pith. sign in

arxiv: 2403.11190 · v2 · submitted 2024-03-17 · ❄️ cond-mat.quant-gas · quant-ph

Dynamical Fermionization and Emergent Bethe Rapidity Structure in the Spatial Density of Cold quenched Lieb-Liniger gas

Sumita Datta , James M Rejcek , Rajasee Datta , Maxim Olshanii This is my paper

Pith reviewed 2026-05-24 03:33 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas quant-ph
keywords Lieb-Liniger gasdynamical fermionizationBethe rapiditiesgeometric quenchquantum Monte Carloballistic expansionone-dimensional Bose gasTonks-Girardeau regime
0
0 comments X

The pith

After a geometric quench the long-time spatial density of a Lieb-Liniger gas scales with velocity x/t and encodes the Bethe rapidity distribution.

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

The paper demonstrates that the nonequilibrium spatial density of a one-dimensional interacting Bose gas following expansion from a smaller hard-wall box directly encodes the underlying momentum (rapidity) distribution. Using an ab initio quantum Monte Carlo method based on the generalized Feynman-Kac representation, the authors compute the many-body time evolution and observe that the density profile acquires a scaling form in the velocity variable x/t. In the long-time limit this profile approaches a stationary distribution whose shape reflects the rapidity structure, with the velocity-space density broadening as interaction strength increases and converging rapidly in the Tonks-Girardeau regime. The result supplies numerical evidence for a direct mapping from real-space observables to the Bethe rapidities of the integrable model via ballistic expansion.

Core claim

In the long-time limit after the quench, the density profile acquires a scaling form in the velocity variable x/t, approaching a stationary distribution whose shape reflects the underlying rapidity structure of the Lieb-Liniger gas.

What carries the argument

The scaling form of the spatial density in the velocity variable x/t that maps directly onto the rapidity distribution through ballistic expansion.

Load-bearing premise

The generalized Feynman-Kac quantum Monte Carlo method accurately captures the exact many-body time evolution without uncontrolled systematic errors that would distort the long-time velocity scaling.

What would settle it

An independent Bethe-ansatz calculation of the rapidity distribution for the same post-quench parameters compared against the simulated long-time density profile in x/t would show a mismatch if the claimed mapping does not hold.

Figures

Figures reproduced from arXiv: 2403.11190 by James M Rejcek, Maxim Olshanii, Rajasee Datta, Sumita Datta.

Figure 1
Figure 1. Figure 1: A plot for the thought experiment of expansion of the gas [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evolution of x-space densities in time; Rapidities for dif [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of x-space densities in time; Rapidities for dif [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of x-space densities in time; Extrapolation and int [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
read the original abstract

We demonstrate that the nonequilibrium spatial density of a one-dimensional interacting Bose gas, following a geometric quench, directly encodes information about the underlying momentum (rapidity) distribution of the system. Starting from the interacting ground state of a Lieb--Liniger gas confined in a hard-wall box of length $L_0$, we study its expansion into a larger box of length $L > L_0$ at fixed interaction strength. Using an ab initio quantum Monte Carlo approach based on the generalized Feynman--Kac representation, we compute the time evolution of the many-body density. We show that, in the long-time limit, the density profile acquires a scaling form in the velocity variable $x/t$, approaching a stationary distribution whose shape reflects the underlying rapidity structure. The velocity-space density broadens systematically with increasing interaction strength and exhibits rapid convergence in the strongly interacting (Tonks--Girardeau) regime. These results provide numerical evidence that ballistic expansion enables a direct mapping between spatial density profiles and the momentum-space structure of the integrable Lieb--Liniger model, offering a practical route to accessing Bethe rapidities through real-space observables.

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 / 0 minor

Summary. The manuscript claims that, following a geometric quench of an interacting Lieb-Liniger Bose gas from a hard-wall box of length L0 into a larger box of length L, the long-time spatial density acquires a stationary scaling form in the velocity variable x/t whose shape directly encodes the initial Bethe rapidity distribution; this is demonstrated via generalized Feynman-Kac quantum Monte Carlo simulations that show systematic broadening with increasing interaction strength and rapid convergence in the Tonks-Girardeau limit.

Significance. If the QMC results are free of uncontrolled long-time biases, the work supplies concrete numerical evidence that ballistic expansion maps real-space density profiles onto the rapidity structure of the integrable model, offering a practical route to infer Bethe rapidities from measurable spatial observables in 1D quantum gases.

major comments (1)
  1. [Abstract and Methods (generalized Feynman-Kac QMC)] The central claim rests on the fidelity of the generalized Feynman-Kac QMC evolution to exact unitary dynamics at long times. The abstract (and, by extension, the methods description) supplies no quantitative convergence tests with respect to Monte Carlo sampling, time discretization, or system size, nor benchmarks against exact limits such as the non-interacting or Tonks-Girardeau cases; such validation is load-bearing for the reported velocity scaling and interaction dependence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the potential significance of the numerical evidence for the velocity-space scaling of the post-quench density. We address the concern regarding validation of the generalized Feynman-Kac QMC method below.

read point-by-point responses

  1. Referee: [Abstract and Methods (generalized Feynman-Kac QMC)] The central claim rests on the fidelity of the generalized Feynman-Kac QMC evolution to exact unitary dynamics at long times. The abstract (and, by extension, the methods description) supplies no quantitative convergence tests with respect to Monte Carlo sampling, time discretization, or system size, nor benchmarks against exact limits such as the non-interacting or Tonks-Girardeau cases; such validation is load-bearing for the reported velocity scaling and interaction dependence.

    Authors: We agree that the abstract and methods description as currently written do not contain explicit quantitative convergence tests or benchmarks, and that such material is necessary to support the central claims. In the revised manuscript we will expand the Methods section (and add a dedicated subsection or supplementary note) with systematic convergence data: (i) statistical error versus number of Monte Carlo samples, (ii) dependence on imaginary-time discretization and projection time, (iii) finite-size scaling with particle number and box lengths L0, L, and (iv) direct comparisons to exact analytic results in the non-interacting (γ = 0) and Tonks-Girardeau (γ → ∞) limits, showing agreement of the long-time velocity profiles within statistical uncertainties. These additions will be referenced from the abstract and will underpin the reported interaction dependence and scaling form. revision: yes

Circularity Check

0 steps flagged

No circularity; result from independent numerical evolution

full rationale

The paper performs ab initio time evolution of the quenched Lieb-Liniger gas via generalized Feynman-Kac QMC and reports an observed long-time scaling form in x/t from the resulting density profiles. This is a direct numerical output, not a quantity fitted or defined in terms of the target rapidity structure. No equations reduce the reported scaling to the inputs by construction, and no load-bearing self-citation chain is invoked to derive the result. The comparison to known Bethe rapidities is an external benchmark, not an internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the integrability of the Lieb-Liniger model (standard in the field) and the accuracy of the chosen QMC representation; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The Lieb-Liniger Hamiltonian is integrable and its eigenstates are labeled by Bethe rapidities
    Invoked when stating that the density shape reflects the underlying rapidity structure
  • domain assumption The generalized Feynman-Kac representation yields the exact many-body density evolution
    Basis of the ab initio QMC approach used to compute the time-dependent density

pith-pipeline@v0.9.0 · 5754 in / 1277 out tokens · 20916 ms · 2026-05-24T03:33:35.114625+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

  1. [1]

    H. A. Bethe, Z Phys,71,205,(1931), https://doi.org/10.1007/bf 01341708

    work page doi:10.1007/bf 1931

  2. [2]

    Gaudin, Phys

    M. Gaudin, Phys. Rev A, 4, 386(1971). DOI:10.1103/PhysRevA.4 .386

    work page doi:10.1103/physreva.4 1971

  3. [3]

    J. M. Wilson, M. Malvania, Y. Le, Y. Zhang, M. Rigol, D. S. Weiss,n. Science, 367, 1461(2020). DOI: 10.1126/science.aaz0242

    work page doi:10.1126/science.aaz0242 2020

  4. [4]

    Girardeau, J

    M. Girardeau, J. Math. Phys. 1, 516(1960) DOI: 10.1063/1.1703687(1960)

    work page doi:10.1063/1.1703687(1960 1960

  5. [5]

    Rigol, A

    M. Rigol, A. Muramatsu, Phys. Rev. Lett., 94, 240403(2005). D OI: 10.1103/PhysRevLett.94.240403

    work page doi:10.1103/physrevlett.94.240403 2005

  6. [6]

    E. H. Lieb, W.Liniger, Phys. Rev. 1963, 130, 1605(1963). DOI: 10.1103/PhysRev.130.1605

    work page doi:10.1103/physrev.130.1605 1963

  7. [7]

    Doyon, and J

    I.Bouchoule, B. Doyon, and J. Dubail, SciPost Phys. 9, 044(2020 ). DOI: 10.21468/scipostphys.9.4.044

    work page doi:10.21468/scipostphys.9.4.044 2020

  8. [8]

    S. S. Alam, T. Skaras, L. Yang, H. Pu, Phys. Rev. Lett., 127, 023002(2021). DOI: 10.1103/PhysRevLett.127.023002

    work page doi:10.1103/physrevlett.127.023002 2021

  9. [9]

    Claverie, M

    P. Claverie, M. Cafferel, J. Chem Phys. 88 , 1088 (1988); 88, 1100 (1988).DOI: 10.1063/1.454227

    work page doi:10.1063/1.454227 1988

  10. [11]

    Datta, Int

    S. Datta, Int. J. of Mod Phys.B, 37, 2350024(2023). DOI: 10.1142/S0217979223500248

    work page doi:10.1142/s0217979223500248 2023

  11. [12]

    M. T. Bachelor, X. W. Guan, N. Oelkers, C. Lee, J. Phys. A, 38, 7787(2005). DOI: 10.1088/0305-4470/38/36/001

    work page doi:10.1088/0305-4470/38/36/001 2005

  12. [13]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, Phys. Rev. Lett., 98, 050405(2007). DOI: 10.1103/PhysRevLett.98.050405 26

    work page doi:10.1103/physrevlett.98.050405 2007

  13. [14]

    Dunjko, V

    V. Dunjko, V. Lorent, and M. Olshanii, Phys. Rev. Lett. , 86, 5413(2001). DOI: 10.1103/physrevlett.86.5413

    work page doi:10.1103/physrevlett.86.5413 2001

  14. [15]

    Bouchoule, J

    I. Bouchoule, J. Dubail, J. Stat. Mech., 2, 14003(2022). DOI: 10.1088/1742-5468/ac3659

    work page doi:10.1088/1742-5468/ac3659 2022

  15. [16]

    Malvania, Y

    N. Malvania, Y. Zhang, Y. Le, J. Dubail, M. Rigol, D. S. Weiss, 373, 1129(2021). DOI: 10.1126/science.abf0147

    work page doi:10.1126/science.abf0147 2021

  16. [17]

    Kac in Proceedings of the Second Berkley Symposium (Berkley Press, California, 1951)

    M. Kac in Proceedings of the Second Berkley Symposium (Berkley Press, California, 1951)

    work page 1951

  17. [18]

    Pezer, T

    D.Juki´c, R. Pezer, T. Gasenzer and H. Buljan, Phys. Rev. A, 78,053602(2008)

    work page 2008

  18. [19]

    Klajn, H

    D.Juki´c, B. Klajn, H. Buljan, Phys. Rev. A, 79, 033612(2009). DOI: 10.1103/PhysRevA.79.033612

    work page doi:10.1103/physreva.79.033612 2009

  19. [20]

    Mei, Z., L.Vidmar, F

    Z. Mei, Z., L.Vidmar, F. Heidrich-Meisner, C. J. Bolech, Phys. Rev . A, 93, 021607(2016)

    work page 2016

  20. [21]

    Olshanii, Phys

    M. Olshanii, Phys. Rev.Lett. , 81, 938(1998). DOI: 10.1103/Ph ys- RevLett.81.938

    work page doi:10.1103/ph 1998

  21. [22]

    Journal of the American Statistical Association , author =

    N. Metropolis and S.J. Ulam, Am. Stat. Assoc.,44, 335(1949),doi:10.1080/01621459.1949.10483310

    work page doi:10.1080/01621459.1949.10483310 1949

  22. [23]

    H. F. Trotter,Proc. Am. Math. Soc., 1959, 10, 545(1959), doi:10.1090/s0002-9939-1959-0108732-6

    work page doi:10.1090/s0002-9939-1959-0108732-6 1959

  23. [24]

    M. D. Donsker, M. Kac, J. Res. Natl. Bur. Stand., 44, 581(195 0). DOI: 10.6028/jres.044.050

    work page doi:10.6028/jres.044.050

  24. [25]

    R. P. Feynman, Statistical Physics:a set of lectures (Reading M ass.: Addison-Wesley, 1998)

    work page 1998

  25. [26]

    R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integ rals. DOI: 10.1063/1.3048320 27

    work page doi:10.1063/1.3048320

  26. [27]

    H. S. Schulman, Techniques and Applications of Path Integratio n. DOI: 10.1063/1.2914703

    work page doi:10.1063/1.2914703

  27. [28]

    Korzeniowski, J

    A. Korzeniowski, J. L. Fry, D.E. Orr, N. G. Fazleev, Phys. Rev. Lett., 69, 893(1992). DOI: 10.1103/PhysRevLett.69.893

    work page doi:10.1103/physrevlett.69.893 1992

  28. [29]

    Korzeniowski, J

    A. Korzeniowski, J. Comp. App. Math.,66,333(1996) DOI: 10.1016/0377-0427(95)00170-0

    work page doi:10.1016/0377-0427(95)00170-0 1996

  29. [30]

    Datta, J

    S. Datta, J. L. Fry, N. G. Fazleev, S. A. Alexander, R. L., Coldw ell, Phys. Rev. A , 61, 030502(2000). DOI: 10.1103/PhysRevA.61.030 502

    work page doi:10.1103/physreva.61.030 2000

  30. [31]

    Datta, V

    S. Datta, V. Dunjko, V., M. Olshanii, 4, 2(2000). DOI: 10.3390/physics401000

    work page doi:10.3390/physics401000 2000

  31. [32]

    Wiener, J

    N. Wiener, J. Math and Phys., 131(1923) doi.org/10.1002/sapm192321131

    work page doi:10.1002/sapm192321131 1923

  32. [33]

    Kato, Commun

    T. Kato, Commun. Pur. and App. Math., 10, 151(1957). DOI: 10.1002/cpa.3160100201

    work page doi:10.1002/cpa.3160100201 1957

  33. [34]

    M. D. Donsker, S. R. S. and Varadhan, ”Proc Intl. Conf. Func . Sp. Int.” 1975, 15-33. DOI: 10.1073/pnas.72.3.780

    work page doi:10.1073/pnas.72.3.780 1975

  34. [35]

    Billingsley, Convergence of Probability Measures ; (Wiley,New York, NY, USA, 1968.)

    P. Billingsley, Convergence of Probability Measures ; (Wiley,New York, NY, USA, 1968.)

    work page 1968

  35. [36]

    Simon, Functional Integrals and Quantum Mechanics (Acade mic Press, NY), 1979,doi:10.1016/S0079-8169(08)60979-4

    B. Simon, Functional Integrals and Quantum Mechanics (Acade mic Press, NY), 1979,doi:10.1016/S0079-8169(08)60979-4

    work page doi:10.1016/s0079-8169(08)60979-4 1979

  36. [37]

    C. A. Tracy, H. Widom, 41, 485204(2008). DOI: 10.1088/1751 - 8113/41/48/485204

    work page doi:10.1088/1751 2008

  37. [38]

    J. C. Zill, T. D. Wright, K. V. Kheruntsyan, T. Gasenzer, M. J. D avis, N. J. Phys.,18, 045010(2016). DOI: 10.1088/1367-2630/18/4/0 45010

    work page doi:10.1088/1367-2630/18/4/0 2016

  38. [39]

    G. Lang, F. Hekking, A. Minguzzi, SciPost Phys., 3, 003(2017). DOI: 10.21468/SciPostPhys.3.1.003 doi.org/10.1002/sapm192321131 28

    work page doi:10.21468/scipostphys.3.1.003 2017

  39. [40]

    F. Ares, S. Scopa and S. J. Wald, Phys. A, 55, 375301(2022)

    work page 2022

  40. [41]

    A. S. Campbell, D. M. Gangardt, K. V. Kheruntsyan, Phys. Rev . Lett., 114, 125302(2015). DOI: 10.1103/physrevlett.114.125302

    work page doi:10.1103/physrevlett.114.125302 2015

  41. [42]

    Minguzzi, and D

    A. Minguzzi, and D. M. Gangardt, Phys. Rev. Lett., 94, 240403 (2005). DOI: 10.1103/physrevlett.94.240404

    work page doi:10.1103/physrevlett.94.240404 2005

  42. [43]

    Sutherland, Phys

    B. Sutherland, Phys. Rev. Lett., 80, 3678(1998). DOI: 10.11 03/phys- revlett.80.3678

    work page 1998

  43. [44]

    Tschischik, M

    W. Tschischik, M. Haque, Preprint at arXiv:1411.554v

    work page

  44. [45]

    Zhang, L

    Y. Zhang, L. Vidmar and M. Rigol, Phys. Rev. A , 99, 063605(201 9). DOI: 10.1103/physreva.104.l031303

    work page doi:10.1103/physreva.104.l031303

  45. [46]

    Karatzas, I and S

    I. Karatzas, I and S. E. Shreve, (Springer-Verleg,NY, 1991)

    work page 1991

  46. [47]

    Dubois, G

    L. Dubois, G. Th´ em`eze, F.,Nogrette, J. Dubail and I. Bouchole, arXiv:2312.15344 v1 [cond-mat.quant-gas] 23 Dec 2023

    work page arXiv 2023

  47. [48]

    Korzeneniwski, Stat

    A. Korzeneniwski, Stat. and Prob. Lett.,8, 229(1989)

    work page 1989

  48. [49]

    Simon, Schr¨ odinger Semigroups, Bull

    B. Simon, Schr¨ odinger Semigroups, Bull. Am. Math. Soc.,7,447(1 982)

    work page

  49. [50]

    Panfil, F

    M. Panfil, F. T. Sant’Ana, J. Stat. Mech. Theory and Experimen t, 073103, 2021

    work page 2021

  50. [51]

    Bezzaz, L

    Y. Bezzaz, L. Dubois and I. Bouchoule, SciPost Phys Core 6, 06 4(2023) 29

    work page 2023