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:2203.06689

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

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

Published Mar 13, 2022• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2203.06689

Related papers

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.

arXiv:2112.12987•Physical review. B./Physical review. B•Apr 13, 2022•47 citations

Entanglement Entropy Growth in Disordered Spin Chains with Tunable Range Interactions

Y. Mohdeb, Javad Vahedi, R. N. Bhatt +2 more

The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $α$. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for $α>α_c$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $α$ as $S(t) = S_p \ln(t)/(2α)$. Here, $S_p= 2 \ln2 -1$. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large $α> α_c \approx 1.8$. For $α<α_c$, we find that the entanglement entropy grows as a power-law with time, $S(t)\sim t^{γ(α)}$ with $0<γ(α)<1$ a decaying function of the interaction exponent $α$.

arXiv:2303.02415•Mar 4, 2023•0 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.

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

Bounds on Rényi entropy growth in many-body quantum systems

Zhengyan Darius Shi

We prove rigorous bounds on the growth of $\ensuremath{\alpha}$-R\'enyi entropies ${S}_{\ensuremath{\alpha}}(t)$ (the von Neumann entropy being the special case $\ensuremath{\alpha}=1$) associated with any subsystem $A$ of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ (with $\ensuremath{\alpha}\ensuremath{\ne}1$) can be exponentially larger than $|{S}_{1}^{\ensuremath{'}}(t)|$ as a function of the subsystem size $|A|$. For $D$-dimensional systems with geometric locality, we prove bounds on $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ that depend on the decay rate of interactions with distance. When $\ensuremath{\alpha}=1$, the bound is $|A|$ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with $w>2D+1$. However, for $\ensuremath{\alpha}>1$, the bound is $|A|$ independent only when the interactions are finite-range or decay faster than $V(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}$ for some $c$ depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on $k$-local systems with or without geometric locality. A central theme of this work is that the value of $\ensuremath{\alpha}$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\ensuremath{\alpha}$-R\'enyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.

arXiv:2212.07444•Physical Review A•Apr 8, 2024•2 citations

Many-Body-Localization Transition in the Strong Disorder Limit: Entanglement Entropy from the Statistics of Rare Extensive Resonances

Cécile Monthus

The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).

arXiv:1510.03711•Entropy•Apr 1, 2016•26 citations

Global quench dynamics and the growth of entanglement entropy in disordered spin chains with tunable range interactions

Y. Mohdeb, Javad Vahedi, R. N. Bhatt +2 more

The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $\ensuremath{\alpha}$. Using a dynamical version of the strong disorder renormalization group we find for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\ensuremath{\alpha}$ as $S(t)={S}_{p}ln(t)/(2\ensuremath{\alpha})$. Here, ${S}_{p}=2ln2\ensuremath{-}1$. We present results of numerical exact diagonalization calculations for system sizes up to $N\ensuremath{\sim}16$ spins, in good agreement with the analytical results for sufficiently large $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.8$. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the entanglement entropy grows as a power law with time, $S(t)\ensuremath{\sim}{t}^{\ensuremath{\gamma}(\ensuremath{\alpha})}$ with $0<\ensuremath{\gamma}(\ensuremath{\alpha})<1$ a decaying function of the interaction exponent $\ensuremath{\alpha}$.

Physical review. B./Physical review. B•Oct 18, 2023•4 citations
We comment on the recent paper by Ghosh and Žnidarič (Phys. Rev. B 105, 144203 (2022)) which studies the growth of the number entropy SNS_NSN​ in the Heisenberg model with random magnetic fields after a quantum quench. The authors present arguments for an intermediate power-law growth in time ttt 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 S~N(L,W)\widetilde{S}_N(L,W)SN​(L,W) for fixed length LLL are only bounded from above by 'the ergodic value' and are already far below this value for W≪1W\ll 1W≪1. Furthermore, the saturation values can show non-monotonic scaling with LLL. (ii) Power-law fits SN(t)∼1/tαS_N(t)\sim 1/t^αSN​(t)∼1/tα -- with α=1α=1α=1 expected based on the resonance model described in the paper -- yield a system-size dependent exponent ααα while fits SN∼1W3ln⁡ln⁡tS_N\sim \frac{1}{W^3}\ln\ln tSN​∼W31​lnlnt 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 study the growth of the number entropy SN{S}_{\mathrm{N}}SN​ 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 SN{S}_{\mathrm{N}}SN​ 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 SN{S}_{\mathrm{N}}SN​ in the disordered isotropic Heisenberg model we first reconfirm that, indeed, there is a small growth of SN{S}_{\mathrm{N}}SN​. 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 SN{S}_{\mathrm{N}}SN​: trivial initial growth at short times, a slow power-law growth at intermediate times, and the scaling of the saturation value of SN{S}_{\mathrm{N}}SN​ with the disorder strength. Because resonances crucially depend on individual disorder realizations, the growth of SN{S}_{\mathrm{N}}SN​ also heavily varies depending on the initial state, and therefore SN{S}_{\mathrm{N}}SN​ and von Neumann entropy can behave rather differently.
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large α>αc≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.
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 prove rigorous bounds on the growth of $\ensuremath{\alpha}$-R\'enyi entropies ${S}_{\ensuremath{\alpha}}(t)$ (the von Neumann entropy being the special case $\ensuremath{\alpha}=1$) associated with any subsystem AAA of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ (with $\ensuremath{\alpha}\ensuremath{\ne}1$) can be exponentially larger than $|{S}_{1}^{\ensuremath{'}}(t)|$ as a function of the subsystem size ∣A∣|A|∣A∣. For DDD-dimensional systems with geometric locality, we prove bounds on $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ that depend on the decay rate of interactions with distance. When $\ensuremath{\alpha}=1$, the bound is ∣A∣|A|∣A∣ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with w>2D+1w>2D+1w>2D+1. However, for $\ensuremath{\alpha}>1$, the bound is ∣A∣|A|∣A∣ independent only when the interactions are finite-range or decay faster than $V(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}$ for some ccc depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on kkk-local systems with or without geometric locality. A central theme of this work is that the value of $\ensuremath{\alpha}$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\ensuremath{\alpha}$-R\'enyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.
The space of one-dimensional disordered interacting quantum models displaying a many-body localization (MBL) transition seems sufficiently rich to produce critical points with level statistics interpolating continuously between the Poisson statistics of the localized phase and the Wigner–Dyson statistics of the delocalized phase. In this paper, we consider the strong disorder limit of the MBL transition, where the level statistics at the MBL critical point is close to the Poisson statistics. We analyze a one-dimensional quantum spin model, in order to determine the statistical properties of the rare extensive resonances that are needed to destabilize the MBL phase. At criticality, we find that the entanglement entropy can grow with an exponent 0 < α < 1 anywhere between the area law α = 0 and the volume law α = 1 , as a function of the resonances properties, while the entanglement spectrum follows the strong multifractality statistics. In the MBL phase near criticality, we obtain the simple value ν = 1 for the correlation length exponent. Independently of the strong disorder limit, we explain why, for the many-body localization transition concerning individual eigenstates, the correlation length exponent ν is not constrained by the usual Harris inequality ν ≥ 2 / d , so that there is no theoretical inconsistency with the best numerical measure ν = 0 . 8 ( 3 ) obtained by Luitz et al. (2015).
The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $\ensuremath{\alpha}$. Using a dynamical version of the strong disorder renormalization group we find for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\ensuremath{\alpha}$ as $S(t)={S}_{p}ln(t)/(2\ensuremath{\alpha})$. Here, ${S}_{p}=2ln2\ensuremath{-}1$. We present results of numerical exact diagonalization calculations for system sizes up to $N\ensuremath{\sim}16$ spins, in good agreement with the analytical results for sufficiently large $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.8$. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the entanglement entropy grows as a power law with time, $S(t)\ensuremath{\sim}{t}^{\ensuremath{\gamma}(\ensuremath{\alpha})}$ with $0<\ensuremath{\gamma}(\ensuremath{\alpha})<1$ a decaying function of the interaction exponent $\ensuremath{\alpha}$.