Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2007.06586 · Bulletin of the American Physical Society

Characterizing the many-body localization transition through correlations

Benjamin Villalonga, Bryan K. Clark

Published Jul 13, 2020• 6 citations
arXivHTMLPDF

Abstract

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 $r$ eventually crossing over to an exponential decay deep in the MBL phase. At the transition, the stretched exponential goes as $e^{-A\sqrt{r}}$, a decay that is reminiscent of the random singlet phase. While the standard deviation of the $\log(QMI)$ has a range dependence, the $\log(QMI)$ 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.

DOI
10.48550/arxiv.2007.06586

Related papers

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

Benjamin Villalonga, Bryan K. Clark

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.

arXiv:2005.13558•May 27, 2020•19 citations

Statistics of correlation functions in the random Heisenberg chain

Luis A. Colmenarez, Paul McClarty, Masud Haque +1 more

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.

arXiv:1906.10701•SciPost Physics•Nov 21, 2019•34 citations

Statistics of correlation functions in the random Heisenberg chain

Colmenarez, Luis, Paul McClarty, Masudul Haque +1 more

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.

•Jan 1, 2019•33 citations

Statistics of systemwide correlations in the random-field XXZ chain: Importance of rare events in the many-body localized phase

Jeanne Colbois, Fabien Alet, Nicolas Laflorencie

Motivated by recent debates around the many-body localization (MBL) problem, and in particular its stability against systemwide resonances, we investigate long-distance spin-spin correlations across the phase diagram of the random-field XXZ model, with a particular focus on the strong disorder regime. Building on state-of-the-art shift-invert diagonalization techniques, we study the high-energy behavior of transverse and longitudinal correlation functions, computed at the largest possible distance, for a broad range of disorder and interaction strengths. Our results show that while transverse correlations display a fairly stable exponential decay over the entire XXZ phase diagram, longitudinal correlations exhibit markedly different behavior, revealing distinct physical regimes. More precisely, we identify an intermediate disorder region where standard observables show well-converged MBL behavior [Colbois et al., Phys. Rev. Lett. 133, 116502 (2024)] while the distributions of longitudinal correlations reveal unexpected fat tails toward large values. These rare events strongly influence the average decay of longitudinal correlations, which we find to be algebraic in a broad region inside the supposed MBL phase, whereas the typical decay remains mostly exponential. At stronger disorder and weaker interactions, this intermediate regime is replaced by a more conventional exponential decay with short correlation lengths for both typical and average correlators, as expected for standard localization. Our findings shed light on the systemwide instabilities and raise important questions about the impact of such rare but large long-range correlations on the stability of the MBL phase. Finally, we discuss the possible fate of the intermediate region in the context of recent perspectives in the field.

arXiv:2410.10325•Physical review. B./Physical review. B•Dec 26, 2024•10 citations

Tracking the many-body localized to ergodic transition via extremal statistics of entanglement eigenvalues

Abhisek Samanta, Kedar Damle, Rajdeep Sensarma

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: $p*>0$ 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.

arXiv:2001.10198•Physical review. B./Physical review. B•Sep 2, 2020•11 citations

Exploring one-particle orbitals in large many-body localized systems

Benjamin Villalonga, Xiongjie Yu, David J. Luitz +1 more

Disorder and interactions can lead to the breakdown of statistical mechanics in quantum systems by transitioning to a many-body localized (MBL) phase. The thermal-to-MBL transition occurs by either increasing the amount of disorder present in a system or by decreasing its energy until it crosses the so-called mobility edge. The authors construct a set of approximate integrals of motion that are quasi-universal across the energy spectrum and, thus, across the mobility edge, too. Interestingly, MBL eigenstates under the mobility edge are aware of the existence of a thermal phase at a higher energy. The phase diagram is probed close to the transition and in the MBL phase. One finds a universal distribution of the strength of the couplings of the integrals of motion in the strong disorder limit.

arXiv:1710.05036•Physical review. B./Physical review. B•Mar 7, 2018•35 citations
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.
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.
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.
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.
Motivated by recent debates around the many-body localization (MBL) problem, and in particular its stability against systemwide resonances, we investigate long-distance spin-spin correlations across the phase diagram of the random-field XXZ model, with a particular focus on the strong disorder regime. Building on state-of-the-art shift-invert diagonalization techniques, we study the high-energy behavior of transverse and longitudinal correlation functions, computed at the largest possible distance, for a broad range of disorder and interaction strengths. Our results show that while transverse correlations display a fairly stable exponential decay over the entire XXZ phase diagram, longitudinal correlations exhibit markedly different behavior, revealing distinct physical regimes. More precisely, we identify an intermediate disorder region where standard observables show well-converged MBL behavior [Colbois et al., Phys. Rev. Lett. 133, 116502 (2024)] while the distributions of longitudinal correlations reveal unexpected fat tails toward large values. These rare events strongly influence the average decay of longitudinal correlations, which we find to be algebraic in a broad region inside the supposed MBL phase, whereas the typical decay remains mostly exponential. At stronger disorder and weaker interactions, this intermediate regime is replaced by a more conventional exponential decay with short correlation lengths for both typical and average correlators, as expected for standard localization. Our findings shed light on the systemwide instabilities and raise important questions about the impact of such rare but large long-range correlations on the stability of the MBL phase. Finally, we discuss the possible fate of the intermediate region in the context of recent perspectives in the field.
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.
Disorder and interactions can lead to the breakdown of statistical mechanics in quantum systems by transitioning to a many-body localized (MBL) phase. The thermal-to-MBL transition occurs by either increasing the amount of disorder present in a system or by decreasing its energy until it crosses the so-called mobility edge. The authors construct a set of approximate integrals of motion that are quasi-universal across the energy spectrum and, thus, across the mobility edge, too. Interestingly, MBL eigenstates under the mobility edge are aware of the existence of a thermal phase at a higher energy. The phase diagram is probed close to the transition and in the MBL phase. One finds a universal distribution of the strength of the couplings of the integrals of motion in the strong disorder limit.