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Register a new userDetecting ergodic bubbles at the crossover to many-body localization using neural networks
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The transition between ergodic and many-body localized phases is expected to occur via an avalanche mechanism, in which emph{ergodic bubbles} that arise due to local fluctuations in system properties thermalize their surroundings leading to delocalization of the system, unless the disorder is sufficiently strong to stop this process. We propose an algorithm based on neural networks that allows to detect the ergodic bubbles using experimentally measurable two-site correlation functions. Investigating time evolution of the system, we observe a logarithmic in time growth of the ergodic bubbles in the MBL regime. The distribution of the size of ergodic bubbles converges during time evolution to an exponentially decaying distribution in the MBL regime, and a power-law distribution with a thermal peak in the critical regime, supporting thus the scenario of delocalization through the avalanche mechanism. Our algorithm permits to pin-point quantitative differences in time evolution of systems with random and quasiperiodic potentials, as well as to identify rare (Griffiths) events. Our results open new pathways in studies of the mechanisms of thermalization of disordered many-body systems and beyond.
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Subsystems of strongly disordered, interacting quantum systems can fail to thermalize because of the phenomenon of many-body localization (MBL). In this article, we explore a tensor network description of the eigenspectra of such systems. Specifically, we will argue that the presence of a complete set of local integrals of motion in MBL implies an efficient representation of the entire spectrum of energy eigenstates with a single tensor network, a emph{spectral} tensor network. Our results are rigorous for a class of idealized systems related to MBL with integrals of motion of finite support. In one spatial dimension, the spectral tensor network allows for the efficient computation of expectation values of a large class of operators (including local operators and string operators) in individual energy eigenstates and in ensembles.
The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the transition point, the logarithmic negativity obeys a scale invariant exponential decay with respect to the ratio of block separation to size, whereas the mutual information obeys a polynomial decay. The observed scale invariance of the quantum correlations in a microscopic model opens the direction to probe the fractal structure in critical eigenstates using tensor network techniques and provide constraints on the theory of the many-body localization transition.
Polynomially filtered exact diagonalization method (POLFED) for large sparse matrices is introduced. The algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix. The memory requirements scale better with system size than in the state-of-the-art shift-invert approach. The potential of POLFED is demonstrated examining many-body localization transition in 1D interacting quantum spin-1/2 chains. We investigate the disorder strength and system size scaling of Thouless time. System size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite-size effects in the system. We discuss possible scenarios regarding the many-body localization transition obtaining estimates for the critical disorder strength.
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 $p* = lim_{lambda_2 rightarrow ln(2)^{+}}P_2(lambda_2)$, where $P_2(lambda_2)$ is the probability distribution of the second lowest entanglement eigenvalue $lambda_2$, behaves as an order-parameter for the MBL phase: $p*> 0$ in the MBL phase, while $p* = 0$ 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.
The many-body localised (MBL) to thermal crossover observed in exact diagonalisation studies remains poorly understood as the accessible system sizes are too small to be in an asymptotic scaling regime. We develop a model of the crossover in short 1D chains in which the MBL phase is destabilised by the formation of many-body resonances. The model reproduces several properties of the numerically observed crossover, including an apparent correlation length exponent $ u=1$, exponential growth of the Thouless time with disorder strength, linear drift of the critical disorder strength with system size, scale-free resonances, apparent $1/omega$ dependence of disorder-averaged spectral functions, and sub-thermal entanglement entropy of small subsystems. In the crossover, resonances induced by a local perturbation are rare at numerically accessible system sizes $L$ which are smaller than a emph{resonance length} $lambda$. For $L gg sqrt{lambda}$, resonances typically overlap, and this model does not describe the asymptotic transition. The model further reproduces controversial numerical observations which Refs. [v{S}untajs et al, 2019] and [Sels & Polkovnikov, 2020] claimed to be inconsistent with MBL. We thus argue that the numerics to date is consistent with a MBL phase in the thermodynamic limit.
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