pith. sign in

arxiv: 2605.20065 · v1 · pith:DWX2RHUInew · submitted 2026-05-19 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· quant-ph

Non-equilibrium quantum dynamics of interacting integrable models by Monte Carlo sampling Lehmann representations

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

Pith reviewed 2026-05-20 03:47 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gasquant-ph
keywords Monte Carlo samplingLehmann representationquantum quenchesintegrable modelsLieb-Liniger modelnon-equilibrium dynamicsQuench Actionorder parameter
0
0 comments X

The pith

A Monte Carlo sampling scheme evaluates the Lehmann representation to compute time-dependent local observables after quantum quenches in integrable models.

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

The paper introduces a Monte Carlo sampling method to numerically evaluate the Lehmann representation for time-dependent expectation values of local operators in interacting integrable quantum systems following homogeneous quenches. This technique supports both the full Lehmann sum and the Quench Action formalism while reaching system sizes and evolution times beyond the limits of prior approaches. Benchmarks confirm agreement with exact results in non-interacting lattice and continuum models as well as short-time weak-interaction cases. The method is then used to follow the order-parameter dynamics in the repulsive Lieb-Liniger gas quenched from a Bose-Einstein condensate across a wide range of interaction strengths. The authors also identify the onset of a sign problem when the same sampling is applied to more general dynamical correlators.

Core claim

We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths.

What carries the argument

Monte Carlo sampling of the Lehmann sum (or Quench Action) applied to local observables

If this is right

  • The method extends the reachable scales for computing local operator dynamics after quenches in integrable models.
  • It yields the order-parameter time evolution in the Lieb-Liniger gas for arbitrary interaction strengths.
  • It identifies regimes where a sign problem appears for broader classes of dynamical correlators.
  • The same sampling framework can be applied to both full Lehmann sums and Quench Action calculations.

Where Pith is reading between the lines

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

  • The technique could be used to test predictions for relaxation toward generalized Gibbs ensembles in systems too large for exact diagonalization.
  • Similar sampling ideas might be adapted to track local observables in quenches that break integrability weakly.
  • Efficiency gains could allow systematic studies of finite-size effects on prethermalization plateaus.

Load-bearing premise

The Monte Carlo sampling of the Lehmann sum remains efficient and sign-problem free for the local observables and interaction regimes studied.

What would settle it

Numerical evidence that the Monte Carlo estimator variance grows exponentially with system size or time for local operators in the Lieb-Liniger model at moderate interactions would falsify the claimed efficiency and scalability.

Figures

Figures reproduced from arXiv: 2605.20065 by Fabian H. L. Essler, Riccardo Senese.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. MC results from QA ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: are well fitted by (see SM) ⟨ϕ(x, t)⟩ ∝ e −iω(c,n)t e −t/τ(c,n) . (12) The frequency ω(c, n) > 0 and decay time τ (c, n) > 0 are expected to have a non-trivial analytical dependence on c and n, cf. the case of ⟨σ x j (t)⟩ in TFIC from Ref. [75]. The MC method provides very useful information on the structure of the spectral sum [41]. As in interacting the￾ories the relevant |F{λ(j)} | are exponentially sma… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Dots show QA and DS spectral weights ( [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Log-scale plots showing convergence with [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Log-log plots showing the decay with [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

Determining the dynamics of interacting integrable many-particle quantum systems at finite times after homogeneous quantum quenches is a long-standing challenge. We present a Monte Carlo sampling scheme that numerically evaluates the Lehmann representation for time-dependent expectation values of local operators, allowing us to access system sizes and times significantly beyond the reach of existing methods. The approach accommodates both the full Lehmann sum and the Quench Action formalism. We benchmark against exact results for non-interacting lattice and continuum models and short-time results at weak interactions, finding excellent agreement. We apply the method to quantum quenches from a Bose-Einstein condensate in the repulsive Lieb-Liniger model and determine the time evolution of the order parameter for a wide range of interaction strengths. We discuss the emergence of a "sign problem" for more general dynamical correlators and setups.

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 / 2 minor

Summary. The manuscript introduces a Monte Carlo sampling scheme to numerically evaluate the Lehmann representation (and Quench Action formalism) for time-dependent expectation values of local operators after homogeneous quenches in interacting integrable models. It reports benchmarks showing excellent agreement with exact results for non-interacting lattice and continuum models as well as short-time weak-interaction regimes, followed by an application to the order-parameter dynamics in the repulsive Lieb-Liniger model after a quench from a Bose-Einstein condensate. Limitations such as the emergence of a sign problem for more general dynamical correlators are explicitly discussed.

Significance. If the central numerical method holds, the work provides a practical route to system sizes and evolution times beyond the reach of existing techniques for local observables in integrable systems. The parameter-free benchmarks against independent exact solutions and the transparent treatment of the sign-problem limitation constitute clear strengths that support the method's utility for the targeted class of problems.

minor comments (2)
  1. The relation between the full Lehmann sum and the Quench Action implementation could be stated more explicitly in the methods section to aid readers unfamiliar with the latter formalism.
  2. Figure captions for the Lieb-Liniger results should include the precise range of interaction strengths and the number of Monte Carlo samples used to allow direct reproducibility assessment.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its strengths in benchmarks and transparent discussion of limitations, and recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a Monte Carlo sampling algorithm to evaluate the Lehmann representation (or Quench Action) for time-dependent local operator expectations after quenches in integrable models. Its central claim is the existence and efficiency of this numerical scheme for the targeted class of observables, which is directly benchmarked against independent exact solutions for non-interacting lattice and continuum cases plus short-time weak-interaction analytics. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz smuggled from prior work by the same authors; the reported quantities are obtained from the sampling procedure itself and cross-validated externally rather than being equivalent to the inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The approach rests on standard properties of integrable models and Monte Carlo sampling rather than new postulates. No free parameters are fitted to the target data; interaction strength is an input varied across runs.

axioms (2)
  • standard math The Lehmann representation converges to the correct time-dependent expectation value when summed over the complete basis of eigenstates.
    Invoked in the description of the Monte Carlo sampling scheme for the full Lehmann sum.
  • domain assumption Integrability permits an efficient representation of the post-quench state via the Quench Action formalism.
    Used to accommodate both the full sum and Quench Action in the sampling procedure.

pith-pipeline@v0.9.0 · 5673 in / 1396 out tokens · 44466 ms · 2026-05-20T03:47:03.684205+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

77 extracted references · 77 canonical work pages · 2 internal anchors

  1. [1]

    Tuned the values of the free parameters in the MC schemes such to always have an average acceptance rate close to or higher than 10%. In particular, settingQ∈[2,10] (relevant for both QA and DS), ∆∈[2,6] (DS) andp ph = [0.7,0.9] (DS) always leads to accurate sampling of the stationary distributionP({λ (j)}) =|F {λ(j)}|/Z

    work page

  2. [2]

    For example, we verified that the cutoffs chosen were sufficiently high that integers in the sampling never reach them

    Verified convergence with the choices of UV cutoffs (needed in LL because it is a field theory) on the maximal allowed (half-odd) integers in the sampling. For example, we verified that the cutoffs chosen were sufficiently high that integers in the sampling never reach them

    work page

  3. [3]

    Once reached, the distribution appears to be stationary

    Verified that irrespective of the initial sets{λ (j)}1 chosen, the Markov chain always converges towards the same distribution after a sufficiently large number of stepsℓ. Once reached, the distribution appears to be stationary. These checks were achieved by monitoring the sampled values ofg {λ(j) } like in Fig. 4 (main text) and Fig. S5, or those for the...

    work page

  4. [4]

    We setℓ burn-in to be 1/11 of the total number of MC stepsℓ burn-in +ℓ max

    Always discarded a “burn-in” number of initial MC stepsℓ, to allow time for the Markov chain to reach the stationary distribution. We setℓ burn-in to be 1/11 of the total number of MC stepsℓ burn-in +ℓ max. 18

    work page

  5. [5]

    lumpiness

    Extracted the statistical uncertainty on the MC results by computing the standard error on the mean (at each timet) associated with the output ofCMarkov chains run in parallel (we verified that the statistical fluctuations are well fitted by normal distributions). Usually we setC∈[50,200]. The uncertainties obtained were negligible in all regimes of physi...

    work page

  6. [6]

    We need to separate the contribution of diagonal matrix elements (⟨λsp|ϕ†(x)ϕ(0)|λsp⟩) from that of off-diagonal ones (⟨µ|ϕ †(x)ϕ(0)|λsp⟩for|µ⟩ ̸=|λ sp⟩). This is achieved by settingP(µ=λ sp) = 0 in the Metropolis acceptance step, and estimating the correlator as ⟨ψ(t)|ϕ†(x)ϕ(0)|ψ(t)⟩ − →e −2|x| + N L −e −2|x| YMC(t) ,(S114) whereY MC(t) indicates the out...

    work page

  7. [7]

    This is very similar to the caseh 0 > hof TFIC, see Section IV.B

    We observe that even thoughf µ,λ(x) can be either positive or negative, it is positive for the vast majority of the eigenstates sampled (no sign problem). This is very similar to the caseh 0 > hof TFIC, see Section IV.B

    work page

  8. [8]

    The saddle-point eigenstate|λ sp⟩is constructed as in Section V.A. However, due to the very slow∼λ −2 decay of the QA saddle-point root density atc=∞(see Section V.A), which must be contrasted with the∼λ −4 decay of the interacting case 0< c <∞, the set of positive integers{J j}defining|λ sp⟩for largeLincludes very large integers. We find that this enhanc...

    work page

  9. [9]

    [63] in terms of a difference of Fredholm determinants can be evaluated by standard numerical methods based on quadrature rules [67]

    The eact result for (S112) of Ref. [63] in terms of a difference of Fredholm determinants can be evaluated by standard numerical methods based on quadrature rules [67]. Fixingx= 5, we employ a Gauss-Legendre quadrature grid with several thousands of point to reach convergence (up ton= 8000). The cutoff is varied in the rangeλ max ∈[200,600] to check conve...

    work page

  10. [10]

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

    work page 1993

  11. [11]

    Takahashi,Thermodynamics of One-Dimensional Solvable Models, 1st ed

    M. Takahashi,Thermodynamics of One-Dimensional Solvable Models, 1st ed. (Cambridge University Press, 1999)

    work page 1999

  12. [12]

    This represents the integrable analogue of the diagonal ETH

    work page

  13. [13]

    C. N. Yang and C. P. Yang, Journal of Mathematical Physics10, 1115 (1969)

    work page 1969

  14. [14]

    Mossel and J.-S

    J. Mossel and J.-S. Caux, Journal of Physics A: Mathematical and Theoretical45, 255001 (2012)

    work page 2012

  15. [15]

    Caux and F

    J.-S. Caux and F. H. L. Essler, Physical Review Letters110, 257203 (2013)

    work page 2013

  16. [16]

    Caux, Journal of Statistical Mechanics: Theory and Experiment2016, 064006 (2016)

    J.-S. Caux, Journal of Statistical Mechanics: Theory and Experiment2016, 064006 (2016)

    work page 2016

  17. [17]

    Brockmann, J

    M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, Journal of Physics A: Mathematical and Theoretical47, 145003 (2014)

    work page 2014

  18. [18]

    De Nardis, B

    J. De Nardis, B. Wouters, M. Brockmann, and J.-S. Caux, Physical Review A89, 033601 (2014)

    work page 2014

  19. [19]

    Pozsgay, M

    B. Pozsgay, M. Mesty´ an, M. Werner, M. Kormos, G. Zar´ and, and G. Tak´ acs, Physical Review Letters113, 117203 (2014)

    work page 2014

  20. [20]

    Wouters, J

    B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol, and J.-S. Caux, Physical Review Letters113, 117202 (2014)

    work page 2014

  21. [21]

    Bertini, E

    B. Bertini, E. Tartaglia, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2017, 103107 (2017)

    work page 2017

  22. [22]

    Bertini, E

    B. Bertini, E. Tartaglia, and P. Calabrese, Journal of Statistical Mechanics: Theory and Experiment2018, 063104 (2018)

    work page 2018

  23. [23]

    F. H. L. Essler and A. J. J. M. De Klerk, Physical Review X14, 031048 (2024)

    work page 2024

  24. [24]

    Kubo, Journal of the Physical Society of Japan12, 570 (1957)

    R. Kubo, Journal of the Physical Society of Japan12, 570 (1957)

    work page 1957

  25. [25]

    Maldacena, S

    J. Maldacena, S. H. Shenker, and D. Stanford, Journal of High Energy Physics2016, 106 (2016)

    work page 2016

  26. [26]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Physical Review X8, 021014 (2018)

    work page 2018

  27. [27]

    C. W. Von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Physical Review X8, 021013 (2018)

    work page 2018

  28. [28]

    Khemani, A

    V. Khemani, A. Vishwanath, and D. A. Huse, Physical Review X8, 031057 (2018)

    work page 2018

  29. [29]

    Rakovszky, F

    T. Rakovszky, F. Pollmann, and C. Von Keyserlingk, Physical Review X8, 031058 (2018)

    work page 2018

  30. [30]

    A. Chan, A. De Luca, and J. Chalker, Physical Review X8, 041019 (2018)

    work page 2018

  31. [31]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, Physical Review X9, 041017 (2019)

    work page 2019

  32. [32]

    Murthy and M

    C. Murthy and M. Srednicki, Physical Review Letters123, 230606 (2019)

    work page 2019

  33. [33]

    Foini and J

    L. Foini and J. Kurchan, Physical Review E99, 042139 (2019)

    work page 2019

  34. [34]

    Despite the absence of a strict, state-independent Lieb-Robinson bound [26] for operator spreading in theories with un- bounded spectra and momenta, the restriction to a finite energy density—imposed by the state ˆρin (S6)—acts as an effective ultraviolet regulator. Consequently, the high-energy modes are suppressed, and contributions to expectation value...

    work page

  35. [35]

    E. H. Lieb and D. W. Robinson, Communications in Mathematical Physics28, 251 (1972)

    work page 1972

  36. [36]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Physical Review Letters97, 050401 (2006)

    work page 2006

  37. [37]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics65, 239 (2016)

    work page 2016

  38. [38]

    Pappalardi, L

    S. Pappalardi, L. Foini, and J. Kurchan, Physical Review Letters129, 170603 (2022)

    work page 2022

  39. [39]

    Rottoli and V

    F. Rottoli and V. Alba, Physical Review B113, 054308 (2026)

    work page 2026

  40. [40]

    Multiscale Structure of Eigenstate Thermalization

    P. Orlov, R. Sharipov, and E. Ilievski, arXiv preprint arXiv:2605.08079 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  41. [41]

    Senese and F

    R. Senese and F. H. L. Essler, Physical Review B113, 165425 (2026)

    work page 2026

  42. [42]

    This is because many of the eigenstates|n 2⟩are expected to be further (in rapidity space) from|n⟩than the eigenstates |n1⟩are

    work page

  43. [43]

    Piroli, B

    L. Piroli, B. Pozsgay, and E. Vernier, Nuclear Physics B925, 362 (2017)

    work page 2017

  44. [44]

    Hartmann, G

    M. Hartmann, G. Mahler, and O. Hess, Letters in Mathematical Physics68, 103 (2004)

    work page 2004

  45. [45]

    L. F. Santos, A. Polkovnikov, and M. Rigol, Physical Review Letters107, 040601 (2011)

    work page 2011

  46. [46]

    Srednicki, Journal of Physics A: Mathematical and General32, 1163 (1999)

    M. Srednicki, Journal of Physics A: Mathematical and General32, 1163 (1999)

    work page 1999

  47. [47]

    Eigenstate thermalization

    R. Patil and M. Rigol, arXiv preprint arXiv:2604.11872 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  48. [48]

    For MFIC with OBC, this is chosen to be on the site just to the left of the center of the chain

    work page

  49. [49]

    Calabrese, F

    P. Calabrese, F. H. L. Essler, and M. Fagotti, Journal of Statistical Mechanics: Theory and Experiment2012, P07016 (2012)

    work page 2012

  50. [50]

    G. B. Mbeng, A. Russomanno, and G. E. Santoro, SciPost Physics Lecture Notes , 82 (2024)

    work page 2024

  51. [51]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics16, 407 (1961)

    work page 1961

  52. [52]

    Ghoshal and A

    S. Ghoshal and A. Zamolodchikov, International Journal of Modern Physics A09, 3841 (1994)

    work page 1994

  53. [53]

    Fioretto and G

    D. Fioretto and G. Mussardo, New Journal of Physics12, 055015 (2010)

    work page 2010

  54. [54]

    Sotiriadis, D

    S. Sotiriadis, D. Fioretto, and G. Mussardo, Journal of Statistical Mechanics: Theory and Experiment2012, P02017 (2012)

    work page 2012

  55. [55]

    Bertini, D

    B. Bertini, D. Schuricht, and F. H. L. Essler, Journal of Statistical Mechanics: Theory and Experiment2014, P10035 (2014)

    work page 2014

  56. [56]

    G. V. Gehlen, N. Iorgov, S. Pakuliak, V. Shadura, and Y. Tykhyy, Journal of Physics A: Mathematical and Theoretical 41, 095003 (2008)

    work page 2008

  57. [57]

    Iorgov, V

    N. Iorgov, V. Shadura, and Y. Tykhyy, Journal of Statistical Mechanics: Theory and Experiment2011, P02028 (2011)

    work page 2011

  58. [58]

    Granet, M

    E. Granet, M. Fagotti, and F. H. L. Essler, SciPost Physics9, 033 (2020)

    work page 2020

  59. [59]

    Metropolis, A

    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, The Journal of Chemical Physics21, 26 1087 (1953)

    work page 1953

  60. [60]

    W. K. Hastings, Biometrika57, 97 (1970)

    work page 1970

  61. [61]

    R. Y. Rubinstein and D. P. Kroese,Simulation and the Monte Carlo Method(Wiley, 2016)

    work page 2016

  62. [62]

    Gilks, S

    W. Gilks, S. Richardson, and D. Spiegelhalter, eds.,Markov Chain Monte Carlo in Practice(Chapman and Hall/CRC, 1995)

    work page 1995

  63. [63]

    < p M chosen in (S47) is arbitrary

    We stress that the orderingp 1 < p 2 < . . . < p M chosen in (S47) is arbitrary. Indeed, swapping two pairs (p i,−p i) and (pj,−p j) fori̸=jin the representation (S38) of|p, h⟩ R leads to no sign changes, and therefore the QA sum (S47) is invariant under permutations in the ordering ofp 1, . . . , pM. This implies that in the MC sampling no time should be...

    work page

  64. [64]

    Brockmann, Journal of Statistical Mechanics: Theory and Experiment2014, P05006 (2014)

    M. Brockmann, Journal of Statistical Mechanics: Theory and Experiment2014, P05006 (2014)

    work page 2014

  65. [65]

    Brockmann, J

    M. Brockmann, J. De Nardis, B. Wouters, and J.-S. Caux, Journal of Physics A: Mathematical and Theoretical47, 345003 (2014)

    work page 2014

  66. [66]

    Kojima, V

    T. Kojima, V. E. Korepin, and N. A. Slavnov, Communications in Mathematical Physics188, 657 (1997)

    work page 1997

  67. [67]

    J.-S. Caux, P. Calabrese, and N. A. Slavnov, Journal of Statistical Mechanics: Theory and Experiment2007, P01008 (2007)

    work page 2007

  68. [68]

    Piroli and P

    L. Piroli and P. Calabrese, Journal of Physics A: Mathematical and Theoretical48, 454002 (2015)

    work page 2015

  69. [69]

    D. B. Creamer, H. Thacker, and D. Wilkinson, Physics Letters B92, 144 (1980)

    work page 1980

  70. [70]

    J. R. Weaver, The American Mathematical Monthly92, 711 (1985)

    work page 1985

  71. [71]

    Kr¨ onke and P

    S. Kr¨ onke and P. Schmelcher, Physical Review A98, 013629 (2018)

    work page 2018

  72. [72]

    De Nardis and J.-S

    J. De Nardis and J.-S. Caux, Journal of Statistical Mechanics: Theory and Experiment2014, P12012 (2014)

    work page 2014

  73. [73]

    Caux and P

    J.-S. Caux and P. Calabrese, Physical Review A74, 031605 (2006)

    work page 2006

  74. [74]

    Caux, Journal of Mathematical Physics50, 095214 (2009)

    J.-S. Caux, Journal of Mathematical Physics50, 095214 (2009)

    work page 2009

  75. [75]

    Panfil and J.-S

    M. Panfil and J.-S. Caux, Physical Review A89, 033605 (2014)

    work page 2014

  76. [76]

    Bornemann, Mathematics of Computation79, 871 (2009)

    F. Bornemann, Mathematics of Computation79, 871 (2009)

    work page 2009

  77. [77]

    Lemm and C

    M. Lemm and C. Rubiliani, arXiv preprint arXiv:2603.26209 (2026)

    work page arXiv 2026