A second order upper bound to the free energy of the two dimensional Bose gas

Florian Haberberger; Lukas Junge

arxiv: 2604.10270 · v1 · submitted 2026-04-11 · 🧮 math-ph · math.MP

A second order upper bound to the free energy of the two dimensional Bose gas

Florian Haberberger , Lukas Junge This is my paper

Pith reviewed 2026-05-10 15:49 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords two-dimensional Bose gasfree energy upper boundBogoliubov theoryBerezinskii-Kosterlitz-Thouless transitiondilute regimequasiparticle dispersion

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The pith

An explicit upper bound on the free energy density of the dilute two-dimensional Bose gas below the Berezinskii-Kosterlitz-Thouless temperature follows from Bogoliubov theory.

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

The paper establishes an explicit upper bound for the free energy density of a two-dimensional Bose gas when the dilute parameter rho a squared is small and the temperature lies below the superfluid transition. The bound is obtained by applying Bogoliubov theory to the many-body system and incorporates the energy contribution of quasiparticles whose dispersion is the square root of p to the fourth plus eight pi rho delta p squared. A sympathetic reader would care because the free energy determines all thermodynamic quantities and because two-dimensional Bose gases exhibit a distinctive transition where conventional mean-field methods fail. The explicit form of the bound supplies a concrete reference point against which more complete calculations or experiments can be checked.

Core claim

We derive an explicit upper bound for the free energy density of the two-dimensional Bose gas in the dilute regime using Bogoliubov theory for temperatures below the Berezinskii-Kosterlitz-Thouless critical temperature. The bound captures the contribution of quasiparticle modes with dispersion relation sqrt(p^4 + 8 pi rho delta p^2) where delta equals 2 divided by the absolute value of log of rho a squared plus the log of the absolute value of that logarithm.

What carries the argument

The Bogoliubov approximation applied to the interaction Hamiltonian, producing an upper bound on the free energy from the spectrum of quasiparticles obeying the dispersion sqrt(p^4 + 8 pi rho delta p^2) with the logarithmically renormalized delta.

Load-bearing premise

Bogoliubov theory remains accurate enough to produce a useful explicit upper bound on the free energy in the stated dilute regime below the BKT temperature.

What would settle it

A direct numerical evaluation of the free energy for a large but finite system in the dilute limit with temperature below the BKT point that lies above the derived upper bound would show the claim is false.

read the original abstract

We consider a two-dimensional Bose gas in the dilute regime where $\rho a^2$ is small. For temperatures below the Berezinskii-Kosterlitz-Thouless critical temperature, we derive an explicit upper bound for the free energy density using Bogoliubov theory. Our result captures the contribution of quasiparticle modes with dispersion relation $\sqrt{p^4 + 8\pi \rho\, \delta\, p^2}$ and where $\delta = 2 / (|\log(\rho a^2)| + \log |\log(\rho a^2)|)$.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit upper bound on the free energy of the 2D Bose gas below BKT via Bogoliubov theory with a double-log delta, but the error control on the approximation needs checking.

read the letter

The punchline is that Haberberger and Junge have produced an explicit second-order upper bound on the free energy of the 2D Bose gas below BKT by means of Bogoliubov theory with a renormalized dispersion involving a double-logarithmic delta. What is new here is the specific choice of delta = 2 / (|log(rho a^2)| + log |log(rho a^2)|) and the resulting bound that incorporates the quasiparticle contribution with dispersion sqrt(p^4 + 8 pi rho delta p^2). This form does not appear in the earlier literature they reference. The paper does well in making the bound explicit and tying it to the physical dilute regime. It gives a clear statement of the result that can be compared against other approximations or simulations. The soft spot lies in the use of Bogoliubov theory for the upper bound. The stress test points out that in 2D the phase fluctuations are delicate, and the quadratic approximation may not automatically give an upper bound with error smaller than the delta term. The derivation must include a quantitative control on the remainder from the full interaction Hamiltonian; if that is absent or insufficient, the claimed accuracy does not hold. This paper targets specialists in mathematical physics working on Bose gases and the BKT transition. A reader focused on deriving bounds for free energies in dilute quantum systems would find the explicit expression valuable for further study or extension. It deserves a serious referee because the claim is technical and specific enough that experts can verify the steps and the error estimates. I recommend sending it for peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript considers the two-dimensional Bose gas in the dilute regime (small ρa²). Below the Berezinskii-Kosterlitz-Thouless critical temperature, it derives an explicit upper bound on the free energy density by applying Bogoliubov theory. The bound incorporates the quasiparticle contribution with dispersion relation √(p⁴ + 8πρ δ p²), where δ = 2 / (|log(ρa²)| + log |log(ρa²)|).

Significance. If the upper bound is rigorously controlled with remainder o(δ), the result would supply a useful explicit second-order correction to the free energy in the 2D dilute Bose gas below the BKT scale. The direct incorporation of the renormalized dispersion and the parameter δ (arising from the logarithmic renormalization) is a clear strength of the claimed expression.

major comments (1)

[Abstract and main derivation (Bogoliubov approximation)] The central claim that Bogoliubov theory produces a controlled upper bound at order δ requires an explicit quantitative estimate that the error from the non-quadratic interaction terms is o(δ) as ρa² → 0, uniformly for T below the BKT temperature. In two dimensions the infrared fluctuations are strong; without such an a priori bound on the remainder (e.g., via a trial state or direct comparison with the full Hamiltonian), the displayed term may be contaminated by uncontrolled contributions of the same order. This issue is load-bearing for the second-order accuracy asserted in the abstract.

minor comments (1)

[Introduction and statement of results] Clarify the precise range of temperatures (relative to the BKT scale) for which the bound is stated to hold, and confirm that all constants in the dispersion are independent of the ultraviolet cutoff.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concern regarding the control of the error in the Bogoliubov approximation below.

read point-by-point responses

Referee: [Abstract and main derivation (Bogoliubov approximation)] The central claim that Bogoliubov theory produces a controlled upper bound at order δ requires an explicit quantitative estimate that the error from the non-quadratic interaction terms is o(δ) as ρa² → 0, uniformly for T below the BKT temperature. In two dimensions the infrared fluctuations are strong; without such an a priori bound on the remainder (e.g., via a trial state or direct comparison with the full Hamiltonian), the displayed term may be contaminated by uncontrolled contributions of the same order. This issue is load-bearing for the second-order accuracy asserted in the abstract.

Authors: We thank the referee for highlighting this key technical point. The upper bound in the manuscript is obtained variationally by evaluating the free-energy functional on the thermal state of the quadratic Bogoliubov Hamiltonian with the renormalized dispersion √(p⁴ + 8πρ δ p²). This automatically supplies a rigorous upper bound to the true free energy. The explicit form we display arises from the quadratic contribution together with the choice of δ that encodes the leading logarithmic renormalization. We agree that a fully quantitative a priori bound showing that the expectation of the non-quadratic remainder is o(δ) uniformly below the BKT temperature would make the second-order character completely rigorous. Such an estimate is not carried out in detail in the present work. In the revised version we will add a dedicated remark clarifying the status of the remainder and sketching how the infrared fluctuations are suppressed by the specific form of δ and the dilute limit. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses standard Bogoliubov quadratic approximation with explicit dispersion correction expressed in physical parameters

full rationale

The paper derives an explicit upper bound on the free energy density below the BKT temperature by applying Bogoliubov theory to the dilute 2D Bose gas. The dispersion relation sqrt(p^4 + 8 pi rho delta p^2) with delta = 2 / (|log(rho a^2)| + log |log(rho a^2)|) is written directly in terms of the physical dilute parameter rho a^2; it is not obtained by fitting to the target free-energy quantity or by renaming a prior result. No self-definitional step, fitted-input prediction, or load-bearing self-citation chain appears in the derivation chain. The central claim remains an independent (if possibly uncontrolled) upper bound constructed from the quadratic Hamiltonian, without reducing by construction to its own inputs. This is the normal non-circular outcome for a paper that applies an established approximation technique to a new regime.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the applicability of Bogoliubov theory to the dilute 2D Bose gas below the BKT temperature and on standard properties of the two-dimensional scattering length.

axioms (1)

domain assumption Bogoliubov theory yields a controlled upper bound on the free energy in the dilute regime below the BKT temperature
Invoked to obtain the explicit bound from the quasiparticle dispersion.

pith-pipeline@v0.9.0 · 5388 in / 1322 out tokens · 31256 ms · 2026-05-10T15:49:10.591746+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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J.Andersen, Ground state pressure and energy density of an interacting homogeneous Bose gas in two dimensions,Eur. Phys. J. B28.4(2002), pp. 389–396

work page 2002

[2] [2]

G.Basti, C.Boccato, S.Cenatiempo, and A.Deuchert, A new upper bound on the specific free energy of dilute Bose gases, arXiv:2507.20877 (2025)

work page arXiv 2025

[3] [3]

G.Basti, S.Cenatiempo, A.Giuliani, A.Olgiati, G.Pasqualetti, and B.Schlein, Upper bound for the ground state energy of a dilute bose gas of hard spheres,Arch. Ration. Mech. Anal.248.6(2024)

work page 2024

[4] [4]

G.Basti, S.Cenatiempo, A.Olgiati, G.Pasqualetti, and B.Schlein, A Sec- ond Order Upper Bound for the Ground State Energy of a Hard-Sphere Gas in the Gross–Pitaevskii Regime,Commun. Math. Phys.399.1(2023), pp. 1–55

work page 2023

[5] [5]

G.Basti, S.Cenatiempo, and B.Schlein, A new second-order upper bound for the ground state energy of dilute Bose gases,Forum of Mathematics, Sigma9(2021), e74

work page 2021

[6] [6]

L.Berezinskii, Destruction of Long-range Order in One-dimensional and Two-dimen- sional Systems Possessing a Continuous Symmetry Group

V. L.Berezinskii, Destruction of Long-range Order in One-dimensional and Two-dimen- sional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems,Sov. Phys. JETP34.3(1972), pp. 610–616

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N.Bose, Plancks Gesetz und Lichtquantenhypothese,Z

S. N.Bose, Plancks Gesetz und Lichtquantenhypothese,Z. Phys.26(1924), pp. 178–181

work page 1924

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Henri Poincar´ e24.8(2023), pp

C.Caraci, S.Cenatiempo, and B.Schlein, The excitation spectrum of two-dimensional bose gases in the gross–pitaevskii regime,Ann. Henri Poincar´ e24.8(2023), pp. 2877–2928

work page 2023

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Y.Chernyand A

A. Y.Chernyand A. A.Shanenko, Dilute Bose gas in two dimensions: Density expan- sions and the Gross-Pitaevskii equation,Phys. Rev. E64.2(2001), 027105

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

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J.Dyson, Ground-state energy of a hard-sphere gas,Phys

F. J.Dyson, Ground-state energy of a hard-sphere gas,Phys. Rev.106.1(1957), pp. 20– 26

work page 1957

[12] [12]

Zweite Abhandlung.Sitzungs- berichte der Preußischen Akademie der Wissenschaften, I(1925), pp

A.Einstein, Quantentheorie des einatomigen idealen Gases. Zweite Abhandlung.Sitzungs- berichte der Preußischen Akademie der Wissenschaften, I(1925), pp. 3–14

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L.Erd ˝os, B.Schlein, and H.-T.Yau, The ground state energy of a low density Bose gas: a second order upper bound,Phys. Rev. A78.5(2008), 053627

work page 2008

[14] [14]

M.F annes, A continuity property of the entropy density for spin lattice systems,Com- munications in Mathematical Physics31(1973), pp. 291–294

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S.Fournais, T.Girardot, L.Junge, L.Morin, and M.Olivieri, The ground state energy of a two-dimensional bose gas,Commun. Math. Phys.405.3(2024), 59

work page 2024

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S.Fournais, T.Girardot, L.Junge, L.Morin, M.Olivieri, and A.Triay, The free energy of dilute Bose gases at low temperatures interacting via strong potentials, arXiv:2408.14222 (2024)

work page arXiv 2024

[17] [17]

The free energy of dilute Bose gases at low temperatures,

F.Haberberger, C.Hainzl, P. T.Nam, R.Seiringer, and A.Triay, The free energy of dilute Bose gases at low temperatures, arXiv:2304.02405 (2023)

work page arXiv 2023

[18] [18]

F.Haberberger, C.Hainzl, B.Schlein, and A.Triay, Upper bound for the free energy of dilute bose gases at low temperature, arXiv:2405.03378 (2024)

work page arXiv 2024

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C.Hohenberg, Existence of long-range order in one and two dimensions,Phys

P. C.Hohenberg, Existence of long-range order in one and two dimensions,Phys. Rev. 158.2(1967), pp. 383–386. 21

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Rev.98.5(1955), pp

R.Jastrow, Many-body problem with strong forces,Phys. Rev.98.5(1955), pp. 1479– 1484

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M.Kosterlitzand D

J. M.Kosterlitzand D. J.Thouless, Ordering, metastability and phase transitions in two-dimensional systems,J. Phys. C: Solid State Phys.6.7(1973), pp. 1181–1203

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H.Lieb, R.Seiringer, J

E. H.Lieb, R.Seiringer, J. P.Solovej, and J.Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach seminars, Birkh¨ auser, (2005)

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H.Lieband J.Yngvason, The ground state energy of a dilute two-dimensional bose gas,Journal of Statistical Physics103.3(2001), pp

E. H.Lieband J.Yngvason, The ground state energy of a dilute two-dimensional bose gas,Journal of Statistical Physics103.3(2001), pp. 509–526

work page 2001

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S.Mayerand R.Seiringer, The free energy of the two-dimensional dilute Bose gas. II. Upper bound,Journal of Mathematical Physics61.6(2020), 061901

work page 2020

[25] [25]

D.Merminand H.W agner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models,Phys

N. D.Merminand H.W agner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models,Phys. Rev. Lett.17.22(1966), pp. 1133–1136

work page 1966

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C.Moraand Y.Castin, Extension of Bogoliubov theory to quasicondensates,Phys. Rev. A67.5(2003), 053615

work page 2003

[27] [27]

C.Moraand Y.Castin, Ground state energy of the two-dimensional weakly interacting bose gas: First correction beyond bogoliubov theory,Phys. Rev. Lett.102.18(2009), 180404

work page 2009

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A.Nielsenand I

M. A.Nielsenand I. L.Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, 1st ed., Cambridge University Press, (2012)

work page 2012

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W.Robinson, The Thermodynamic Pressure in Quantum Statistical Mechanics,9, Lecture Notes in Physics, Berlin, Heidelberg: Springer Berlin Heidelberg, (1971)

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D.Ruelle, Statistical mechanics: Rigorous results, Mathematical physics monograph se- ries, New York: W. A. Benjamin, (1969)

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M.Schick, Two-dimensional system of hard-core bosons,Phys. Rev. A3.3(1971), 1067– 1073

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R.Seiringer, Free energy of a dilute bose gas: Lower bound,Commun. Math. Phys. 279.3(2008), pp. 595–636

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R.Seiringer, The Thermodynamic Pressure of a Dilute Fermi Gas,Commun. Math. Phys.261.3(2006), pp. 729–757

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P.Solovej, Mathematical physics of dilute Bose gases,Comptes Rendus

J. P.Solovej, Mathematical physics of dilute Bose gases,Comptes Rendus. Physique26 (G1 2025), pp. 339–348

work page 2025

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S.Vladimirov, Equations of mathematical physics, Illustrated reprint,3, Monographs and textbooks in pure and applied mathematics, New York: M.Dekker, (1971)

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J.Yin, Free energies of dilute bose gases: Upper bound,Journal of Statistical Physics 141.4(2010), pp. 683–726. 22

work page 2010