pith. sign in

arxiv: 2603.04373 · v2 · submitted 2026-03-04 · 🪐 quant-ph · cond-mat.quant-gas

Dynamical Behaviour of Density Correlations Across the Chaotic Phase for Interacting Bosons

\'Oscar Due\~nas , Alberto Rodr\'iguez This is my paper

Pith reviewed 2026-05-15 16:44 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gas
keywords Bose-Hubbard modeldensity correlationsquantum chaoscorrelation transport distanceballistic propagationone-dimensional bosonsintegrable-chaotic crossover
0
0 comments X

The pith

Density correlation fronts in the Bose-Hubbard model propagate ballistically for all interaction strengths, including the chaotic regime.

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

The work studies the spreading of two-point density correlations in the one-dimensional Bose-Hubbard model in the thermodynamic limit, tracked through the correlation transport distance. Integrable points of the model produce ballistic growth of this distance, while the onset of chaos produces a clear sub-ballistic regime. Close inspection of the full space-time profiles shows that the leading front edge continues to advance at constant speed regardless of interaction strength. The measured slowdown of the transport distance is instead traced to the growth of long-time, distance-dependent correlation tails and to a stronger decay of the front amplitude once chaos appears. The results refine the light-cone picture by separating front motion from the overall spreading measure.

Core claim

In the thermodynamic limit of the Bose-Hubbard Hamiltonian the correlation transport distance grows ballistically in the integrable limits but becomes sub-ballistic once the chaotic phase is entered. The front of the correlation profile nevertheless advances ballistically at all interaction strengths. The sub-ballistic slowdown originates from the appearance of long-time distance-dependent correlation tails together with an enhanced decay of the correlation-front amplitude.

What carries the argument

The correlation transport distance (CTD), a measurable quantity that tracks the spatial extent of the two-point density-correlation profile as a function of time.

If this is right

  • The correlation front velocity stays constant across the integrable-to-chaotic crossover.
  • Sub-ballistic CTD growth is produced by distance-dependent tails that persist at long times.
  • The front amplitude decays faster once chaotic dynamics set in.
  • These features hold in the thermodynamic limit and are visible in experimentally accessible correlation profiles.

Where Pith is reading between the lines

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

  • The same separation of front speed from overall spreading may appear in other chaotic lattice models when similar correlation measures are examined.
  • Experiments that resolve both the front position and the tail shape at late times could directly test the proposed mechanism.
  • The result suggests that light-cone bounds remain sharp even when global transport quantities look sub-ballistic.

Load-bearing premise

The thermodynamic-limit analysis and numerical extraction of long-time tails cleanly separate integrable and chaotic regimes without finite-size effects controlling the observed sub-ballistic behavior.

What would settle it

A simulation or experiment in which the leading-edge velocity extracted from the correlation profile itself becomes sub-ballistic once the system enters the chaotic phase, or in which the long-time tails fail to appear while the CTD remains sub-ballistic.

Figures

Figures reproduced from arXiv: 2603.04373 by Alberto Rodr\'iguez, \'Oscar Due\~nas.

Figure 1
Figure 1. Figure 1: FIG. 1. Optimal parameters used in the iTEBD simulations, (a) maximum local occupation number [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Time evolution of the CTD for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Pseudo-distribution [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Analytical spatio-temporal correlation profiles for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Numerically obtained spatio-temporal correlation profiles for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. (a) Time evolution of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Velocity of correlation front for [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Decay of the correlation front with distance for [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) Pseudo-distribution [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Representation of the pseudo-distribution [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
read the original abstract

We investigate the propagation of two-point density correlations in the one-dimensional Bose-Hubbard Hamiltonian in the thermodynamic limit in terms of the correlation transport distance (CTD), an experimentally measurable magnitude that characterizes the spatial spreading of correlations in time. We confirm that the integrable limits of the model exhibit CTD ballistic growth, while the onset of the chaotic phase leads to the emergence of a pronounced sub-ballistic regime, in agreement with previous results for finite systems. By a meticulous analysis of the spatio-temporal correlation profiles, we show that the correlation front nonetheless propagates ballistically for all interaction strengths, and that the chaos-induced slowdown of the CTD originates from the emergence of long-time distance-dependent correlation tails, together with an enhanced decay of the correlation front amplitude. Our results thus provide a detailed characterization of correlation transport that goes beyond a simple light-cone picture.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript examines the propagation of two-point density correlations in the one-dimensional Bose-Hubbard Hamiltonian in the thermodynamic limit via the correlation transport distance (CTD). It reports ballistic CTD growth in integrable limits and the emergence of a sub-ballistic regime in the chaotic phase, in agreement with prior finite-system results. The correlation front is shown to propagate ballistically for all interaction strengths, with the CTD slowdown attributed to the appearance of long-time distance-dependent correlation tails together with enhanced decay of the front amplitude, thereby going beyond a simple light-cone description.

Significance. If the thermodynamic-limit extraction of front velocities and tail exponents proves robust, the work supplies a concrete mechanistic account of how chaos modifies correlation spreading—via tails and amplitude decay—while preserving ballistic front propagation. This refines the light-cone paradigm and offers testable signatures for experiments on ultracold bosons, strengthening the link between many-body chaos and observable transport quantities.

major comments (2)
  1. [Thermodynamic-limit analysis and numerical extrapolation] The central separation between ballistic front propagation and sub-ballistic CTD (arising from tails plus amplitude decay) rests on numerical extrapolation to the thermodynamic limit. The manuscript must demonstrate explicit convergence of both the extracted front velocity and the tail exponents with system size L, including error estimates or scaling plots, to rule out finite-size artifacts that could contaminate the integrable-versus-chaotic distinction.
  2. [Spatio-temporal correlation profiles] The claim that the front remains strictly ballistic for all U while CTD slows sub-ballistically requires quantitative support for the front-tail decomposition (e.g., fitted velocities independent of L after subtraction). Without shown convergence data or sensitivity checks on fitting windows, the sub-ballistic CTD could still reflect residual finite-size effects rather than genuine chaotic signatures.
minor comments (2)
  1. [Abstract] The abstract states agreement with previous finite-system results but provides no details on error bars, data exclusion criteria, or fitting procedures; a short methods summary or supplementary note would improve verifiability.
  2. [Introduction] Notation for the CTD and correlation profiles should be defined more explicitly at first use, including any normalization conventions, to aid readers unfamiliar with the prior literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments on the numerical robustness of our thermodynamic-limit claims. We agree that additional explicit convergence data will strengthen the presentation and plan to incorporate them in the revised version.

read point-by-point responses

  1. Referee: [Thermodynamic-limit analysis and numerical extrapolation] The central separation between ballistic front propagation and sub-ballistic CTD (arising from tails plus amplitude decay) rests on numerical extrapolation to the thermodynamic limit. The manuscript must demonstrate explicit convergence of both the extracted front velocity and the tail exponents with system size L, including error estimates or scaling plots, to rule out finite-size artifacts that could contaminate the integrable-versus-chaotic distinction.

    Authors: We agree that explicit convergence checks are essential. In the revised manuscript we will add scaling plots of the extracted front velocity v_f and tail exponent versus 1/L (for several representative U values spanning integrable and chaotic regimes), together with error bars obtained from the fitting procedure. These plots confirm convergence to well-defined thermodynamic-limit values, with the ballistic-versus-sub-ballistic distinction between integrable and chaotic regimes remaining sharp. revision: yes

  2. Referee: [Spatio-temporal correlation profiles] The claim that the front remains strictly ballistic for all U while CTD slows sub-ballistically requires quantitative support for the front-tail decomposition (e.g., fitted velocities independent of L after subtraction). Without shown convergence data or sensitivity checks on fitting windows, the sub-ballistic CTD could still reflect residual finite-size effects rather than genuine chaotic signatures.

    Authors: We will augment the revision with a quantitative front-tail decomposition analysis. This will include (i) a table/figure of fitted front velocities after explicit subtraction of the long-time tails, demonstrating L-independence within error bars, and (ii) sensitivity tests obtained by systematically varying the space-time fitting windows. The resulting velocities remain ballistic and stable across the full range of U, confirming that the sub-ballistic CTD growth originates from the distance-dependent tails and front-amplitude decay rather than finite-size artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on numerical analysis independent of inputs

full rationale

The paper performs a numerical study of density correlations in the Bose-Hubbard model, extracting CTD from spatio-temporal profiles in the thermodynamic limit. It reports ballistic front propagation for all U and attributes sub-ballistic CTD to distance-dependent tails plus front-amplitude decay. No equations define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and no uniqueness theorem or ansatz is imported via self-citation to force the result. The reference to prior finite-system results is external corroboration rather than a load-bearing self-referential step. The derivation chain is self-contained against the numerical data and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard Bose-Hubbard Hamiltonian and conventional many-body quantum dynamics; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard quantum mechanics governs the time evolution of the Bose-Hubbard Hamiltonian in the thermodynamic limit.
    The model and correlation definitions presuppose conventional Schrödinger evolution and lattice boson statistics.

pith-pipeline@v0.9.0 · 5443 in / 1142 out tokens · 28781 ms · 2026-05-15T16:44:15.213291+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

  1. [1]

    Haake, S

    Haake F, Gnutzmann S and Kuś M 2018Quantum Signatures of ChaosFourth ed. ed (Springer International Publishing) ISBN 978-3-319-97579-5 URLhttp://link.springer.com/10.1007/978-3-319-97580-1

    work page doi:10.1007/978-3-319-97580-1

  2. [2]

    & Love, W

    Izrailev F M 1990 Simple models of quantum chaos: Spectrum and eigenfunctionsPhys. Rep.196(5-6) 299–392 URL https://linkinghub.elsevier.com/retrieve/pii/037015739090067C

    work page arXiv 1990

  3. [3]

    Brody T A, Flores J, French J B, Mello P A, Pandey A and Wong S S M 1981 Random-matrix physics: spectrum and strength fluctuationsRev. Mod. Phys.53385–479 URLhttps://link.aps.org/doi/10.1103/RevModPhys.53.385

    work page doi:10.1103/revmodphys.53.385 1981

  4. [4]

    Rep.299189–425 URLhttps://linkinghub.elsevier.com/retrieve/pii/S0370157397000884

    Guhr T, Müller-Groeling A and Weidenmüller H A 1998 Random-matrix theories in quantum physics: common concepts Phys. Rep.299189–425 URLhttps://linkinghub.elsevier.com/retrieve/pii/S0370157397000884

    work page 1998

  5. [5]

    Deutsch J M 1991 Quantum statistical mechanics in a closed systemPhys. Rev. A43(4) 2046–2049 URLhttps://link. aps.org/doi/10.1103/PhysRevA.43.2046

    work page doi:10.1103/physreva.43.2046 1991

  6. [6]

    Rigol M, Dunjko V and Olshanii M 2008 Thermalization and its mechanism for generic isolated quantum systemsNature 452(7189) 854–858 URLhttp://dx.doi.org/10.1038/nature06838

    work page doi:10.1038/nature06838 2008

  7. [7]

    Rep.6261–58 URLhttp://linkinghub.elsevier.com/retrieve/pii/S0370157316000831

    Borgonovi F, Izrailev F M, Santos L F and Zelevinsky V G 2016 Quantum chaos and thermalization in isolated systems of interacting particlesPhys. Rep.6261–58 URLhttp://linkinghub.elsevier.com/retrieve/pii/S0370157316000831

    work page 2016

  8. [8]

    Cheneau M, Barmettler P, Poletti D, Endres M, Schauß P, Fukuhara T, Gross C, Bloch I, Kollath C and Kuhr S 2012 Light-cone-like spreading of correlations in a quantum many-body systemNature481484–487 URLhttps://doi.org/ 10.1038/nature10748

    work page doi:10.1038/nature10748 2012

  9. [9]

    org/10.1126/science.1224953 16

    Gring M, Kuhnert M, Langen T, Kitagawa T, Rauer B, Schreitl M, Mazets I, Smith D A, Demler E and Schmiedmayer J 2012 Relaxation and prethermalization in an isolated quantum systemScience337(6100) 1318–1322 URLhttps://doi. org/10.1126/science.1224953 16

    work page doi:10.1126/science.1224953 2012

  10. [10]

    com/articles/nphys2232

    Trotzky S, Chen Y A, Flesch A, McCulloch I P, Schollwöck U, Eisert J and Bloch I 2012 Probing the relaxation towards equilibriuminanisolatedstronglycorrelatedone-dimensionalbosegasNat .Phys.8(4)325–330URLhttps://www.nature. com/articles/nphys2232

    work page 2012

  11. [11]

    Ronzheimer J P, Schreiber M, Braun S, Hodgman S S, Langer S, McCulloch I P, Heidrich-Meisner F, Bloch I and Schneider U 2013 Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensionsPhys. Rev. Lett. 110(20) 205301 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.110.205301

    work page doi:10.1103/physrevlett.110.205301 2013

  12. [12]

    Phys.9(10) 640–643 URLhttp://www.nature.com/articles/nphys2739

    Langen T, Geiger R, Kuhnert M, Rauer B and Schmiedmayer J 2013 Local emergence of thermal correlations in an isolated quantum many-body systemNat. Phys.9(10) 640–643 URLhttp://www.nature.com/articles/nphys2739

    work page 2013

  13. [13]

    Meinert F, Mark M J, Kirilov E, Lauber K, Weinmann P, Gröbner M and Nägerl H C 2014 Interaction-induced quantum phase revivals and evidence for the transition to the quantum chaotic regime in 1d atomic bloch oscillationsPhys. Rev. Lett.112(19) 193003 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.112.193003

    work page doi:10.1103/physrevlett.112.193003 2014

  14. [14]

    Langen, S

    Langen T, Erne S, Geiger R, Rauer B, Schweigler T, Kuhnert M, Rohringer W, Mazets I E, Gasenzer T and Schmiedmayer J 2015 Experimental observation of a generalized gibbs ensembleScience348(6231) 207–211 URLhttps://www.science. org/doi/10.1126/science.1257026

    work page doi:10.1126/science.1257026 2015

  15. [15]

    Kaufman A M, Tai M E, Lukin A, Rispoli M, Schittko R, Preiss P M and Greiner M 2016 Quantum thermalization through entanglement in an isolated many-body systemScience353794–800 URLhttps://www.science.org/doi/abs/10.1126/ science.aaf6725

    work page 2016

  16. [16]

    Bordia P, Lüschen H P, Hodgman S S, Schreiber M, Bloch I and Schneider U 2016 Coupling identical one-dimensional many-body localized systemsPhys. Rev. Lett.116(14) 140401 URLhttps://link.aps.org/doi/10.1103/PhysRevLett. 116.140401

    work page doi:10.1103/physrevlett 2016

  17. [17]

    Choi J, Hild S, Zeiher J, Schauß P, Rubio-Abadal A, Yefsah T, Khemani V, Huse D A, Bloch I and Gross C 2016 Exploring the many-body localization transition in two dimensionsScience3521547–1552 URLhttps://www.science.org/doi/ abs/10.1126/science.aaf8834

    work page doi:10.1126/science.aaf8834 2016

  18. [18]

    Lukin A, Rispoli M, Schittko R, Tai M E, Kaufman A M, Choi S, Khemani V, Léonard J and Greiner M 2019 Probing entanglement in a many-body–localized systemScience364256–260 URLhttps://www.science.org/doi/abs/10.1126/ science.aau0818

    work page 2019

  19. [19]

    Rubio-Abadal A, Choi J, Zeiher J, Hollerith S, Rui J, Bloch I and Gross C 2019 Many-body delocalization in the presence of a quantum bathPhys. Rev. X9(4) 041014 URLhttps://link.aps.org/doi/10.1103/PhysRevX.9.041014

    work page doi:10.1103/physrevx.9.041014 2019

  20. [20]

    Rispoli M, Lukin A, Schittko R, Kim S, Tai M E, Léonard J and Greiner M 2019 Quantum critical behaviour at the many- body localization transitionNature573(7774) 385–389 URLhttp://www.nature.com/articles/s41586-019-1527-2

    work page 2019

  21. [21]

    Phys.19481–485 URLhttps://doi.org/10.1038/ s41567-022-01887-3

    Léonard J, Kim S, Rispoli M, Lukin A, Schittko R, Kwan J, Demler E, Sels D and Greiner M 2023 Probing the on- set of quantum avalanches in a many-body localized systemNat. Phys.19481–485 URLhttps://doi.org/10.1038/ s41567-022-01887-3

    work page 2023

  22. [22]

    Phys.URLhttps://www.nature.com/ articles/s41567-024-02611-z

    Wienand J F, Karch S, Impertro A, Schweizer C, McCulloch E, Vasseur R, Gopalakrishnan S, Aidelsburger M and Bloch I 2024 Emergence of fluctuating hydrodynamics in chaotic quantum systemsNat. Phys.URLhttps://www.nature.com/ articles/s41567-024-02611-z

    work page 2024

  23. [23]

    Srednicki M 1996 Thermal fluctuations in quantized chaotic systemsJ. Phys. A: Math. Gen.29L75 URLhttps://doi. org/10.1088/0305-4470/29/4/003

    work page doi:10.1088/0305-4470/29/4/003 1996

  24. [24]

    Srednicki M 1999 The approach to thermal equilibrium in quantized chaotic systemsJ. Phys. A: Math. Gen.321163 URL https://doi.org/10.1088/0305-4470/32/7/007

    work page doi:10.1088/0305-4470/32/7/007 1999

  25. [25]

    Pausch L, Carnio E G, Buchleitner A and Rodríguez A 2025 How to seed ergodic dynamics of interacting bosons under conditions of many-body quantum chaosRep. Prog. Phys.88057602 URLhttps://doi.org/10.1088/1361-6633/add0de

    work page doi:10.1088/1361-6633/add0de 2025

  26. [26]

    Dueñas O, Peña D and Rodríguez A 2025 Propagation of two-particle correlations across the chaotic phase for interacting bosonsPhys. Rev. Res.7(1) L012031 URLhttps://link.aps.org/doi/10.1103/PhysRevResearch.7.L012031

    work page doi:10.1103/physrevresearch.7.l012031 2025

  27. [27]

    Läuchli A M and Kollath C 2008 Spreading of correlations and entanglement after a quench in the one-dimensional Bose-Hubbard modelJ. Stat. Mech. Theory Exp.2008P05018 URLhttps://iopscience.iop.org/article/10.1088/ 1742-5468/2008/05/P05018

    work page 2008

  28. [28]

    Barmettler P, Poletti D, Cheneau M and Kollath C 2012 Propagation front of correlations in an interacting bose gasPhys. Rev. A85(5) 053625 URLhttps://link.aps.org/doi/10.1103/PhysRevA.85.053625

    work page doi:10.1103/physreva.85.053625 2012

  29. [29]

    Rep.94135 URLhttps://doi.org/10.1038/s41598-019-40679-3

    Despres J, Villa L and Sanchez-Palencia L 2019 Twofold correlation spreading in a strongly correlated lattice bose gasSci. Rep.94135 URLhttps://doi.org/10.1038/s41598-019-40679-3

    work page doi:10.1038/s41598-019-40679-3 2019

  30. [30]

    Fisher M P A, Weichman P B, Grinstein G and Fisher D S 1989 Boson localization and the superfluid-insulator transition Phys. Rev. B40546–570 URLhttp://link.aps.org/doi/10.1103/PhysRevB.40.546

    work page doi:10.1103/physrevb.40.546 1989

  31. [31]

    Phys.56243–379 URLhttps://doi.org/10.1080/00018730701223200

    Lewenstein M, Sanpera A, Ahufinger V, Damski B, Sen A and Sen U 2007 Ultracold atomic gases in optical lattices: mim- icking condensed matter physics and beyondAdv. Phys.56243–379 URLhttps://doi.org/10.1080/00018730701223200

    work page doi:10.1080/00018730701223200 2007

  32. [32]

    Bloch I, Dalibard J and Zwerger W 2008 Many-body physics with ultracold gasesRev. Mod. Phys.80885–964 URL https://link.aps.org/doi/10.1103/RevModPhys.80.885

    work page doi:10.1103/revmodphys.80.885 2008

  33. [33]

    Cazalilla M A, Citro R, Giamarchi T, Orignac E and Rigol M 2011 One dimensional bosons: From condensed matter systems to ultracold gasesRev. Mod. Phys.831405–1466 URLhttps://doi.org/10.1103/RevModPhys.83.1405

    work page doi:10.1103/revmodphys.83.1405 2011

  34. [34]

    Rep.6071–101 URL https://doi.org/10.1016/j.physrep.2015.10.004

    Krutitsky K V 2016 Ultracold bosons with short-range interaction in regular optical latticesPhys. Rep.6071–101 URL https://doi.org/10.1016/j.physrep.2015.10.004

    work page doi:10.1016/j.physrep.2015.10.004 2016

  35. [35]

    Lett.68632–638 URL https://doi.org/10.1209/epl/i2004-10265-7 17

    Kolovsky A R and Buchleitner A 2004 Quantum chaos in the Bose-Hubbard modelEurophys. Lett.68632–638 URL https://doi.org/10.1209/epl/i2004-10265-7 17

    work page doi:10.1209/epl/i2004-10265-7 2004

  36. [36]

    Biroli G, Kollath C and Läuchli A M 2010 Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems Phys. Rev. Lett.105250401 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.105.250401

    work page doi:10.1103/physrevlett.105.250401 2010

  37. [37]

    Kollath C, Roux G, Biroli G and Läuchli A M 2010 Statistical properties of the spectrum of the extended bose–hubbard modelJ. Stat. Mech. Theory Exp.2010P08011 URLhttps://doi.org/10.1088/1742-5468/2010/08/P08011

    work page doi:10.1088/1742-5468/2010/08/p08011 2010

  38. [38]

    Beugeling W, Moessner R and Haque M 2014 Finite-size scaling of eigenstate thermalizationPhys. Rev. E89042112 URL https://link.aps.org/doi/10.1103/PhysRevE.89.042112

    work page doi:10.1103/physreve.89.042112 2014

  39. [39]

    Beugeling W, Moessner R and Haque M 2015 Off-diagonal matrix elements of local operators in many-body quantum systemsPhys. Rev. E91012144 URLhttp://journals.aps.org/pre/abstract/10.1103/PhysRevE.91.012144

    work page doi:10.1103/physreve.91.012144 2015

  40. [40]

    Beugeling W, Andreanov A and Haque M 2015 Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement entropyJ. Stat. Mech. Theory Exp.2015P02002 URLhttp://stacks.iop.org/ 1742-5468/2015/i=2/a=P02002?key=crossref.a5120d9b4aa2a319c7830bb028289735

    work page 2015

  41. [41]

    Phys.18033009 URLhttps: //doi.org/10.1088/1367-2630/18/3/033009

    Dubertrand R and Müller S 2016 Spectral statistics of chaotic many-body systemsNew J. Phys.18033009 URLhttps: //doi.org/10.1088/1367-2630/18/3/033009

    work page doi:10.1088/1367-2630/18/3/033009 2016

  42. [42]

    Beugeling W, Bäcker A, Moessner R and Haque M 2018 Statistical properties of eigenstate amplitudes in complex quantum systemsPhys. Rev. E98022204 URLhttps://link.aps.org/doi/10.1103/PhysRevE.98.022204

    work page doi:10.1103/physreve.98.022204 2018

  43. [43]

    de la Cruz J, Lerma-Hernández S and Hirsch J G 2020 Quantum chaos in a system with high degree of symmetriesPhys. Rev. E102032208 URLhttps://link.aps.org/doi/10.1103/PhysRevE.102.032208

    work page doi:10.1103/physreve.102.032208 2020

  44. [44]

    Russomanno A, Fava M and Fazio R 2020 Nonergodic behavior of the clean Bose-Hubbard chainPhys. Rev. B102144302 URLhttps://link.aps.org/doi/10.1103/PhysRevB.102.144302

    work page doi:10.1103/physrevb.102.144302 2020

  45. [45]

    Pausch L, Carnio E G, Rodríguez A and Buchleitner A 2021 Chaos and Ergodicity across the Energy Spectrum of Interacting BosonsPhys. Rev. Lett.126150601 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.126.150601

    work page doi:10.1103/physrevlett.126.150601 2021

  46. [46]

    Phys.23123036 URLhttps://iopscience.iop.org/article/10.1088/1367-2630/ac3c0d

    Pausch L, Carnio E G, Buchleitner A and Rodríguez A 2021 Chaos in the Bose-Hubbard model and random two-body HamiltoniansNew J. Phys.23123036 URLhttps://iopscience.iop.org/article/10.1088/1367-2630/ac3c0d

    work page doi:10.1088/1367-2630/ac3c0d 2021

  47. [47]

    Pausch L, Buchleitner A, Carnio E G and Rodríguez A 2022 Optimal route to quantum chaos in the Bose-Hubbard model J. Phys. A Math. Theor.55324002 URLhttps://iopscience.iop.org/article/10.1088/1751-8121/ac7e0b

    work page doi:10.1088/1751-8121/ac7e0b 2022

  48. [48]

    Kollath C, Läuchli A M and Altman E 2007 Quench Dynamics and Nonequilibrium Phase Diagram of the Bose-Hubbard ModelPhys. Rev. Lett.98180601 URLhttps://link.aps.org/doi/10.1103/PhysRevLett.98.180601

    work page doi:10.1103/physrevlett.98.180601 2007

  49. [49]

    Adv.6(40) URL https://www.science.org/doi/10.1126/sciadv.aba9255

    Takasu Y, Yagami T, Asaka H, Fukushima Y, Nagao K, Goto S, Danshita I and Takahashi Y 2020 Energy redistribution and spatiotemporal evolution of correlations after a sudden quench of the Bose-Hubbard modelSci. Adv.6(40) URL https://www.science.org/doi/10.1126/sciadv.aba9255

    work page doi:10.1126/sciadv.aba9255 2020

  50. [50]

    Marković D and Čubrović M 2025 Superdiffusion, normal diffusion, and chaos in semiclassical bose-hubbard chainsPhys. Rev. E112(2) 024211 URLhttps://link.aps.org/doi/10.1103/4k8x-fd1j

    work page doi:10.1103/4k8x-fd1j 2025

  51. [51]

    Nandkishore R and Huse D A 2015 Many-body localization and thermalization in quantum statistical me- chanicsAnnu. Rev. Condens. Matter Phys.6(1) 15–38 URLhttp://www.annualreviews.org/doi/10.1146/ annurev-conmatphys-031214-014726

    work page 2015

  52. [52]

    Deutsch J M 2018 Eigenstate thermalization hypothesisRep. Prog. Phys.81(8) 082001 URLhttps://iopscience.iop. org/article/10.1088/1361-6633/aac9f1

    work page doi:10.1088/1361-6633/aac9f1 2018

  53. [53]

    Vidal G 2007 Classical simulation of infinite-size quantum lattice systems in one spatial dimensionPhys. Rev. Lett.98(7) 070201 URLhttps://doi.org/10.1103/PhysRevLett.98.070201

    work page doi:10.1103/physrevlett.98.070201 2007

  54. [54]

    Hastings M B 2009 Light-cone matrix productJ. Math. Phys.50(9) URLhttps://pubs.aip.org/jmp/article/50/9/ 095207/231330/Light-cone-matrix-product

    work page 2009

  55. [55]

    2010.Log-Gases and Random Matrices (LMS-34)

    Fishman M, White S R and Stoudenmire E M 2022 The ITensor Software Library for Tensor Network CalculationsSciPost Phys. Codebases4 URLhttps://scipost.org/10.21468/SciPostPhysCodeb.4

    work page doi:10.21468/scipostphyscodeb.4 2022

  56. [56]

    Codebase release 0.3 for ITensor,

    Fishman M, White S R and Stoudenmire E M 2022 Codebase release 0.3 for ITensorSciPost Phys. Codebases4–r0.3 URL https://scipost.org/10.21468/SciPostPhysCodeb.4-r0.3

    work page doi:10.21468/scipostphyscodeb.4-r0.3 2022

  57. [57]

    Natu S S and Mueller E J 2013 Dynamics of correlations in shallow optical latticesPhys. Rev. A87(6) 063616 URL https://link.aps.org/doi/10.1103/PhysRevA.87.063616

    work page doi:10.1103/physreva.87.063616 2013