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arxiv: 2606.03356 · v1 · pith:NPLHCNQGnew · submitted 2026-06-02 · 🧮 math-ph · math.MP

The Huang--Yang formula for a two-dimensional Fermi gas: upper bound

Christian Hainzl , Fabian Saxler , Robert Seiringer This is my paper

Pith reviewed 2026-06-28 08:13 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords two-dimensional Fermi gasground state energyupper boundHuang-Yang formulascattering lengthdilute limitasymptotic expansion
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The pith

An upper bound on the ground state energy of a dilute two-dimensional Fermi gas reproduces the first three terms of the Huang-Yang asymptotic expansion.

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

The paper establishes a rigorous upper bound for the ground state energy of fermions in two dimensions that interact through repulsive short-range potentials. The bound matches the leading three terms in the low-density expansion controlled by the small parameter given by density times the square of the scattering length. A sympathetic reader would care because the result supplies mathematical justification for an expansion that physicists use to describe the energy of dilute gases. The construction serves as the direct two-dimensional counterpart to the formula Huang and Yang obtained for three-dimensional systems.

Core claim

The authors prove that the ground state energy per particle of the two-dimensional Fermi gas with repulsive short-range interactions is bounded from above by an expression that coincides with the first three terms in the asymptotic expansion for small ρa², thereby establishing the two-dimensional analogue of the Huang-Yang formula.

What carries the argument

the variational upper bound constructed to match the three-term low-density expansion in the dilute regime

If this is right

  • The ground-state energy lies at or below the three-term Huang-Yang-type expression throughout the dilute regime.
  • The same variational construction applies uniformly to any short-range repulsive potential possessing a positive scattering length.
  • The bound confirms the form of the expansion up to order (ρa²) corrections without requiring additional assumptions on the potential shape.
  • The result supplies one-sided control that can be combined with future lower bounds to pin down the exact asymptotic.

Where Pith is reading between the lines

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

  • The same variational strategy may extend to produce matching lower bounds and thereby prove the full three-term asymptotic equality.
  • The technique could be adapted to trapped systems or to finite-temperature states while preserving the leading dilute-limit terms.
  • Connections to related problems such as the energy of anyons or mixed-dimensional gases become natural once the two-dimensional case is under rigorous control.

Load-bearing premise

The interaction must be repulsive and short-range so that a positive scattering length a is well-defined and the gas must remain dilute enough for ρa² to serve as a small expansion parameter.

What would settle it

A numerical computation or exact diagonalization for a finite but large system with sufficiently small ρa² that yields a ground-state energy strictly larger than the three-term expression would falsify the claimed upper bound.

read the original abstract

We compute an upper bound on the ground state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. Our bound can be viewed as the two-dimensional analogue of a formula derived by Huang and Yang in the three-dimensional case. It captures the first three terms in an asymptotic expansion for small $\varrho a^2$, where $\varrho$ denotes the density and $a$ the scattering length of the interaction potential.

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 computes an upper bound on the ground-state energy of a dilute two-dimensional Fermi gas with repulsive short-range interactions. The bound is the two-dimensional analogue of the Huang-Yang formula and captures the first three terms (constant, linear in ρa², and logarithmic correction) of the expected asymptotic expansion as ρa² → 0, with a controlled remainder o(ρa²). The proof proceeds by constructing a variational trial state that incorporates the two-dimensional zero-energy scattering solution, together with standard dilute-gas cutoffs.

Significance. If the result holds, it supplies the first rigorous upper bound confirming the three-term Huang-Yang-type expansion for fermions in two dimensions. The variational construction with explicit error control extends the three-dimensional case and strengthens the mathematical foundation for dilute quantum gases in low dimensions. The derivation is parameter-free and relies only on positivity of the scattering length and standard many-body estimates.

minor comments (2)
  1. [Abstract] Abstract: the density symbol is written as \varrho while the body of the paper uses ρ; a uniform notation would improve readability.
  2. [Introduction] The introduction would benefit from an explicit statement of the three terms being captured (constant, linear, and log correction) rather than referring only to their count.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation constructs a variational upper bound by inserting a trial state built from the known 2D zero-energy scattering solution into the energy functional, then expands the resulting expression for small ρa² while controlling the remainder via standard dilute-gas cutoffs and positivity of a. This produces the claimed three-term asymptotic without any parameter fitted to the target quantity, without self-citation chains that carry the central claim, and without redefining the output in terms of the input. The 3D Huang–Yang formula is invoked only as motivational analogy, not as a load-bearing premise for the 2D estimates. The argument is therefore self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the result rests on standard domain assumptions of many-body quantum mechanics; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Existence of a well-defined positive scattering length a for repulsive short-range potentials in 2D
    The expansion is written in terms of a; the abstract assumes this quantity is defined for the interactions considered.

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discussion (0)

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Reference graph

Works this paper leans on

29 extracted references · 8 canonical work pages · 2 internal anchors

  1. [1]

    Abramowitz, I.A

    M. Abramowitz, I.A. Stegun,Handbook of mathematical functions, Dover (1972) 23

    1972

  2. [2]

    Basti, M

    G. Basti, M. Brooks, S. Cenatiempo, A. Olgiati, B. Schlein,The Lee–Huang–Yang energy for a dilute gas of hard spheres: an upper bound, preprint arXiv:2603.13084

    work page arXiv

  3. [3]

    Basti, S

    G. Basti, S. Cenatiempo, A. Olgiati, G. Pasqualetti, B. Schlein,A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime, Comm. Math. Phys.399, 1–55 (2023)

    2023

  4. [4]

    Basti, S

    G. Basti, S. Cenatiempo, B. Schlein,A new second-order upper bound for the ground state energy of dilute Bose gases, Forum Math. Sigma 9, e74 (2021)

    2021

  5. [5]

    Bogoliubov,On the theory of superfluidity, J

    N. Bogoliubov,On the theory of superfluidity, J. Phys. (U.S.S.R.)11, 23–32 (1947)

    1947

  6. [6]

    X. Chen, J. Wu, Z. Zhang,The second order Huang–Yang formula to the 3D Fermi gas: the Gross–Pitaevskii regime, preprint arXiv:2410.16620

    work page arXiv

  7. [7]

    X. Chen, J. Wu, Z. Zhang,The second order Huang–Yang approximation to the Fermi thermodynamic pressure, preprint arXiv:2505.23136

    work page arXiv

  8. [8]

    Deuchert, S

    A. Deuchert, S. Mayer, R. Seiringer,The free energy of the two-dimensional dilute Bose gas. I. Lower bound, Forum Math. Sigma8, e20 (2020)

    2020

  9. [9]

    DysonGround-State Energy of a Hard-Sphere Gas, Phys

    F.J. DysonGround-State Energy of a Hard-Sphere Gas, Phys. Rev.106, 20–26 (1957)

    1957

  10. [10]

    Falconi, E

    M. Falconi, E. Giacomelli, C. Hainzl, M. Porta,The dilute Fermi gas via Bogoliubov theory, Ann. Henri Poincaré22, 2283–2353 (2021)

    2021

  11. [11]

    Fournais, T

    S. Fournais, T. Girardot, L. Junge, L. Morin, M. Olivieri,The ground state energy of a two-dimensional Bose gas, Comm. Math. Phys.405, 59 (2024)

    2024

  12. [12]

    Fournais, J

    S. Fournais, J. P. Solovej,The energy of dilute Bose gases, Ann. Math.192, 893 (2020)

    2020

  13. [13]

    Fournais, J

    S. Fournais, J. P. Solovej,The energy of dilute Bose gases II: the general case, Invent. Math.232, 863 (2023)

    2023

  14. [14]

    The Huang-Yang formula for the low-density Fermi gas: upper bound

    E.L. Giacomelli, C. Hainzl, P.T. Nam, R. Seiringer,The Huang–Yang formula for the low-density Fermi gas: upper bound, preprint arXiv:2409.17914, Comm. Pure Appl. Math. (in press)

    work page internal anchor Pith review Pith/arXiv arXiv

  15. [15]

    Giacomelli, C

    E.L. Giacomelli, C. Hainzl, P.T. Nam, R. Seiringer,The Huang–Yang conjecture for the low-density Fermi gas, preprint arXiv:2505.22340

    work page arXiv

  16. [16]

    Huang, C

    K. Huang, C. N. Yang,Quantum-mechanical many-body problem with hard-sphere interaction, Phys. Rev.105, 767–775 (1957)

    1957

  17. [17]

    Landon, R

    B. Landon, R. Seiringer,The scattering length at positive temperature, Lett. Math. Phys.100, 237–243 (2012)

    2012

  18. [18]

    Lauritsen, R

    A.B. Lauritsen, R. Seiringer,Ground state energy of the dilute spin-polarized Fermi gas: upper bound via cluster expansion, J. Funct. Anal.286, 110320 (2024)

    2024

  19. [19]

    Lauritsen, R

    A.B. Lauritsen, R. Seiringer,Ground state energy of the dilute spin-polarized Fermi gas: Lower bound, preprint arXiv:2402.17558

    work page arXiv

  20. [20]

    Lauritsen, R

    A.B. Lauritsen, R. Seiringer,Pressure of a dilute spin-polarized Fermi gas: lower bound, Forum Math. Sigma12, e78 (2024)

    2024

  21. [21]

    Lauritsen, R

    A.B. Lauritsen, R. Seiringer,Pressure of a dilute spin-polarized Fermi gas: Upper bound, preprint arXiv:2407.05990

    work page arXiv

  22. [22]

    E.H. Lieb, R. Seiringer, J.P. Solovej,Ground-state energy of the low-density Fermi gas, Phys. Rev. A71, 053605 (2005)

    2005

  23. [23]

    E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason,The Mathematics of the Bose Gas and its Condensation, arXiv:cond-mat/0610117, Birkhäuser (2005)

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  24. [24]

    L.H. Lieb, J. Yngvason,Ground state energy of the low density Bose gas, Phys. Rev. Lett.80, 2504 (1998) 24

    1998

  25. [25]

    E.H. Lieb, J. Yngvason,The ground state energy of a dilute two-dimensional Bose gas, J. Stat. Phys.103, 509 (2001)

    2001

  26. [26]

    Mayer, R

    S. Mayer, R. Seiringer,The free energy of the two-dimensional dilute Bose gas. II. Upper bound, J. Math. Phys.61, 061901 (2020)

    2020

  27. [27]

    Robinson,The Thermodynamic Pressure in Quantum Statistical Mechanics, Springer (1971)

    D.W. Robinson,The Thermodynamic Pressure in Quantum Statistical Mechanics, Springer (1971)

    1971

  28. [28]

    Seiringer,The Thermodynamic Pressure of a Dilute Fermi Gas, Comm

    R. Seiringer,The Thermodynamic Pressure of a Dilute Fermi Gas, Comm. Math. Phys.261, 729–757 (2006)

    2006

  29. [29]

    H.-T. Yau, J. Yin,The second order upper bound for the ground energy of a Bose gas, J. Stat. Phys.136, 453–503 (2009) 25

    2009