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arXiv:2005.13558

Eigenstates hybridize on all length scales at the many-body localization transition

Benjamin Villalonga, Bryan K. Clark

Published May 27, 2020• 19 citations
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Abstract

An interacting quantum system can transition from an ergodic to a many-body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian. Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in the MBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstates of the disordered Heisenberg model as the disorder strength, $W$, is varied adiabatically. We focus on the probability that two `colliding' eigenstates hybridize as a function of both the range $R$ at which they differ as well as the strength of their hybridization. We find, in the MBL phase, that the probability of a colliding eigenstate hybridizing strongly at range $R$ decays as $Pr(R)\propto \exp [-R/η]$, with a length scale $η(W) = 1 / (B \log(W / W_c) )$ which diverges at the critical disorder strength $W_c$. This leads to range-invariance at the transition, suggesting the formation of resonating cat states at all ranges. This range invariance does not survive to the ergodic phase, where hybridization is exponentially more likely at large range, a fact that can be understood with simple combinatorial arguments. In fact, compensating for these combinatorial effects allows us to define an additional correlation length $ξ$ in the MBL phase which is in excellent agreement with previous works and which takes the critical value $1 / \log(2)$ at the transition, found in previous works to destabilize the MBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-level collisions of eigenstates close in energy.

DOI
10.48550/arxiv.2005.13558

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An interacting quantum system can transition from an ergodic to a many-body localized (MBL) phase under the presence of sufficiently large disorder. Both phases are radically different in their dynamical properties, which are characterized by highly excited eigenstates of the Hamiltonian. Each eigenstate can be characterized by the set of quantum numbers over the set of (local, in the MBL phase) integrals of motion of the system. In this work we study the evolution of the eigenstates of the disordered Heisenberg model as the disorder strength, WWW, is varied adiabatically. We focus on the probability that two `colliding' eigenstates hybridize as a function of both the range RRR at which they differ as well as the strength of their hybridization. We find, in the MBL phase, that the probability of a colliding eigenstate hybridizing strongly at range RRR decays as Pr(R)∝exp⁡[−R/η]Pr(R)\propto \exp [-R/η]Pr(R)∝exp[−R/η], with a length scale η(W)=1/(Blog⁡(W/Wc))η(W) = 1 / (B \log(W / W_c) )η(W)=1/(Blog(W/Wc​)) which diverges at the critical disorder strength WcW_cWc​. This leads to range-invariance at the transition, suggesting the formation of resonating cat states at all ranges. This range invariance does not survive to the ergodic phase, where hybridization is exponentially more likely at large range, a fact that can be understood with simple combinatorial arguments. In fact, compensating for these combinatorial effects allows us to define an additional correlation length ξξξ in the MBL phase which is in excellent agreement with previous works and which takes the critical value 1/log⁡(2)1 / \log(2)1/log(2) at the transition, found in previous works to destabilize the MBL phase. Finally, we show that deep in the MBL phase hybridization is dominated by two-level collisions of eigenstates close in energy.
Closed, interacting, quantum systems have the potential to transition to a many-body localized (MBL) phase under the presence of sufficiently strong disorder, hence breaking ergodicity and failing to thermalize. In this work we study the distribution of correlations throughout the ergodic-MBL phase diagram. We find the typical correlations in the MBL phase decay as a stretched exponential with range rrr eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as e−Are^{-A\sqrt{r}}e−Ar​, a decay that is reminiscent of the random singlet phase. While the standard deviation of the log⁡(QMI)\log(QMI)log(QMI) has a range dependence, the log⁡(QMI)\log(QMI)log(QMI) converges to a range-invariant distribution on all other moments (i.e., the skewness and higher) at the transition. The universal nature of these distributions provides distinct phenomenology of the transition different from both the ergodic and MBL phenomenologies. In addition to the typical correlations, we study the extreme correlations in the system, finding that the probability of strong long-range correlations is maximal at the transition, suggesting the proliferation of resonances there. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, finding that this probability is non-zero deep in the MBL phase but vanishes at moderate disorder well above the transition.
The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).
Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) – at least in one dimensional systems – leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder – in the thermal phase – we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in S_i^z S_{i+r}^z S i z S i + r z correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the S_i^zS_{i+r}^z S i z S i + r z correlator exhibits striking differences between the MBL and Anderson insulator cases.
Understanding the nature of the transition from the delocalized to the many-body localized (MBL) phase is an important unresolved issue. To probe the nature of the MBL transition, we investigate the universal properties of single-particle excitations produced in highly excited many-body eigenstates of a disordered interacting quantum many-body system. In a class of one-dimensional spinless fermionic models with random disorder, we study the finite-size scaling of the ratio of typical to average values of the single-particle local density of states and the scattering rates across the MBL transition. Our results indicate that the MBL transition in this class of one-dimensional models of spinless fermions is continuous in nature. For various ranges of interactions in the system, the critical exponent $\ensuremath{\nu}$ with which the correlation length $\ensuremath{\xi}$ diverges at the transition point ${W}_{c}, \ensuremath{\xi}\ensuremath{\sim}|W\ensuremath{-}{W}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, satisfies the Chayes-Chayes-Fisher-Spencer (CCFS) bound $\ensuremath{\nu}\ensuremath{\ge}2/d$ where ddd is the physical dimension of the system. We also discuss why the critical exponent obtained from finite-size scaling of the conventional diagnostic of many-body localization, the level spacing ratio, strongly violates the CCFS bound while the single-particle density of states and scattering rates are consistent with the CCFS criterion.
Ergodic quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH). However, strong disorder can destroy ergodicity through many-body localization (MBL) - at least in one dimensional systems - leading to a clear signal of the MBL transition in the probability distributions of energy eigenstate expectation values of local operators. For a paradigmatic model of MBL, namely the random-field Heisenberg spin chain, we consider the full probability distribution of eigenstate correlation functions across the entire phase diagram. We find gaussian distributions at weak disorder, as predicted by pure ETH. At intermediate disorder - in the thermal phase - we find further evidence for anomalous thermalization in the form of heavy tails of the distributions. In the MBL phase, we observe peculiar features of the correlator distributions: a strong asymmetry in (SiSi+rz)-S-z correlators skewed towards negative values; and a multimodal distribution for spin-flip correlators. A quantitative quasi-degenerate perturbation theory calculation of these correlators yields a surprising agreement of the full distribution with the exact results, revealing, in particular, the origin of the multiple peaks in the spin-flip correlator distribution as arising from the resonant and off-resonant admixture of spin configurations. The distribution of the (SiSi+rz)-S-z correlator exhibits striking differences between the MBL and Anderson insulator cases.
We propose a multiscale diagonalization scheme to study disordered one-dimensional chains, in particular, the transition between many-body localization (MBL) and the ergodic phase, expected to be governed by resonant spots. Our scheme focuses on the dichotomy of MBL versus validity of the eigenstate thermalization hypothesis. We show that a few natural assumptions imply that the system is localized with probability one at criticality. On the ergodic side, delocalization is induced by a quantum avalanche seeded by large ergodic spots, whose size diverges at the transition. On the MBL side, the typical localization length tends to the inverse of the maximal entropy density at the transition, but there is a divergent length scale related to the response to an inclusion of large ergodic spots. A mean-field approximation analytically illustrates these results and predicts a power-law distribution for thermal inclusions at criticality.