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An emerging many-body ergodic state between two classes of many-body localized states

Wai Pang Sze, Tai Kai Ng

Published May 24, 2020• 0 citations
OpenAlex

Abstract

In this paper, we report our numerical analysis of energy level spacing statistics for one-dimensional spin-$1/2$ XXZ model with random on-site longitudinal magnetic fields $B_i$. We concentrate on the strong disorder limit where the system is expected to be in either a paramagnetic many-body localized (MBL) state or a spin-glass MBL state. By analyzing the energy-level spacing statistics as a function of strength of random magnetic field $h/J_{\perp}$ ($-h\leq B_i\leq h$), interaction strength $J_z/J_{\perp}$, energy of the many-body state $E$ and the number of spin-$\uparrow$ particles in the system $M=\sum_i(s_i^z+{1\over2})$, we show that there exists a small region between the paramagnetic and spin-glass MBL phases where an ergodic phase emerges. The ergodic phase emerges at the middle of the many-body energy spectrum when $M\sim{N\over2}$ and is not adiabatically connected to the ergodic phase that occurs in the weak-disorder, weak-interaction limit.

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We report in this paper our numerical analysis of energy level spacing statistics for the one-dimensional spin-1/21/21/2 XXZ model in random on-site longitudinal magnetic fields BiB_iBi​ (−h≤Bi≤h-h\leq B_i\leq h−h≤Bi​≤h)). We concentrate on the strong disorder limit $J_{\perp}<
We study two different classes of strongly interacting many-body quantum systems in this thesis. The first concerns interacting quantum system with strong disorder. The second is a quantum spin liquid problem. Since many-body quantum systems with strong interaction are difficult in analytic approaches, numerical methods will be used to study these systems. In the first part of this thesis, we investigate the many-body localization of the 1D spin-1/2 XXZ model with random magnetic field by Exact Diagonalization. We confirm previous results that there exists a mobility edge between ergodic and localized phases at weak disorder. Furthermore, we study the difference between two many-body localization phases in the XXZ model and confirm that there exists a phase transition between two kinds of many-body localization. Surprisingly, we observe a narrow region of ergodic phase that exists between the two MBL phases. We call the phase Many-Body Ergodic phase since the phase is not analytically connected to the trivial ergodic phase at weak order. In the second part of this thesis, we study the spin -1/2 XXZ spin model on the triangular lattice by mean of Variational Monte Carlo simulation on a class of resonant valence bond (RVB) trial wave-functions. We find that as a function of J⊥/Jz, there exists a transition point where magnetic order vanishes (at J⊥/Jz ~ 0.45). Moreover, two kinds of RVB states competed in the Ising regime when J⊥/Jz < 0.23.
Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.
Abstract We study the many-body localization (MBL) properties of the Heisenberg XXZ spin- 12\frac 1221​ chain in a random magnetic field. We prove that the system exhibits localization in any given energy interval at the bottom of the spectrum in a nontrivial region of the parameter space. This region, which includes weak interaction and strong disorder regimes, is independent of the size of the system and depends only on the energy interval. Our approach is based on the reformulation of the localization problem as an expression of quasi-locality for functions of the random many-body XXZ Hamiltonian. This allows us to extend the fractional moment method for proving localization, previously derived in a single-particle localization context, to the many-body setting.
We investigate the occurrence of many-body localization (MBL) on a spin-1/2 transverse-field Ising model defined on a Chimera connectivity graph with random exchange interactions and longitudinal fields. We observe a transition from an ergodic phase to a nonthermal phase for individual energy eigenstates induced by a critical disorder strength for the Ising parameters. Our result follows from the analysis of both the mean half-system block entanglement and the energy-level statistics. We identify the critical point associated with this transition using the maximum variance of the block entanglement over the disorder ensemble as a function of the disorder strength. The calculated energy density phase diagram shows the existence of a mobility edge in the energy spectrum. In terms of the energy-level statistics, the system changes from the Gaussian orthogonal ensemble for weak disorder to a Poisson distribution limit for strong randomness, which implies localization behavior. We then realize the time-independent disordered Ising Hamiltonian experimentally using a reverse annealing quench-pause-quench protocol on a D\mathrm{D}D-wave 2000Q programmable quantum annealer. We characterize the transition from the thermal to the localized phase through magnetization measurements at the end of the annealing dynamics, and the results are compatible with our theoretical prediction for the critical point. However, the same behavior can be reproduced using a classical spin-vector Monte Carlo simulation, which suggests that genuine quantum signatures of the phase transition remain out of reach using this experimental platform and protocol.
Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.