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arxiv: 2606.30762 · v1 · pith:OFTCUB5Inew · submitted 2026-06-29 · ❄️ cond-mat.stat-mech · quant-ph

Athermality of generalized Gibbs ensembles

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

Pith reviewed 2026-07-01 01:49 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords athermalitygeneralized Gibbs ensemblequantum quenchintegrable systemcriticalityrelative entropythermalization
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The pith

Generalized Gibbs ensembles often have anomalously small athermality when the post-quench Hamiltonian is critical.

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

The paper computes the athermality of GGEs, defined as relative entropy to the closest thermal state, for quenches in several integrable models. It finds that this quantity is often anomalously small when the final Hamiltonian has a critical ground state, even though the energy density is finite, and proves that athermality always develops a singularity at that critical point coming from the GGE entropy. A reader would care because the result gives a concrete number for how far these long-time states depart from ordinary thermal equilibrium and shows an unexpected link between integrability and ground-state criticality.

Core claim

In integrable quantum systems after a quench, the GGE relaxes to a state whose athermality, measured by relative entropy distance to the nearest thermal state, becomes anomalously small whenever the post-quench Hamiltonian is critical in its ground state; the same quantity develops a singularity at criticality that is inherited directly from the entropy of the GGE. This holds in the XY chain, Lieb-Liniger gas, XXZ chain, and harmonic chain.

What carries the argument

Athermality, defined as the relative entropy between the GGE and the closest thermal state.

If this is right

  • Athermality drops below the scale set by finite energy density in critical cases.
  • The singularity at criticality appears systematically across free and interacting models.
  • The effect is inherited from the GGE entropy rather than from the choice of reference thermal state.

Where Pith is reading between the lines

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

  • The same singularity might appear in other distance measures between GGEs and thermal states.
  • Numerical or experimental access to the GGE entropy could be used to locate the critical point without ground-state calculations.
  • The result suggests that criticality can mask the non-thermal character of conserved quantities in quench protocols.

Load-bearing premise

The relative entropy to the closest thermal state is the appropriate quantitative measure of how far a GGE departs from ordinary thermal equilibrium.

What would settle it

A direct computation of the GGE entropy or relative entropy in the XY chain showing no singularity as the anisotropy parameter is tuned through the critical value.

Figures

Figures reproduced from arXiv: 2606.30762 by Bruno Bertini, Katja Klobas, Pasquale Calabrese, Riccardo Senese.

Figure 1
Figure 1. Figure 1: T = 0 quantum phase diagram of the XY chain in a transverse field with Hamiltonian HXY(γ, h) from Eq. (19). See text for description of (i) Ising critical line (blue dashed) and (ii) XX critical line (red). (i) Each point on the line (γ ̸= 0, h = 1) is a quantum critical point in the Ising universality class, separating the ferromagnetic ordered phase of h < 1 (order along ˆx for γ > 0 and along ˆy for γ <… view at source ↗
Figure 2
Figure 2. Figure 2: Density plots for the density of athermality [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows cuts where γ = γ0 and hence the quench is entirely driven by h0 → h [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: α as a function of h, for fixed γ ̸= γ0 (h- and γ-driven quench) and h0. The blue dashed lines indicates the critical Ising point h = 1. The inset in (a) shows the same data in log-scale. around the critical line (γ > 0, h = 1) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: α as a function of γ, for fixed h ̸= h0 (h- and γ-driven quench) and γ0. The blue dashed lines indicates the critical XX point γ = 0. where sgn(x) is the sign function. Therefore nGGE(k) is subject to a discontinuous jump from 0 to 1 whenever h or h0 cross the critical points h = 1 or h0 = 1. Exactly on the critical Ising line we find lim k→0 h lim h→1 nGGE(k) i = 1 2 for h0 ̸= 1, γ ̸= 0 , (31) and analogo… view at source ↗
Figure 6
Figure 6. Figure 6: Log-scale comparison between the occupation functions [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) for γ0 > 0 and h0 < 1) emerges from the interplay (i.e. the difference) between sYY[ϱβ∗ ] and sYY[ϱGGE] (cf. same parameter region in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Densities of athermality α (a), GGE entropy sYY[ϱGGE] (b) and GE entropy sYY[ϱβ∗ ] (c) as a function of ∆ in the quench from the N´eel state (Eq. (38) for θ = 0). The dashed blue lines in the interval 0 < ∆ < 1 are just guides to the eye that connect results for the anisotropies in Eq. (36). The vertical dashed line indicates the critical point ∆ = 1. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Log-scale plots for the GE (dashed lines) and GGE (continuous lines) root densities [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Density of athermality α (a), GGE and GE entropy densities (b) and temperatures 1/β∗ (c) as a function of ∆ in the quench from the tilted-N´eel state (38) with θ = π/5. In (c) we also plot for comparison the temperature of the N´eel quench (θ = 0). The continuous lines are just guides to the eye. angles θ ̸= 0 for the initial states (38). From [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Density of athermality α associated with the BEC-to-repulsive-LL quench. (a) α as a function of the postquench interaction c > 0 at fixed particle density d. (b) α as a function of d at fixed postquench c. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison between ϱβ∗,µ∗ (λ) (continuous line) and ϱGGE(λ) (dashed line) in the repulsive regime for different values of c > 0 and density of particles d. 6.2 Results In the context of the BEC to LL quench, we can vary the postquench Hamiltonian by tuning the interaction c at fixed |ψ(0)⟩, or change the initial state at fixed postquench c by varying the initial density of particles d. For c > 0, we plot … view at source ↗
Figure 13
Figure 13. Figure 13: GGE entropy density for repulsive c > 0 (s[ϱGGE]) and attractive c < 0 (s[ϱGGE]) interactions in the LL quench from the BEC state (42) with density of particles d = 1. The point c = 0 corresponds a trivial zero due to absence of the quench. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Densities of athermality α (a), GGE and GE entropy (b) and energy e (c) as a function of m in the quench from the GS with m0 = 1. The vertical dashed lines indicate the position of the trivial zero m = m0. The inset in (a) shows the same data in log-scale. With the knowledge of nGGE(k) and nβ∗ we can use Eq. (48) to compute the GE and GGE entropy densities and hence the athermality α. In [PITH_FULL_IMAGE… view at source ↗
Figure 15
Figure 15. Figure 15: GGE and thermal occupation functions nGGE(k) and nβ∗ (k) for a few values of the postquench mass m in the quench from the GS with m0 = 1. (a) m = 0.02 close to criticality; (b) m = 1.37, corresponding to the non-trivial minimum in the athermality of [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Density plots for e, sYY[ϱβ∗ ] and sYY[ϱGGE], associated with the athermality plots of [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Log-scale comparison between the occupation functions [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Density plots and cuts for e, sYY[ϱβ∗ ] and sYY[ϱGGE], associated with the athermality plots of [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Log-scale plots for the GE (dashed lines) and GGE (continuous lines) root densities [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Log-scale plots for the GE (dashed lines) and GGE (continuous lines) hole densities [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Log-scale plots for the GE (dashed lines) and GGE (continuous lines) filling functions [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Thermal and GGE entropy densities for the BEC to repulsive ( [PITH_FULL_IMAGE:figures/full_fig_p034_22.png] view at source ↗
read the original abstract

Integrable quantum systems evolving from non-equilibrium initial states do not thermalize to conventional Gibbs ensembles (GE). Instead, at long times they relax to generalized Gibbs ensembles (GGEs), which incorporate the full set of local and quasi-local conserved quantities. While GGEs have been extensively studied in the literature, a quantitative characterization of how different they are from ordinary GEs is still lacking. In this work, we address this question by employing the concept of athermality, which we define within quantum resource theory as the relative entropy between a given state and the closest thermal state. We compute the athermality for several quantum quenches in paradigmatic integrable models, including the free XY spin chain, the interacting Lieb-Liniger model, the XXZ spin chain, and the harmonic chain. We find that often the athermality becomes anomalously small when the post-quench Hamiltonian is critical in its ground state, despite probing physics at a finite energy density. We also prove that it systematically develops a singularity at criticality, which is inherited from the entropy of the GGE.

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 defines athermality of a generalized Gibbs ensemble (GGE) as the relative entropy to the closest thermal state and computes this quantity for several integrable models after quantum quenches (free XY chain, Lieb-Liniger, XXZ chain, harmonic chain). It reports that athermality is often anomalously small when the post-quench Hamiltonian is critical in its ground state (despite finite energy density) and proves that athermality develops a singularity at criticality inherited from the GGE entropy.

Significance. If the computations and proof hold, the work supplies a concrete, resource-theoretic metric for the departure of GGEs from ordinary thermal states. The multi-model numerical evidence together with the analytic demonstration of a singularity at criticality (even at finite energy density) constitutes a substantive advance; the explicit inheritance from GGE entropy is a strength that distinguishes the result from purely numerical observations.

minor comments (3)
  1. The abstract and introduction should briefly indicate whether the singularity proof explicitly verifies that the maximization over thermal states does not cancel the non-analyticity inherited from S(ρ_GGE); while the claim is made, a one-sentence pointer to the relevant step would aid readability.
  2. Figure captions for the numerical results on the XY, Lieb-Liniger, XXZ and harmonic chains should state the system sizes, quench parameters, and fitting procedures used to identify the anomalously small athermality regime.
  3. A short paragraph discussing why the relative-entropy distance (rather than, e.g., trace distance or fidelity) is the natural choice within the resource-theory framework would clarify the scope of the conclusions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript, including the recognition of the substantive advance provided by the multi-model evidence and the analytic proof of the singularity at criticality. The referee recommends minor revision, but no specific major comments were raised in the report. We will incorporate minor improvements to presentation and clarity in the revised version.

Circularity Check

0 steps flagged

No circularity detected; athermality and singularity derived from explicit computation

full rationale

The paper defines athermality as the relative entropy D(ρ_GGE || σ_thermal) minimized over thermal states and computes this quantity directly for quenches in the XY, Lieb-Liniger, XXZ, and harmonic models. The reported anomalous smallness at criticality and the singularity are obtained from these evaluations, with the proof that the non-analyticity is inherited from S(ρ_GGE) following from the explicit form of the minimized expression. No steps reduce by construction to fitted parameters, self-citations, or ansatzes; the central claims rest on model-specific calculations of the defined distance rather than on any renaming or self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of relative entropy and on the established fact that integrable systems relax to GGEs; no free parameters, ad-hoc axioms or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Integrable quantum systems evolving from non-equilibrium initial states relax to generalized Gibbs ensembles at long times.
    Stated in the first sentence of the abstract as background.
  • domain assumption Athermality is defined as the relative entropy between a given state and the closest thermal state.
    Introduced in the abstract as the quantitative measure employed.

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discussion (0)

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Works this paper leans on

211 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    J. M. Deutsch,Quantum statistical mechanics in a closed system, Phys. Rev. A43, 2046 (1991)

    2046

  2. [2]

    Srednicki,Chaos and quantum thermalization, Phys

    M. Srednicki,Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994)

    1994

  3. [3]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore,Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys.83, 863 (2011)

    2011

  4. [4]

    Eisert, M

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

    2015

  5. [5]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,From quantum chaos and eigenstate thermal- ization to statistical mechanics and thermodynamics, Adv. Phys.65, 239 (2016)

    2016

  6. [6]

    Calabrese, F

    P. Calabrese, F. H. Essler, and G. Mussardo,Introduction to ‘Quantum integrability in out of equi- librium systems’, J. Stat. Mech. 064001 (2016)

    2016

  7. [7]

    Bastianello, B

    A. Bastianello, B. Bertini, B. Doyon, and R. Vasseur,Introduction to the Special Issue on Emergent Hydrodynamics in Integrable Many-Body Systems, J. Stat. Mech. 014001 (2022)

    2022

  8. [8]

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

    2016

  9. [9]

    F. H. L. Essler and M. Fagotti,Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech. 064002 (2016)

    2016

  10. [10]

    Alba and P

    V. Alba and P. Calabrese,Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Scien.114, 7947 (2017)

    2017

  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). 34

    2020

  12. [12]

    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,Quantum Inverse Scattering Method and Cor- relation Functions, 1st ed. (Cambridge University Press, Aug. 1993)

    1993

  13. [13]

    Moeckel and S

    M. Moeckel and S. Kehrein,Interaction Quench in the Hubbard Model, Phys. Rev. Lett.100, 175702 (2008)

    2008

  14. [14]

    Kollar, F

    M. Kollar, F. A. Wolf, and M. Eckstein,Generalized Gibbs ensemble prediction of prethermalization plateaus and their relation to nonthermal steady states in integrable systems, Phys. Rev. B84, 054304 (2011)

    2011

  15. [15]

    Marcuzzi, J

    M. Marcuzzi, J. Marino, A. Gambassi, and A. Silva,Prethermalization in a Nonintegrable Quantum Spin Chain after a Quench, Phys. Rev. Lett.111, 197203 (2013)

    2013

  16. [16]

    F. H. L. Essler, S. Kehrein, S. R. Manmana, and N. J. Robinson,Quench dynamics in a model with tuneable integrability breaking, Phys. Rev. B89, 165104 (2014)

    2014

  17. [17]

    Bertini, F

    B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson,Prethermalization and Thermalization in Models with Weak Integrability Breaking, Phys. Rev. Lett.115, 180601 (2015)

    2015

  18. [18]

    G. P. Brandino, J.-S. Caux, and R. M. Konik,Glimmers of a Quantum KAM Theorem: Insights from Quantum Quenches in One-Dimensional Bose Gases, Phys. Rev. X5, 041043 (2015)

    2015

  19. [19]

    Bertini and M

    B. Bertini and M. Fagotti,Pre-relaxation in weakly interacting models, J. Stat. Mech. P07012 (2015)

    2015

  20. [20]

    Babadi, E

    M. Babadi, E. Demler, and M. Knap,Far-from-Equilibrium Field Theory of Many-Body Quantum Spin Systems: Prethermalization and Relaxation of Spin Spiral States in Three Dimensions, Phys. Rev. X5, 041005 (2015)

    2015

  21. [21]

    Bertini, F

    B. Bertini, F. H. L. Essler, S. Groha, and N. J. Robinson,Thermalization and light cones in a model with weak integrability breaking, Phys. Rev. B94, 245117 (2016)

    2016

  22. [22]

    Universal prethermalization dynamics of entanglement entropies after a global quench

    M. Fagotti and M. Collura,Universal prethermalization dynamics of entanglement entropies after a global quench, arXiv:1507.02678 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  23. [23]

    Alba and M

    V. Alba and M. Fagotti,Prethermalization at Low Temperature: The Scent of Long-Range Order, Phys. Rev. Lett.119, 010601 (2017)

    2017

  24. [24]

    Mallayya, M

    K. Mallayya, M. Rigol, and W. De Roeck,Prethermalization and Thermalization in Isolated Quantum Systems, Phys. Rev. X9, 021027 (2019)

    2019

  25. [25]

    Bertini and P

    B. Bertini and P. Calabrese,Prethermalization and thermalization in entanglement dynamics, Phys. Rev. B102, 094303 (2020)

    2020

  26. [26]

    Durnin, M

    J. Durnin, M. J. Bhaseen, and B. Doyon,Nonequilibrium Dynamics and Weakly Broken Integrability, Phys. Rev. Lett.127, 130601 (2021)

    2021

  27. [27]

    Lopez-Piqueres, B

    J. Lopez-Piqueres, B. Ware, S. Gopalakrishnan, and R. Vasseur,Hydrodynamics of nonintegrable systems from a relaxation-time approximation, Phys. Rev. B103, L060302 (2021)

    2021

  28. [28]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss,A quantum Newton’s cradle, Nature440, 900 (2006)

    2006

  29. [29]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer,Relaxation and Prethermalization in an Isolated Quantum System, Science337, 1318 (2012)

    2012

  30. [30]

    Langen, S

    T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer, and J. Schmiedmayer,Experimental observation of a generalized Gibbs ensemble, Science 348, 207 (2015)

    2015

  31. [31]

    Vidmar and M

    L. Vidmar and M. Rigol,Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 064007 (2016)

    2016

  32. [32]

    Ilievski, M

    E. Ilievski, M. Medenjak, T. Prosen, and L. Zadnik,Quasilocal charges in integrable lattice systems, J. Stat. Mech. 064008 (2016)

    2016

  33. [33]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii,Relaxation in a Completely Integrable Many- Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons, Phys. Rev. Lett.98, 050405 (2007)

    2007

  34. [34]

    Caux and R

    J.-S. Caux and R. M. Konik,Constructing the Generalized Gibbs Ensemble after a Quantum Quench, Phys. Rev. Lett.109, 175301 (2012)

    2012

  35. [35]

    Mossel and J.-S

    J. Mossel and J.-S. Caux,Generalized TBA and generalized Gibbs, J. Phys. A45, 255001 (2012)

    2012

  36. [36]

    Doyon,Thermalization and Pseudolocality in Extended Quantum Systems, Commun

    B. Doyon,Thermalization and Pseudolocality in Extended Quantum Systems, Commun. Math. Phys. 351, 155 (2017)

    2017

  37. [37]

    Dymarsky and K

    A. Dymarsky and K. Pavlenko,Generalized Gibbs Ensemble of 2d CFTs at large central charge in the thermodynamic limit, JHEP 01 (2019) 098. 35

    2019

  38. [38]

    Ilievski and E

    E. Ilievski and E. Quinn,The equilibrium landscape of the Heisenberg spin chain, SciPost Phys.7, 033 (2019)

    2019

  39. [39]

    L. F. Santos, A. Polkovnikov, and M. Rigol,Weak and strong typicality in quantum systems, Phys. Rev. E86, 010102(R) (2012)

    2012

  40. [40]

    Collura, M

    M. Collura, M. Kormos, and P. Calabrese,Stationary entanglement entropies following an interaction quench in 1D Bose gas, J. Stat. Mech. P01009 (2014)

    2014

  41. [41]

    Kormos, L

    M. Kormos, L. Bucciantini, and P. Calabrese,Stationary entropies after a quench from excited states in the Ising chain, Europhys. Lett.107, 40002 (2014)

    2014

  42. [42]

    Gurarie,Global large time dynamics and the generalized Gibbs ensemble, J

    V. Gurarie,Global large time dynamics and the generalized Gibbs ensemble, J. Stat. Mech. P02014 (2013)

    2013

  43. [43]

    Fagotti,Finite-size corrections versus relaxation after a sudden quench, Phys

    M. Fagotti,Finite-size corrections versus relaxation after a sudden quench, Phys. Rev. B87, 165106 (2013)

    2013

  44. [44]

    D´ ora,Escort distribution function of work done and diagonal entropies in quenched Luttinger liquids, Phys

    B. D´ ora,Escort distribution function of work done and diagonal entropies in quenched Luttinger liquids, Phys. Rev. B90, 245132 (2014)

    2014

  45. [45]

    Piroli, E

    L. Piroli, E. Vernier, P. Calabrese, and M. Rigol,Correlations and diagonal entropy after quantum quenches in XXZ chains, Phys. Rev. B95, 054308 (2017)

    2017

  46. [46]

    Alba and P

    V. Alba and P. Calabrese,Quench action and R´ enyi entropies in integrable systems, Phys. Rev. B 96, 115421 (2017)

    2017

  47. [47]

    Bertini, E

    B. Bertini, E. Tartaglia, and P. Calabrese,Entanglement and diagonal entropies after a quench with no pair structure, J. Stat. Mech. 063104 (2018)

    2018

  48. [48]

    Kormos, Y.-Z

    M. Kormos, Y.-Z. Chou, and A. Imambekov,Exact Three-Body Local Correlations for Excited States of the 1D Bose Gas, Phys. Rev. Lett.107, 230405 (2011)

    2011

  49. [49]

    Pozsgay,Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain, J

    B. Pozsgay,Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain, J. Stat. Mech. P11017 (2011)

    2011

  50. [50]

    Pozsgay,Mean values of local operators in highly excited Bethe states, J

    B. Pozsgay,Mean values of local operators in highly excited Bethe states, J. Stat. Mech. P01011 (2011)

    2011

  51. [51]

    Negro and F

    S. Negro and F. Smirnov,On one-point functions for sinh-Gordon model at finite temperature, Nucl. Phys. B875, 166 (2013)

    2013

  52. [52]

    Mesty´ an and B

    M. Mesty´ an and B. Pozsgay,Short distance correlators in the XXZ spin chain for arbitrary string distributions, J. Stat. Mech. P09020 (2014)

    2014

  53. [53]

    Bertini, L

    B. Bertini, L. Piroli, and P. Calabrese,Quantum quenches in the sinh-Gordon model: steady state and one-point correlation functions, J. Stat. Mech. 063102 (2016)

    2016

  54. [54]

    Pozsgay,Excited state correlations of the finite Heisenberg chain, J

    B. Pozsgay,Excited state correlations of the finite Heisenberg chain, J. Phys. A50, 074006 (2017)

    2017

  55. [55]

    Bastianello, L

    A. Bastianello, L. Piroli, and P. Calabrese,Exact Local Correlations and Full Counting Statistics for Arbitrary States of the One-Dimensional Interacting Bose Gas, Phys. Rev. Lett.120, 190601 (2018)

    2018

  56. [56]

    Bastianello and L

    A. Bastianello and L. Piroli,From the sinh-Gordon field theory to the one-dimensional Bose gas: exact local correlations and full counting statistics, J. Stat. Mech. 113104 (2018)

    2018

  57. [57]

    Calabrese, F

    P. Calabrese, F. H. L. Essler, and M. Fagotti,Quantum quenches in the transverse field Ising chain: II. Stationary state properties, J. Stat. Mech. P07022 (2012)

    2012

  58. [58]

    K. K. Kozlowski,On the thermodynamic limit of form factor expansions of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain, J. Math. Phys.59, 091408 (2018)

    2018

  59. [59]

    Granet, M

    E. Granet, M. Fagotti, and F. H. L. Essler,Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions, SciPost Phys.9, 033 (2020)

    2020

  60. [60]

    Granet,Low-density limit of dynamical correlations in the Lieb–Liniger model, J

    E. Granet,Low-density limit of dynamical correlations in the Lieb–Liniger model, J. Phys. A54, 154001 (2021)

    2021

  61. [61]

    Granet and F

    E. Granet and F. H. L. Essler,A systematic1/c-expansion of form factor sums for dynamical corre- lations in the Lieb-Liniger model, SciPost Phys.9, 082 (2020)

    2020

  62. [62]

    De Nardis, B

    J. De Nardis, B. Doyon, M. Medenjak, and M. Panfil,Correlation functions and transport coefficients in generalised hydrodynamics, J. Stat. Mech. 014002 (2022)

    2022

  63. [63]

    Doyon, G

    B. Doyon, G. Perfetto, T. Sasamoto, and T. Yoshimura,Emergence of Hydrodynamic Spatial Long- Range Correlations in Nonequilibrium Many-Body Systems, Phys. Rev. Lett.131, 027101 (2023)

    2023

  64. [64]

    Rossini, S

    D. Rossini, S. Suzuki, G. Mussardo, G. E. Santoro, and A. Silva,Long time dynamics following a quench in an integrable quantum spin chain: Local versus nonlocal operators and effective thermal behavior, Phys. Rev. B82, 144302 (2010). 36

    2010

  65. [65]

    Rossini, A

    D. Rossini, A. Silva, G. Mussardo, and G. E. Santoro,Effective Thermal Dynamics Following a Quantum Quench in a Spin Chain, Phys. Rev. Lett.102, 127204 (2009)

    2009

  66. [66]

    Fagotti and F

    M. Fagotti and F. H. L. Essler,Reduced density matrix after a quantum quench, Phys. Rev. B87, 245107 (2013)

    2013

  67. [67]

    Fagotti, M

    M. Fagotti, M. Collura, F. H. L. Essler, and P. Calabrese,Relaxation after quantum quenches in the spin-1/2 Heisenberg XXZ chain, Phys. Rev. B89, 125101 (2014)

    2014

  68. [68]

    Pozsgay,The generalized Gibbs ensemble for Heisenberg spin chains, J

    B. Pozsgay,The generalized Gibbs ensemble for Heisenberg spin chains, J. Stat. Mech. P07003 (2013)

    2013

  69. [69]

    F. G. S. L. Brand˜ ao, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens,Resource Theory of Quantum States Out of Thermal Equilibrium, Phys. Rev. Lett.111, 250404 (2013)

    2013

  70. [70]

    Lostaglio, D

    M. Lostaglio, D. Jennings, and T. Rudolph,Description of quantum coherence in thermodynamic processes requires constraints beyond free energy, Nat. Commun.6, 6383 (2015)

    2015

  71. [71]

    Brand˜ ao, M

    F. Brand˜ ao, M. Horodecki, N. Ng, J. Oppenheim, and S. Wehner,The second laws of quantum thermodynamics, Proc. Nat. Acad. Scien.112, 3275 (2015)

    2015

  72. [72]

    G. Gour, M. P. M¨ uller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern,The resource theory of informational nonequilibrium in thermodynamics, Phys. Rep.583, 1 (2015)

    2015

  73. [73]

    Coecke, T

    B. Coecke, T. Fritz, and R. W. Spekkens,A mathematical theory of resources, Inform. Comput.250, 59 (2016)

    2016

  74. [75]

    Sels and M

    D. Sels and M. Wouters,Stationary ensemble approximations of dynamic quantum states: Optimizing the generalized Gibbs ensemble, Phys. Rev. E92, 022123 (2015)

    2015

  75. [76]

    Sachdev,Quantum Phase Transitions, 2nd ed

    S. Sachdev,Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Apr. 2011)

    2011

  76. [77]

    Araki,Gibbs states of a one dimensional quantum lattice, Commun

    H. Araki,Gibbs states of a one dimensional quantum lattice, Commun. Math. Phys.14, 120 (1969)

    1969

  77. [78]

    Porta, N

    S. Porta, N. T. Ziani, D. M. Kennes, F. M. Gambetta, M. Sassetti, and F. Cavaliere,Effective metal- insulator nonequilibrium quantum phase transition in the Su-Schrieffer-Heeger model, Phys. Rev. B 98, 214306 (2018)

    2018

  78. [79]

    Fagotti and P

    M. Fagotti and P. Calabrese,Evolution of entanglement entropy following a quantum quench: Analytic results for the XY chain in a transverse magnetic field, Phys. Rev. A78, 010306 (2008)

    2008

  79. [80]

    Russomanno, G

    A. Russomanno, G. E. Santoro, and R. Fazio,Entanglement entropy in a periodically driven Ising chain, J. Stat. Mech. 073101 (2016)

    2016

  80. [81]

    S. Paul, P. Titum, and M. Maghrebi,Hidden quantum criticality and entanglement in quench dynam- ics, Phys. Rev. Res.6, L032003 (2024)

    2024

Showing first 80 references.