pith. sign in

arxiv: 2605.28956 · v1 · pith:6NMIHR3Unew · submitted 2026-05-27 · ❄️ cond-mat.quant-gas · math-ph· math.MP· quant-ph

Symmetry and integrability in the anyon-Hubbard model

Martin Bonkhoff , Grennon Gurney , Friethjof Theel , Peter Schmelcher , Nathan L. Harshman , Thore Posske This is my paper

Pith reviewed 2026-06-29 09:01 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas math-phmath.MPquant-ph
keywords anyon-Hubbard modelintegrabilitysymmetry classesboundary conditionsone-dimensional anyonsspectral degeneraciescold atomsdoublon states
0
0 comments X

The pith

Two anyons in the anyon-Hubbard model are integrable only under periodic boundary conditions.

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

The paper establishes that the anyon-Hubbard model for one-dimensional particles with tunable statistics exhibits symmetries that switch among classes AI, BDI, and CI according to system size, particle number, and boundary conditions. It shows that two anyons become integrable with periodic boundaries but lose integrability with open boundaries, yielding exact solutions for a doublon state hidden in scattering states and for the nullspace of noninteracting anyons. These features resolve spectral signatures across the full range of limits from interacting bosons to pseudofermions. A sympathetic reader would care because the distinctions directly shape the energy levels and degeneracies observable in cold-atom realizations of anyons.

Core claim

The anyon-Hubbard Hamiltonian of finite length displays a switching between symmetry classes AI, BDI, and CI that depends on system size, particle number, and boundary conditions; two anyons are integrable with periodic boundaries but not with open boundaries; the model admits an exactly solvable doublon state within the continuum and an exact nullspace solution for two noninteracting anyons; all limits, including bosons and pseudofermions, are analyzed for their spectral properties.

What carries the argument

The anyon-Hubbard Hamiltonian with a statistics parameter and either periodic or open boundary conditions, whose symmetries and integrability determine the spectrum.

If this is right

  • The spectrum of two anyons under periodic boundaries contains degeneracies fixed by the symmetry class.
  • Limits to bosons and to pseudofermions each produce characteristic, fully solvable spectra.
  • A doublon bound state remains exactly solvable even when embedded in the scattering continuum.
  • The nullspace of two noninteracting anyons admits an exact closed-form solution.
  • Symmetry class determines which states are accessible or protected in finite-size systems.

Where Pith is reading between the lines

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

  • The boundary-condition dependence of integrability may allow experiments to switch between chaotic and regular dynamics by changing trap geometry.
  • The exact doublon solution could be used as a benchmark for numerical methods applied to larger particle numbers.
  • Symmetry switching with particle number suggests that adding more anyons may restore or destroy integrability in a predictable pattern.

Load-bearing premise

The anyon-Hubbard Hamiltonian with the chosen boundary conditions and statistics parameter captures the low-energy physics of the anyons realized in the cold-atom experiments.

What would settle it

Measure the energy spectrum of two anyons in a finite chain: under periodic boundaries the levels must exhibit the degeneracies required by integrability, while under open boundaries those degeneracies must be absent.

Figures

Figures reproduced from arXiv: 2605.28956 by Friethjof Theel, Grennon Gurney, Martin Bonkhoff, Nathan L. Harshman, Peter Schmelcher, Thore Posske.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The anyon-Hubbard model describes [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The two-particle anyon-Hubbard model’s integrability depends on its parameters and boundary conditions. (a) and [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
read the original abstract

Recent cold atom experiments have realized one-dimensional anyons and enabled the tuning of 1D~statistics between bosons and fermions. Here, we analyze the symmetries, integrability, and resulting degeneracies of the underlying anyon-Hubbard model of finite length. Our results reveal a switching between symmetry classes AI, BDI, and CI in dependence on system size, particle number, and boundary conditions, and show that two anyons with periodic boundaries are integrable, while two anyons with open boundary conditions are not. We include a comprehensive analysis of all model limits, especially of interacting bosons and pseudofermions and resolve spectral signatures. We additionally reveal an exactly solvable doublon state that hides in the continuum of scattering states and the exact solution of the nullspace of two noninteracting anyons. The uncovered symmetries shape the fundamental properties of the one-dimensional anyons at hand, and the predicted states are accessible in state-of-the-art experiments.

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 analyzes the symmetries, integrability, and degeneracies of the finite-size anyon-Hubbard model. It reports a switching between symmetry classes AI, BDI, and CI that depends on system size, particle number, and boundary conditions. Central results include the integrability of two anyons under periodic boundary conditions (but not open), an exactly solvable doublon state embedded in the scattering continuum, the exact nullspace solution for two noninteracting anyons, and a resolution of spectral signatures across the interacting-boson and pseudofermion limits.

Significance. If the derivations hold, the work supplies concrete, parameter-free results on symmetry-class switching and exact solvability for a model directly tied to recent cold-atom realizations of 1D anyons. The identification of boundary-dependent integrability and hidden exact states offers falsifiable predictions for finite-system spectra and degeneracies that experiments can test.

minor comments (2)
  1. The abstract states that two anyons with periodic boundaries are integrable while those with open boundaries are not; a brief statement in the main text clarifying whether this distinction survives the two-body reduction or requires the full many-body Hilbert space would aid readability.
  2. Notation for the statistics parameter and the precise implementation of periodic versus open boundary conditions (e.g., how the anyonic phase is incorporated into the hopping terms) should be introduced once in a dedicated paragraph rather than piecemeal across sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report contains no enumerated major comments, so we provide no point-by-point rebuttals below. We remain ready to incorporate any minor clarifications or corrections once they are specified.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives symmetry classes (AI/BDI/CI), integrability distinctions (PBC vs OBC for two anyons), and exact solutions (doublon state, nullspace) directly from the anyon-Hubbard Hamiltonian operators, finite-size spectrum, and boundary conditions. These are internal mathematical statements obtained by solving the model's Schrödinger equation and identifying conserved quantities, with no reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The model limits (bosons, pseudofermions) are analyzed as special cases of the same Hamiltonian without circular renaming or smuggling of ansatze.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty pending full text.

pith-pipeline@v0.9.1-grok · 5719 in / 1026 out tokens · 24099 ms · 2026-06-29T09:01:56.373690+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

146 extracted references · 6 canonical work pages · 3 internal anchors

  1. [1]

    Im(k 1) = 0 = Im(k2),(67)

  2. [2]

    Re(k 1),= 0 = Re(k 2),(68)

  3. [3]

    CUI: Advanced Imaging of Matter

    Im(k 1,2),= 0 = Re(k 2,1),(69) 4)k ∗ 1 =k 2.(70) Physically, these choices correspond to different scenarios of bound and dispersive behavior, a distinction that only 11 becomes unambiguous in the thermodynamic limitL→ ∞[62, 64, 114, 130, 132, 133], where the scattering states become continuous and bound states can emerge at the edges of this continuum [6...

    2056

  4. [4]

    J. Kwan, P. Segura, Y. Li, S. Kim, A. V. Gorshkov, A. Eckardt, B. Bakkali-Hassani, and M. Greiner, Real- ization of one-dimensional anyons with arbitrary statis- tical phase, Science386, 1055 (2024)

    2024

  5. [5]

    G. T. Horowitz, Exactly soluble diffeomorphism invari- ant theories, Commun. Math. Phys.125, 417 (1989)

    1989

  6. [6]

    Fr¨ olian, C

    A. Fr¨ olian, C. S. Chisholm, E. Neri, C. R. Cabrera, R. Ramos, A. Celi, and L. Tarruell, Realizing a 1D topo- logical gauge theory in an optically dressed BEC, Nature 608, 293 (2022)

    2022

  7. [7]

    B. Wang, A. Vashisht, Y. Guo, S. Dhar, M. Landini, H.-C. N¨ agerl, and N. Goldman, Anyonization of bosons in one dimension: An effective swap model (2025), arXiv:2504.21208

    work page arXiv 2025

  8. [8]

    Y. Zeng, A. Bastianello, S. Dhar, Z. Wang, X. Yu, M. Horvath, G. E. Astrakharchik, Y. Guo, H.-C. N¨ agerl, and M. Landini, Realization of fractional Fermi seas (2026), arXiv:2602.17657 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  9. [9]

    Bakkali-Hassani, J

    B. Bakkali-Hassani, J. Kwan, P. Segura, Y. Li, I. Tes- faye, G. Valent´ ı-Rojas, A. Eckardt, and M. Greiner, Re- vealing Pseudo-Fermionization and Chiral Binding of 14 One-Dimensional Anyons using Adiabatic State Prepa- ration (2026), arXiv:2602.20421 [cond-mat.quant-gas]

    work page arXiv 2026

  10. [10]

    C. S. Chisholm, A. Fr¨ olian, E. Neri, R. Ramos, L. Tar- ruell, and A. Celi, Encoding a one-dimensional topo- logical gauge theory in a Raman-coupled Bose-Einstein condensate, Phys. Rev. Res.4, 043088 (2022)

    2022

  11. [11]

    Keilmann, S

    T. Keilmann, S. Lanzmich, I. McCulloch, and M. Roncaglia, Statistically induced phase transitions and anyons in 1D optical lattices, Nat. Commun.2, 361 (2011)

    2011

  12. [12]

    Greschner, G

    S. Greschner, G. Sun, D. Poletti, and L. Santos, Density-Dependent Synthetic Gauge Fields Using Peri- odically Modulated Interactions, Phys. Rev. Lett.113, 215303 (2014)

    2014

  13. [13]

    Greschner and L

    S. Greschner and L. Santos, Anyon Hubbard model in one-dimensional optical lattices, Phys. Rev. Lett.115, 053002 (2015)

    2015

  14. [14]

    Str¨ ater, S

    C. Str¨ ater, S. C. L. Srivastava, and A. Eckardt, Floquet Realization and Signatures of One-Dimensional Anyons in an Optical Lattice, Phys. Rev. Lett.117, 205303 (2016)

    2016

  15. [15]

    Greschner, L

    S. Greschner, L. Cardarelli, and L. Santos, Probing the exchange statistics of one-dimensional anyon models, Phys. Rev. A97, 053605 (2018)

    2018

  16. [16]

    S. Dhar, B. Wang, M. Horvath, A. Vashisht, Y. Zeng, M. B. Zvonarev, N. Goldman, Y. Guo, M. Landini, and H.-C. N¨ agerl, Observing anyonization of bosons in a quantum gas, Nature642, 53 (2025)

    2025

  17. [17]

    Nagies, B

    S. Nagies, B. Wang, A. C. Knapp, A. Eckardt, and N. L. Harshman, Beyond braid statistics: Constructing a lat- tice model for anyons with exchange statistics intrinsic to one dimension, SciPost Phys.16, 086 (2024)

    2024

  18. [18]

    Kundu, Exact Solution of Double delta Function Bose Gas through an Interacting Anyon Gas, Phys

    A. Kundu, Exact Solution of Double delta Function Bose Gas through an Interacting Anyon Gas, Phys. Rev. Lett.83, 1275 (1999)

    1999

  19. [19]

    Bonkhoff, K

    M. Bonkhoff, K. J¨ agering, S. Eggert, A. Pelster, M. Thorwart, and T. Posske, Bosonic Continuum The- ory of One-Dimensional Lattice Anyons, Phys. Rev. Lett.126, 163201 (2021)

    2021

  20. [20]

    O. I. Patu, V. E. Korepin, and D. V. Averin, Correla- tion functions of one-dimensional Lieb–Liniger anyons, J. Phys. A: Math. Theor.40, 14963 (2007)

    2007

  21. [21]

    M. T. Batchelor, X.-W. Guan, and N. Oelkers, One- Dimensional Interacting Anyon Gas: Low-Energy Prop- erties and Haldane Exclusion Statistics, Phys. Rev. Lett.96, 210402 (2006)

    2006

  22. [22]

    Aglietti, L

    U. Aglietti, L. Griguolo, R. Jackiw, S.-Y. Pi, and D. Seminara, Anyons and chiral solitons on a line, Phys. Rev. Lett.77, 4406 (1996)

    1996

  23. [23]

    Posske, B

    T. Posske, B. Trauzettel, and M. Thorwart, Second quantization of Leinaas-Myrheim anyons in one dimen- sion and their relation to the Lieb-Liniger model, Phys. Rev. B96, 195422 (2017)

    2017

  24. [24]

    N. L. Harshman and A. C. Knapp, Topological exchange statistics in one dimension, Phys. Rev. A105, 052214 (2022)

    2022

  25. [25]

    Hao and S

    Y. Hao and S. Chen, Dynamical properties of hard-core anyons in one-dimensional optical lattices, Phys. Rev. A86, 043631 (2012)

    2012

  26. [26]

    Bonkhoff, K

    M. Bonkhoff, K. J¨ agering, S. Hu, A. Pelster, S. Eg- gert, and I. Schneider, Anyonic Phase Transitions in the 1D Extended Hubbard Model with Fractional Statistics, Phys. Rev. Lett.135, 036601 (2025)

    2025

  27. [27]

    N. L. Harshman and A. C. Knapp, Anyons from three- body hard-core interactions in one dimension, Ann. Phys.412, 168003 (2020)

    2020

  28. [29]

    O. W. Greenberg and A. M. L. Messiah, Selection Rules for Parafields and the Absence of Para Particles in Na- ture, Phys. Rev.138, B1155 (1965)

    1965

  29. [30]

    Fredenhagen, K

    K. Fredenhagen, K. H. Rehren, and B. Schroer, Supers- election sectors with braid group statistics and exchange algebras - I. General theory, Commun. Math. Phys125, 201 (1989)

    1989

  30. [31]

    Wang and K

    Z. Wang and K. R. A. Hazzard, Particle exchange statistics beyond fermions and bosons, Nature637, 314 (2025)

    2025

  31. [32]

    J. M. Leinaas and J. Myrheim, On the theory of identi- cal particles, Nuov. Cim. Soc. Ital. Fis.37B, 1 (1977)

    1977

  32. [33]

    Wilczek, Quantum Mechanics of Fractional-Spin Par- ticles, Phys

    F. Wilczek, Quantum Mechanics of Fractional-Spin Par- ticles, Phys. Rev. Lett.49, 957 (1982)

    1982

  33. [34]

    Kitaev, Anyons in an exactly solved model and be- yond, Ann

    A. Kitaev, Anyons in an exactly solved model and be- yond, Ann. Phys.321, 2 (2006)

    2006

  34. [35]

    Fractional statistics

    F. D. M. Haldane, “Fractional statistics” in arbitrary di- mensions: A generalization of the Pauli principle, Phys. Rev. Lett.67, 937 (1991)

    1991

  35. [36]

    Theel, M

    F. Theel, M. Bonkhoff, P. Schmelcher, T. Posske, and N. L. Harshman, Chirally Protected State Manipulation by Tuning One-Dimensional Statistics, Phys. Rev. Lett. 135, 063401 (2025)

    2025

  36. [37]

    Nayak, S

    C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Non-Abelian anyons and topologi- cal quantum computation, Rev. Mod. Phys.80, 1083 (2008)

    2008

  37. [38]

    Wigner, Grundlagen der Quantenmechanik, inGrup- pentheorie und ihre Anwendung auf die Quanten- mechanik der Atomspektren, edited by E

    E. Wigner, Grundlagen der Quantenmechanik, inGrup- pentheorie und ihre Anwendung auf die Quanten- mechanik der Atomspektren, edited by E. Wigner (Vieweg+Teubner Verlag, Wiesbaden, 1931) pp. 34–43

    1931

  38. [39]

    Cartan, Sur une classe remarquable d’espaces de Rie- mann, Bull

    E. Cartan, Sur une classe remarquable d’espaces de Rie- mann, Bull. Soc. Math. Fr.54, 214 (1926)

    1926

  39. [41]

    M. R. Zirnbauer, Particle-hole symmetries in condensed matter, J. Math. Phys.62, 10.1063/5.0035358 (2020)

    work page doi:10.1063/5.0035358 2020

  40. [42]

    Bethe, Zur Theorie der Metalle: I

    H. Bethe, Zur Theorie der Metalle: I. Eigenwerte und Eigenfunktionen der linearen Atomkette, Z. Phys.71, 205 (1931)

    1931

  41. [43]

    Essler, H

    F. Essler, H. Frahm, F. G¨ ohmann, A. Kl¨ umper, and V. Korepin, The One-Dimensional Hubbard model, The One-Dimensional Hubbard Model, Cambridge, UK: Cambridge University Press (2010)

    2010

  42. [44]

    Baxter,Exactly Solved Models in Statistical Mechan- ics(Academic Press, 2016)

    R. Baxter,Exactly Solved Models in Statistical Mechan- ics(Academic Press, 2016)

    2016

  43. [45]

    E. H. Lieb, Exact analysis of an interacting bose gas. II. the excitation spectrum, Phys. Rev.130, 1616 (1963)

    1963

  44. [46]

    Solidification

    F. D. M. Haldane, “Solidification” in a soluble model of bosons on a one-dimensional lattice: The “Boson- Hubbard chain”, Phys. Lett. A80, 281 (1980)

    1980

  45. [47]

    T. C. Choy, Some exact results for a degenerate Hub- bard model in one dimension, Phys. Lett. A80, 49 (1980)

    1980

  46. [48]

    T. C. Choy and F. D. M. Haldane, Failure of Bethe- Ansatz solutions of generalisations of the Hubbard chain to arbitrary permutation symmetry, Phys. Lett. A90, 15 83 (1982)

    1982

  47. [49]

    A. R. Kolovsky and A. Buchleitner, Quantum chaos in the Bose-Hubbard model, Europhys. Lett.68, 632 (2004)

    2004

  48. [50]

    Kollath, A

    C. Kollath, A. M. L¨ auchli, and E. Altman, Quench dy- namics and nonequilibrium phase diagram of the Bose- Hubbard model, Phys. Rev. Lett.98, 180601 (2007)

    2007

  49. [51]

    Kollath, G

    C. Kollath, G. Roux, G. Biroli, and A. M. L¨ auchli, Statistical properties of the spectrum of the extended Bose–Hubbard model, J. Stat. Mech.: Theory Exp. 2010(08), P08011

    2010

  50. [52]

    de la Cruz, S

    J. de la Cruz, S. Lerma-Hern´ andez, and J. G. Hirsch, Quantum chaos in a system with high degree of symme- tries, Phys. Rev. E102, 032208 (2020)

    2020

  51. [53]

    Russomanno, M

    A. Russomanno, M. Fava, and R. Fazio, Nonergodic be- havior of the clean Bose-Hubbard chain, Phys. Rev. B 102, 144302 (2020)

    2020

  52. [54]

    Links, The Yang–Baxter paradox, J

    J. Links, The Yang–Baxter paradox, J. Phys. A: Math. Theor.54, 254001 (2021)

    2021

  53. [55]

    Pausch, E

    L. Pausch, E. G. Carnio, A. Rodr´ ıguez, and A. Buchleit- ner, Chaos and ergodicity across the energy spectrum of interacting bosons, Phys. Rev. Lett.126, 150601 (2021)

    2021

  54. [56]

    Pausch, A

    L. Pausch, A. Buchleitner, E. G. Carnio, and A. Rodr´ ıguez, Optimal route to quantum chaos in the Bose–Hubbard model, J. Phys. A: Math. Theor.55, 324002 (2022)

    2022

  55. [57]

    M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, One dimensional bosons: From condensed matter systems to ultracold gases, Rev. Mod. Phys.83, 1405 (2011)

    2011

  56. [58]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari,Bose-Einstein Condensa- tion and Superfluidity(Oxford University Press, 2016)

    2016

  57. [59]

    Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys

    M. Olshanii, Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons, Phys. Rev. Lett.81, 938 (1998)

    1998

  58. [60]

    Hofstetter, I

    W. Hofstetter, I. Affleck, D. Nelson, and U. Schollw¨ ock, Non-hermitian Luttinger liquids and vortex physics, Eu- rophys. Lett.66, 178 (2004)

    2004

  59. [61]

    Affleck, W

    I. Affleck, W. Hofstetter, D. R. Nelson, and U. Schollw¨ ock, Non-hermitian Luttinger liquids and flux line pinning in planar superconductors, J. Stat. Mech.: Theory Exp.2004(10), P10003

    2004

  60. [62]

    Oelkers and J

    N. Oelkers and J. Links, Ground-state properties of the attractive one-dimensional Bose-Hubbard model, Phys. Rev. B75, 115119 (2007)

    2007

  61. [64]

    Valiente and D

    M. Valiente and D. Petrosyan, Scattering resonances and two-particle bound states of the extended Hub- bard model, J. Phys. B: At. Mol. Opt. Phys.42, 121001 (2009)

    2009

  62. [66]

    C. D. E. Boschi, E. Ercolessi, L. Ferrari, P. Naldesi, F. Ortolani, and L. Taddia, Bound states and expansion dynamics of interacting bosons on a one-dimensional lattice, Phys. Rev. A90, 043606 (2014)

    2014

  63. [67]

    Longhi and G

    S. Longhi and G. D. Valle, Tamm–Hubbard surface states in the continuum, J. Phys.: Condens. Matter25, 235601 (2013)

    2013

  64. [68]

    J. Polo, P. Naldesi, A. Minguzzi, and L. Amico, Exact results for persistent currents of two bosons in a ring lattice, Phys. Rev. A101, 043418 (2020)

    2020

  65. [69]

    Y. Li, D. Schneble, and T.-C. Wei, Two-particle states in one-dimensional coupled Bose-Hubbard mod- els, Phys. Rev. A105, 053310 (2022)

    2022

  66. [70]

    Zhang, S

    W. Zhang, S. Greschner, E. Fan, T. C. Scott, and Y. Zhang, Ground-state properties of the one- dimensional unconstrained pseudo-anyon Hubbard model, Phys. Rev. A95, 053614 (2017)

    2017

  67. [71]

    Longhi and G

    S. Longhi and G. Della Valle, Anyonic Bloch oscilla- tions, Phys. Rev. B85, 165144 (2012)

    2012

  68. [72]

    Longhi and G

    S. Longhi and G. D. Valle, Anyons in one-dimensional lattices: A photonic realization, Opt. Lett.37, 2160 (2012)

    2012

  69. [73]

    L. L. Lau and S. Dutta, Quantum walk of two anyons across a statistical boundary, Phys. Rev. Res.4, L012007 (2022)

    2022

  70. [74]

    Zheng, J

    C. Zheng, J. Xie, M. Zhang, Y. Chen, and Y. Zhang, Necessity of orthogonal basis vectors for the two-anyon problem in a one-dimensional lattice, Commun. Theor. Phys.76, 125103 (2024)

    2024

  71. [75]

    Bonkhoff, S

    M. Bonkhoff, S. B. J¨ ager, I. Schneider, A. Pelster, and S. Eggert, Coherence properties of the repulsive anyon- Hubbard dimer, Phys. Rev. B108, 155134 (2023)

    2023

  72. [76]

    M. R. Zirnbauer, Symmetry classes, inThe Oxford Handbook of Random Matrix Theory, edited by G. Ake- mann, J. Baik, and P. Di Francesco (Oxford University Press, 2015)

    2015

  73. [77]

    Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys

    L. Tonks, The complete equation of state of one, two and three-dimensional gases of hard elastic spheres, Phys. Rev.50, 955 (1936)

    1936

  74. [78]

    Girardeau, Relationship between systems of impen- etrable bosons and fermions in one dimension, J

    M. Girardeau, Relationship between systems of impen- etrable bosons and fermions in one dimension, J. Math. Phys.1, 516 (1960)

    1960

  75. [80]

    F. H. L. Essler, H. Frahm, F. G¨ ohmann, A. Kl¨ umper, and V. E. Korepin,The One-Dimensional Hubbard Model(Cambridge University Press, 2005)

    2005

  76. [81]

    M. T. Batchelor, A. Foerster, X.-W. Guan, J. Links, and H.-Q. Zhou, The quantum inverse scattering method with anyonic grading, J. Phys. A: Math. Theor.41, 465201 (2008)

    2008

  77. [82]

    Kundu, Quantum Integrable 1D anyonic Models: Construction through Braided Yang-Baxter Equation, SIGMA

    A. Kundu, Quantum Integrable 1D anyonic Models: Construction through Braided Yang-Baxter Equation, SIGMA. Symmetry, Integrability and Geometry: Meth- ods and Applications6, 080 (2010)

    2010

  78. [83]

    O. I. Pˆ atu, V. E. Korepin, and D. V. Averin, Correla- tion functions of one-dimensional Lieb–Liniger anyons, J. Phys. A: Math. Theor.40, 14963 (2007)

    2007

  79. [84]

    M. T. Batchelor, X.-W. Guan, and J.-S. He, The Bethe ansatz for 1D interacting anyons, J. Stat. Mech.: Theory Exp.2007(03), P03007

    2007

  80. [85]

    Serbyn, D

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

    2021

Showing first 80 references.