The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sublinear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field. By a finite size scaling analysis of the peak position and crossover point from log to linear scaling we collect evidence that ergodicity is broken before the MBL transition in this model.
The intriguing phenomenon of many-body localization (MBL) has attracted significant interest recently, but a complete characterization is still lacking. In this work we introduce the total correlations, a concept from quantum information theory capturing multipartite correlations, to the study of this phenomenon. We demonstrate that the total correlations of the diagonal ensemble provides a meaningful diagnostic tool to pin-down, probe, and better understand the MBL transition and ergodicity breaking in quantum systems. In particular, we show that the total correlations has sublinear dependence on the system size in delocalized, ergodic phases, whereas we find that it scales extensively in the localized phase developing a pronounced peak at the transition. We exemplify the power of our approach by means of an exact diagonalization study of a Heisenberg spin chain in a disordered field. By a finite size scaling analysis of the peak position and crossover point from log to linear scaling we collect evidence that ergodicity is broken before the MBL transition in this model.
The diagonal ensemble is the infinite time average of a quantum state following unitary dynamics in systems without degeneracies. In analogy to the time average of a classical phase-space dynamics, it is intimately related to the ergodic properties of the quantum system giving information on the spreading of the initial state in the eigenstates of the Hamiltonian. In this work we apply a concept from quantum information, known as total correlations, to the diagonal ensemble. Forming an upper bound on the multipartite entanglement, it quantifies the combination of both classical and quantum correlations in a mixed state. We generalize the total correlations of the diagonal ensemble to more general $\ensuremath{\alpha}$-Renyi entropies and focus on the cases $\ensuremath{\alpha}=1$ and $\ensuremath{\alpha}=2$ with further numerical extensions in mind. Here we show that the total correlations of the diagonal ensemble is a generic indicator of ergodicity breaking, displaying a subextensive behavior when the system is ergodic. We demonstrate this by investigating its scaling in a range of spin chain models focusing not only on the cases of integrability breaking but also emphasize its role in understanding the transition from an ergodic to a many-body localized phase in systems with disorder or quasiperiodicity.
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
We analyze the finite-size scaling of the average gap ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin chain with quasiperiodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of the Berezinskii-Kosterlitz-Thouless transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sublinear drift with the system size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.
In this work, we use exact matrix diagonalization to explore the many-body localization (MBL) transition in a random-field Heisenberg chain. We demonstrate that the fidelity and fidelity susceptibility can be utilized to characterize the interaction-driven many-body localization transition in this closed spin system which is in agreement with previous analytical and numerical results [S. Garnerone, N. T. Jacobson, S. Haas, and P. Zanardi, Phys. Rev. Lett. 102, 057205 (2009)PRLTAO0031-900710.1103/PhysRevLett.102.057205; P. Zanardi and N. Paunkovic, Phys. Rev. E 74, 031123 (2006)PLEEE81539-375510.1103/PhysRevE.74.031123]. In particular, instead of ground-state fidelity, we test the fidelity between two diagonal ensembles related by a small parameter perturbation δh, it is special that here the parameter perturbation δh_{i} for each site are random variables like h_{i}. It shows that fidelity of the diagonal ensemble develop a pronounced drop at the transition. We utilize fidelity to estimate the critical disorder strength h_{c} for different system size, we get h_{c}∈ [2.5,3.9] and get a power-law decay with an exponent of roughly -1.49(2) for system size N, and can extrapolate h_{c}^{inf} of the infinite system is about 2.07 which all agree with a recent work by Huse and Pal, in which the MBL transition in the same model was predicted to be hc [2,4]. We also estimate the scaling of maximum of averaged fidelity susceptibility as a function of system size N, it shows a power law increase with an exponent of about 5.05(1).
We investigate the phenomenon of spatial many-body localization (MBL) through pairwise correlation measures based on one- and two-point correlation functions. The system considered is the Heisenberg spin-1/2 chain with exchange interaction J and random on-site disorder of strength h. As a representative pairwise correlation measure obtained from one-point functions, we use global entanglement. Through its finite-size-scaling analysis, we locate the MBL critical point at hc/J=3.8. As for measures involving two-point functions, we analyze pairwise geometric classical, quantum, and total correlations. Similarly to what happens for continuous quantum phase transitions, it is the derivatives of these two-point correlation measures that identify the MBL critical point, which is found to be in the range ${h}_{c}/J\ensuremath{\in}[3,4]$. Our approach relies on very simple measures that do not require access to multipartite entanglement or large portions of the system.
Many-body localization (MBL) is a novel prototype of ergodicity breaking due to the emergence of local integrals of motion (LIOMs) in a disordered interacting quantum system. To better understand the role played by the existence of such LIOMs, we explore and study some of their structural properties across the MBL transition. We first consider a one-dimensional XXZ spin chain in a disordered magnetic field and introduce and implement a nonperturbative, fast, and accurate method of constructing LIOMs. In contrast to already existing methods, our scheme allows obtaining LIOMs not only in the deep MBL phase but, rather, near the transition point too. Then, we take the matrix representation of LIOM operators as an adjacency matrix of a directed graph whose elements describe the connectivity of ordered eigenbasis in the Hilbert space. Our cluster-size analysis for this graph shows that the MBL transition coincides with a percolation transition in the Hilbert space. By performing finite-size scaling, we compare the critical disorder and correlation exponent $\ensuremath{\nu}$ both in the presence and absence of interactions. Finally, we also discuss how the distribution of diagonal elements of LIOM operators in a typical cluster signals the transition.