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arxiv: 2606.20445 · v1 · pith:XLBJT3K3new · submitted 2026-06-18 · ❄️ cond-mat.stat-mech · hep-th· quant-ph

Space-time duality approach to (inhomogeneous) integrable quenches

Riccardo Travaglino , Pasquale Calabrese , Katja Klobas , Bruno Bertini This is my paper

Pith reviewed 2026-06-26 15:30 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-thquant-ph
keywords space-time dualityintegrable quenchesentanglement growthcharge fluctuationsnon-equilibrium dynamicsquantum many-body systemsinhomogeneous quenches
0
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The pith

The space-time duality approach resolves its ambiguity and now applies to inhomogeneous quenches and non-symmetric initial states.

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

The paper establishes that an intrinsic ambiguity previously restricting the space-time duality approach to homogeneous quenches and symmetric initial states can be resolved from first principles. This resolution yields closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches in integrable systems. A sympathetic reader would care because the duality exchanges space and time roles, allowing equilibrium statistical mechanics tools to address a broader range of non-equilibrium dynamics. The work benchmarks these predictions against exact solutions and simulations while showing consistency with the quasiparticle picture for entanglement entropy.

Core claim

By extending the space-time duality approach unambiguously from first principles, the authors derive closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches, including inhomogeneous cases and non-symmetric initial states, with benchmarks against the Rule 54 cellular automaton and the XXZ chain.

What carries the argument

Space-time duality approach (SDA), which relates non-equilibrium quantities such as entanglement growth rates and charge fluctuations to equilibrium properties by exchanging the roles of space and time.

If this is right

  • Closed-form expressions become available for entanglement growth after general integrable quenches.
  • Charge fluctuations can now be predicted for initial states lacking spatial symmetry.
  • The framework covers inhomogeneous quenches without additional restrictions.
  • Specialization to entanglement entropy reproduces the quasiparticle picture results.

Where Pith is reading between the lines

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

  • The approach may enable direct comparison with experiments that inherently include spatial inhomogeneities.
  • If the first-principles resolution generalizes, similar dualities could apply to a wider class of non-integrable models.
  • This could connect non-equilibrium dynamics more tightly to thermodynamic fluctuation theorems.

Load-bearing premise

The space-time duality extends unambiguously to inhomogeneous quenches and non-symmetric initial states from first principles without new inconsistencies or adjustments.

What would settle it

A mismatch between the derived closed-form predictions and either the exact analytical solution of the Rule 54 quantum cellular automaton or TEBD simulations of the XXZ chain for an inhomogeneous quench starting from a non-symmetric state.

Figures

Figures reproduced from arXiv: 2606.20445 by Bruno Bertini, Katja Klobas, Pasquale Calabrese, Riccardo Travaglino.

Figure 1
Figure 1. Figure 1: (a): A tensor-network diagram of a local expectation value after a quantum quench under circuit evolution. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Instantaneous slope of the Rényi entropies of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Instantanoeus slope of the FCS of the left half [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sketch of a possible reparametrisation f(µ) that is non-monotonic but injective. but not monotonic, such as a piecewise function given in [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Instantaneous slope of Rényi entropies in two quenches in the XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Instantaneous slope of Rényi entropies in two quenches in the XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Instantaneous slope of Rényi entropies in two quenches in the XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Dynamics of the FCS in the XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dynamics of the FCS in the XXZ chain with [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
read the original abstract

Characterising the universal aspects of non-equilibrium quantum many-body dynamics is one of the key goals of this century's physics research. Progress, however, is hindered by the lack of general theoretical frameworks for studying interacting quantum matter far from equilibrium. A recent breakthrough has been the realization that several key non-equilibrium quantities, such as the rate of growth of entanglement or the fluctuations of conserved charges within finite subsystems, can be related to equilibrium properties through a space-time duality that effectively exchanges the roles of space and time. This observation effectively enables the study of non-equilibrium phenomena using tools and concepts borrowed from equilibrium statistical mechanics and thermodynamics. A first proof of principle of this framework, dubbed space-time duality approach (SDA), was provided by interacting integrable systems, where thermodynamic properties can often be characterized exactly, while dynamical quantities typically remain beyond analytical reach. Subsequent developments, however, revealed that the SDA suffered from an intrinsic ambiguity, restricting its applicability to homogeneous quenches and to charge fluctuations arising from symmetric initial states. Here we resolve this ambiguity from first principles and derive closed-form predictions for entanglement growth and charge fluctuations after general quantum quenches. We benchmark our results against the exact analytical solution of the Rule 54 quantum cellular automaton and extensive TEBD simulations of the XXZ chain. Moreover we show that, when specialised to the entanglement entropy, our framework naturally reproduces the predictions of the quasiparticle 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

0 major / 3 minor

Summary. The manuscript extends the space-time duality approach (SDA) to inhomogeneous integrable quenches and general initial states. It claims to resolve an intrinsic ambiguity in the prior SDA framework from first principles, yielding closed-form predictions for entanglement growth and charge fluctuations after arbitrary quantum quenches. The results are benchmarked against the exact analytical solution of the Rule 54 quantum cellular automaton and TEBD simulations of the XXZ chain; when specialized to entanglement entropy, the framework reproduces the quasiparticle picture.

Significance. If the first-principles resolution holds, the work substantially broadens the SDA's applicability beyond homogeneous quenches and symmetric states, allowing equilibrium thermodynamic tools to address a wider range of non-equilibrium observables in integrable systems. The explicit benchmarks against an exact solution and numerical data, together with recovery of the quasiparticle picture, constitute concrete validation. The absence of free parameters or ad-hoc adjustments in the central construction is a notable strength.

minor comments (3)
  1. The abstract and introduction refer to 'closed-form predictions' and 'first-principles derivation'; the main text should include an explicit statement of the steps that remove the prior ambiguity (e.g., the choice of contour or regularization) so that readers can verify the resolution without external references.
  2. Figure captions and the benchmark sections should report quantitative error measures (e.g., relative deviation from Rule 54 or TEBD data) rather than qualitative agreement statements.
  3. Notation for the space-time duality map and the resulting effective partition functions should be introduced once with a clear table or equation list to avoid repeated redefinitions across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript, the detailed summary, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The paper presents an extension of the space-time duality approach resolved from first principles, with explicit benchmarks to independent exact solutions (Rule 54 cellular automaton) and numerical TEBD simulations of the XXZ chain. The reproduction of the quasiparticle picture for entanglement entropy is shown as a consistency check rather than a definitional input. No quoted equation reduces a prediction to a fitted parameter or prior self-citation by construction; the central claims remain externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, invented entities, or ad-hoc axioms are described. The framework rests on the pre-existing space-time duality concept.

axioms (1)
  • domain assumption Space-time duality exchanges roles of space and time to relate non-equilibrium quantities to equilibrium properties in integrable systems
    Invoked as the foundation of the SDA framework throughout the abstract.

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Reference graph

Works this paper leans on

86 extracted references

  1. [1]

    Eisert, M

    J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many- body systems out of equilibrium, Nat. Phys.11, 124 (2015)

    2015

  2. [2]

    Calabrese, F

    P. Calabrese, F. H. Essler, and G. Mussardo, Introduc- tion to ‘Quantum integrability in out of equilibrium sys- tems’, J. Stat. Mech.: Theory Exp.2016, 064001

    2016

  3. [3]

    Serbyn, D

    M. Serbyn, D. A. Abanin, and Z. Papić, Quantum many- body scars and weak breaking of ergodicity, Nat. Phys. 17, 675 (2021)

    2021

  4. [4]

    Bastianello, B

    A. Bastianello, B. Bertini, B. Doyon, and R. Vasseur, Introduction to the special issue on emergent hydrody- namics in integrable many-body systems, J. Stat. Mech.: Theory Exp.2022, 014001

    2022

  5. [5]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    2019

  6. [6]

    Calabrese and J

    P. Calabrese and J. Cardy, Evolution of entanglement en- tropy in one-dimensional systems, J. Stat. Mech.: Theory Exp.2005, P04010

    2005

  7. [7]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X7, 031016 (2017)

    2017

  8. [8]

    S. H. Shenker and D. Stanford, Multiple shocks, J. High Energy Phys.12, 046

  9. [9]

    Kitaev, A simple model of quantum holography, in KITP Program: Entanglement in Strongly-Correlated Quantum Matter(2015)

    A. Kitaev, A simple model of quantum holography, in KITP Program: Entanglement in Strongly-Correlated Quantum Matter(2015)

    2015

  10. [10]

    Hosur, X.-L

    P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, J. High Energy Phys.2016(2), 004

    2016

  11. [11]

    Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Phys

    P. Calabrese, Entanglement spreading in non-equilibrium integrable systems, SciPost Phys. Lect. Notes, 20 (2020)

    2020

  12. [12]

    Zhou and A

    T. Zhou and A. Nahum, Entanglement membrane in chaotic many-body systems, Phys. Rev. X10, 031066 (2020)

    2020

  13. [13]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- induced phase transitions in the dynamics of entangle- ment, Phys. Rev. X9, 031009 (2019)

    2019

  14. [14]

    Y. Li, X. Chen, and M. P. A. Fisher, Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

    2019

  15. [15]

    Sacha and J

    K. Sacha and J. Zakrzewski, Time crystals: a review, Rep. Prog. Phys.81, 016401 (2017)

    2017

  16. [16]

    Khemani, R

    V. Khemani, R. Moessner, and S. L. Sondhi, A brief his- tory of time crystals, arXiv:1910.10745 (2019)

    arXiv 1910

  17. [17]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum2, 79 (2018)

    2018

  18. [18]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, C. Chin, B. DeMarco, S. E. Economou, M. A. Eriks- son, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Kollár, B. L. Lev, M. D. Lukin, R. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier- Smith, I. Siddiqi, R. Simmonds, M...

    2021

  19. [19]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854 (2008)

    2008

  20. [20]

    Carleo, F

    G. Carleo, F. Becca, M. Schiró, and M. Fabrizio, Lo- calization and glassy dynamics of many-body quantum systems, Sci. Rep.2, 243 (2012)

    2012

  21. [21]

    A. J. A. James and R. M. Konik, Quantum quenches in two spatial dimensions using chain array matrix product states, Phys. Rev. B92, 161111 (2015)

    2015

  22. [22]

    R. P. Feynman, Simulating physics with computers (cRc Press, 2018) pp. 133–153

    2018

  23. [23]

    Murciano, P

    S. Murciano, P. Calabrese, and R. M. Konik, Postquantum-quench growth of Rényi entropies in low- dimensionalcontinuumbosonicsystems,Phys.Rev.Lett. 129, 106802 (2022)

    2022

  24. [24]

    S.-H. Lin, M. P. Zaletel, and F. Pollmann, Efficient simu- lation of dynamics in two-dimensional quantum spin sys- tems with isometric tensor networks, Phys. Rev. B106, 245102 (2022)

    2022

  25. [25]

    Krinitsin, N

    W. Krinitsin, N. Tausendpfund, M. Heyl, M. Rizzi, and M. Schmitt, Time evolution of the quantum Ising model in two dimensions using tree tensor networks, Phys. Rev. B112, 134310 (2025)

    2025

  26. [26]

    Tindall, A

    J. Tindall, A. F. Mello, M. Fishman, E. M. Stoudenmire, and D. Sels, Dynamics of disordered quantum systems with two- and three-dimensional tensor networks, Science 392, 868 (2026)

    2026

  27. [27]

    Berges, S

    J. Berges, S. Borsányi, and C. Wetterich, Prethermaliza- tion, Phys. Rev. Lett.93, 142002 (2004)

    2004

  28. [28]

    Sotiriadis and J

    S. Sotiriadis and J. Cardy, Quantum quench in interact- ing field theory: A self-consistent approximation, Phys. Rev. B81, 134305 (2010)

    2010

  29. [29]

    Schiró and M

    M. Schiró and M. Fabrizio, Time-dependent mean field theory for quench dynamics in correlated electron sys- tems, Phys. Rev. Lett.105, 076401 (2010)

    2010

  30. [30]

    Gambassi and P

    A. Gambassi and P. Calabrese, Quantum quenches as classical critical films, Europhys. Lett.95, 66007 (2011). 8

    2011

  31. [31]

    Chiocchetta, M

    A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Short-time universal scaling in an isolated quantum sys- tem after a quench, Phys. Rev. B91, 220302(R) (2015)

    2015

  32. [32]

    H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys.86, 779 (2014)

    2014

  33. [33]

    Dalfovo, S

    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys.71, 463 (1999)

    1999

  34. [34]

    Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys

    G. Vidal, Efficient classical simulation of slightly entan- gled quantum computations, Phys. Rev. Lett.91, 147902 (2003)

    2003

  35. [35]

    A. J. Daley, C. Kollath, U. Schollwöck, and G. Vidal, Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces, J. Stat. Mech.: Theory Exp.2004, P04005

    2004

  36. [36]

    Phys.326, 96 (2011)

    U.Schollwöck,Thedensity-matrixrenormalizationgroup in the age of matrix product states, Ann. Phys.326, 96 (2011)

    2011

  37. [37]

    S. R. White and A. E. Feiguin, Real-time evolution us- ing the density matrix renormalization group, Phys. Rev. Lett.93, 076401 (2004)

    2004

  38. [38]

    Hauschild and F

    J. Hauschild and F. Pollmann, Efficient numerical sim- ulations with tensor networks: Tensor network Python (TeNPy), SciPost Phys. Lect. Notes, 5 (2018)

    2018

  39. [39]

    Fishman, S

    M. Fishman, S. R. White, and E. M. Stoudenmire, The iTensor software library for tensor network calculations, SciPost Phys. Codebases, 4 (2022)

    2022

  40. [40]

    Dubail, Entanglement scaling of operators: a confor- mal field theory approach, with a glimpse of simulability of long-time dynamics in1 + 1d, J

    J. Dubail, Entanglement scaling of operators: a confor- mal field theory approach, with a glimpse of simulability of long-time dynamics in1 + 1d, J. Phys. A50, 234001 (2017)

    2017

  41. [41]

    A. Rath, V. Vitale, S. Murciano, M. Votto, J. Dubail, R. Kueng, C. Branciard, P. Calabrese, and B. Vermer- sch, Entanglement barrier and its symmetry resolution: Theory and experimental observation, PRX Quantum4, 010318 (2023)

    2023

  42. [42]

    Bertini, K

    B. Bertini, K. Klobas, V. Alba, G. Lagnese, and P. Cal- abrese, Growth of Rényi entropies in interacting inte- grable models and the breakdown of the quasiparticle picture, Phys. Rev. X12, 031016 (2022)

    2022

  43. [43]

    Bertini, P

    B. Bertini, P. W. Claeys, and T. Prosen, Exactly solvable quantum many-body dynamics from space-time duality, Rev. Mod. Phys.98, 025001 (2026)

    2026

  44. [44]

    M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Matrix product states for dynamical simulation of infinite chains, Phys. Rev. Lett.102, 240603 (2009)

    2009

  45. [45]

    Müller-Hermes, J

    A. Müller-Hermes, J. Ignacio Cirac, and M. C. Bañuls, Tensor network techniques for the computation of dy- namical observables in one-dimensional quantum spin systems, New J. Phys.14, 075003 (2012)

    2012

  46. [46]

    Ippoliti, T

    M. Ippoliti, T. Rakovszky, and V. Khemani, Fractal, log- arithmic, and volume-law entangled nonthermal steady states via spacetime duality, Phys. Rev. X12, 011045 (2022)

    2022

  47. [47]

    A.Lerose, M.Sonner,andD.A.Abanin,Influencematrix approach to many-body Floquet dynamics, Phys. Rev. X 11, 021040 (2021)

    2021

  48. [48]

    Bertini, K

    B. Bertini, K. Klobas, and T.-C. Lu, Entanglement neg- ativity and mutual information after a quantum quench: Exact link from space-time duality, Phys. Rev. Lett.129, 140503 (2022)

    2022

  49. [49]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Exact spectral form factor in a minimal model of many-body quantum chaos, Phys. Rev. Lett.121, 264101 (2018)

    2018

  50. [50]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Exact correlation func- tions for dual-unitary lattice models in1 + 1dimensions, Phys. Rev. Lett.123, 210601 (2019)

    2019

  51. [51]

    C. N. Yang and C. P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta- function interaction, J. Mat. Phys.10, 1115 (1969)

    1969

  52. [52]

    Takahashi,Thermodynamics of One-Dimensional Solvable Models(Cambridge University Press, 1999)

    M. Takahashi,Thermodynamics of One-Dimensional Solvable Models(Cambridge University Press, 1999)

    1999

  53. [53]

    Caux and F

    J.-S. Caux and F. H. L. Essler, Time evolution of local observables after quenching to an integrable model, Phys. Rev. Lett.110, 257203 (2013)

    2013

  54. [54]

    Ilievski, J

    E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler, and T. Prosen, Complete generalized Gibbs en- sembles in an interacting theory, Phys. Rev. Lett.115, 157201 (2015)

    2015

  55. [55]

    Caux, The quench action, J

    J.-S. Caux, The quench action, J. Stat. Mech.: Theory Exp.2016, 064006

    2016

  56. [56]

    Alba and P

    V. Alba and P. Calabrese, Quench action and rényi en- tropies in integrable systems, Phys. Rev. B96, 115421 (2017)

    2017

  57. [57]

    O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X6, 041065 (2016)

    2016

  58. [58]

    Bertini, M

    B. Bertini, M. Collura, J. De Nardis, and M. Fagotti, Transport in out-of-equilibriumXXZchains: Exact profiles of charges and currents, Phys. Rev. Lett.117, 207201 (2016)

    2016

  59. [59]

    Doyon, Lecture notes on generalised hydrodynamics, SciPost Phys

    B. Doyon, Lecture notes on generalised hydrodynamics, SciPost Phys. Lect. Notes, 18 (2020)

    2020

  60. [60]

    V. Alba, B. Bertini, M. Fagotti, L. Piroli, and P. Rug- giero, Generalized-hydrodynamic approach to inhomoge- neousquenches: correlations, entanglementandquantum effects, J. Stat. Mech.: Theory Exp.2021, 114004

    2021

  61. [61]

    F. H. Essler, A short introduction to generalized hydro- dynamics, Physica A631, 127572 (2023)

    2023

  62. [62]

    Alba and P

    V. Alba and P. Calabrese, Entanglement and thermody- namics after a quantum quench in integrable systems, Proc. Natl. Acad. Sci. U.S.A.114, 7947 (2017)

    2017

  63. [63]

    Alba and P

    V. Alba and P. Calabrese, Entanglement dynamics after quantum quenches in generic integrable systems, SciPost Phys.4, 017 (2018)

    2018

  64. [64]

    Bertini, P

    B. Bertini, P. Calabrese, M. Collura, K. Klobas, and C. Rylands, Nonequilibrium full counting statistics and symmetry-resolved entanglement from space-time dual- ity, Phys. Rev. Lett.131, 140401 (2023)

    2023

  65. [65]

    Piroli, B

    L. Piroli, B. Pozsgay, and E. Vernier, What is an inte- grable quench?, Nucl. Phys. B925, 362 (2017)

    2017

  66. [66]

    Bertini, K

    B. Bertini, K. Klobas, M. Collura, P. Calabrese, and C. Rylands, Dynamics of charge fluctuations from asym- metric initial states, Phys. Rev. B109, 184312 (2024)

    2024

  67. [67]

    D. X. Horváth and C. Rylands, Full counting statistics of charge in quenched quantum gases, Phys. Rev. A109, 043302 (2024)

    2024

  68. [68]

    D. X. Horvath, B. Doyon, and P. Ruggiero, Full count- ing statistics after quantum quenches as hydrodynamic fluctuations, arXiv:2411.14406 (2025)

    arXiv 2025

  69. [69]

    D. X. Horvath, B. Doyon, and P. Ruggiero, A hydrody- namic theory for non-equilibrium full counting statistics in one-dimensional quantum systems, arXiv:2507.05954 (2025)

    arXiv 2025

  70. [70]

    B.Bertini, M.Fagotti, L.Piroli,andP.Calabrese,Entan- glement evolution and generalised hydrodynamics: non- interacting systems, J. Phys. A51, 39LT01 (2018). 9

    2018

  71. [71]

    K.Klobas, B.Bertini,andL.Piroli,Exactthermalization dynamics in the “Rule 54” quantum cellular automaton, Phys. Rev. Lett.126, 160602 (2021)

    2021

  72. [72]

    Klobas and B

    K. Klobas and B. Bertini, Entanglement dynamics in Rule 54: Exact results and quasiparticle picture, SciPost Phys.11, 107 (2021)

    2021

  73. [73]

    Klobas, Non-equilibrium dynamics of symmetry- resolved entanglement and entanglement asymmetry: Exact asymptotics in Rule 54, J

    K. Klobas, Non-equilibrium dynamics of symmetry- resolved entanglement and entanglement asymmetry: Exact asymptotics in Rule 54, J. Phys. A57, 505001 (2024)

    2024

  74. [74]

    V. Alba, B. Bertini, and M. Fagotti, Entanglement evo- lution and generalised hydrodynamics: Interacting inte- grable systems, SciPost Phys.7, 005 (2019)

    2019

  75. [75]

    V. E. Korepin, N. M. Bogoliubov, and A. G. Izer- gin,Quantum Inverse Scattering Method and Correla- tion Functions,CambridgeMonographsonMathematical Physics (Cambridge University Press, 1993)

    1993

  76. [76]

    A. B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models: Scaling three state Potts and Lee- Yang models, Nucl. Phys. B342, 695 (1990)

    1990

  77. [77]

    See the Supplemental Material for: (i) More details on the treatment of inhomogeneous quenches; (ii) Further numerical results

  78. [78]

    Orbach, Linear antiferromagnetic chain with anisotropic coupling, Phys

    R. Orbach, Linear antiferromagnetic chain with anisotropic coupling, Phys. Rev.112, 309 (1958)

    1958

  79. [79]

    Light- cone

    L. Bonnes, F. H. L. Essler, and A. M. Läuchli, “Light- cone” dynamics after quantum quenches in spin chains, Phys. Rev. Lett.113, 187203 (2014)

    2014

  80. [80]

    Klobas and B

    K. Klobas and B. Bertini, Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular au- tomaton Rule 54, SciPost Phys.11, 106 (2021)

    2021

Showing first 80 references.