[Paper Review] From Anderson localization on Random Regular Graphs to Many-Body localization

K. S. Tikhonov, A. D. Mirlin|arXiv (Cornell University)|Feb 11, 2021

Quantum many-body systemsPhysics and Astronomy122 references103 citations

TL;DR

This paper establishes a deep connection between Anderson localization on random regular graphs (RRG) and many-body localization (MBL) in disordered quantum systems. By mapping MBL models with short-range and power-law interactions onto RRG, the authors show that key features—ergodic delocalized phase, localized critical point, fractal eigenstate scaling, and power-law dynamical correlations—emerge in both, providing a unified framework for understanding MBL transitions despite the RRG model being a simplified toy model of Fock space.

ABSTRACT

The article reviews the physics of Anderson localization on random regular graphs (RRG) and its connections to many-body localization (MBL) in disordered interacting systems. Properties of eigenstate and energy level correlations in delocalized and localized phases, as well at criticality, are discussed. In the many-body part, models with short-range and power-law interactions are considered, as well as the quantum-dot model representing the limit of the "most long-range" interaction. Central themes -- which are common to the RRG and MBL problems -- include ergodicity of the delocalized phase, localized character of the critical point, strong finite-size effects, and fractal scaling of eigenstate correlations in the localized phase.

Motivation & Objective

• . The paper aims to clarify the connections between Anderson localization on random regular graphs (RRG) and many-body localization (MBL) in disordered interacting systems.
• It seeks to identify common physical features—such as ergodicity in the delocalized phase, localized nature of the critical point, and fractal eigenstate scaling—that are shared between RRG and MBL models.
• The authors aim to demonstrate that the RRG model serves as a tractable toy model for studying MBL, capturing essential critical behavior despite its simplifications.
• They investigate how different interaction types (short-range, power-law, long-range) in MBL systems map onto the RRG framework and whether key observables like the inverse participation ratio (IPR) and level statistics exhibit universal scaling.

Proposed method

• . The authors employ field-theoretical methods to describe Anderson localization on RRG, particularly focusing on the localization transition and critical behavior.
• They analyze eigenstate and energy level statistics using tools such as the inverse participation ratio (IPR), eigenfunction correlations, and return probability.
• For MBL systems, they consider three classes: quantum-dot models (representing long-range interactions), power-law interaction models (with 1 < α < 2), and short-range interaction models (e.g., spin chains).
• Analytical scaling arguments are used to predict the behavior of the MBL transition, including the critical disorder Wc and IPR scaling in the localized phase.
• Numerical exact diagonalization (ED) is employed to validate analytical predictions, particularly for 1D spin chains with random fields and power-law interactions.
• The study compares dynamical correlations of eigenstates (β(ω)) across RRG and MBL systems, revealing universal power-law scaling in the localized phase.

Experimental results

Research questions

• RQ1 . To what extent does the RRG model capture the critical behavior of many-body localization in disordered quantum systems?
• RQ2 How do eigenstate correlations and the inverse participation ratio (IPR) scale in the localized phase of MBL systems, and how do they compare to RRG?
• RQ3 What is the role of rare regions and spectral diffusion in the MBL transition, and how do they affect finite-size scaling?
• RQ4 How does the critical disorder Wc scale with system size L in MBL models with short-range and power-law interactions, and does it converge to a finite value in the thermodynamic limit?
• RQ5 To what extent do dynamical correlations of eigenstates (β(ω)) exhibit universal power-law scaling across RRG and MBL systems?

Key findings

• . The delocalized phase in both RRG and MBL systems is ergodic, as confirmed by the IPR scaling as P2 ∼ N^0 in the RRG case and P2 ∼ N^0 in the MBL case with short-range or power-law interactions.
• The critical point in both RRG and MBL systems is localized, with the critical disorder Wc increasing slowly with system size L, consistent with a logarithmic or slower-than-power-law dependence.
• In the localized phase of MBL systems with short-range or power-law interactions, the IPR scales as P2 ∼ N^{-τ(W)} with τ(W) ∼ 1/W, indicating multifractal eigenstate structure due to rare resonances.
• For MBL systems with power-law interactions (d < α < 2d), the critical disorder Wc(L) scales as predicted by the RRG mapping, and numerical results confirm the analytical scaling of Wc(L) and the ergodicity of the delocalized phase.
• Dynamical correlations β(ω) in both RRG and MBL systems (e.g., spin chains) exhibit power-law scaling β(ω) ∝ ω^{-μ(W)} in the localized phase, with disorder-dependent exponent μ(W), mirroring the same behavior in RRG.
• Despite differences in Hilbert space structure and the absence of rare ergodic regions in RRG, the model captures the essential physics of MBL, including criticality and eigenstate statistics, making it a valuable toy model for understanding the MBL transition.