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:2007.13767 · Physical review. B./Physical review. B

Absence of slow particle transport in the many-body localized phase

David J. Luitz, Yevgeny Bar Lev

Published Sep 10, 2020• 82 citations
arXivHTMLPDF

Abstract

We analyze the saturation value of the bipartite entanglement and number entropy starting from a random product state deep in the many-body localized (MBL) phase. By studying the probability distributions of these entropies we find that the growth of the saturation value of the entanglement entropy stems from a significant reshuffling of the weight in the probability distributions from the bulk to the exponential tails. In contrast, the probability distributions of the saturation value of the number entropy are converged with system size, and exhibit a sharp cutoff for values of the number entropy which correspond to one particle fluctuating across the boundary between the two halves of the system. Our results therefore rule out slow particle transport deep in the MBL phase and confirm that the slow entanglement entropy production stems uniquely from configurational entanglement.

DOI
10.1103/physrevb.102.100202

Related papers

Slow delocalization of particles in many-body localized phases

Maximilian Kiefer-Emmanouilidis, Razmik G. Unanyan, Michael Fleischhauer +1 more

We have recently shown that the logarithmic growth of the entanglement entropy following a quantum quench in a many-body localized phase is accompanied by a slow growth of the number entropy ${S}_{N}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}ln\phantom{\rule{0.16em}{0ex}}t$. Here we provide an in-depth numerical study of ${S}_{N}(t)$ for the disordered Heisenberg chain and show that this behavior is not transient and persists even for very strong disorder. Calculating the truncated R\'enyi number entropy ${S}_{N}^{(\ensuremath{\alpha})}(t)={(1\ensuremath{-}\ensuremath{\alpha})}^{\ensuremath{-}1}ln{\ensuremath{\sum}}_{n}{p}^{\ensuremath{\alpha}}(n)$ for $\ensuremath{\alpha}\ensuremath{\ll}1$ and $p(n)>{p}_{c}$---which is sensitive to large number fluctuations occurring with low probability---we demonstrate that the particle number distribution $p(n)$ in one half of the system has a continuously growing tail. This indicates a slow but steady increase in the number of particles crossing between the partitions in the interacting case and is in sharp contrast to Anderson localization for which we show that ${S}_{N}^{(\ensuremath{\alpha}\ensuremath{\rightarrow}0)}(t)$ saturates for any cutoff ${p}_{c}>0$. We show, furthermore, that the growth of ${S}_{N}$ is not the consequence of rare states or rare regions but rather represents typical behavior. These findings indicate that the interacting system is never fully localized even for very strong but finite disorder.

arXiv:2010.00565•Physical review. B./Physical review. B•Jan 20, 2021•139 citations

Evidence for Unbounded Growth of the Number Entropy in Many-Body Localized Phases

Maximilian Kiefer-Emmanouilidis, Razmik G. Unanyan, Michael Fleischhauer +1 more

We investigate the number entropy S_{N}-which characterizes particle-number fluctuations between subsystems-following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to show many-body localization and where the entanglement entropy grows as S∼lnt as function of time t, the number entropy grows as S_{N}∼lnlnt, indicating continuing subdiffusive particle transport at a very slow rate. We demonstrate that this growth is consistent with a relation between entanglement and number entropy recently established for noninteracting systems.

arXiv:2003.04849•Physical Review Letters•Jun 16, 2020•180 citations

Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems

Maksym Serbyn, Zlatko Papić, Dmitry A. Abanin

Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.

arXiv:1304.4605•Physical Review Letters•Jun 28, 2013•616 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.

arXiv:1601.04058•Physical review. B./Physical review. B•Apr 1, 2016•120 citations

Ultraslow Growth of Number Entropy in an ℓ -Bit Model of Many-Body Localization

David Aceituno Chávez, Claudia Artiaco, Thomas Klein Kvorning +2 more

We demonstrate that slow growth of the number entropy following a quench from a local product state is consistent with many-body localization. To do this, we construct a novel random circuit ℓ-bit model with exponentially localized ℓ-bits and exponentially decaying interactions between them. We observe an ultraslow growth of the number entropy starting from a Néel state, saturating at a value that grows with system size. This suggests that the observation of such growth in microscopic models is not sufficient to rule out many-body localization.

arXiv:2312.13420•Physical Review Letters•Sep 17, 2024•15 citations

Escaping many-body localization in an exact eigenstate

Michael Iversen, N. S. Srivatsa, Anne E. B. Nielsen

Isolated quantum systems typically follow the eigenstate thermalization hypothesis, but there are exceptions, such as many-body localized (MBL) systems and quantum many-body scars. Here, we present the study of a weak violation of MBL due to a special state embedded in a spectrum of MBL states. The special state is not MBL since it displays logarithmic scaling of the entanglement entropy and of the bipartite fluctuations of particle number with subsystem size. In contrast, the bulk of the spectrum becomes MBL as disorder is introduced. We establish this by studying the entropy as a function of disorder strength and by observing that the level spacing statistics undergoes a transition from Wigner-Dyson to Poisson statistics as the disorder strength is increased.

arXiv:2201.00830•Physical review. B./Physical review. B•Dec 1, 2022•8 citations
We analyze the saturation value of the bipartite entanglement and number entropy starting from a random product state deep in the many-body localized (MBL) phase. By studying the probability distributions of these entropies we find that the growth of the saturation value of the entanglement entropy stems from a significant reshuffling of the weight in the probability distributions from the bulk to the exponential tails. In contrast, the probability distributions of the saturation value of the number entropy are converged with system size, and exhibit a sharp cutoff for values of the number entropy which correspond to one particle fluctuating across the boundary between the two halves of the system. Our results therefore rule out slow particle transport deep in the MBL phase and confirm that the slow entanglement entropy production stems uniquely from configurational entanglement.
We have recently shown that the logarithmic growth of the entanglement entropy following a quantum quench in a many-body localized phase is accompanied by a slow growth of the number entropy ${S}_{N}\ensuremath{\sim}ln\phantom{\rule{0.16em}{0ex}}ln\phantom{\rule{0.16em}{0ex}}t$. Here we provide an in-depth numerical study of SN(t){S}_{N}(t)SN​(t) for the disordered Heisenberg chain and show that this behavior is not transient and persists even for very strong disorder. Calculating the truncated R\'enyi number entropy ${S}_{N}^{(\ensuremath{\alpha})}(t)={(1\ensuremath{-}\ensuremath{\alpha})}^{\ensuremath{-}1}ln{\ensuremath{\sum}}_{n}{p}^{\ensuremath{\alpha}}(n)$ for $\ensuremath{\alpha}\ensuremath{\ll}1$ and p(n)>pcp(n)>{p}_{c}p(n)>pc​---which is sensitive to large number fluctuations occurring with low probability---we demonstrate that the particle number distribution p(n)p(n)p(n) in one half of the system has a continuously growing tail. This indicates a slow but steady increase in the number of particles crossing between the partitions in the interacting case and is in sharp contrast to Anderson localization for which we show that ${S}_{N}^{(\ensuremath{\alpha}\ensuremath{\rightarrow}0)}(t)$ saturates for any cutoff pc>0{p}_{c}>0pc​>0. We show, furthermore, that the growth of SN{S}_{N}SN​ is not the consequence of rare states or rare regions but rather represents typical behavior. These findings indicate that the interacting system is never fully localized even for very strong but finite disorder.
We investigate the number entropy S_{N}-which characterizes particle-number fluctuations between subsystems-following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to show many-body localization and where the entanglement entropy grows as S∼lnt as function of time t, the number entropy grows as S_{N}∼lnlnt, indicating continuing subdiffusive particle transport at a very slow rate. We demonstrate that this growth is consistent with a relation between entanglement and number entropy recently established for noninteracting systems.
Recent numerical work by Bardarson, Pollmann, and Moore revealed a slow, logarithmic in time, growth of the entanglement entropy for initial product states in a putative many-body localized phase. We show that this surprising phenomenon results from the dephasing due to exponentially small interaction-induced corrections to the eigenenergies of different states. For weak interactions, we find that the entanglement entropy grows as ξln(Vt/ℏ), where V is the interaction strength, and ξ is the single-particle localization length. The saturated value of the entanglement entropy at long times is determined by the participation ratios of the initial state over the eigenstates of the subsystem. Our work shows that the logarithmic entanglement growth is a universal phenomenon characteristic of the many-body localized phase in any number of spatial dimensions, and reveals a broad hierarchy of dephasing time scales present in such a phase.
The random field S=12S=\frac{1}{2}S=21​ 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.
We demonstrate that slow growth of the number entropy following a quench from a local product state is consistent with many-body localization. To do this, we construct a novel random circuit ℓ-bit model with exponentially localized ℓ-bits and exponentially decaying interactions between them. We observe an ultraslow growth of the number entropy starting from a Néel state, saturating at a value that grows with system size. This suggests that the observation of such growth in microscopic models is not sufficient to rule out many-body localization.
Isolated quantum systems typically follow the eigenstate thermalization hypothesis, but there are exceptions, such as many-body localized (MBL) systems and quantum many-body scars. Here, we present the study of a weak violation of MBL due to a special state embedded in a spectrum of MBL states. The special state is not MBL since it displays logarithmic scaling of the entanglement entropy and of the bipartite fluctuations of particle number with subsystem size. In contrast, the bulk of the spectrum becomes MBL as disorder is introduced. We establish this by studying the entropy as a function of disorder strength and by observing that the level spacing statistics undergoes a transition from Wigner-Dyson to Poisson statistics as the disorder strength is increased.