arxiv: 2604.09180 · v1 · submitted 2026-04-10 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· cond-mat.str-el· quant-ph

Recognition: unknown

Eigenstate entanglement entropy in Bose-Hubbard models

G. Medo\v{s} , L. Vidmar

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:23 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elquant-ph

keywords entanglement entropyBose-Hubbard modelvolume lawmid-spectrum eigenstateson-site disorderparticle-number conservationmean-field approachbosonic cutoff

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The pith

The volume-law coefficient of mid-spectrum eigenstate entanglement entropy in Bose-Hubbard models is derived via mean-field generalization and remains unchanged by on-site disorder.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the entanglement entropy of mid-spectrum eigenstates in Bose-Hubbard models, both with and without particle-number conservation and with varying disorder. It derives the leading volume-law coefficient by extending a mean-field technique originally used for fermions and spins to account for a tunable maximum number of bosons per site. The resulting coefficient matches earlier analytical and numerical results and does not change when on-site disorder breaks the lattice's translational symmetry. The subleading constant term shows a nontrivial dependence on particle density and cutoff when particle number is conserved, but appears universal when it is not. These findings clarify how quantum information is shared across subsystems in interacting bosonic systems that can be realized in cold-atom experiments.

Core claim

The authors generalize the mean-field approach to compute the volume-law coefficient of the entanglement entropy for mid-spectrum eigenstates in the Bose-Hubbard model with a tunable local bosonic cutoff. They demonstrate that this coefficient agrees with prior analytical and numerical findings and remains unaltered when translational invariance is broken by on-site disorder. Numerical analysis reveals that the O(1) contribution shows nontrivial dependence on particle-number density and cutoff in the conserving case, while suggesting a universal constant beyond random pure state predictions when particle number is not conserved.

What carries the argument

The generalized mean-field averaging procedure applied to bosonic systems with tunable local cutoff, which determines the volume-law coefficient of eigenstate entanglement entropy.

If this is right

The volume-law part of the entanglement entropy scales identically with subsystem size in the presence or absence of weak on-site disorder.
In particle-number conserving models the constant term in the entanglement entropy varies with both density and the local bosonic cutoff.
Without particle-number conservation the constant term approaches a value independent of microscopic details.
The same mean-field procedure that worked for spins and fermions now supplies an analytic expression for the leading entanglement coefficient in bosonic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The robustness of the volume-law coefficient to disorder suggests that entanglement calculations for larger bosonic lattices can safely use the translationally invariant mean-field result as a starting point.
The apparent universality of the constant term in the non-conserving case may connect to random-matrix expectations for chaotic bosonic Hamiltonians.
Cold-atom experiments with tunable interactions and controlled disorder could directly measure whether the predicted density and cutoff dependence of the constant term holds.

Load-bearing premise

That mid-spectrum eigenstates in the Bose-Hubbard model can be approximated by the same mean-field averaging used for fermions and spins, without bosonic-specific corrections that would change the volume-law coefficient.

What would settle it

Exact numerical diagonalization of a disordered Bose-Hubbard chain that yields a volume-law coefficient different from the mean-field prediction would falsify the claim that the coefficient is unchanged by disorder.

Figures

Figures reproduced from arXiv: 2604.09180 by G. Medo\v{s}, L. Vidmar.

**Figure 2.** Figure 2: FIG. 2. The distribution [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The distribution [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Subtracted means [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

read the original abstract

While the eigenstate entanglement entropy has been extensively studied for fermionic systems, much less is known about bosonic systems. Here, we study the entanglement entropy of mid-spectrum eigenstates of Bose-Hubbard models, focusing on weakly disordered models with and without particle-number conservation, and contrasting them with the translationally-invariant model. We analyze the volume-law and O(1) contributions to the entanglement entropy via the averages over mid-spectrum eigenstates and the corresponding distributions. We derive the volume-law coefficient of the entanglement entropy by generalizing the mean-field approach from [Phys. Rev. Lett. 119, 220603 (2017)] to many-body systems with a tunable local bosonic cutoff, which agrees with previous analytical and numerical results from [Phys. Rev. B 110, 235154 (2024)]. We show that the volume-law contribution to the entanglement entropy does not change upon breaking translational invariance via on-site disorder. We then numerically study the role of the subleading O(1) contribution to the entanglement entropy. We find that, in the particle-number conserving case, it exhibits a nontrivial dependence on the particle-number density and the local bosonic cutoff, while without particle-number conservation, results suggest the emergence of a universal O(1) contribution beyond the random pure state predictions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The volume-law coefficient for mid-spectrum entanglement in Bose-Hubbard models is derived by generalizing a 2017 mean-field method and stays unchanged by disorder, with new numerical details on the O(1) term.

read the letter

The main things to take away are that the volume-law coefficient for the entanglement entropy of mid-spectrum states in Bose-Hubbard models can be derived by generalizing a mean-field approach, and that this coefficient stays the same when translational invariance is broken by disorder. They do a solid job adapting the 2017 mean-field method to bosons with a tunable local cutoff. The resulting volume-law expression matches earlier analytical and numerical findings from the 2024 paper. The numerical work on the O(1) correction shows a clear dependence on particle density and the cutoff in the conserving case, while the non-conserving case points toward a universal value different from random pure states. These observations are new for the bosonic setting. The approach builds on established techniques and cross-checks with prior independent results, so the central volume-law claim holds up without obvious circularity. The disorder independence follows directly from the averaging procedure. One area that needs scrutiny is the validity of the mean-field generalization for bosons. The original derivation was for systems with different statistics, and the hard cutoff on occupation might affect the single-site reduced density matrix in ways not captured by simply swapping the local dimension. The paper reports agreement with numerics, but the details of how bosonic commutation relations are handled in the ensemble average would benefit from explicit discussion. This work is for researchers focused on entanglement entropy in quantum lattice models, particularly those dealing with bosonic particles and disorder. It provides a practical analytical tool and some fresh numerical insights that can guide further studies in ultracold atoms. I recommend sending it for peer review. The claims are specific enough that referees can check the derivation and the numerics without too much trouble.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates eigenstate entanglement entropy (EE) of mid-spectrum states in Bose-Hubbard models, with and without particle-number conservation and with/without on-site disorder. The central claims are: (i) the volume-law coefficient is derived by generalizing the mean-field averaging procedure of Phys. Rev. Lett. 119, 220603 (2017) to a local bosonic Hilbert space truncated at finite cutoff N_b, yielding a result that agrees with prior analytical/numerical work in Phys. Rev. B 110, 235154 (2024); (ii) this volume-law contribution is invariant under breaking of translational invariance by disorder; (iii) the subleading O(1) term shows nontrivial dependence on density and cutoff in the conserving case and approaches a universal value (beyond random pure-state predictions) in the non-conserving case.

Significance. If the mean-field generalization is valid, the work supplies an analytical expression for the volume-law EE coefficient in bosonic systems, extending the 2017 fermionic/spin result to tunable local dimension and providing a concrete check against independent 2024 numerics. The reported invariance of the volume-law term under weak disorder is a clear, falsifiable statement with implications for eigenstate thermalization and many-body localization in bosonic models. Numerical exploration of the O(1) term highlights differences between conserving and non-conserving dynamics. Credit is due for the explicit cutoff dependence in the derivation and for the direct comparison to prior independent results rather than self-consistent fitting.

major comments (1)

The derivation of the volume-law coefficient (generalizing the 2017 PRL mean-field procedure) assumes that the ensemble average over mid-spectrum eigenstates factorizes identically once the local dimension is replaced by (N_b + 1). Because bosons obey different commutation relations and the cutoff imposes a hard occupation bound, the single-site reduced density matrix may acquire cutoff-dependent correlations absent in the original fermionic derivation. The cited numerical agreement with Phys. Rev. B 110, 235154 (2024) does not yet demonstrate that such corrections vanish in the thermodynamic limit; an explicit expansion or additional numerical test varying N_b at fixed filling would be required to confirm the assumption is load-bearing only for the reported coefficient.

minor comments (1)

The abstract and introduction would benefit from a one-sentence statement of the precise functional form obtained for the volume-law coefficient (e.g., in terms of N_b and filling) to allow immediate comparison with the cited 2024 result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive major comment. We address the point below and will incorporate clarifications and additional checks in a revised version.

read point-by-point responses

Referee: The derivation of the volume-law coefficient (generalizing the 2017 PRL mean-field procedure) assumes that the ensemble average over mid-spectrum eigenstates factorizes identically once the local dimension is replaced by (N_b + 1). Because bosons obey different commutation relations and the cutoff imposes a hard occupation bound, the single-site reduced density matrix may acquire cutoff-dependent correlations absent in the original fermionic derivation. The cited numerical agreement with Phys. Rev. B 110, 235154 (2024) does not yet demonstrate that such corrections vanish in the thermodynamic limit; an explicit expansion or additional numerical test varying N_b at fixed filling would be required to confirm the assumption is load-bearing only for the reported coefficient.

Authors: We appreciate this careful scrutiny of the mean-field generalization. The procedure replaces the local dimension with N_b + 1 while averaging the reduced density matrix over mid-spectrum eigenstates treated as ergodic in the truncated many-body Hilbert space; the bosonic commutation relations govern the Hamiltonian but do not alter the structure of the local occupation basis used for the ensemble average of the single-site reduced density matrix. The reported agreement with the independent 2024 numerics (which already scan multiple N_b at fixed filling) indicates that cutoff-induced corrections remain subleading for the volume-law coefficient. To strengthen the presentation, we will add an explicit expansion of the leading term in the revised manuscript and include a supplementary numerical test of the coefficient versus N_b at fixed density, confirming robustness in the thermodynamic limit. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation generalizes external mean-field method and agrees with independent benchmarks

full rationale

The central derivation obtains the volume-law coefficient by generalizing the mean-field procedure of the cited 2017 PRL (external to this work) to a bosonic Hilbert space with tunable cutoff N_b. The paper then reports numerical agreement with independent prior results from the 2024 PRB. No equation or step reduces the claimed coefficient to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain whose validity is presupposed by the present manuscript. The effect of disorder is analyzed as a separate numerical observation. Because the load-bearing step rests on an external reference plus external benchmarks rather than on the paper's own inputs, the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; the mean-field generalization is the primary assumption, but no explicit free parameters, new entities, or additional axioms are identifiable without the full text.

pith-pipeline@v0.9.0 · 5541 in / 1191 out tokens · 37858 ms · 2026-05-10T17:23:55.935801+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Typical entanglement entropy with charge conservation
quant-ph 2026-04 unverdicted novelty 7.0

Typical entanglement entropy with fixed global charge is given by the local thermal entropy at fixed charge density for both U(1) and SU(2) symmetries in the thermodynamic limit.

Reference graph

Works this paper leans on

64 extracted references · 4 canonical work pages · cited by 1 Pith paper

[1]

Srednicki, Entropy and area, Phys

M. Srednicki, Entropy and area, Phys. Rev. Lett.71, 666 (1993)

1993
[2]

Eisert, M

J. Eisert, M. Cramer, and M. B. Plenio,Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys. 82, 277 (2010)

2010
[3]

M. B. Hastings, An area law for one-dimensional quan- tum systems, J. Stat. Mech.2007, P08024 (2007)

2007
[4]

Calabrese and J

P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech.2005, P04010 (2005)

2005
[5]

Calabrese and J

P. Calabrese and J. Cardy, Entanglement and correla- tion functions following a local quench: a conformal field theory approach, J. Stat. Mech.2007, P10004 (2007)

2007
[6]

Eisler and I

V. Eisler and I. Peschel, Evolution of entanglement after a local quench, J. Stat. Mech.2007, P06005 (2007)

2007
[7]

LeBlond, K

T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, Entanglement and matrix elements of observables in in- teracting integrable systems, Phys. Rev. E100, 062134 (2019)

2019
[8]

D. N. Page, Average Entropy of a Subsystem, Phys. Rev. Lett.71, 1291 (1993). ArXiv:gr-qc/9305007

work page Pith review arXiv 1993
[9]

Beugeling, A

W. Beugeling, A. Andreanov, and M. Haque, Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement en- tropy, J. Stat. Mech. (2015), P02002

2015
[10]

Z.-C. Yang, C. Chamon, A. Hamma, and E. R. Mucciolo, Two-Component Structure in the Entanglement Spec- trum of Highly Excited States, Phys. Rev. Lett.115, 267206 (2015)

2015
[11]

L.VidmarandM.Rigol, EntanglementEntropyofEigen- states of Quantum Chaotic Hamiltonians, Phys. Rev. Lett.119, 220603 (2017)

2017
[12]

J. R. Garrison and T. Grover, Does a Single Eigenstate Encode the Full Hamiltonian?, Phys. Rev. X8, 021026 (2018)

2018
[13]

Dymarsky, N

A. Dymarsky, N. Lashkari, and H. Liu, Subsystem eigen- state thermalization hypothesis, Phys. Rev. E97, 012140 (2018)

2018
[14]

Y. O. Nakagawa, M. Watanabe, H. Fujita, and S. Sug- iura, Universality in volume-law entanglement of scram- bled pure quantum states, Nat. Comm.9, 1635 (2018)

2018
[15]

Huang, Universal eigenstate entanglement of chaotic local Hamiltonians, Nuc

Y. Huang, Universal eigenstate entanglement of chaotic local Hamiltonians, Nuc. Phys. B938, 594 (2019)

2019
[16]

Murthy and M

C. Murthy and M. Srednicki, Structure of chaotic eigen- states and their entanglement entropy, Phys. Rev. E100, 022131 (2019)

2019
[17]

Miao and T

Q. Miao and T. Barthel, Eigenstate Entanglement: Crossover from the Ground State to Volume Laws, Phys. Rev. Lett.127, 040603 (2021)

2021
[18]

Bianchi, L

E. Bianchi, L. Hackl, M. Kieburg, M. Rigol, and L. Vid- mar, Volume-LawEntanglementEntropyofTypicalPure Quantum States, PRX Quantum3, 030201 (2022)

2022
[19]

Vidmar, L

L. Vidmar, L. Hackl, E. Bianchi, and M. Rigol, Entan- glement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians, Phys. Rev. Lett.119, 020601 (2017)

2017
[20]

C. Liu, X. Chen, and L. Balents, Quantum entangle- ment of the Sachdev-Ye-Kitaev models, Phys. Rev. B97, 245126 (2018)

2018
[21]

Hackl, L

L. Hackl, L. Vidmar, M. Rigol, and E. Bianchi, Aver- age eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models, Phys. Rev. B99, 075123 (2019)

2019
[22]

Łydżba, M

P. Łydżba, M. Rigol, and L. Vidmar, Eigenstate En- tanglement Entropy in Random Quadratic Hamiltonians, Phys. Rev. Lett.125, 180604 (2020)

2020
[23]

Łydżba, M

P. Łydżba, M. Rigol, and L. Vidmar, Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians, Phys. Rev. B103, 104206 (2021)

2021
[24]

Storms and R

M. Storms and R. R. P. Singh, Entanglement in ground and excited states of gapped free-fermion systems and theirrelationshipwithFermisurfaceandthermodynamic equilibrium properties, Phys. Rev. E89, 012125 (2014)

2014
[25]

Swietek, M

R. Swietek, M. Kliczkowski, L. Vidmar, and M. Rigol, Eigenstate entanglement entropy in the integrable spin- 1/2 XYZ model, Phys. Rev. E109, 024117 (2024)

2024
[26]

Bianchi and P

E. Bianchi and P. Donà, Typical entanglement entropy in the presence of a center: Page curve and its variance, Phys. Rev. D100, 105010 (2019)

2019
[27]

Y. Yauk, R. Patil, Y. Zhang, M. Rigol, and L. Hackl, Typical entanglement entropy in systems with particle- number conservation, Phys. Rev. B110, 235154 (2024)

2024
[28]

S. C. Morampudi, A. Chandran, and C. Laumann, Uni- versal Entanglement of Typical States in Constrained Systems, Phys. Rev. Lett.124, 050602 (2020)

2020
[29]

Patil, L

R. Patil, L. Hackl, G. R. Fagan, and M. Rigol, Aver- age pure-state entanglement entropy in spin systems with 11 SU(2) symmetry, Phys. Rev. B108, 245101 (2023)

2023
[30]

Bianchi, P

E. Bianchi, P. Dona, and R. Kumar, Non-Abelian symmetry-resolved entanglement entropy, SciPost Phys. 17, 127 (2024)

2024
[31]

Haque, P

M. Haque, P. A. McClarty, and I. M. Khaymovich, En- tanglement of midspectrum eigenstates of chaotic many- body systems: Reasons for deviation from random en- sembles, Phys. Rev. E105, 014109 (2022)

2022
[32]

Kliczkowski, R

M. Kliczkowski, R. Świętek, L. Vidmar, and M. Rigol, Average entanglement entropy of midspectrum eigen- states of quantum-chaotic interacting Hamiltonians, Phys. Rev. E107, 064119 (2023)

2023
[33]

Huang, Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians, IEEE Journal on Selected Areas in Information Theory5, 694 (2024)

Y. Huang, Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians, IEEE Journal on Selected Areas in Information Theory5, 694 (2024)

2024
[34]

J. F. Rodriguez-Nieva, C. Jonay, and V. Khemani, Quan- tifying Quantum Chaos through Microcanonical Dis- tributions of Entanglement, Phys. Rev. X14, 031014 (2024)

2024
[35]

C. M. Langlett and J. F. Rodriguez-Nieva, Entanglement PatternsofQuantumChaoticHamiltonianswithaScalar U(1) Charge, Phys. Rev. Lett.134, 230402 (2025)

2025
[36]

C. M. Langlett, C. Jonay, V. Khemani, and J. F. Rodriguez-Nieva, Quantum chaos at finite temperature in local spin Hamiltonians (2025). ArXiv:2501.13164 [cond-mat]

work page arXiv 2025
[37]

Kollath, G

C. Kollath, G. Roux, G. Biroli, and A. M. Läuchli, Sta- tistical properties of the spectrum of the extended Bose– Hubbard model, J. Stat. Mech.2010, P08011 (2010)

2010
[38]

M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed mat- ter systems to ultracold gases, Rev. Mod. Phys.83, 1405 (2011)

2011
[39]

Lukin, M

A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kauf- man, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science364, 256 (2019)

2019
[40]

H. P. Lüschen, P. Bordia, S. Scherg, F. Alet, E. Alt- man, U. Schneider, and I. Bloch, Observation of Slow Dynamics near the Many-Body Localization Transition in One-Dimensional Quasiperiodic Systems, Phys. Rev. Lett.119, 260401 (2017)

2017
[41]

Greiner and S

M. Greiner and S. Fölling, Optical lattices., Nature453, 736 (2008)

2008
[42]

Rispoli, A

M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, and M. Greiner, Quantum critical behaviour at the many-body localization transition, Nature573, 385 (2019)

2019
[43]

S. S. Kondov, W. R. McGehee, W. Xu, and B. DeMarco, Disorder-Induced Localization in a Strongly Correlated Atomic Hubbard Gas, Phys. Rev. Lett.114, 083002 (2015)

2015
[44]

A. H. Karamlou, I. T. Rosen, S. E. Muschinske, C. N. Barrett, A. Di Paolo, L. Ding, P. M. Harrington, M. Hays, R. Das, D. K. Kim, B. M. Niedzielski, M. Schuldt, K. Serniak, M. E. Schwartz, J. L. Yoder, S. Gustavsson, Y. Yanay, J. A. Grover, and W. D. Oliver, Probing entanglement in a 2D hard-core Bose–Hubbard lattice, Nature629, 561 (2024)

2024
[45]

J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio- Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch, and C. Gross, Exploring the many-body localization transition in two dimensions, Science352, 1547 (2016)

2016
[46]

Roushan, C

P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas, A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jef- frey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quin- tana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis, Spectro- scopic signatures of localizat...

2017
[47]

Schreiber, S

M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lüschen, M. H. Fischer, R. Vosk, E. Altman, U. Schneider, and I. Bloch, Observation of many-body localization of inter- acting fermions in a quasirandom optical lattice, Science 349, 842 (2015)

2015
[48]

Greiner, O

M. Greiner, O. Mandel, T. Esslinger, and T. W. Ha, Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms, Nature415, 39 (2002)

2002
[49]

Greiner, O

M. Greiner, O. Mandel, T. W. Hänsch, and I. Bloch, Collapse and revival of the matter wave field of a Bose– Einstein condensate, Nature419, 51 (2002)

2002
[50]

I. B. Spielman, W. D. Phillips, and J. V. Porto, The Mott insulator transition in two dimensions, Phys. Rev. Lett. 98, 080404 (2007). ArXiv:cond-mat/0606216

work page arXiv 2007
[51]

Trotzky, Y.-A

S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollwöck, J. Eisert, and I. Bloch, Probing the re- laxation towards equilibrium in an isolated strongly cor- related one-dimensional Bose gas, Nature Phys8, 325 (2012)

2012
[52]

J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr, Single-Atom Resolved Fluores- cence Imaging of an Atomic Mott Insulator, Nature467, 68 (2010). ArXiv:1006.3799 [cond-mat]

work page arXiv 2010
[53]

A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R.Schittko, P.M.Preiss, andM.Greiner, Quantumther- malization through entanglement in an isolated many- body system, Science353, 794 (2016)

2016
[54]

Islam, R

R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement en- tropy in a quantum many-body system, Nature528, 77 (2015)

2015
[55]

A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring Entanglement Growth in Quench Dynamics of Bosons in an Optical Lattice, Phys. Rev. Lett.109, 020505 (2012)

2012
[56]

Sierant and J

P. Sierant and J. Zakrzewski, Many-body localization of bosons in optical lattices, New J. Phys.20, 043032 (2018)

2018
[57]

Sierant, D

P. Sierant, D. Delande, and J. Zakrzewski, Many-Body Localization for Randomly Interacting Bosons, Acta Phys. Pol. A132, 1707 (2017)

2017
[58]

M.HopjanandF.Heidrich-Meisner, Many-bodylocaliza- tionfromaone-particleperspectiveinthedisorderedone- dimensional Bose-Hubbard model, Phys. Rev. A101, 063617 (2020)

2020
[59]

Pausch, E

L. Pausch, E. G. Carnio, A. Rodríguez, and A. Buchleit- ner, Chaos and Ergodicity across the Energy Spectrum of Interacting Bosons, Phys. Rev. Lett.126, 150601 (2021)

2021
[60]

Sierant, M

P. Sierant, M. Lewenstein, A. Scardicchio, L. Vidmar, and J. Zakrzewski, Many-body localization in the age of classical computing, Reports on Progress in Physics88, 026502 (2025)

2025
[61]

Sierant, M

P. Sierant, M. Lewenstein, and J. Zakrzewski, Polyno- mially filtered exact diagonalization approach to many- body localization, Phys. Rev. Lett.125, 156601 (2020). 12

2020
[62]

Bianchi, L

E. Bianchi, L. Hackl, and M. Kieburg, Page curve for fermionic Gaussian states, Phys. Rev. B103, L241118 (2021)

2021
[63]

Lisiecki, L

M. Lisiecki, L. Vidmar, and P. Łydżba, Localization transitions in quadratic systems without quantum chaos, Phys. Rev. E111, 054110 (2025)

2025
[64]

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