertices as basis states and links as resonant transitions.
• They introduce a chemical potential μ for 3-cycles to control network topology, simulating the formation of clusters via spontaneous structural disorder.
• Spectral analysis of the Laplacian matrix reveals a two-zonal structure with a large gap at μ > μ_cr, indicating a mobility edge.
• Level spacing statistics are used to classify states: Wigner-Dyson for delocalized, Poisson for localized, enabling identification of phase boundaries.
• Numerical visualization of the adjacency matrix structure supports the existence of non-ergodic delocalized states with memory dependence.
• The study draws parallels with replica symmetry breaking and eigenvalue instantons, suggesting a modular structure underlying the localization transition.

Experimental results

Research questions

• RQ1 Does the presence of 3-cycles in RRG and CERN networks induce a localization transition in the Fock space without external disorder?
• RQ2 Can a mobility edge emerge in the spectrum due to structural disorder from 3-cycles, separating delocalized and localized phases?
• RQ3 Are the delocalized states above μ_cr ergodic or non-ergodic, and what evidence supports this classification?
• RQ4 How do resonant triples (3-cycles) in the network correspond to interacting many-body states in the Fock space?
• RQ5 What is the role of memory dependence in the delocalized phase, and does it support the existence of non-ergodic delocalized states?

Key findings

• Above a critical chemical potential μ_cr for 3-cycles, the Laplacian spectrum splits into two zones: one with delocalized states (Wigner-Dyson statistics) and one with localized states (Poisson statistics), separated by a mobility edge.
• The localized states correspond to clusters formed by the most densely connected subgraphs, indicating structural localization without diagonal disorder.
• The delocalized phase exhibits strong memory dependence in wave functions, supporting the existence of non-ergodic delocalized states rather than ergodic ones.
• Resonant triples (3-cycles) in the network are interpreted as interacting many-body states in Fock space, with their number critically influencing localization behavior.
• The transition is interpreted as 'localization without disorder' in a translationally invariant system, driven by spontaneous structural organization via 3-cycle formation.
• Numerical evidence suggests that the delocalized phase may correspond to one-step replica symmetry breaking, consistent with non-ergodicity and multifractal wave function structure.