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arXiv:1707.08845 · Acta Physica Polonica A

Many-Body Localization for Randomly Interacting Bosons

Piotr Sierant, Dominique Delande, Jakub Zakrzewski

Published Dec 1, 2017• 37 citations
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Abstract

We study many-body localization in a one dimensional optical lattice filled with bosons. The interaction between bosons is assumed to be random, which can be realized for atoms close to a microchip exposed to a spatially fluctuating magnetic field. Close to a Feshbach resonance, such controlled fluctuations can be transfered to the interaction strength. We show that the system reveals an inverted mobility edge, with mobile particles at the lower edge of the spectrum. A statistical analysis of level spacings allows us to characterize the transition between localized and excited states. The existence of the mobility edge is confirmed in large systems, by time dependent numerical simulations using tDMRG. A simple analytical model predicts the long time behavior of the system.

DOI
10.12693/aphyspola.132.1707

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We study many-body localization in a one dimensional optical lattice filled with bosons. The interaction between bosons is assumed to be random, which can be realized for atoms close to a microchip exposed to a spatially fluctuating magnetic field. Close to a Feshbach resonance, such controlled fluctuations can be transfered to the interaction strength. We show that the system reveals an inverted mobility edge, with mobile particles at the lower edge of the spectrum. A statistical analysis of level spacings allows us to characterize the transition between localized and excited states. The existence of the mobility edge is confirmed in large systems, by time dependent numerical simulations using tDMRG. A simple analytical model predicts the long time behavior of the system.
Many-body localization for a system of bosons trapped in a one-dimensional lattice is discussed. Two models that may be realized for cold atoms in optical lattices are considered. The model with a random on-site potential is compared with previously introduced random interactions model. While the origin and character of the disorder in both systems is different they show interesting similar properties. In particular, many-body localization appears for a sufficiently large disorder as verified by a time evolution of initial density wave states as well as using statistical properties of energy levels for small system sizes. Starting with different initial states, we observe that the localization properties are energy-dependent which reveals an inverted many-body localization edge in both systems (that finding is also verified by statistical analysis of energy spectrum). Moreover, we consider computationally challenging regime of transition between many body localized and extended phases where we observe a characteristic algebraic decay of density correlations which may be attributed to subdiffusion (and Griffiths-like regions) in the studied systems. Ergodicity breaking in the disordered Bose–Hubbard models is compared with the slowing-down of the time evolution of the clean system at large interactions.
We construct a solvable spin chain model of many-body localization (MBL) with a tunable mobility edge. This simple model not only demonstrates analytically the existence of mobility edges in interacting one-dimensional (1D) disordered systems, but also allows us to study their physics. By establishing a connection between MBL and a quantum central limit theorem (QCLT), we show that many-body localization-delocalization transitions can be visualized as tuning a mobility edge in the energy spectrum. Since the effective disorder strength for individual eigenstates depends on energy density, we identify "energy-resolved disorder strength" as a physical mechanism for the appearance of mobility edges, and support the universality of this mechanism by arguing its presence in a large class of models including the random-field Heisenberg chain. We also construct models with multiple mobility edges. All our constructions can be made translationally invariant.
Whether the many-body mobility edges can exist in a one-dimensional interacting quantum system is a controversial problem, mainly hampered by the limited system sizes amenable to numerical simulations. We investigate the transition from chaos to localization by constructing a combined random matrix, which has two extremes, one of Gaussian orthogonal ensemble and the other of Poisson statistics, drawn from different distributions. We find that by fixing a scaling parameter, the mobility edges can exist while increasing the matrix dimension D→∞D\rightarrow\inftyD→∞, depending on the distribution of matrix elements of the diagonal uncorrelated matrix. By applying those results to a specific one-dimensional isolated quantum system of random diagonal elements, we confirm the existence of a many-body mobility edge, connecting it with results on the onset of level repulsion extracted from ensembles of mixed random matrices.
Motivated by recent experiments on interacting bosons in a quasi-one-dimensional optical lattice [M. Rispoli et al., Nature (London) 573, 385 (2019)], we analyze theoretically properties of the system in the crossover between delocalized and localized regimes. Comparison of time dynamics for uniform and density-wave-like initial states enables a demonstration of the existence of the mobility edge. To this end we define a different observable, the mean speed of transport at long times. It gives us an efficient estimate of the critical disorder for the crossover. We also show that the mean velocity growth of occupation fluctuations close to the edges of the system carries similar information. Using the quantum quench procedure, we show that it is possible to probe the mobility edge for different energies.
We study many-body localized quantum systems subject to periodic driving. We find that the presence of a mobility edge anywhere in the spectrum is enough to lead to delocalization for any driving strength and frequency. By contrast, for a fully localized many-body system, a delocalization transition occurs at a finite driving frequency. We present numerical studies on a system of interacting one-dimensional bosons and the quantum random energy model, as well as simple physical pictures accounting for those results.
We study many-body localization (MBL) in an interacting one-dimensional system with a deterministic aperiodic potential. Below the threshold potential h<hch<{h}_{c}h<hc​, the noninteracting system has single particle mobility edges (MEs) at $\ifmmode\pm\else\textpm\fi{}{E}_{c}$ while for h>hch>{h}_{c}h>hc​ all the single particle states are localized. We demonstrate that even in the presence of single particle MEs, interactions do not always delocalize the system and the interacting system can have MBL. Our numerical calculation of energy level spacing statistics, participation ratio in the Fock space, and Shannon entropy shows that for some regime of particle densities, even for h<hch<{h}_{c}h<hc​, many-body states at the top (with E>E2E>{E}_{2}E>E2​) and the bottom of the spectrum (with E<E1E<{E}_{1}E<E1​) remain localized, though states in the middle of the spectrum are delocalized. Variance of entanglement entropy (EE) also diverges at E1,2{E}_{1,2}E1,2​, indicating a transition from MBL to a delocalized regime, though transitions from volume to area law scaling for EE and from thermal to nonthermal behavior occur inside the MBL regime much below E1{E}_{1}E1​ and above E2{E}_{2}E2​.