pith. sign in

arxiv: 2508.17908 · v2 · pith:ASZUOI3Lnew · submitted 2025-08-25 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· quant-ph

Finite temperature single-particle Green's function in the Lieb-Liniger model

Riccardo Senese , Fabian H. L. Essler This is my paper

Pith reviewed 2026-05-21 22:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gasquant-ph
keywords Lieb-Liniger modelGreen's functionMonte Carlo samplingspectral functionfinite temperatureintegrable systemsbosonic gasesLehmann representation
0
0 comments X

The pith

A Monte Carlo algorithm computes the finite-temperature Green's function in the Lieb-Liniger model across all parameters.

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

The authors introduce a Monte Carlo sampling procedure to evaluate the Lehmann representation of the single-particle Green's function at finite temperature in the repulsive Lieb-Liniger model. This representation expresses the Green's function as a thermal sum over matrix elements connecting the system's energy eigenstates. The method thereby yields the spectral function for any temperature, any interaction strength, and any generalized Gibbs ensemble. A sympathetic reader would care because analytic solutions for dynamics in this integrable system are available only in special limits, so a controlled numerical route fills a practical gap. The paper confirms the approach by matching known exact results at infinite repulsion and for static correlators.

Core claim

The authors develop a Monte Carlo sampling algorithm to numerically evaluate the Lehmann representation for the finite temperature single-particle Green's function in the repulsive Lieb-Liniger model. This allows them to determine the spectral function in the full range of temperatures and interactions, as well as in generalized Gibbs ensembles. They test the results against known results for dynamics at infinite interaction strength and static correlators, finding excellent agreement.

What carries the argument

Monte Carlo sampling of the Lehmann representation, which sums thermal Boltzmann-weighted matrix elements of the field operator between eigenstates of the Lieb-Liniger Hamiltonian.

If this is right

  • The spectral function is now obtainable for arbitrary finite temperatures and interaction parameters in the model.
  • The same procedure applies directly to non-equilibrium states described by generalized Gibbs ensembles.
  • Static correlation functions are recovered as special cases of the Green's function computation.
  • Results in the infinite-repulsion limit reproduce known exact dynamical quantities for free fermions.

Where Pith is reading between the lines

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

  • The algorithm could be used to extract transport coefficients or response functions that are hard to obtain by other means in one-dimensional quantum gases.
  • Similar Monte Carlo sampling of Lehmann representations might be adapted to related integrable models such as the XXZ spin chain.
  • Direct comparison of the computed spectral functions with data from ultracold-atom experiments at finite temperature becomes feasible.

Load-bearing premise

The Monte Carlo sampling procedure converges without large systematic biases or uncontrolled errors across the full range of temperatures, interaction strengths, and generalized Gibbs ensembles.

What would settle it

A statistically significant deviation between the numerically obtained spectral function and the exact analytic result known for the Tonks-Girardeau limit at a chosen finite temperature would show that the sampling algorithm fails to be reliable.

Figures

Figures reproduced from arXiv: 2508.17908 by Fabian H. L. Essler, Riccardo Senese.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Average [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same MCMC plots as in Fig [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Density plots for [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: we show Ce(k, ω) in GGEs (8). By choosing β3 ̸= 0 we break the parity, which leads to marked differences compared to the thermal case of [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
read the original abstract

We develop a Monte Carlo sampling algorithm to numerically evaluate the Lehmann representation for the finite temperature single-particle Green's function in the repulsive Lieb-Liniger model. This allows us to determine the spectral function in the full range of temperatures and interactions, as well as in generalized Gibbs ensembles. We test our results against known results for dynamics at infinite interaction strength and static correlators, and find excellent agreement.

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

1 major / 2 minor

Summary. The manuscript develops a Monte Carlo sampling algorithm to numerically evaluate the Lehmann representation for the finite-temperature single-particle Green's function in the repulsive Lieb-Liniger model. This enables computation of the spectral function across the full range of temperatures, interaction strengths, and generalized Gibbs ensembles. The approach is tested against known results for dynamics at infinite interaction strength (Tonks-Girardeau limit) and for static correlators, with reported excellent agreement.

Significance. If the numerical convergence and error control hold across the parameter space, the work would provide a useful computational tool for dynamical correlation functions in integrable one-dimensional Bose gases at finite temperature and in GGEs, where analytic methods are limited. The use of exact Bethe-ansatz form factors within the sampling procedure is a methodological strength that could be extended to other integrable models.

major comments (1)
  1. [Numerical results and validation] The central claim that the Monte Carlo algorithm determines the spectral function over the entire (T, c, GGE) space rests on the assumption of controlled statistical and truncation errors for finite interaction strengths. However, the reported validations are restricted to the c=∞ limit and static correlators; these do not probe the full complexity of finite-c form-factor matrix elements or the low-T spectral weight distribution. Additional quantitative convergence tests (e.g., dependence of results on sample size and energy cutoff) for intermediate c and finite T are required to substantiate the broad applicability.
minor comments (2)
  1. The abstract and introduction would benefit from a brief statement of the typical system sizes and number of Bethe states sampled in the Monte Carlo procedure.
  2. Clarify the precise definition of the generalized Gibbs ensemble parameters used in the numerical examples.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive suggestion for strengthening the numerical validation. We respond to the major comment below.

read point-by-point responses

  1. Referee: The central claim that the Monte Carlo algorithm determines the spectral function over the entire (T, c, GGE) space rests on the assumption of controlled statistical and truncation errors for finite interaction strengths. However, the reported validations are restricted to the c=∞ limit and static correlators; these do not probe the full complexity of finite-c form-factor matrix elements or the low-T spectral weight distribution. Additional quantitative convergence tests (e.g., dependence of results on sample size and energy cutoff) for intermediate c and finite T are required to substantiate the broad applicability.

    Authors: We agree that additional quantitative convergence tests for finite c and finite T would strengthen the presentation. The current manuscript validates the implementation against exact analytic results available in the Tonks-Girardeau limit (c=∞) for dynamics and against known static correlators for finite c. The algorithm itself employs the exact Bethe-ansatz form factors for any finite c, so the underlying representation is controlled; however, we acknowledge that explicit demonstrations of statistical and truncation error control at intermediate c are valuable to support the broad applicability claim. In the revised manuscript we will add figures that show the dependence of the computed spectral function on Monte Carlo sample size and on the energy cutoff, for representative intermediate values such as c=1 and c=2 at finite temperatures (including low-T regimes). These tests will be performed both in the thermal ensemble and in selected GGEs to address the full scope of the referee's concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity in numerical Monte Carlo evaluation

full rationale

The paper develops a Monte Carlo sampling algorithm to evaluate the Lehmann representation of the finite-temperature single-particle Green's function in the repulsive Lieb-Liniger model. The central procedure is a direct numerical computation whose results are validated against independent external benchmarks (Tonks-Girardeau limit at infinite interaction and static correlators). No load-bearing step reduces by definition or self-citation to the target quantities themselves; the method is self-contained as an algorithmic implementation with stated convergence checks against known analytic cases.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the method rests on standard quantum-statistical assumptions with no additional free parameters or invented entities mentioned.

axioms (1)
  • standard math The Lehmann representation correctly expresses the finite-temperature Green's function in terms of energy eigenstates and matrix elements.
    Standard spectral decomposition of correlation functions in quantum mechanics.

pith-pipeline@v0.9.0 · 5589 in / 1214 out tokens · 43730 ms · 2026-05-21T22:14:25.260412+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

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

  1. Realization of fractional Fermi seas

    cond-mat.quant-gas 2026-02 unverdicted novelty 7.0

    Experimental realization of fractional Fermi seas in excited 1D Bose gas via interaction-strength ramps, with Friedel oscillations as signatures.

Reference graph

Works this paper leans on

71 extracted references · 71 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

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

    work page 1993

  2. [2]

    E. H. Lieb and W. Liniger, Exact Analysis of an Interact- ing Bose Gas. I. The General Solution and the Ground State, Phys. Rev. 130, 1605 (1963)

    work page 1963

  3. [3]

    E. H. Lieb, Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum, Phys. Rev. 130, 1616 (1963)

    work page 1963

  4. [4]

    C. N. Yang and C. P. Yang, Thermodynamics of a One- Dimensional System of Bosons with Repulsive Delta- Function Interaction, Journal of Mathematical Physics 10, 1115 (1969)

    work page 1969

  5. [5]

    N. A. Slavnov, Calculation of scalar products of wave functions and form factors in the framework of the alcebraic Bethe ansatz, Theoretical and Mathematical Physics 79, 502 (1989)

    work page 1989

  6. [6]

    V. E. Korepin and N. A. Slavnov, The time depen- dent correlation function of an Impenetrable Bose gas as a Fredholm minor.I, Communications in Mathemati- cal Physics 129, 103 (1990)

    work page 1990

  7. [7]

    N. A. Slavnov, Nonequal-time current correlation func- tion in a one-dimensional Bose gas, Theoretical and Mathematical Physics 82, 273 (1990)

    work page 1990

  8. [8]

    Caux and P

    J.-S. Caux and P. Calabrese, Dynamical density-density correlations in the one-dimensional Bose gas, Phys. Rev. A 74, 031605 (2006)

    work page 2006

  9. [9]

    J.-S. Caux, P. Calabrese, and N. A. Slavnov, One-particle dynamical correlations in the one-dimensional Bose gas, Journal of Statistical Mechanics: Theory and Experi- ment 2007, P01008 (2007)

    work page 2007

  10. [10]

    Kitanine, K

    N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras, Form factor approach to dy- namical correlation functions in critical models, Journal of Statistical Mechanics: Theory and Experiment 2012, P09001 (2012)

    work page 2012

  11. [11]

    Imambekov, T

    A. Imambekov, T. L. Schmidt, and L. I. Glazman, One- dimensional quantum liquids: Beyond the Luttinger liq- uid paradigm, Rev. Mod. Phys. 84, 1253 (2012)

    work page 2012

  12. [12]

    Kormos, A

    M. Kormos, A. Shashi, Y.-Z. Chou, J.-S. Caux, and A. Imambekov, Interaction quenches in the one- dimensional Bose gas, Phys. Rev. B 88, 205131 (2013)

    work page 2013

  13. [13]

    De Nardis, B

    J. De Nardis, B. Wouters, M. Brockmann, and J.-S. Caux, Solution for an interaction quench in the Lieb- Liniger Bose gas, Phys. Rev. A 89, 033601 (2014)

    work page 2014

  14. [14]

    K. K. Kozlowski, Large-Distance and Long-Time Asymp- totic Behavior of the Reduced Density Matrix in the Non- Linear Schr¨ odinger Model, Annales Henri Poincar´ e16, 437 (2015)

    work page 2015

  15. [15]

    Piroli, P

    L. Piroli, P. Calabrese, and F. H. L. Essler, Multiparticle Bound-State Formation following a Quantum Quench to the One-Dimensional Bose Gas with Attractive Interac- tions, Phys. Rev. Lett. 116, 070408 (2016)

    work page 2016

  16. [16]

    Doyon and H

    B. Doyon and H. Spohn, Drude Weight for the Lieb- Liniger Bose Gas, SciPost Phys. 3, 039 (2017)

    work page 2017

  17. [17]

    J. D. Nardis, D. Bernard, and B. Doyon, Diffusion in generalized hydrodynamics and quasiparticle scattering, SciPost Phys. 6, 049 (2019)

    work page 2019

  18. [18]

    Granet and F

    E. Granet and F. H. L. Essler, A systematic 1 /c- expansion of form factor sums for dynamical correlations in the Lieb-Liniger model, SciPost Phys. 9, 082 (2020)

    work page 2020

  19. [19]

    Granet, Low-density limit of dynamical correlations in the Lieb–Liniger model, Journal of Physics A: Mathe- matical and Theoretical 54, 154001 (2021)

    E. Granet, Low-density limit of dynamical correlations in the Lieb–Liniger model, Journal of Physics A: Mathe- matical and Theoretical 54, 154001 (2021)

    work page 2021

  20. [20]

    Granet and F

    E. Granet and F. H. L. Essler, Systematic strong coupling expansion for out-of-equilibrium dynamics in the Lieb- Liniger model, SciPost Phys. 11, 068 (2021)

    work page 2021

  21. [21]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, Observation of a One-Dimensional Tonks-Girardeau Gas, Science 305, 1125 (2004)

    work page 2004

  22. [22]

    Paredes, A

    B. Paredes, A. Widera, V. Murg, O. Mandel, S. F¨ olling, I. Cirac, G. V. Shlyapnikov, T. W. H¨ ansch, and I. Bloch, Tonks–Girardeau gas of ultracold atoms in an optical lat- tice, Nature 429, 277 (2004)

    work page 2004

  23. [23]

    Fabbri, M

    N. Fabbri, M. Panfil, D. Cl´ ement, L. Fallani, M. Ingus- cio, C. Fort, and J.-S. Caux, Dynamical structure fac- tor of one-dimensional Bose gases: Experimental signa- tures of beyond-Luttinger-liquid physics, Phys. Rev. A 91, 043617 (2015)

    work page 2015

  24. [24]

    Meinert, M

    F. Meinert, M. Panfil, M. J. Mark, K. Lauber, J.-S. Caux, and H.-C. N¨ agerl, Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling, Phys. Rev. Lett. 115, 085301 (2015)

    work page 2015

  25. [25]

    B. Fang, A. Johnson, T. Roscilde, and I. Bouchoule, Momentum-Space Correlations of a One-Dimensional Bose Gas, Phys. Rev. Lett. 116, 050402 (2016)

    work page 2016

  26. [26]

    Kinoshita, T

    T. Kinoshita, T. Wenger, and D. S. Weiss, A quantum Newton’s cradle, Nature 440, 900 (2006)

    work page 2006

  27. [27]

    Hofferberth, I

    S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Non-equilibrium coherence dy- namics in one-dimensional Bose gases, Nature 449, 324 (2007)

    work page 2007

  28. [28]

    Cheneau, P

    M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara, C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of correlations in a quantum many-body system, Nature 481, 484 (2012)

    work page 2012

  29. [29]

    Bouchoule and J

    I. Bouchoule and J. Dubail, Generalized hydrodynam- ics in the one-dimensional Bose gas: theory and experi- ments, Journal of Statistical Mechanics: Theory and Ex- 6 periment 2022, 014003 (2022)

    work page 2022

  30. [30]

    Girardeau, Relationship between Systems of Impene- trable Bosons and Fermions in One Dimension, Journal of Mathematical Physics 1, 516 (1960)

    M. Girardeau, Relationship between Systems of Impene- trable Bosons and Fermions in One Dimension, Journal of Mathematical Physics 1, 516 (1960)

    work page 1960

  31. [31]

    D. B. Creamer, H. Thacker, and D. Wilkinson, Quantum Gel’fand-Levitan method as a generalized Jordan-Wigner transformation, Physics Letters B 92, 144 (1980)

    work page 1980

  32. [32]

    Higher conservation laws for the quantum non-linear Schroedinger equation

    B. Davies and V. E. Korepin, Higher conservation laws for the quantum non-linear Schr¨ odinger equation, arXiv preprint arXiv:1109.6604 (2011)

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  33. [33]

    Caux and F

    J.-S. Caux and F. H. L. Essler, Time Evolution of Lo- cal Observables After Quenching to an Integrable Model, Phys. Rev. Lett. 110, 257203 (2013)

    work page 2013

  34. [34]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Re- laxation 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)

    work page 2007

  35. [35]

    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)

    work page 2015

  36. [36]

    Kojima, V

    T. Kojima, V. E. Korepin, and N. A. Slavnov, Deter- minant Representation for Dynamical Correlation Func- tions of the Quantum Nonlinear Schr¨ odinger Equa- tion, Communications in Mathematical Physics 188, 657 (1997)

    work page 1997

  37. [37]

    Piroli and P

    L. Piroli and P. Calabrese, Exact formulas for the form factors of local operators in the Lieb–Liniger model, Journal of Physics A: Mathematical and Theoretical 48, 454002 (2015)

    work page 2015

  38. [38]

    Srednicki, The approach to thermal equilibrium in quantized chaotic systems, Journal of Physics A: Math- ematical and General 32, 1163 (1999)

    M. Srednicki, The approach to thermal equilibrium in quantized chaotic systems, Journal of Physics A: Math- ematical and General 32, 1163 (1999)

    work page 1999

  39. [39]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics 65, 239 (2016)

    work page 2016

  40. [40]

    F. H. L. Essler and A. J. J. M. de Klerk, Statistics of Ma- trix Elements of Local Operators in Integrable Models, Phys. Rev. X 14, 031048 (2024)

    work page 2024

  41. [41]

    Caux, Correlation functions of integrable models: A description of the ABACUS algorithm, Journal of Math- ematical Physics 50, 095214 (2009)

    J.-S. Caux, Correlation functions of integrable models: A description of the ABACUS algorithm, Journal of Math- ematical Physics 50, 095214 (2009)

    work page 2009

  42. [42]

    A. J. J. M. de Klerk and J.-S. Caux, Improved Hilbert space exploration algorithms for finite temperature cal- culations, SciPost Phys. Core 6, 039 (2023)

    work page 2023

  43. [43]

    Cheng, Y.-Y

    S. Cheng, Y.-Y. Chen, X.-W. Guan, W.-L. Yang, and H.- Q. Lin, One-body dynamical correlation function of the Lieb-Liniger model at finite temperature, Phys. Rev. A 111, L010802 (2025)

    work page 2025

  44. [44]

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

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

    work page 1999

  45. [45]

    In the following we consider only macrostates character- ized by root densities that vanish faster than any power of 1/λ when λ → ∞

    work page

  46. [46]

    GGE expectation values are dominated (exponentially with L) by eigenstates belonging to the macrostate ρ(λ) of highest entropy density under the constraintsR dλ ρ(λ)λn = ⟨Q(n)⟩GGE /L

    work page

  47. [47]

    F. H. L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrable quantum spin chains, Journal of Statistical Mechanics: Theory and Experi- ment 2016, 064002 (2016)

    work page 2016

  48. [48]

    In particular, we plot the average Mλ,µ for 100 eigen- states |µ⟩ randomly generated by solving the Bethe equa- tions (3) with integers {J (µ) j = ˜I (λ) j ±1/2}N −1 j=1 (the signs ± are drawn at random), where {˜I (λ) j } are N −1 of the N half-odd integers I (λ) j that the determine the rapidities λj

    work page

  49. [49]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calcula- tions by fast computing machines, The journal of chem- ical physics 21, 1087 (1953)

    work page 1953

  50. [50]

    W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57, 97 (1970)

    work page 1970

  51. [51]

    R. Y. Rubinstein and D. P. Kroese, Simulation and the Monte Carlo method (John Wiley & Sons, 2016)

    work page 2016

  52. [52]

    Gilks, S

    W. Gilks, S. Richardson, and D. Spiegelhalter, Markov Chain Monte Carlo in Practice (CRC Press, 1995)

    work page 1995

  53. [53]

    Feller, An introduction to probability theory and its applications, Vol

    W. Feller, An introduction to probability theory and its applications, Vol. 1 (John Wiley & Sons, 1957)

    work page 1957

  54. [54]

    Bouchoule, B

    I. Bouchoule, B. Doyon, and J. Dubail, The effect of atom losses on the distribution of rapidities in the one- dimensional Bose gas, SciPost Phys. 9, 044 (2020)

    work page 2020

  55. [55]

    Gritsev, T

    V. Gritsev, T. Rostunov, and E. Demler, Exact meth- ods in the analysis of the non-equilibrium dynamics of integrable models: application to the study of correla- tion functions for non-equilibrium 1D Bose gas, Journal of Statistical Mechanics: Theory and Experiment 2010, P05012 (2010)

    work page 2010

  56. [56]

    We verified that a polynomial decay α/Lβ fits well σm(L) for all the pairs ( τ, γ) in Fig. 1

    work page

  57. [57]

    Lenard, Momentum Distribution in the Ground State of the One-Dimensional System of Impenetrable Bosons, Journal of Mathematical Physics 5, 930 (1964)

    A. Lenard, Momentum Distribution in the Ground State of the One-Dimensional System of Impenetrable Bosons, Journal of Mathematical Physics 5, 930 (1964)

    work page 1964

  58. [58]

    Lenard, One-Dimensional Impenetrable Bosons in Thermal Equilibrium, Journal of Mathematical Physics 7, 1268 (1966)

    A. Lenard, One-Dimensional Impenetrable Bosons in Thermal Equilibrium, Journal of Mathematical Physics 7, 1268 (1966)

    work page 1966

  59. [59]

    O. I. Pˆ at ¸u, Exact spectral function of the Tonks- Girardeau gas at finite temperature, Phys. Rev. A 106, 053306 (2022)

    work page 2022

  60. [60]

    Bornemann, On the numerical evaluation of Fred- holm determinants, Mathematics of Computation 79, 871 (2010)

    F. Bornemann, On the numerical evaluation of Fred- holm determinants, Mathematics of Computation 79, 871 (2010)

    work page 2010

  61. [61]

    Panfil and J.-S

    M. Panfil and J.-S. Caux, Finite-temperature correlations in the Lieb-Liniger one-dimensional Bose gas, Phys. Rev. A 89, 033605 (2014)

    work page 2014

  62. [62]

    N. J. Robinson, A. J. J. M. de Klerk, and J.-S. Caux, On computing non-equilibrium dynamics following a quench, SciPost Phys. 11, 104 (2021)

    work page 2021

  63. [63]

    O. I. Pˆ at ¸u, V. E. Korepin, and D. V. Averin, One- dimensional impenetrable anyons in thermal equilibrium: II. Determinant representation for the dynamic correla- tion functions, Journal of Physics A: Mathematical and Theoretical 41, 255205 (2008)

    work page 2008

  64. [64]

    F. W. King, Hilbert Transforms, Encyclopedia of Mathe- matics and its Applications (Cambridge University Press, 2009)

    work page 2009

  65. [65]

    F. A. Smirnov, Form factors in completely integrable models of quantum field theory, Vol. 14 (World Scientific, 1992)

    work page 1992

  66. [66]

    Brenes, J

    M. Brenes, J. Goold, and M. Rigol, Low-frequency be- havior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic 7 XXZ chain, Phys. Rev. B 102, 075127 (2020)

    work page 2020

  67. [67]

    Zhang, L

    Y. Zhang, L. Vidmar, and M. Rigol, Statistical proper- ties of the off-diagonal matrix elements of observables in eigenstates of integrable systems, Phys. Rev. E 106, 014132 (2022)

    work page 2022

  68. [68]

    LeBlond, K

    T. LeBlond, K. Mallayya, L. Vidmar, and M. Rigol, En- tanglement and matrix elements of observables in in- teracting integrable systems, Phys. Rev. E 100, 062134 (2019)

    work page 2019

  69. [69]

    LeBlond and M

    T. LeBlond and M. Rigol, Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems, Phys. Rev. E 102, 062113 (2020)

    work page 2020

  70. [70]

    frequency

    F. Rottoli and V. Alba, Eigenstate Thermalization Hy- pothesis (ETH) for off-diagonal matrix elements in in- tegrable spin chains, arXiv preprint arXiv:2505.23602 (2025). 8 SUPPLEMENTARY MATERIAL DET AILS OF MARKOV CHAIN MONTE CARLO SAMPLING We aim to sample |µ⟩ eigenstates according to the probability distribution Pλ(µ) = exp( −Mλ,µ)/Zλ, where |λ⟩ is a f...

    work page arXiv 2025

  71. [71]

    intermediate

    visible at ω > 0. This zero-temperature dispersion ω2(k) is associated with energies ω2 = Eµ − Eλ and momenta k = Pµ −Pλ where |λ⟩ is the ground state of N particles and |µ⟩ is constructed by starting from the ground state with N − 1 particles and implementing a particle-hole excitation that creates a hole in the Fermi sea and adds a particle at the posit...

    work page