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arXiv:2201.04673 · Physical review. B./Physical review. B

Subdiffusive Thouless time scaling in the Anderson model on random regular graphs

Luis Colmenárez, David J. Luitz, Ivan M. Khaymovich, Giuseppe De Tomasi

Published May 20, 2022• 36 citations
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Abstract

Strong disorder usually gives rise to slow transport regimes in quantum many-body systems. The Thouless time scaling with system size, probed by long-range spectral correlations, allows distinguishing diffusive, subdiffusive, and localized dynamics regimes. Here, the authors investigate the Thouless time in the Anderson model on regular random graphs, a proxy for many-body localization. These results support the existence of a subdiffusive regime anticipating the localized phase.

DOI
10.1103/physrevb.105.174207

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Strong disorder usually gives rise to slow transport regimes in quantum many-body systems. The Thouless time scaling with system size, probed by long-range spectral correlations, allows distinguishing diffusive, subdiffusive, and localized dynamics regimes. Here, the authors investigate the Thouless time in the Anderson model on regular random graphs, a proxy for many-body localization. These results support the existence of a subdiffusive regime anticipating the localized phase.
We study the finite-time dynamics of an initially localized wave packet in the Anderson model on the random regular graph (RRG) and show the presence of a subdiffusion phase coexisting both with ergodic and putative nonergodic phases. The full probability distribution $\mathrm{\ensuremath{\Pi}}(x,t)$ of a particle to be at some distance xxx from the initial state at time ttt is shown to spread subdiffusively over a range of disorder strengths. The comparison of this result with the dynamics of the Anderson model on Zd{\mathbb{Z}}^{d}Zd lattices, d>2d>2d>2, which is subdiffusive only at the critical point implies that the limit $d\ensuremath{\rightarrow}\ensuremath{\infty}$ is highly singular in terms of the dynamics. A detailed analysis of the propagation of $\mathrm{\ensuremath{\Pi}}(x,t)$ in space-time (x,t)(x,t)(x,t) domain identifies four different regimes determined by the position of a wave front Xfront(t){X}_{\text{front}}(t)Xfront​(t), which moves subdiffusively to the most distant sites ${X}_{\text{front}}(t)\ensuremath{\sim}{t}^{\ensuremath{\beta}}$ with an exponent $\ensuremath{\beta}<1$. Importantly, the Anderson model on the RRG can be considered as proxy of the many-body localization transition (MBL) on the Fock space of a generic interacting system. In the final discussion, we outline possible implications of our findings for MBL.
The Anderson transition on random graphs is a paradigm for understanding high-dimensional quantum phase transitions driven by disorder and closely mirrors several features of many-body localization. In this Letter, we introduce a unitary Anderson model on small-world graphs, enabling large-scale, long-time simulations of wave-packet dynamics. This allows us to uncover logarithmically slow critical dynamics, two distinct localization times, and a crossover to ergodic diffusion. Finite-time scaling reveals distinct critical exponents, establishing a dynamical universality class for this exotic transition. Our model opens the way to an experimental realization of the Anderson transition on random graphs in quantum simulators. By combining disorder, high dimensionality, and dynamics, our approach provides a general framework for probing universal features of out-of-equilibrium quantum phase transitions, including those relevant to many-body localization.
A disordered quantum system can be driven, e.g., by increasing disorder, through the Anderson transition between a delocalized and localized phase. Recently, Anderson localization on treelike graphs has attracted much attention in view of its connections with problems of many-body localization. One of the central questions in this context is that of ergodicity of eigenstates. The authors of this paper perform a numerical study of Anderson transition on random regular graphs (RRGs) with diagonal disorder. The problem can be described as a tight-binding model on a lattice with NNN sites that is locally a tree with constant connectivity. Focusing on the delocalized side of the transition, they show that the data can be interpreted in terms of the finite-size crossover from a small ($N\ensuremath{\ll}{N}_{c}$) to a large ($N\ensuremath{\gg}{N}_{c}$) system, where Nc{N}_{c}Nc​ is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wave-function statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly nontrivial. The results support an analytical prediction that states in the delocalized phase (and at $N\ensuremath{\gg}{N}_{c}$) are ergodic.
We study the real-time dynamics of a two-dimensional Anderson-Hubbard model using nonequilibrium self-consistent perturbation theory within the second-Born approximation. When compared with exact diagonalization performed on small clusters, we demonstrate that for strong disorder this technique approaches the exact result on all available timescales, while for intermediate disorder, in the vicinity of the many-body localization transition, it produces quantitatively accurate results up to nontrivial times. Our method allows for the treatment of system sizes inaccessible by any numerically exact method and for the complete elimination of finite-size effects for the times considered. We show that for a sufficiently strong disorder the system becomes nonergodic, while for intermediate disorder strengths and for all accessible timescales transport in the system is strictly subdiffusive. We argue that these results are incompatible with a simple percolation picture, but are consistent with the heuristic random resistor network model where subdiffusion may be observed for long times until a crossover to diffusion occurs. The prediction of slow finite-time dynamics in a two-dimensional interacting and disordered system can be directly verified in future cold-atoms experiments.
We study transport in a one-dimensional boundary-driven Anderson insulator (the XX spin chain with onsite disorder) with randomly positioned onsite dephasing, observing a transition from diffusive to subdiffusive spin transport below a critical density of sites with dephasing. This model is intended to mimic the passage of an excitation through (many-body) insulating regions or ergodic bubbles, therefore providing a toy model for the diffusion-subdiffusion transition observed in the disordered Heisenberg model by \ifmmode \check{Z}\else \v{Z}\fi{}nidari\ifmmode \check{c}\else \v{c}\fi{} et al. [Phys. Rev. Lett. 117, 040601 (2016)]. We also present the exact solution of a semiclassical model of conductors and insulators introduced by Agarwal et al. [Phys. Rev. Lett. 114, 160401 (2015)], which exhibits both diffusive and subdiffusive phases, and qualitatively reproduces the results of the quantum system. The critical properties of both models, when passing from diffusion to subdiffusion, are interpreted in terms of ``Griffiths effects.'' We show that the finite-size scaling comes from the interplay of three characteristic lengths: one associated with disorder (the localization length), one with dephasing, and the third with the percolation problem defining large, rare, insulating regions. We conjecture that the latter, which grows logarithmically with system size, may potentially be responsible for the fact that heavy-tailed resistance distributions typical of Griffiths effects have not been observed in subdiffusive interacting systems.