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arXiv:1405.5780 · Journal of Statistical Mechanics Theory and Experiment

Signatures of many-body localisation in a system without disorder and the relation to a glass transition

James M Hickey, Sam Genway, Juan P Garrahan

Published May 1, 2016• 77 citations
arXivHTMLPDF

Abstract

Abstract We study a quantum spin system—adapted from a facilitated spin model for classical glasses—with local bilinear interactions and without quenched disorder which seems to display characteristic signatures of a many-body localisation (MBL) transition. From direct diagonalisation of small systems, we find a change in certain dynamical and spectral properties at a critical value of a coupling, from those characteristic of a thermalising phase to those characteristic of a MBL phase. The system we consider is known to have a quantum phase transition in its ground-state in the limit of large size, related to a first-order active-to-inactive phase transition in the stochastic trajectories of an associated classical model of glasses. Our results here suggest that this first-order transition in the low-lying spectrum may influence the rest of the spectrum of the system in the large size limit. These findings may help understand the connection between MBL and structural glass transitions.

DOI
10.1088/1742-5468/2016/05/054047

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Abstract We study a quantum spin system—adapted from a facilitated spin model for classical glasses—with local bilinear interactions and without quenched disorder which seems to display characteristic signatures of a many-body localisation (MBL) transition. From direct diagonalisation of small systems, we find a change in certain dynamical and spectral properties at a critical value of a coupling, from those characteristic of a thermalising phase to those characteristic of a MBL phase. The system we consider is known to have a quantum phase transition in its ground-state in the limit of large size, related to a first-order active-to-inactive phase transition in the stochastic trajectories of an associated classical model of glasses. Our results here suggest that this first-order transition in the low-lying spectrum may influence the rest of the spectrum of the system in the large size limit. These findings may help understand the connection between MBL and structural glass transitions.
We present a detailed analysis of the length-and timescales needed to approach the critical region of MBL from the delocalised phase, studying both eigenstates and the time evolution of an initial state.For the eigenstates we show that in the delocalised region there is a single length, which is a function of disorder strength, controlling the finite-size flow.Small systems look localised, and only for larger systems do resonances develop which restore ergodicity in the form of the eigenstate thermalisation hypothesis.For the transport properties, we study the time necessary to transport a single spin across a domain wall, showing how this grows quickly with increasing disorder, and compare it with the Heisenberg time.For a sufficiently large system the Heisenberg time is always larger than the transport time, but for a smaller system this is not necessarily the case.We conclude that the properties of the MBL transition cannot be explored using the system sizes or times available to current numerical and experimental studies.
Since the seminal work of Anderson, localisation has been recognised as a standard mechanism allowing quantum many-body systems to escape ergodicity. This idea acquired even more prominence in the last decade as it has been argued that localisation -- dubbed many-body localisation (MBL) in this context -- can sometimes survive local interactions in the presence of sufficiently strong disorder. A conventional signature of localisation is in the statistical properties of the spectrum -- spectral statistics -- which differ qualitatively from those in the ergodic phase. Although features of the spectral statistics are routinely used as numerical diagnostics for localisation, they have never been derived from first principles in the presence of non-trivial interactions. Here we fill this gap and provide the example of a simple class of quantum many-body systems -- which we dub strongly localised quantum circuits -- that are interacting, localised, and where the spectral statistics can be characterised exactly. Furthermore, we show that these systems exhibit a cascade of three different regimes for spectral correlations depending on the energy scale: at small, intermediate, and large scales they behave as disconnected patches of three decreasing sizes. We argue that these features appear in generic MBL systems, with the difference that only at the smallest scale they become Poissonian.
Many-body localization occurs in isolated quantum systems when Anderson localization persists in the presence of finite interactions. Despite strong evidence for the existence of a many-body localization transition, a reliable extraction of the critical disorder strength is difficult due to a large drift with system size in the studied quantities. In this Letter, we explore two entanglement properties that are promising for the study of the many-body localization transition: the variance of the half-chain entanglement entropy of exact eigenstates and the long time change in entanglement after a local quench from an exact eigenstate. We investigate these quantities in a disordered quantum Ising chain and use them to estimate the critical disorder strength and its energy dependence. In addition, we analyze a spin-glass transition at large disorder strength and provide evidence for it being a separate transition. We, thereby, give numerical support for a recently proposed phase diagram of many-body localization with localization protected quantum order [Huse et al., Phys. Rev. B 88, 014206 (2013).
We consider the random-field Heisenberg model, a paradigmatic model for many-body localization (MBL), and add a Markovian dephasing bath coupled to the Anderson orbitals of the model's noninteracting limit. We map this system to a classical facilitated hopping model that is computationally tractable for large system sizes and investigate its dynamics. The classical model exhibits a robust crossover between an ergodic (thermal) phase and a frozen (localized) phase. The frozen phase is destabilized by thermal subregions (bubbles), which thermalize surrounding sites by providing a fluctuating interaction energy and so enable off-resonance particle transport. Investigating steady-state transport, we observe that the interplay between thermal and frozen bubbles leads to a clear transition between diffusive and subdiffusive regimes. This phenomenology both describes the MBL system coupled to a bath and provides a classical analog for the MBL transition in the corresponding quantum model, in that the classical model displays long local memory times. It also highlights the importance of the details of the bath coupling in studies of MBL systems coupled to thermal environments.
Some interacting disordered many-body systems are unable to thermalize when the quenched disorder becomes larger than a threshold value. Although several properties of nonzero energy density eigenstates (in the middle of the many-body spectrum) exhibit a qualitative change across this many-body localization (MBL) transition, many of the commonly used diagnostics only do so over a broad transition regime. Here we provide evidence that the transition can be located precisely even at modest system sizes by sharply defined changes in the distribution of extremal eigenvalues of the reduced density matrix of subsystems. In particular, our results suggest that $p*={lim}_{{\ensuremath{\lambda}}_{2}\ensuremath{\rightarrow}ln{(2)}^{+}}{P}_{2}({\ensuremath{\lambda}}_{2})$, where ${P}_{2}({\ensuremath{\lambda}}_{2})$ is the probability distribution of the second lowest entanglement eigenvalue ${\ensuremath{\lambda}}_{2}$, behaves as an ``order parameter'' for the MBL phase: p∗>0p*>0p∗>0 in the MBL phase, while p∗=0p*=0p∗=0 in the ergodic phase with thermalization. Thus, in the MBL phase, there is a nonzero probability that a subsystem is entangled with the rest of the system only via the entanglement of one subsystem qubit with degrees of freedom outside the region. In contrast, this probability vanishes in the thermal phase.
In this paper, we theoretically investigate the many-body localization properties of one-dimensional Ising spin-1 chains by using the methods of exact matrix diagonalization. We compare it with the MBL properties of the Ising spin-1/2 chains. The results indicate that the one-dimensional Ising spin-1 chains can also undergo MBL phase transition. There are various forms of disorder, and we compare the effects of different forms of quasi-disorder and random disorder on many-body localization in this paper. First, we calculate the exctied-state fidelity to study the MBL phase transtion. By changing the form of the quasi-disorder, we study the MBL transition of the system with different forms of quasi-disorder and compare them with those of the random disordered system. The results show that both random disorder and quasi-disorder can cause the MBL phase transition in the one-dimensional Ising spin-1 chains. In order to study the effect of spin interactions, we compare Ising spin-1 chains and spin-1/2 chains with the next-nearest-neighbour(N-N) two-body interactions and the next-next-nearest-neighbour (N-N-N)interactions. The results show that the critical point increases with the addition of the interaction. Then we study the dynamical properties of the model by the dynamical behavior of diagonal entropy (DE), local magnetization and the time evolution of fidelity to further prove the occurrence of MBL phase transition in the disordered Ising spin-1 chains with the (N-N) coupling term and distinguish the ergodic phase (thermal phase) and the many-body localized phase. Lastly, we delve into the impact of periodic driving on one-dimensional Ising spin-1 chains. And we compare it with the results obtained from the Ising spin-1/2 chains. It shows that periodic driving can cause Ising spin-1 chains and Ising spin-1/2 chains to occur the MBL transition.