arXiv:2010.00565 · Physical review. B./Physical review. B

Slow delocalization of particles in many-body localized phases

Maximilian Kiefer-Emmanouilidis, Razmik G. Unanyan, Michael Fleischhauer, Jesko Sirker

Published Jan 20, 2021• 139 citations

Abstract

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.

DOI: 10.1103/physrevb.103.024203

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

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Many-body localization in a disorder Bose-Hubbard chain

Jie Chen, Xiaoqun Wang

Apr 17, 20211 citation

Signatures of slow particle transport in the interacting Aubry-André chain

S. Sharifan, M. Kiefer-Emmanouilidis, J. Sirker

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Resonance-induced growth of number entropy in strongly disordered systems

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arXiv:2010.00565 · Physical review. B./Physical review. B

Slow delocalization of particles in many-body localized phases

Maximilian Kiefer-Emmanouilidis, Razmik G. Unanyan, Michael Fleischhauer, Jesko Sirker

Published Jan 20, 2021• 139 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.103.024203

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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.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.

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arXiv:2007.13767Physical review. B./Physical review. BSep 10, 202082 citations

Many-body localization in a disorder Bose-Hubbard chain

Jie Chen, Xiaoqun Wang

Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.

Apr 17, 20211 citation

Signatures of slow particle transport in the interacting Aubry-André chain

S. Sharifan, M. Kiefer-Emmanouilidis, J. Sirker

Slow delocalization in strongly disordered many-body quantum chains can originate from two distinct mechanisms: rare resonances and avalanches, rooted in the energy statistics, or an energy-independent combinatorial path-entropy obstruction in operator space. Quasiperiodic potentials eliminate rare regions with similar potentials that are thought to be responsible for avalanche instabilities and modify the statistics of resonances, thus allowing us to delineate the two mechanisms. To do so, we numerically study the entanglement entropy $S(t)$ , number entropy $S_N(t)$ , and Hartley number entropy $S_H(t)$ following a quantum quench in the interacting Aubry-André chain with disorder strength $D$ . We find $S(t)\simμ\ln t$ , $S_N(t)\simν\ln\ln t$ with growth rates $μ\simν\sim D^{-γ}$ and $γ\approx 3$ , the same behavior that was previously found in the case of quenched disorder. Furthermore, the saturation values $S_H(t\to\infty)$ continue to grow with system size $L$ as $\ln L$ , a growth that persists for $D\gtrsim 8$ where MBL was argued to be stable. The similarity of the quasi-periodic and quenched disorder cases points to a common origin of the slow particle transport that does not rely on rare-region avalanches and persists despite strongly modified resonance statistics.Slow delocalization in strongly disordered many-body quantum chains can originate from two distinct mechanisms: rare resonances and avalanches, rooted in the energy statistics, or an energy-independent combinatorial path-entropy obstruction in operator space. Quasiperiodic potentials eliminate rare regions with similar potentials that are thought to be responsible for avalanche instabilities and modify the statistics of resonances, thus allowing us to delineate the two mechanisms. To do so, we numerically study the entanglement entropy $S(t)$, number entropy $S_N(t)$, and Hartley number entropy $S_H(t)$ following a quantum quench in the interacting Aubry-André chain with disorder strength $D$. We find $S(t)\simμ\ln t$, $S_N(t)\simν\ln\ln t$ with growth rates $μ\simν\sim D^{-γ}$ and $γ\approx 3$, the same behavior that was previously found in the case of quenched disorder. Furthermore, the saturation values $S_H(t\to\infty)$ continue to grow with system size $L$ as $\ln L$, a growth that persists for $D\gtrsim 8$ where MBL was argued to be stable. The similarity of the quasi-periodic and quenched disorder cases points to a common origin of the slow particle transport that does not rely on rare-region avalanches and persists despite strongly modified resonance statistics.

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Resonance-induced growth of number entropy in strongly disordered systems

Roopayan Ghosh, Marko Žnidarič

We study the growth of the number entropy ${S}_{\mathrm{N}}$ in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of ${S}_{\mathrm{N}}$ has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying ${S}_{\mathrm{N}}$ in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of ${S}_{\mathrm{N}}$ . However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained ${S}_{\mathrm{N}}$ : trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of ${S}_{\mathrm{N}}$ with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of ${S}_{\mathrm{N}}$ also heavily varies depending on the initial state, and therefore ${S}_{\mathrm{N}}$ and von Neumann entropy can behave rather differently.We study the growth of the number entropy ${S}_{\mathrm{N}}$ in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of ${S}_{\mathrm{N}}$ has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying ${S}_{\mathrm{N}}$ in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of ${S}_{\mathrm{N}}$. However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained ${S}_{\mathrm{N}}$: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of ${S}_{\mathrm{N}}$ with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of ${S}_{\mathrm{N}}$ also heavily varies depending on the initial state, and therefore ${S}_{\mathrm{N}}$ and von Neumann entropy can behave rather differently.

arXiv:2112.12987Physical review. B./Physical review. BApr 13, 202247 citations

Extended slow dynamical regime close to the many-body localization transition

David J. Luitz, Nicolas Laflorencie, Fabien Alet

Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to $L=28$ sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth $z=1$ of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.Many-body localization is characterized by a slow logarithmic growth of the entanglement entropy after a global quantum quench while the local memory of an initial density imbalance remains at infinite time. We investigate how much the proximity of a many-body localized phase can influence the dynamics in the delocalized ergodic regime where thermalization is expected. Using an exact Krylov space technique, the out-of-equilibrium dynamics of the random-field Heisenberg chain is studied up to $L=28$ sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth $z=1$ of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.

arXiv:1511.05141Physical review. B./Physical review. BFeb 17, 2016319 citations

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)

{S}_{N}^{(\ensuremath{\alpha}\ensuremath{\rightarrow}0)}(t)

saturates for any cutoff

{p}_{c}>0

. We show, furthermore, that the growth of

{S}_{N}

S(t)

, number entropy

S_N(t)

, and Hartley number entropy

S_H(t)

following a quantum quench in the interacting Aubry-André chain with disorder strength

D

. We find

S(t)\simμ\ln t

S_N(t)\simν\ln\ln t

with growth rates

μ\simν\sim D^{-γ}

and

γ\approx 3

, the same behavior that was previously found in the case of quenched disorder. Furthermore, the saturation values

S_H(t\to\infty)

continue to grow with system size

L

\ln L

, a growth that persists for

D\gtrsim 8

where MBL was argued to be stable. The similarity of the quasi-periodic and quenched disorder cases points to a common origin of the slow particle transport that does not rely on rare-region avalanches and persists despite strongly modified resonance statistics.

We study the growth of the number entropy

{S}_{\mathrm{N}}

in one-dimensional number-conserving interacting systems with strong disorder, which are believed to display many-body localization. Recently a slow and small growth of

{S}_{\mathrm{N}}

has been numerically reported, which, if holding at asymptotically long times in the thermodynamic limit, would imply ergodicity and therefore the absence of true localization. By numerically studying

{S}_{\mathrm{N}}

in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of

{S}_{\mathrm{N}}

. However, we show that such growth is fully compatible with localization. To be specific, using a simple model that accounts for expected rare resonances we can analytically predict several main features of numerically obtained

{S}_{\mathrm{N}}

: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of

{S}_{\mathrm{N}}

with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of

{S}_{\mathrm{N}}

also heavily varies depending on the initial state, and therefore

{S}_{\mathrm{N}}

and von Neumann entropy can behave rather differently.

L=28

sites, starting from an initially unentangled high-energy product state. Within most of the delocalized phase, we find a sub-ballistic entanglement growth $S(t)\ensuremath{\propto}{t}^{1/z}$ with a disorder-dependent exponent $z\ensuremath{\ge}1$, in contrast with the pure ballistic growth

z=1

of clean systems. At the same time, anomalous relaxation is also observed for the spin imbalance $\mathcal{I}(t)\ensuremath{\propto}{t}^{\ensuremath{-}\ensuremath{\zeta}}$ with a continuously varying disorder-dependent exponent $\ensuremath{\zeta}$, vanishing at the transition. This provides a clear experimental signature for detecting this nonconventional regime.