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REVIEW 2 major objections 4 minor 57 references

Ground-state energy of a trapped attractive Fermi gas converges to the Thomas-Fermi energy in one and two dimensions, and approximate ground states converge to Vlasov minimizers.

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2026-08-02 20:57 UTC pith:KEVSON4E

load-bearing objection The d=1 result looks plausible, but Lemma 3.2 mis-scales in d=2 and the abstract promises a repulsive case that isn't proven; the paper needs a fix before it can be used as stated. the 2 major comments →

arxiv 2602.21640 v2 pith:KEVSON4E submitted 2026-02-25 math-ph cond-mat.quant-gasmath.MP

Semi-classical limit of an attractive Fermi gas in one or two dimensions

Thomas Gamet (UMPA-ENSL) This is my paper
classification math-ph cond-mat.quant-gasmath.MP MSC 81Q1081V7046N5082B10 PACS 05.30.Fk03.65.Sq71.10.Ca
keywords Fermi gasattractive interactionsemi-classical limitThomas-Fermi functionalHusimi functionsDiaconis-Freedman theoremground-state energyVlasov energy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies a trapped gas of N spin-polarized fermions with attractive short-range interactions in one or two dimensions. It shows that, for a certain scaling of the interaction range, the ground-state energy is asymptotically equal to N times the Thomas-Fermi energy, a simple functional of the spatial density. It also proves that the Husimi functions (quantum phase-space distributions) of approximate ground states converge to minimizers of the corresponding Vlasov energy. The result extends the known mean-field semi-classical limit to attractive interactions, where standard methods for repulsive potentials fail. A key step is a quantitative mean-field approximation using the Diaconis-Freedman theorem, after averaging empirical measures to restore the Pauli principle.

Core claim

The central claim is that for d=1,2 and 0<β<2/(d(2d+1)), the ground-state energy E(N) of the Hamiltonian with N fermions, a confining potential V, and attractive interaction w_N = N^{dβ} w(N^β·) satisfies E(N)=N E_TF + o(N), where E_TF is the minimum of the Thomas-Fermi functional c_TF ∫ρ^{1+2/d} + ∫Vρ - I_w∫ρ². Furthermore, the one-body Husimi functions of approximate ground states converge, up to extraction, to a probability measure on phase-space measures that is concentrated on minimizers of the Vlasov energy; the limiting measures satisfy the Pauli principle 0≤m≤(2π)^{-d}. The proof uses an upper bound via Lieb's variational principle and a lower bound built from semi-classical approxim

What carries the argument

The argument rests on the Thomas-Fermi functional (1.20) as the effective energy, and on the Husimi functions m^{(k)}_{Ψ_N} (smoothed phase-space densities) for the lower bound. The lower bound rewrites the N-body energy as an integral over empirical measures using the Diaconis-Freedman theorem, then averages these measures over small phase-space cells to restore the Pauli principle, and finally compares the resulting energy to the Thomas-Fermi functional. The parameter β controlling the interaction range is constrained by the validity of the a priori one-body energy bound in Lemma 3.2.

Load-bearing premise

The a priori bound (3.5) that energy-bounded states have one-body energy at most N^{1+βd/2} is the load-bearing estimate; if it fails, the lower bound construction collapses and the allowed range of β shrinks toward zero.

What would settle it

If one could construct a sequence of N-fermion states with energy of order N but with one-body energy growing faster than N^{1+βd/2}, the a priori bound would be false, and the lower bound argument would fail. Numerically, one could simulate the N-body Schrödinger equation for d=1 with a short-range attractive potential and check whether the ground-state energy divided by N approaches the Thomas-Fermi value for β up to 1/d.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The ground-state energy of a trapped attractive Fermi gas is asymptotically given by a simple density functional, enabling quantitative predictions for experiments with Feshbach-tuned attractive interactions.
  • The convergence of Husimi functions means that quantum ground states become semiclassical measures supported on the phase-space minimizer, validating the Vlasov equation description for such systems.
  • The repulsive case with positive Fourier transform is also covered, showing the method is not specific to attraction.
  • The result extends the semi-classical limit to interaction scalings beyond the mean-field regime, where the interaction becomes local in the limit.

Where Pith is reading between the lines

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

  • The restrictive condition β<2/[d(2d+1)] appears technical, and one might expect the optimal threshold to be β<1/d; the author's own remark suggests improvement is plausible with stronger a priori bounds.
  • The 1D case has non-unique Thomas-Fermi minimizers (e.g., double-well potentials can populate one well), so any quantitative rate or uniqueness claim would require additional structure.
  • The method could likely be adapted to spin-1/2 fermions, with the spin degeneracy merely altering the constant c_TF, as the author notes.
  • A testable extension is to compute the next-order correction to the energy (order N^{1-?}) to see whether the Thomas-Fermi functional is indeed the full leading term in this attractive scaling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies N spin-polarized fermions in d=1,2 with Hamiltonian (1.12), semiclassical scaling ℏ=N^{-1/d}, and short-range attractive interactions w_N=N^{dβ}w(N^β·). It claims E(N)=N E_TF+o(N) (Theorem 1.8) for β<2/[d(2d+1)], and convergence of Husimi functions of approximate ground states to minimizers of the Vlasov energy (Theorem 1.17). The strategy is an upper bound via the Hartree/Lieb variational principle, and a lower bound via Husimi functions, the Diaconis–Freedman theorem, averaging of empirical measures, and an approximate Pauli principle. The d=1 case is treated with a relaxed functional due to non-uniqueness/discontinuity of minimizers.

Significance. If correct, the paper would provide a rigorous derivation of the Thomas–Fermi energy for attractive Fermi gases with nonlocal short-range interactions, extending the well-developed repulsive mean-field theory to a physically relevant attractive setting, and including convergence of states. The paper is carefully structured, with many detailed estimates and a self-contained treatment of the 1D minimizer problem. However, the d=2 proof contains a scaling error that propagates through the a priori bounds and parameter choices; as it stands, Theorem 1.8 for d=2 is not established.

major comments (2)
  1. [§3.1, Eq. (3.14), Lemma 3.2] The scaling in (3.14) is incorrect for d=2. For w_N=N^{dβ}w(N^β·) and p=1+d/2, ||w_N||_p = N^{dβ(1-1/p)}||w||_p, so ||w_N||_p^{1+d/2} = N^{βd^2/2}||w||_p^{1+d/2}, not N^{βd/2}. Thus (3.16) and (3.5) should be O(N^{1+βd^2/2}), which is O(N^{1+2β}) for d=2. This changes (3.20)-(3.21) and propagates to Lemma 3.3: for d=2 the interaction error in (3.39)-(3.42) becomes O(N^{1+3β}√ℏ_x), requiring ℏ_x≪N^{-6β}. Compatibility with ℏ_xℏ_p=ℏ^2=N^{-1} and ℏ_p≪1 only allows ℏ_x≫N^{-1}, so the lemma fails for β≥1/6. This is a load-bearing error, not a typo.
  2. [§3.5, Eqs. (3.102), (3.129), (3.130)] With the corrected Lemma 3.2, the Markov bound (3.102) becomes P(Ξ^c) ≤ C τ^{-1} N^{βd^2/2}, so the last error term in (3.129) is N^{dβ}P(Ξ^c) ≤ C N^{βd(1+d/2)} τ^{-1}, i.e. N^{4β}τ^{-1} for d=2. The parameter choice (3.130) then requires τ≫N^{4β} (to make this error vanish) and τ≪N^{4β} (from the τ^2N^{dβ}/L^4 term), so the window closes. The lower bound gives at best liminf ≥ E_TF - C, not ≥ E_TF. Hence Proposition 3.1 and Theorem 1.8 for d=2 are not proved as stated, and Theorem 1.17 inherits the gap through (4.1).
minor comments (4)
  1. [Abstract] The abstract states that the results 'extend to the case of a repulsive interaction of positive Fourier transform', but no theorem, proposition, or example in the text addresses repulsive interactions. The proof relies on the attractive sign and on Assumption 1.7 (Iw < cTF in d=2). This claim should be removed or substantiated.
  2. [Definition 1.11, Eq. (1.37)] The prefactor in the definition of the semiclassical Fourier transform appears to read (2πℏx)^{-d/2}; it should presumably be (2πℏ)^{-d/2}.
  3. [Appendix title] Typo: 'Additionnal material' should be 'Additional material'.
  4. [Definition 4.3] The 1-Wasserstein distance is defined with sup over ∥φ∥_{Lip}≤1; the Lipschitz seminorm is usually denoted |φ|_{Lip}. Please clarify notation.

Circularity Check

0 steps flagged

No circularity found: the Thomas–Fermi limit is derived from independent a priori bounds and semi-classical estimates; ETF is not assumed in the proof.

full rationale

The paper's central result, Theorem 1.8, asserts E(N)=N ETF+o(N), where ETF is the infimum of the Thomas–Fermi functional (1.20) built directly from the model constants cTF, V and Iw=∫w. The upper bound is obtained by constructing one-body density matrices via Lieb's variational principle and approximating their Hartree energy by the Vlasov energy; it does not use the target energy. The lower bound proceeds through a priori estimates (Lemma 3.2), a semi-classical rewriting in terms of Husimi functions (Lemma 3.3), a Diaconis–Freedman de Finetti-type reduction, and averaging arguments (Lemmas 3.19, 3.21, 3.23, 3.25). None of these steps fits a parameter to the quantity being predicted, nor defines an object in terms of the result. The only self-citation, [25], is contextual ('In a previous work [25], we have studied...') and is not load-bearing for Theorem 1.8 or Theorem 1.17. Cited results such as [21], [31], [18], [42], [44] are external mathematical tools or standard theorems, not a substitute for the paper's own estimates. The convergence of Husimi functions in Theorem 1.17 likewise follows from the independently established energy bounds and lower semi-continuity arguments. The reader's identified scaling issue in Lemma 3.2 -- whether (3.14) and consequently (3.5), (3.102), or the parameter choices in (3.130) are correct for d=2 -- is a mathematical correctness concern about the proof of the stated β-range, not a circularity in the derivation. Even if that exponent were wrong, the argument would not be assuming its conclusion; it would simply be an invalid or incomplete proof step. Accordingly, no circular step is exhibited, and the circularity score is 0.

Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

The proof uses only standard mathematical tools from prior literature (Lieb-Thirring inequality, Diaconis-Freedman theorem, bath-tube principle, Fatou lemma) and the stated conditions on V and w. There are no fitted parameters; the constants c_TF and I_w are inputs fixed by the model. The auxiliary alpha in Appendix A.1 is explicitly fixed as I_w^2/(4c_TF), not tuned to data.

axioms (9)
  • standard math Lieb-Thirring inequality for fermions
    Used in Lemma 3.2, equation (3.12), to bound kinetic energy in terms of density.
  • standard math Diaconis-Freedman theorem for exchangeable sequences
    Theorem 3.7 is the basis for rewriting N-body Husimi functions as mixtures of product measures; cited from [18] and [53].
  • standard math Bath-tube principle (Lieb-Loss)
    Used in Remark 1.4 and Lemma 3.19 to relate Vlasov minimization to Thomas-Fermi minimization.
  • standard math Lieb's variational principle
    Proposition 2.6, from [42], used to reduce N-body problem to Hartree energy.
  • standard math Approximate Pauli principle for empirical measures (Girardot-Rougerie [31, Thm 4.4])
    Proposition 3.14 is quoted directly from [31]; it is a proven theorem, not a new postulate.
  • standard math Fatou's lemma for weakly converging probabilities [20]
    Used in Lemma 4.8 to pass to the limit in the lower bound.
  • domain assumption Assumption 1.5 on the trapping potential V (coercive, locally Lipschitz gradient)
    Controls tightness and the size of error terms in Lemmas 3.3 and 3.16.
  • domain assumption Assumption 1.6: level sets of V have Lebesgue measure zero (d=1 only)
    Needed to prove existence of Thomas-Fermi minimizers and to show relaxed minimizers are true minimizers (Appendix A.1).
  • domain assumption Assumption 1.7 on the interaction w: w>=0, w in L1 cap L-infinity, grad w in L1 cap L^{1+d/2}, and in d=2 I_w<c_TF
    Ensures the Thomas-Fermi functional is bounded below and the scaling estimates hold.

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0 comments
read the original abstract

We study the ground-state of a Fermi gas with short range attrative interactions in one or two dimensions. N fermions are placed in a confining potential, and interact with each other through a negative potential, whose range is larger than the typical distance between particles. We show the convergence of the ground state energy of the Hamiltonian to a Thomas-Fermi energy in the large N limit. Furthermore, we prove convergence of the ground states, in the sense of their Husimi functions. These results extend to the case of a repulsive interaction of positive Fourier transform.

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Reference graph

Works this paper leans on

57 extracted references · 10 linked inside Pith

  1. [1]

    Agerskov, R

    J. Agerskov, R. Reuvers, and J. P. Solovej. Ground state ene rgy of dilute fermi gases in 1d, 2025. arXiv:2509.00536 [math-ph]

    Pith/arXiv arXiv 2025

  2. [2]

    Benedikter, P

    N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer. Optimal upper bound for the correlation energy of a fermi gas in the mean-field regime. Communications in Mathematical Physics , 374(3):2097–2150, 2020

    2097

  3. [3]

    Benedikter, P

    N. Benedikter, P. T. Nam, M. Porta, B. Schlein, and R. Seiringer. Correlation energy of a weakly interacting fermi gas. Inventiones mathematicae, 225(3):885–979, 2021

    2021

  4. [4]

    Benedikter, M

    N. Benedikter, M. Porta, C. Saffirio, and B. Schlein. From the Har tree Dynamics to the Vlasov Equation. Archive for Rational Mechanics and Analysis , 221(1):273–334, July 2016

    2016

  5. [5]

    Benedikter, M

    N. Benedikter, M. Porta, and B. Schlein. Mean–Field Evolution of F ermionic Systems. Communications in Mathematical Physics , 331(3):1087–1131, Nov. 2014

    2014

  6. [6]

    Benedikter, M

    N. Benedikter, M. Porta, and B. Schlein. Effective Evolution Equa tions from Quantum Dynamics, Feb

  7. [7]

    Benedikter, M

    N. Benedikter, M. Porta, B. Schlein, and R. Seiringer. Correlatio n energy of a weakly interacting fermi gas with large interaction potential. Archive for Rational Mechanics and Analysis , 247(4):65, 2023

    2023

  8. [8]

    Benguria

    R. Benguria. The von Weizsacker and exchange corrections in Thomas-Ferm i theory . PhD thesis, Princeton University, June 1979

    1979

  9. [9]

    H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differenti al Equations . Springer New York, 2011

    2011

  10. [10]

    Casadei and S

    G. Casadei and S. Lill. The Ground State Energy of a Mean-Field F ermi Gas in Two Dimensions, Jan

  11. [11]

    X. Chen, J. Wu, and Z. Zhang. The second order huang-yang f ormula to the 3d fermi gas: the gross- pitaevskii regime, 2024. arXiv:2410.16620 [math-ph]

    Pith/arXiv arXiv 2024

  12. [12]

    X. Chen, J. Wu, and Z. Zhang. The second order Huang-Yang a pproximation to the Fermi thermody- namic pressure, June 2025. arXiv:2505.23136 [math-ph]

    Pith/arXiv arXiv 2025

  13. [13]

    M. R. Christiansen, C. Hainzl, and P. T. Nam. The Gell-Mann–Brue ckner Formula for the Correlation Energy of the Electron Gas: A Rigorous Upper Bound in the Mean-Fie ld Regime. Communications in Mathematical Physics, 401(2):1469–1529, July 2023

    2023

  14. [14]

    M. R. Christiansen, C. Hainzl, and P. T. Nam. The random phase a pproximation for interacting fermi gases in the mean-field regime. Forum of Mathematics, Pi , 11, 2023

    2023

  15. [15]

    M. R. Christiansen, C. Hainzl, and P. T. Nam. The Correlation Ene rgy of the Electron Gas in the Mean-Field Regime, May 2024. arXiv:2405.01386 [math-ph]

    Pith/arXiv arXiv 2024

  16. [16]

    C´ ardenas

    E. C´ ardenas. Norm convergence of confined fermionic syste ms at zero temperature. Letters in Mathe- matical Physics, 114(2):38, Mar. 2024

    2024

  17. [17]

    C´ ardenas

    E. C´ ardenas. The quantitative semi-classical limit of a large Fer mi system at zero temperature, May

  18. [18]

    Diaconis and D

    P. Diaconis and D. Freedman. Finite Exchangeable Sequences. The Annals of Probability , 8(4), Aug

  19. [19]

    Falconi, E

    M. Falconi, E. L. Giacomelli, C. Hainzl, and M. Porta. The dilute ferm i gas via bogoliubov theory. Annales Henri Poincar´ e, 22(7):2283–2353, 2021

    2021

  20. [20]

    E. A. Feinberg, P. O. Kasyanov, and N. V. Zadoianchuk. Fatou ’s lemma for weakly converging proba- bilities. Theory of Probability & Its Applications , 58(4):683–689, 2014. 34

    2014

  21. [21]

    Fournais, M

    S. Fournais, M. Lewin, and J. P. Solovej. The semi-classical limit o f large fermionic systems. Calculus of Variations and Partial Differential Equations , 57(4):105, 2018

    2018

  22. [22]

    Fournais and P

    S. Fournais and P. S. Madsen. Semi-classical Limit of Confined Fe rmionic Systems in Homogeneous Magnetic Fields. Annales Henri Poincar´ e, 21(5):1401–1449, May 2020

    2020

  23. [23]

    Fournais and S

    S. Fournais and S. Mikkelsen. An optimal semiclassical bound on c ommutators of spectral projections with position and momentum operators. Letters in Mathematical Physics , 110(12):3343–3373, Dec. 2020

    2020

  24. [24]

    Fournais, B

    S. Fournais, B. Ruba, and J. P. Solovej. Ground state energy of dense gases of strongly interacting fermions. Annales Henri Poincar´ e, 2024

    2024

  25. [25]

    T. Gamet. Ground state energy of a dilute inhomogeneous Ferm i gas, Oct. 2025. arXiv:2510.21268 [math-ph]

    arXiv 2025

  26. [26]

    E. L. Giacomelli. An optimal upper bound for the dilute fermi gas in three dimensions. Journal of Functional Analysis, 285(8):110073, 2023

    2023

  27. [27]

    E. L. Giacomelli. An optimal lower bound for the low density fermi g as in three dimensions, 2024. arXiv:2410.08904 [math-ph]

    arXiv 2024

  28. [28]

    E. L. Giacomelli, C. Hainzl, P. T. Nam, and R. Seiringer. The huang- yang formula for the low-density fermi gas: upper bound, 2024. arXiv:2409.17914 [math-ph]

    Pith/arXiv arXiv 2024

  29. [29]

    E. L. Giacomelli, C. Hainzl, P. T. Nam, and R. Seiringer. The huang- yang conjecture for the low-density fermi gas, 2025. arXiv:2505.22340 [math-ph]

    Pith/arXiv arXiv 2025

  30. [30]

    Girardot

    T. Girardot. Approximation de champ moyen pour un gaz d’anyons . These de doctorat, Universit´ e Grenoble Alpes, Sept. 2021

    2021

  31. [31]

    Girardot and N

    T. Girardot and N. Rougerie. Semiclassical Limit for Almost Fermio nic Anyons. Communications in Mathematical Physics, 387(1):427–480, Oct. 2021

    2021

  32. [32]

    Girardot and N

    T. Girardot and N. Rougerie. A Lieb-Thirring inequality for exten ded anyons. Letters in Mathematical Physics, 113(1), Feb. 2023

    2023

  33. [33]

    Hainzl, M

    C. Hainzl, M. Porta, and F. Rexze. On the correlation energy of interacting fermionic systems in the mean-field regime. Communications in Mathematical Physics , 374(2):485–524, 2020

    2020

  34. [34]

    Hewitt and L

    E. Hewitt and L. J. Savage. Symmetric Measures on Cartesian P roducts. Transactions of the American Mathematical Society, 80(2):470–501, 1955. Publisher: American Mathematical Society

    1955

  35. [35]

    Huang and C

    K. Huang and C. N. Yang. Quantum-mechanical many-body pro blem with hard-sphere interaction. Physical Review, 105(3):767–775, 1957

    1957

  36. [36]

    A. Kafle, R. Senaratne, D. Cavazos-Cavazos, H.-Y. Cui, T. Gia marchi, H. Pu, X.-W. Guan, and R. G. Hulet. Low Energy Excitations of a 1D Fermi Gas with Attractiv e Interactions, Dec. 2025. arXiv:2512.08866 [cond-mat]

    arXiv 2025

  37. [37]

    A. B. Lauritsen. Almost optimal upper bound for the ground st ate energy of a dilute fermi gas via cluster expansion. Annales Henri Poincar´ e, 26(1):203–243, 2025

    2025

  38. [38]

    A. B. Lauritsen and R. Seiringer. Ground state energy of the d ilute spin-polarized Fermi gas: Lower bound, Feb. 2024. arXiv:2402.17558 [math-ph]

    Pith/arXiv arXiv 2024

  39. [39]

    A. B. Lauritsen and R. Seiringer. Ground state energy of the d ilute spin-polarized Fermi gas: Upper bound via cluster expansion. Journal of Functional Analysis , 286(7):110320, Apr. 2024

    2024

  40. [40]

    Lewin, P

    M. Lewin, P. S. Madsen, and A. Triay. Semi-classical limit of large f ermionic systems at positive temperature. Journal of Mathematical Physics , 60(9):091901, 2019

    2019

  41. [41]

    E. H. Lieb. Thomas-fermi and related theories of atoms and mo lecules. Reviews of Modern Physics , 53(4):603–641, Oct. 1981. 35

    1981

  42. [42]

    E. H. Lieb. Variational Principle for Many-Fermion Systems. Physical Review Letters , 46(7):457–459, Feb. 1981. Publisher: American Physical Society

    1981

  43. [43]

    E. H. Lieb and M. Loss. Analysis. Graduate studies in mathematics. American Mathematical Society , 1997

    1997

  44. [44]

    E. H. Lieb and R. Seiringer. The Stability of Matter in Quantum Mechanics . Cambridge University Press, 2009

    2009

  45. [45]

    E. H. Lieb, R. Seiringer, and J. P. Solovej. Ground-state ener gy of the low-density fermi gas. Physical Review A , 71(5):053605, 2005

    2005

  46. [46]

    E. H. Lieb and B. Simon. The Hartree-Fock theory for Coulomb s ystems. Communications in Mathe- matical Physics, 53(3):185–194, Feb. 1977

    1977

  47. [47]

    E. H. Lieb and B. Simon. The Thomas-Fermi theory of atoms, mo lecules and solids. Advances in Mathematics, 23(1):22–116, Jan. 1977

    1977

  48. [48]

    E. H. Lieb, J. P. Solovej, and J. Yngvason. Heavy atoms in the s trong magnetic field of a neutron star. Physical Review Letters , 69(5):749–752, Aug. 1992

    1992

  49. [49]

    E. H. Lieb, J. P. Solovej, and J. Yngvason. Asymptotics of hea vy atoms in high magnetic fields: I. Lowest landau band regions. Communications on Pure and Applied Mathematics , 47(4):513–591, Apr. 1994

    1994

  50. [50]

    E. H. Lieb, J. P. Solovej, and J. Yngvason. Asymptotics of hea vy atoms in high magnetic fields: II. Semiclassical regions. Communications in Mathematical Physics , 161(1):77–124, Mar. 1994

    1994

  51. [51]

    E. H. Lieb, J. P. Solovej, and J. Yngvason. Ground states of la rge quantum dots in magnetic fields. Physical Review B , 51(16):10646–10665, Apr. 1995

    1995

  52. [52]

    P´ erice

    D. P´ erice. Multiple Landau level filling for a large magnetic field limit o f 2D fermions. Journal of Mathematical Physics, 65(2):021902, Feb. 2024

    2024

  53. [53]

    Rougerie

    N. Rougerie. De Finetti theorems, mean-field limits and Bose-Ein stein condensation, June 2020. arXiv:1506.05263 [math-ph]. 36

    Pith/arXiv arXiv 2020

  54. [1980]

    Publisher: Institute of Mathematical Statistics

  55. [2015]

    arXiv:1502.02498 [math-ph]

    Pith/arXiv arXiv

  56. [2025]

    arXiv:2505.24706 [math-ph]

    Pith/arXiv arXiv

  57. [2026]

    arXiv:2601.00750 [math-ph]

    arXiv