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Physical Review Research

Characterizing the many-body localization transition by the dynamics of diagonal entropy

Zheng‐Hang Sun, Jian Cui, Heng Fan

Published Feb 18, 2020• 18 citations
Publisher

Abstract

The authors present a method of charactering many-body localization transition via the quench dynamics of diagonal entropy, as an experimentally feasible quantity. The critical point of many-body localization transition can be efficiently detected. The adopted scaling ansatz respects the Harris-Luck bound, and the scaling exponent can provide information for the universality class of many-body localization transition.

DOI
10.1103/physrevresearch.2.013163

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The authors present a method of charactering many-body localization transition via the quench dynamics of diagonal entropy, as an experimentally feasible quantity. The critical point of many-body localization transition can be efficiently detected. The adopted scaling ansatz respects the Harris-Luck bound, and the scaling exponent can provide information for the universality class of many-body localization transition.
Considering ultracold atoms in optical lattices, we propose experimental protocols to study many-body-localization (MBL) length and criticality in quench dynamics. Through numerical simulations with exact diagonalization, we show that in the MBL phase the perturbed density profile following a local quench remains exponentially localized in postquench dynamics. The size of this density profile after long-time-dynamics defines a localization length, which tends to diverge at the MBL-to-ergodic transition as we increase the system size. The determined localization transition point agrees with previous exact diagonalization calculations using other diagnostics. Our numerical results provide evidence for violation of the Harris-Chayes bound for the MBL criticality. The critical exponent $\ensuremath{\nu}$ can be extracted from our proposed dynamical procedure, which can then be used directly in experiments to determine whether the Harris-Chayes-bound holds for the MBL transition. These proposed protocols to detect localization criticality are justified by benchmarking to the well-established results for the noninteracting three-dimensional Anderson localization.
The precise nature of many-body localization (MBL) transitions in both random and quasiperiodic (QP) systems remains elusive so far. In particular, whether MBL transitions in QP and random systems belong to the same universality class or two distinct ones has not been decisively resolved. Here, we investigate MBL transitions in one-dimensional (d=1) QP systems as well as in random systems by state-of-the-art real-space renormalization group (RG) calculation. Our real-space RG shows that MBL transitions in 1D QP systems are characterized by the critical exponent ν≈2.4, which respects the Harris-Luck bound (ν>1/d) for QP systems. Note that ν≈2.4 for QP systems also satisfies the Harris-Chayes-Chayes-Fisher-Spencer bound (ν>2/d) for random systems, which implies that MBL transitions in 1D QP systems are stable against weak quenched disorder since randomness is Harris irrelevant at the transition. We shall briefly discuss experimental means to measure ν of QP-induced MBL transitions.
As strength of disorder enhances beyond a threshold value in many-body systems, a fundamental transformation happens through which the entire spectrum localizes, a phenomenon known as many-body localization. This has profound implications as it breaks down fundamental principles of statistical mechanics, such as thermalization and ergodicity. Due to the complexity of the problem, the investigation of the many-body localization transition has remained a big challenge. The experimental exploration of the transition point is even more challenging as most of the proposed quantities for studying such an effect are practically infeasible. Here, we experimentally implement a scalable protocol for detecting the many-body localization transition point, using the dynamics of an N=12N=12N=12 superconducting qubit array. We show that the sensitivity of the dynamics to random samples becomes maximum at the transition point, which leaves its fingerprints in all spatial scales. By exploiting three quantities, each with a different spatial resolution, we identify the transition point with an excellent match between simulation and experiment. In addition, one can detect the evidence of a mobility edge through slight variation of the transition point as the initial state varies. The protocol is easily scalable and can be performed across various physical platforms.
The authors study the phase transition between thermal and many-body localized phases in one-dimensional systems with quenched randomness and short-range interactions. The approach uses a renormalization group model, very similar to those of previous works, that puts the avalanche instability of the many-body localized phase at center stage. Within that model, the universal critical behavior is determined analytically, and the phase transition is found to belong to a new universality class that differs from the previously found Kosterlitz-Thouless result in important ways.
A key feature of many-body localization (MBL) is the breaking of ergodicity and consequently the emergence of local memory, revealed as the local preservation of information over time. As memory is necessarily a time-dependent concept, it has been partially captured by a few extant studies of dynamical quantities. However, these quantities suffer from a variety of issues which limit their value as true quantifiers of memory; thus, a fundamental and complete information-theoretic understanding of local memory in the context of MBL remains elusive. We outline these issues in detail and introduce the dynamical Holevo quantity to address them. We find that it shows clear scaling behavior across the MBL transition, and we determine a family of two-parameter scaling Ans\"atze which capture this behavior. We perform a comprehensive finite-sized scaling analysis to extract the transition point and scaling exponents.
The many-body localization (MBL) transition is a quantum phase transition involving highly excited eigenstates of a disordered quantum many-body Hamiltonian, which evolve from “extended/ergodic" (exhibiting extensive entanglement entropies and fluctuations) to “localized" (exhibiting area-law scaling of entanglement and fluctuations). The MBL transition can be driven by the strength of disorder in a given spectral range, or by the energy density at fixed disorder – if the system possesses a many-body mobility edge. Here we propose to explore the latter mechanism by using “quantum-quench spectroscopy", namely via quantum quenches of variable width which prepare the state of the system in a superposition of eigenstates of the Hamiltonian within a controllable spectral region. Studying numerically a chain of interacting spinless fermions in a quasi-periodic potential, we argue that this system has a many-body mobility edge; and we show that its existence translates into a clear dynamical transition in the time evolution immediately following a quench in the strength of the quasi-periodic potential, as well as a transition in the scaling properties of the quasi-stationary state at long times. Our results suggest a practical scheme for the experimental observation of many-body mobility edges using cold-atom setups.