[Paper Review] Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble

V. E. Kravtsov, Ivan M. Khaymovich+2 more|arXiv (Cornell University)|Feb 7, 2020

Quantum many-body systems40 references55 citations

TL;DR

The paper introduces a log-normal Rosenzweig-Porter (LN-RP) random matrix ensemble with symmetry parameter p that interpolates Gaussian RP and Lévy ensembles, identifies a tricritical point at p=1 corresponding to Anderson localization on random regular graphs, and shows the multifractal phase collapses there but is unstable to truncation of the log-normal tail.

ABSTRACT

Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robust multifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phase in GRP model is oversimplified: the spectrum of fractal dimensions is degenerate and the mini-band in the local spectrum is not multifractal. In this paper we suggest an extension of the GRP model by adopting a logarithmically-normal (LN) distribution of off-diagonal matrix elements. A family of such LN-RP models is parametrized by a symmetry parameter $p$ and it interpolates between the GRP at $p ightarrow 0$ and Levy ensembles at $p ightarrow\infty$. A special point $p=1$ is shown to be the simplest approximation to the Anderson localization model on a random regular graph.We study in detail the phase diagram of LN-RP model and show that $p=1$ is a tricritical point where the multifractal phase first collapses. This collapse is shown to be unstable with respect to the truncation of the log-normal distribution. We suggest a new criteria of stability of the non-ergodic phases and prove that the Anderson transition in LN-RP model is discontinuous at all $p>0$.

Motivation & Objective

• Motivate the study of Anderson localization on random regular graphs as a proxy for many-body localization.
• Extend the Rosenzweig-Porter model with a log-normal distribution of off-diagonal elements to better capture RRG features.
• Map the LN-RP phase diagram and locate the tricritical point where multifractality collapses.
• Develop criteria for stability of non-ergodic phases and assess ergodic transition robustness.

Proposed method

• Define LN-RP model with a log-normal distribution of off-diagonal matrix elements and a symmetry parameter p.
• Relate LN-RP to RRG through effective long-range hopping and a fat-tailed hopping distribution.
• Derive phase boundaries for ergodic, multifractal, and localized phases using localization (S1) and Mott ergodicity (S2) criteria.
• Use Kullback-Leibler divergences KL1 and KL2 to characterize eigenfunction statistics and locate transitions.
• Perform finite-size scaling and numerics to extract critical points gamma_ET and gamma_AT and exponents nu1, nu2.

Experimental results

Research questions

• RQ1 What is the phase diagram of LN-RP as a function of p and gamma, and how does it interpolate between Gaussian RP and Lévy ensembles?
• RQ2 Does a tricritical point at p=1 exist, and is multifractality stable or unstable near this point?
• RQ3 How do KL1 and KL2 detect ergodic, localized, and multifractal phases in LN-RP?
• RQ4 What is the role of truncating the log-normal tail on the stability of non-ergodic phases?
• RQ5 What are the finite-size scaling properties and critical exponents at the localization and ergodic transitions in LN-RP?

Key findings

• The LN-RP model exhibits three phases (ergodic, multifractal, localized) with transitions that merge at a tricritical point p=1 and gamma=4.
• The multifractal phase collapses at p=1 but this collapse is unstable to truncation of the log-normal distribution tail.
• Ergodic transition becomes fragile for p>=1 and can be displaced to smaller gamma by truncation, reintroducing the multifractal phase.
• KL1 is sensitive to localization transitions and KL2 to ergodic transitions, with clear finite-size scaling behavior observed.
• For p>1, numerical and analytical results show gamma_ET and gamma_AT values consistent with the derived expressions; exponents nu1 and nu2 depend on p, approaching mean-field values for p>=1.
• The Anderson transition on LN-RP is discontinuous for all p>0 according to the proposed criteria and analysis.