arXiv:2112.12987 · Physical review. B./Physical review. B

Resonance-induced growth of number entropy in strongly disordered systems

Roopayan Ghosh, Marko Žnidarič

Published Apr 13, 2022• 47 citations

Abstract

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.

DOI: 10.1103/physrevb.105.144203

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Comment on "Resonance-induced growth of number entropy in strongly disordered systems"

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

arXiv:2203.06689Mar 13, 20222 citations

Slow delocalization of particles in many-body localized phases

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

arXiv:2010.00565Physical review. B./Physical review. BJan 20, 2021139 citations

Universal Slow Growth of Entanglement in Interacting Strongly Disordered Systems

Maksym Serbyn, Zlatko Papić, Dmitry A. Abanin

arXiv:1304.4605Physical Review LettersJun 28, 2013616 citations

Many-body localization in infinite chains

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Entanglement growth in many-body localized systems with long-range interactions

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Many-body localization in a disorder Bose-Hubbard chain

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Apr 17, 20211 citation

arXiv:2112.12987 · Physical review. B./Physical review. B

Resonance-induced growth of number entropy in strongly disordered systems

Roopayan Ghosh, Marko Žnidarič

Published Apr 13, 2022• 47 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.105.144203

Related papers

Comment on "Resonance-induced growth of number entropy in strongly disordered systems"

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

We comment on the recent paper by Ghosh and Žnidarič (Phys. Rev. B 105, 144203 (2022)) which studies the growth of the number entropy $S_N$ in the Heisenberg model with random magnetic fields after a quantum quench. The authors present arguments for an intermediate power-law growth in time $t$ and a sub-ergodic saturation value, claiming consistency of their results with many-body localization (MBL) for strong disorder. We show that these interpretations are inconsistent with other recent studies and discuss specific issues with the analysis of the numerical data. We point out, in particular, that (i) the saturation values $\widetilde{S}_N(L,W)$ for fixed length $L$ are only bounded from above by 'the ergodic value' and are already far below this value for $W\ll 1$ . Furthermore, the saturation values can show non-monotonic scaling with $L$ . (ii) Power-law fits $S_N(t)\sim 1/t^α$ -- with $α=1$ expected based on the resonance model described in the paper -- yield a system-size dependent exponent $α$ while fits $S_N\sim \frac{1}{W^3}\ln\ln t$ do hold independent of system size and over several orders of magnitude in time. (iii) We also argue that for the cases where the effective resonance model works best and predicts a saturation of the number entropy, the same applies to the von-Neumann entropy, i.e.~the dynamics at the considered scales is of single particle type and unrelated to MBL.We comment on the recent paper by Ghosh and Žnidarič (Phys. Rev. B 105, 144203 (2022)) which studies the growth of the number entropy $S_N$ in the Heisenberg model with random magnetic fields after a quantum quench. The authors present arguments for an intermediate power-law growth in time $t$ and a sub-ergodic saturation value, claiming consistency of their results with many-body localization (MBL) for strong disorder. We show that these interpretations are inconsistent with other recent studies and discuss specific issues with the analysis of the numerical data. We point out, in particular, that (i) the saturation values $\widetilde{S}_N(L,W)$ for fixed length $L$ are only bounded from above by 'the ergodic value' and are already far below this value for $W\ll 1$. Furthermore, the saturation values can show non-monotonic scaling with $L$. (ii) Power-law fits $S_N(t)\sim 1/t^α$ -- with $α=1$ expected based on the resonance model described in the paper -- yield a system-size dependent exponent $α$ while fits $S_N\sim \frac{1}{W^3}\ln\ln t$ do hold independent of system size and over several orders of magnitude in time. (iii) We also argue that for the cases where the effective resonance model works best and predicts a saturation of the number entropy, the same applies to the von-Neumann entropy, i.e.~the dynamics at the considered scales is of single particle type and unrelated to MBL.

arXiv:2203.06689Mar 13, 20222 citations

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.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:2010.00565Physical review. B./Physical review. BJan 20, 2021139 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.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:1304.4605Physical Review LettersJun 28, 2013616 citations

Many-body localization in infinite chains

Tilman Enss, F. Andraschko, Jesko Sirker

We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin- $1/2$ Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order ${m}_{s}(t)$ and the growth of entanglement entropy ${S}_{\mathrm{ent}}(t)$ during unitary time evolution. Near the phase transition we find that ${m}_{s}(t)$ decays exponentially to its asymptotic value ${m}_{s}(\ensuremath{\infty})\ensuremath{\ne}0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ${m}_{s}(\ensuremath{\infty})$ shows an exponential sensitivity on disorder with a critical exponent $\ensuremath{\nu}\ensuremath{\sim}0.9$. The entanglement entropy in the ergodic phase grows subballistically, ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}\ensuremath{\le}1$, with $\ensuremath{\alpha}$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-$1/2$ Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order ${m}_{s}(t)$ and the growth of entanglement entropy ${S}_{\mathrm{ent}}(t)$ during unitary time evolution. Near the phase transition we find that ${m}_{s}(t)$ decays exponentially to its asymptotic value ${m}_{s}(\ensuremath{\infty})\ensuremath{\ne}0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ${m}_{s}(\ensuremath{\infty})$ shows an exponential sensitivity on disorder with a critical exponent $\ensuremath{\nu}\ensuremath{\sim}0.9$. The entanglement entropy in the ergodic phase grows subballistically, ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}\ensuremath{\le}1$, with $\ensuremath{\alpha}$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.

arXiv:1608.05733Physical review. B./Physical review. BJan 17, 201790 citations

Entanglement growth in many-body localized systems with long-range interactions

M. Pino

We show that von Neumann entropy grows algebraically in time for many-body localized systems with long-range interactions. The dephasing time of far-apart localized degrees of freedom is studied first. For a disentangled high-energy initial state, our analysis indicates that the entropy of one half of the system evolves as ${t}^{1/(p+1)}$ , where $p$ controls the spatial decay of interactions. Numerical simulations of the entropy evolution in the Coulomb and dipolar cases also exhibit an algebraic growth. We find that the saturation value of von Neumann entropy is extensive but small for an ergodic state, and it does not depend on the shape of the interactions. Due to the algebraic entanglement spread, the signature of an extensive entropy should be much easier to observe in experiments with nonlocal interactions than in those with local ones.We show that von Neumann entropy grows algebraically in time for many-body localized systems with long-range interactions. The dephasing time of far-apart localized degrees of freedom is studied first. For a disentangled high-energy initial state, our analysis indicates that the entropy of one half of the system evolves as ${t}^{1/(p+1)}$, where $p$ controls the spatial decay of interactions. Numerical simulations of the entropy evolution in the Coulomb and dipolar cases also exhibit an algebraic growth. We find that the saturation value of von Neumann entropy is extensive but small for an ergodic state, and it does not depend on the shape of the interactions. Due to the algebraic entanglement spread, the signature of an extensive entropy should be much easier to observe in experiments with nonlocal interactions than in those with local ones.

arXiv:1403.5974Physical Review BNov 20, 201441 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

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}}

{S}_{\mathrm{N}}

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

{S}_{\mathrm{N}}

{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 comment on the recent paper by Ghosh and Žnidarič (Phys. Rev. B 105, 144203 (2022)) which studies the growth of the number entropy

S_N

in the Heisenberg model with random magnetic fields after a quantum quench. The authors present arguments for an intermediate power-law growth in time

t

and a sub-ergodic saturation value, claiming consistency of their results with many-body localization (MBL) for strong disorder. We show that these interpretations are inconsistent with other recent studies and discuss specific issues with the analysis of the numerical data. We point out, in particular, that (i) the saturation values

\widetilde{S}_N(L,W)

for fixed length

L

are only bounded from above by 'the ergodic value' and are already far below this value for

W\ll 1

. Furthermore, the saturation values can show non-monotonic scaling with

L

. (ii) Power-law fits

S_N(t)\sim 1/t^α

-- with

α=1

expected based on the resonance model described in the paper -- yield a system-size dependent exponent

α

while fits

S_N\sim \frac{1}{W^3}\ln\ln t

do hold independent of system size and over several orders of magnitude in time. (iii) We also argue that for the cases where the effective resonance model works best and predicts a saturation of the number entropy, the same applies to the von-Neumann entropy, i.e.~the dynamics at the considered scales is of single particle type and unrelated to MBL.

{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.

We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-

1/2

Heisenberg chains with binary disorder. Starting from the N\'eel state, we analyze the decay of antiferromagnetic order

{m}_{s}(t)

and the growth of entanglement entropy

{S}_{\mathrm{ent}}(t)

during unitary time evolution. Near the phase transition we find that

{m}_{s}(t)

decays exponentially to its asymptotic value ${m}_{s}(\ensuremath{\infty})\ensuremath{\ne}0$ in the localized phase while the data are consistent with a power-law decay at long times in the ergodic phase. In the localized phase, ${m}_{s}(\ensuremath{\infty})$ shows an exponential sensitivity on disorder with a critical exponent $\ensuremath{\nu}\ensuremath{\sim}0.9$. The entanglement entropy in the ergodic phase grows subballistically, ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, $\ensuremath{\alpha}\ensuremath{\le}1$, with $\ensuremath{\alpha}$ varying continuously as a function of disorder. Exact diagonalizations for small systems, on the other hand, do not show a clear scaling with system size and attempts to determine the phase boundary from these data seem to overestimate the extent of the ergodic phase.

{t}^{1/(p+1)}

, where

p

controls the spatial decay of interactions. Numerical simulations of the entropy evolution in the Coulomb and dipolar cases also exhibit an algebraic growth. We find that the saturation value of von Neumann entropy is extensive but small for an ergodic state, and it does not depend on the shape of the interactions. Due to the algebraic entanglement spread, the signature of an extensive entropy should be much easier to observe in experiments with nonlocal interactions than in those with local ones.