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:2010.06678 · Physical review. B./Physical review. B

Entanglement dynamics in the three-dimensional Anderson model

Yang Zhao, Dingyi Feng, Yongbo Hu, Shutong Guo, Jesko Sirker

Published Nov 18, 2020• 15 citations
arXivHTMLPDF

Abstract

The authors study the metal-insulator transition in the three-dimensional Anderson model from the entanglement perspective. They show that at the transition point the entanglement entropy after a quantum quench grows logarithmically in time, while the number entropy shows a double logarithmic growth. This behavior is consistent with exact bounds. It is also the same scaling recently found in many-body localized (MBL) phases, suggesting that the MBL phase might be more akin to an extended critical regime, rather than a fully localized phase.

DOI
10.1103/physrevb.102.195132

Related papers

Entanglement entropy and computational complexity of the Anderson impurity model out of equilibrium: Quench dynamics

Zhuoran He, Andrew J. Millis

We study the growth of entanglement entropy in density-matrix renormalization-group calculations of the real-time quench dynamics of the Anderson impurity model. We find that with an appropriate choice of basis, the entropy growth is logarithmic in both the interacting and noninteracting single-impurity models. The logarithmic entropy growth is understood from a noninteracting chain model as a critical behavior separating regimes of linear growth and saturation of entropy, which correspond respectively to overlapping and gapped energy spectra of the set of bath states. We find that a logarithmic entropy growth is the generic behavior of quenched impurity models when the bath orbitals in the matrix product state are ordered in energy. A noninteracting calculation of the double-impurity Anderson model supports the conclusion in the multi-impurity case. The logarithmic growth of entanglement entropy enables studies of quench dynamics to very long times.

arXiv:1704.01180•Physical review. B./Physical review. B•Aug 7, 2017•20 citations

Logarithmic entanglement growth in two-dimensional disordered fermionic systems

Yang Zhao, Jesko Sirker

We investigate the growth of the entanglement entropy ${S}_{\mathrm{ent}}$ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case, we show that an intermediate weak localization regime exists in which ${S}_{\mathrm{ent}}(t)$ grows logarithmically in time $t$ before Anderson localization sets in. For the case of binary bond disorder near the percolation transition, we find additive logarithmic corrections to area and volume laws as well as a scaling at long times, which is consistent with an infinite randomness fixed point.

arXiv:1906.03503•Physical review. B./Physical review. B•Jul 22, 2019•7 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

Scaling of entanglement entropy in a quantum phase transition in the transverse Ising model induced by a quench

Priyadarshi Majumdar, Pratul Bandyopadhyay

It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of $L$ spins with the rest of the system scales logarithmically with $L$ with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with $L$ but the prefactor depends on the quench time. This puts a constraint on the block size $L$ so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.

Physical Review A•Jan 19, 2010•8 citations

Growth of entanglement entropy under local projective measurements

Michele Coppola, Emanuele Tirrito, Dragi Karevski +1 more

Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.

arXiv:2109.10837•Physical review. B./Physical review. B•Mar 9, 2022•118 citations
The authors study the metal-insulator transition in the three-dimensional Anderson model from the entanglement perspective. They show that at the transition point the entanglement entropy after a quantum quench grows logarithmically in time, while the number entropy shows a double logarithmic growth. This behavior is consistent with exact bounds. It is also the same scaling recently found in many-body localized (MBL) phases, suggesting that the MBL phase might be more akin to an extended critical regime, rather than a fully localized phase.
We study the growth of entanglement entropy in density-matrix renormalization-group calculations of the real-time quench dynamics of the Anderson impurity model. We find that with an appropriate choice of basis, the entropy growth is logarithmic in both the interacting and noninteracting single-impurity models. The logarithmic entropy growth is understood from a noninteracting chain model as a critical behavior separating regimes of linear growth and saturation of entropy, which correspond respectively to overlapping and gapped energy spectra of the set of bath states. We find that a logarithmic entropy growth is the generic behavior of quenched impurity models when the bath orbitals in the matrix product state are ordered in energy. A noninteracting calculation of the double-impurity Anderson model supports the conclusion in the multi-impurity case. The logarithmic growth of entanglement entropy enables studies of quench dynamics to very long times.
We investigate the growth of the entanglement entropy Sent{S}_{\mathrm{ent}}Sent​ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case, we show that an intermediate weak localization regime exists in which Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) grows logarithmically in time ttt before Anderson localization sets in. For the case of binary bond disorder near the percolation transition, we find additive logarithmic corrections to area and volume laws as well as a scaling at long times, which is consistent with an infinite randomness fixed point.
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.
It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of LLL spins with the rest of the system scales logarithmically with LLL with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with LLL but the prefactor depends on the quench time. This puts a constraint on the block size LLL so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.
Nonequilibrium dynamics of many-body quantum systems under the effect of measurement protocols is attracting an increasing amount of attention. It has been recently revealed that measurements may induce an abrupt change in the scaling law of the bipartite entanglement entropy, thus suggesting the existence of different nonequilibrium regimes. However, our understanding of how these regimes appear and whether they survive in the thermodynamic limit is much less established. Here we investigate these questions on a one-dimensional quadratic fermionic model: this allows us to reach system sizes relevant in the thermodynamic sense. We show that local projective measurements induce a qualitative modification of the time growth of the entanglement entropy which changes from linear to logarithmic. However, in the stationary regime, the logarithmic behavior of the entanglement entropy does not survive in the thermodynamic limit and, for any finite value of the measurement rate, we numerically show the existence of a single area-law phase for the entanglement entropy. Finally, exploiting the quasiparticle picture, we further support our results by analyzing the fluctuations of the stationary entanglement entropy and its scaling behavior.