Statistics of correlation functions in the random Heisenberg chain

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 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:1906.10701SciPost PhysicsNov 21, 201934 citations

Long tail distributions near the many-body localization transition

David J. Luitz

The random field $S=\frac{1}{2}$ Heisenberg chain exhibits a dynamical many body localization transition at a critical disorder strength, which depends on the energy density. At weak disorder, the eigenstate thermalization hypothesis (ETH) is fulfilled on average, making local observables smooth functions of energy, whose eigenstate-to-eigenstate fluctuations decrease exponentially with system size. We demonstrate the validity of ETH in the thermal phase as well as its breakdown in the localized phase and show that rare states exist which do not strictly follow ETH, becoming more frequent closer to the transition. Similarly, the probability distribution of the entanglement entropy at intermediate disorder develops long tails all the way down to zero entanglement. We propose that these low entanglement tails stem from localized regions at the subsystem boundaries which were recently discussed as a possible mechanism for subdiffusive transport in the ergodic phase.The random field $S=\frac{1}{2}$ Heisenberg chain exhibits a dynamical many body localization transition at a critical disorder strength, which depends on the energy density. At weak disorder, the eigenstate thermalization hypothesis (ETH) is fulfilled on average, making local observables smooth functions of energy, whose eigenstate-to-eigenstate fluctuations decrease exponentially with system size. We demonstrate the validity of ETH in the thermal phase as well as its breakdown in the localized phase and show that rare states exist which do not strictly follow ETH, becoming more frequent closer to the transition. Similarly, the probability distribution of the entanglement entropy at intermediate disorder develops long tails all the way down to zero entanglement. We propose that these low entanglement tails stem from localized regions at the subsystem boundaries which were recently discussed as a possible mechanism for subdiffusive transport in the ergodic phase.

arXiv:1601.04058Physical review. B./Physical review. BApr 1, 2016120 citations

Large deviations in the many-body localization transition: The case of the random-field XXZ chain

Greivin Alfaro Miranda, Fabien Alet, Giulio Biroli +3 more

The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.The effect of rare system-wide resonances in the many-body localization (MBL) transition has recently attracted significant attention. They are expected to play a prominent role in the stability of the MBL phase, prompting the development of new theoretical frameworks to properly account for their statistical weight. We employ a method based on an analogy with mean-field disordered glassy systems to characterize the statistics of transmission amplitudes between distant many-body configurations in Hilbert space, and apply it to the random-field XXZ spin chain. By introducing a Lagrange multiplier, which formally plays the role of an effective temperature controlling the influence of extreme outliers in the heavy-tailed distribution of propagators, we identify three distinct regimes: (i) an ergodic phase with uniform spreading in Hilbert space, (ii) an intermediate regime where delocalization is driven by rare, disorder-dependent long-range resonances, and (iii) a robust MBL phase where such resonances cannot destabilize localization. We derive a finite-size phase diagram in the disorder--interaction plane both in the spin and in the Anderson basis that quantitatively agrees with recent numerical results based on real-space spin-spin correlation functions. We further demonstrate that even infinitesimal interactions can destroy the Anderson insulator at finite disorder, with the critical disorder remaining finite down to small interaction strengths. By visualizing resonant transmission pathways on the Hilbert space graph, we provide a complementary perspective to real-space and spectral probes, revealing how the destabilization of the MBL phase at finite sizes stems from the emergence of resonant paths that become progressively rarer and shorter-ranged deep in the localized phase.

arXiv:2510.18545Physical review. B./Physical review. BJul 10, 20261 citation

Interaction-Driven Instabilities in the Random-Field X X Z Chain

Jeanne Colbois, Fabien Alet, Nicolas Laflorencie

Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.Despite enormous efforts devoted to the study of the many-body localization (MBL) phenomenon, the nature of the high-energy behavior of the Heisenberg spin chain in a strong random magnetic field is lacking consensus. Here, we take a step back by exploring the weak interaction limit starting from the Anderson localized (AL) insulator. Through shift-invert diagonalization, we find that, below a certain disorder threshold h^{*}, weak interactions necessarily lead to an ergodic instability, whereas at strong disorder the AL insulator directly turns into MBL, in agreement with a simple interpretation of the avalanche theory for restoration of ergodicity. We further map the phase diagram for the generic XXZ model in the disorder h-interaction Δ plane. Taking advantage of the total magnetization conservation, our results unveil the remarkable behavior of the spin-spin correlation functions: in the regime indicated as MBL by standard observables, their exponential decay undergoes an inversion of orientation ξ_{z}>ξ_{x}. We find that the longitudinal length ξ_{z} is a key quantity for capturing ergodic instabilities, as it increases with system size near the thermal phase, in sharp contrast to its transverse counterpart.

arXiv:2403.09608Physical Review LettersSep 12, 202427 citations

Thermalization and Thouless Energy in the Ergodic Side of the Many-Body Localization Transition

Ángel L. Corps, Rafael A. Molina, Armando Relaño

Within the Thouless energy scale, many-body Hamiltonians behave like random matrices. We study the Thouless energy of the disordered Heisenberg quantum spin chain. This model has become the standard model for the study of many-body localization and the transition to an ergodic phase for weak disorder. We focus on the ergodic side of the Many-Body Localization transition. The Thouless energy extracted from long range spectral statistics and from the power spectrum of the fluctuations of the full momentum distribution coincide. This scale behaves non-monotonically and is peaked at an intermediate value of the disorder. An analysis of the scaling behavior of the Thouless energy and its comparison with the number of populated eigenstates after a quench suggests that this scale is not large enough to guarantee thermalization in the thermodynamic limit.Within the Thouless energy scale, many-body Hamiltonians behave like random matrices. We study the Thouless energy of the disordered Heisenberg quantum spin chain. This model has become the standard model for the study of many-body localization and the transition to an ergodic phase for weak disorder. We focus on the ergodic side of the Many-Body Localization transition. The Thouless energy extracted from long range spectral statistics and from the power spectrum of the fluctuations of the full momentum distribution coincide. This scale behaves non-monotonically and is peaked at an intermediate value of the disorder. An analysis of the scaling behavior of the Thouless energy and its comparison with the number of populated eigenstates after a quench suggests that this scale is not large enough to guarantee thermalization in the thermodynamic limit.

Jan 12, 20201 citation

False signatures of non-ergodic behavior in disordered quantum many-body systems

Adith Sai Aramthottil, Lev Vidmar, Jakub Zakrzewski

Ergodic isolated quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH), i.e. the expectation values of local observables in the system's eigenstates approach the predictions of the microcanonical ensemble.However, the ETH does not specify what happens to expectation values of local observables within an energy window when the average over disorder realizations is taken.As a result, the expectation values of local observables can be distributed over a relatively wide interval and may exhibit nontrivial structure, as shown in Aramthottil et al (2021 Phys.Rev. B 104 214201) for a quasiperiodic disordered system for site-resolved magnetization.We argue that the non-Gaussian form of this distribution may falsely suggest non-ergodicity and a breakdown of ETH.By considering various types of disorder, we find that the functional forms of the distributions of matrix elements of the site-resolved magnetization operator mirror the distribution of the onsite disorder.We argue that this distribution is a direct consequence of the local observable having a finite overlap with moments of the Hamiltonian.We then demonstrate how to adjust the energy window when analyzing expectation values of local observables in disordered quantum many-body systems to correctly assess the system's adherence to ETH, and provide a link between the distribution of expectation values in eigenstates and the outcomes of quench experiments.Ergodic isolated quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH), i.e. the expectation values of local observables in the system's eigenstates approach the predictions of the microcanonical ensemble.However, the ETH does not specify what happens to expectation values of local observables within an energy window when the average over disorder realizations is taken.As a result, the expectation values of local observables can be distributed over a relatively wide interval and may exhibit nontrivial structure, as shown in Aramthottil et al (2021 Phys.Rev. B 104 214201) for a quasiperiodic disordered system for site-resolved magnetization.We argue that the non-Gaussian form of this distribution may falsely suggest non-ergodicity and a breakdown of ETH.By considering various types of disorder, we find that the functional forms of the distributions of matrix elements of the site-resolved magnetization operator mirror the distribution of the onsite disorder.We argue that this distribution is a direct consequence of the local observable having a finite overlap with moments of the Hamiltonian.We then demonstrate how to adjust the energy window when analyzing expectation values of local observables in disordered quantum many-body systems to correctly assess the system's adherence to ETH, and provide a link between the distribution of expectation values in eigenstates and the outcomes of quench experiments.

Dec 1, 20250 citations

The random field

S=\frac{1}{2}

Heisenberg chain exhibits a dynamical many body localization transition at a critical disorder strength, which depends on the energy density. At weak disorder, the eigenstate thermalization hypothesis (ETH) is fulfilled on average, making local observables smooth functions of energy, whose eigenstate-to-eigenstate fluctuations decrease exponentially with system size. We demonstrate the validity of ETH in the thermal phase as well as its breakdown in the localized phase and show that rare states exist which do not strictly follow ETH, becoming more frequent closer to the transition. Similarly, the probability distribution of the entanglement entropy at intermediate disorder develops long tails all the way down to zero entanglement. We propose that these low entanglement tails stem from localized regions at the subsystem boundaries which were recently discussed as a possible mechanism for subdiffusive transport in the ergodic phase.