Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures

Maksim D. Tomchenko

REVIEW 4 major objections 4 minor 38 references

Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper argues that a finite periodic system of interacting spinless bosons at temperatures far above the ideal-gas Bose-Einstein condensation temperature has zero-momentum occupation N0 ~ 1, so no condensate of zero-momentum atoms forms.

desk verdict The conclusion is true, but the proof is not: the key approximation for N-quasiparticle states is wrong even in the ideal-gas limit. read the letter →

arxiv 2501.03029 v1 pith:IL54ZQ63 submitted 2025-01-06 cond-mat.other

Maksim D. Tomchenko This is my paper · ORCID

classification cond-mat.other

keywords Bose-Einsteincondensateultrahightemperaturesinteractingbosonszero-momentumoccupationelementaryquasiparticlesdensityfluctuationoperatorscanonicalensembleperiodicboundaryconditions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a spatially uniform system of interacting spinless bosons, held at temperatures far above the ideal-gas Bose-Einstein condensation temperature, cannot sustain a condensate of zero-momentum atoms. The author starts from the exact canonical-ensemble formula for the zero-momentum occupation number and claims that at such ultrahigh temperatures the states that dominate the thermal average are those containing the maximum number N of elementary quasiparticles, each with large momentum. In the zero-order approximation used here, those N-quasiparticle states have negligible zero-momentum occupation, of order 1 rather than order N. If the argument holds, it would supply the missing interacting-system counterpart of the well-known ideal-gas result that N0 vanishes above Tc^(i).

What carries the argument

The load-bearing object is the exact occupation formula (7), together with the zero-order wavefunction approximation (15)-(19) in which the ground state is treated as a constant and an N-quasiparticle state is approximated by the product ρ_{-p1}ρ_{-p2}···ρ_{-pN} of density-fluctuation operators. This approximation, originally motivated for a dilute gas, converts the statement that the statistical sum is dominated by N-quasiparticle states into the statement that each dominant state has zero-momentum occupation near unity. The paper also relies on the claim, taken from earlier work, that the maximum possible number of elementary quasiparticles is N, and on the standard quasiparticle dispersion law to argue that quasiparticles at large momenta behave like free atoms.

What would settle it

A direct numerical computation of the one-body density matrix for a finite periodic system of interacting bosons, for example via path-integral Monte Carlo for N = 256 particles at T = 10 Tc^(i) with a short-range repulsive potential, would settle the claim: if the zero-momentum occupation grows with N instead of staying of order 1, the proposed mechanism fails.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the exact thermal average N0 = (1/Z) Σ_℘ $e^{{-E_℘/k_BT}}$ <Ψ_℘|a0^+ a0|Ψ_℘> can be evaluated by identifying which stationary states contribute at ultrahigh temperatures. The paper claims that for T >> Tc^(i) the dominant states contain N elementary quasiparticles with large momenta, and that in the zero-order approximation these states are represented by wavefunctions built from products of density-fluctuation operators acting on a constant ground state, so each such state has N0^(℘) ~ 1. Since the majority of significant terms in the sum are of this type, the weighted average N0 remains of order 1, not of order N, implying the absence of a Bose-Einstein condensate of zero-momentum atoms.

Load-bearing premise

The argument's load-bearing assumption is that the zero-order approximation for highly excited states, developed for a dilute gas, remains valid for arbitrary interactions at ultrahigh temperatures, so that N-quasiparticle states indeed have zero-momentum occupation of order 1 rather than order N.

Editorial extensions

If this is right

At ultrahigh temperatures the momentum distribution of an interacting Bose gas should approach a Maxwell-Boltzmann form with no macroscopic zero-momentum peak.
The crossover from condensate to no condensate is smooth, driven by the increasing occupation of N-quasiparticle states, rather than by a sharp transition.
The same reasoning, if extended to the grand canonical ensemble as the paper suggests, would predict the same absence of a condensate.
The 'blurring' of the condensate is monotone in temperature: as the number of quasiparticles rises toward N, the zero-momentum occupation drops from order N to order 1.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A stricter test of the zero-order approximation, such as solving the few-quasiparticle wavefunctions beyond the dilute-gas limit, would determine whether the extrapolation to arbitrary interactions is justified; this is a gap the paper leaves open.
The claim that N0 ~ 1 for finite periodic systems suggests that in a finite box the condensate never strictly vanishes but becomes a microscopic occupation; experimental probes of the momentum distribution in trapped gases at high temperature could look for this residue.
If the mechanism is right, it may also apply to other quasiparticle condensates, such as magnons or polaritons, at temperatures far above their 'ideal-gas' transition scales, though the paper does not make that extension.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The paper claims an approximate proof that interacting bosons at T >> Tc have no BEC. The conclusion is certainly true, but the proof does not hold up. The exact formula (7) is fine, and it is a fair point that a rigorous proof for interacting systems is missing. The author deserves credit for being explicit that the argument is approximate and sketchy.

The load-bearing step is the zero-order wavefunction (19): a state with N quasiparticles is written as a product of N density fluctuation operators acting on a constant ground state. This is not a controlled approximation, and it is actually wrong in the limit where it should be exact — the noninteracting gas. The exact N-boson state with distinct momenta is a permanent of plane waves. Applying N density operators produces a sum in which many terms have two or more operators acting on the same particle, meaning fewer than N excited particles and some particles with combined momenta. For N operators on N particles, the permutation terms that give the correct permanent are a vanishing fraction of all terms (N!/N^N ~ e^{-N}). So the state in (19) is dominated by multiple-occupancy configurations and is not the state of N well-defined quasiparticles. Consequently the assertion that N0 ~ 1 for these states is unsupported. The argument that interactions 'blur' the condensate further does not repair this; the zero-order basis is already wrong without interactions.

There are softer problems as well. The dominance of N-quasiparticle states in the partition function is asserted by counting, not proven. The extrapolation from a dilute gas to arbitrary interactions is hand-wavy, as the author admits. The author's prior papers are cited for the maximum quasiparticle number and the complete wavefunction set; self-citation is not automatically a flaw, but here those premises are exactly what is not established independently.

So the paper is honest, clearly written, and addresses a genuine gap in the literature. But the central derivation is not a proof, and the error in Eq. (19) is fatal to the argument. I would not send this to a referee; it would be more appropriate as an explicitly heuristic preprint than as a research paper. My recommendation is to reject with no invitation to revise, unless the author can provide a correct treatment of the N-quasiparticle states.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes an approximate mathematical proof that in a finite, nonrelativistic, periodic system of N interacting spinless bosons at temperatures T >> T_c^(i) (the ideal-gas Bose-Einstein condensation temperature), the zero-momentum occupation N0 satisfies N0 ~ 1 << N, so that no Bose-Einstein condensate of zero-momentum atoms exists. The argument starts from the exact canonical-ensemble expression (7) for N0, asserts that at ultrahigh temperatures the dominant contribution to the partition function comes from states containing N elementary quasiparticles, and then uses the zero-order wavefunctions (15)-(19) to claim that such states have N0 ~ 1. The paper concludes that an ultrahigh temperature 'blurs' the condensate. The author explicitly acknowledges that the reasoning is not rigorous.

Significance. If the claim were established, the paper would fill a recognized gap in the literature, since the absence of BEC at ultrahigh temperatures is widely believed but, according to the author, not rigorously proven for interacting systems. The manuscript correctly identifies the exact formula (7) as a useful starting point and is transparent about the approximate nature of the proof. However, the central derivation rests on unproven and questionable assumptions, and the presented arguments do not constitute a sound proof of the main claim. The paper is thus more a research sketch than a complete proof, and its significance as a citable result is limited by the lack of rigor in the load-bearing steps.

major comments (4)

[Sec. II, Eq. (19)] The zero-order wavefunction for N quasiparticles is not correct even in the ideal-gas limit, which is the limit in which the approximation should be most accurate. For an ideal gas, the N-particle state with momenta p1,...,pN is a symmetrized permanent of plane waves. The product rho_{-p1}...rho_{-pN} (with Psi0=1) contains terms in which two or more density operators act on the same particle, for example e^{-i(p1+p2)r_j}; such terms are absent from the permanent. For N=2, this is readily verified explicitly. Consequently, the expectation value of N0 computed from (19) does not represent the true N0 of the corresponding eigenstates. Since the central conclusion that N-quasiparticle states have N0 ~ 1 is derived directly from (19), the proof fails at its key step.
[Sec. II, around Eq. (7) and the paragraph after properties (i)-(iii)] The assertion that at T >> T_c^(i) the main contribution to the partition function comes from states containing exactly N quasiparticles is not derived. The paper presents a qualitative counting argument ('such states constitute the majority among all possible states') but provides no quantitative estimate of the number of states or of the Boltzmann weights e^{-E/kT}. Without such an estimate, the exact formula (7) cannot be reduced to the N-quasiparticle sector, and the necessity/sufficiency argument for the conclusion is not established.
[Sec. II, paragraph 'The formulae (15)-(19) do not take into account the interatomic interaction'] The extrapolation from a dilute gas to arbitrary interaction strength is uncontrolled. The zero-order wavefunctions (15)-(19) are justified only in the dilute-gas limit where Sg ≈ 0. The claim that interactions cause only 'additional blurring' of the condensate is qualitative and is not supported by any estimate of N0 for strongly interacting states. Since the abstract states the result for interacting bosons in general, this gap is material to the paper's central claim.
[Sec. II, discussion of maximum quasiparticle number] Two load-bearing premises—the maximum possible number of elementary quasiparticles being N, and the form of multi-quasiparticle wavefunctions—are cited from the author's prior works (refs [18,19,29]) and are not derived or independently verified in this manuscript. While citing prior work is common, these premises are the substantive content of the proof; without their derivation or an explicit statement that they are assumptions, the argument is not self-contained. The paper's own admission that the reasoning is 'not rigorous' (Conclusion) is consistent with this limitation.

minor comments (4)

[Title and header] There are typographical errors in the title/header, such as 'in teracting' and 'condensa te', which appear to be hyphenation artifacts and should be corrected.
[Throughout] The notation for the state-dependent occupation number, written as 'N (℘ ) 0', is typeset inconsistently and is hard to read; a consistent notation such as N0^(wp) would improve clarity.
[Sec. II, paragraph after properties (i)-(iii)] The sentence 'properties (i), (ii) and (iii) jointly imply...' reads as a derivation, but the text actually presents a heuristic argument. The authors should label this as an assumption or conjecture to avoid overstating the logical status of the claim.
[References] Ref. [19] is an arXiv preprint; if the proof depends on results from that preprint, the manuscript should indicate whether those results have undergone peer review, given their centrality to the argument.

Circularity Check

3 steps flagged · score 7.0 of 10

The claimed no-condensate proof reduces to the author's prior quasiparticle maximum-N result and the zero-order ansatz (19); Eq. (7) is exact but does not independently force N0 ~ 1.

self definitional [Section II, paragraph following Eq. (19)]
"Finally, the state (19) represents N quasiparticles with momenta p1,..., pN ; in this case, N atoms have the same momenta p1,..., pN . Thus, with an increase in the number of elementary quasiparticles, the BE condensate of zero-momentum atoms gradually fades away. And when the number of quasiparticles becomes close to N , this condensate disappears completely."

The conclusion that N-quasiparticle states have no zero-momentum condensate is made true by definition here: Eq. (19) is interpreted as placing all N atoms into the nonzero momenta p1,...,pN. This is not a computed value of N0^(℘) from Eq. (19); it is simply the reading of the ansatz. The later inference that gas states containing N elementary quasiparticles have N0^(℘) ~ 1 therefore restates the input quasiparticle picture as the output of the thermal average (7), rather than deriving it from the exact formula.
self citation load bearing [Section II, paragraph beginning 'There are two ways of showing...']
"There are two ways of showing that the maximum possible number of elementary quasiparticles is N (see appendix 1 in [18] and section 7 in [29]). For a Bose gas, this can be seen without using formulae: quasiparticles with large momentum |p| have the energy ǫ(p) ≈ ℏ2p2/2m [15] and are similar to free atoms; the number of the latter is equal to N , hence the maximum possible number of elementary quasiparticles with large |p| is also equal to N ."

The proof requires that at T >> Tc the dominant states contain N quasiparticles, so that only N-quasiparticle states need be considered in Eq. (7). The maximum quasiparticle number N is justified by pointing to the author's own prior papers ([18] and [29], with [19] cited nearby). No independent, externally verified, or machine-checked derivation is given in the present paper. Thus a load-bearing premise of the argument is imported from the author's earlier quasiparticle framework rather than established from the exact thermal average.

1 more flagged steps

ansatz smuggled in via citation [Section II, Eqs. (17)-(19)]
"Similarly, the state containing a quasiparticle with momentum p1 and another quasiparticle with momentum p2, in the zero-order approximation can be described by the WF [18, 28] Ψ p1p2 (r1,..., rN ) ≈ρ−p1ρ−p2, and a state with N quasiparticles with momenta p1, p2,..., pN is described in the same approximation by the WF Ψ p1p2... pN (r1,..., rN ) ≈ρ−p1ρ−p2 · · ·ρ−pN. (19)"

The N-quasiparticle wavefunction (19) is introduced as the 'same approximation' as the two-quasiparticle form, citing the author's own [18] while [28] provides only the single-quasiparticle case. This zero-order ansatz already encodes the desired conclusion: each ρ factor is interpreted as moving one atom out of k = 0, so an N-quasiparticle state has all atoms in nonzero momentum states and hence N0 ~ 1. The ansatz is not derived from Eq. (7) and is not benchmarked; for an ideal gas it does not reproduce the exact symmetrized permanent of plane waves. The central claim therefore depends on an unsupported and self-cited ansatz rather than on the exact average (7).

full rationale

The paper's exact starting point, Eq. (7), is merely the canonical thermal average of N0 and does no work by itself. The nontrivial assertions are (i) that the dominant states at T >> Tc contain exactly N quasiparticles and (ii) that such states have N0 ~ 1. Assertion (i) is supported by the author's own prior results [18,19,29] for the maximum quasiparticle number, which are not re-derived or independently verified here. Assertion (ii) follows from the zero-order wavefunction ansatz (19), in which N density-fluctuation operators are interpreted as placing every atom into a nonzero momentum state; the claimed absence of condensate is built into that interpretation. The paper itself concedes 'This reasoning is not rigorous,' and no external benchmark, code verification, or independent theorem is supplied. Consequently, the derivation does not establish the no-condensate result from first principles; it unpacks the author's quasiparticle picture in which the conclusion is already present. This warrants a circularity score of 7 rather than the maximum, because the exact thermal-average formalism is stated and the physical claim may well be true, but the paper's proof chain is not independent of its own assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The argument uses no fitted constants. It rests on domain assumptions about which thermal states dominate and on wavefunction approximations taken from the author's earlier papers. No new physical entities are introduced.

assumptions (4)

domain assumption The dominant contribution to the canonical partition function at T >> Tc comes from states containing N quasiparticles with large momenta and free-particle-like dispersion.
Asserted in Section II based on Maxwell-Boltzmann statistics and physical considerations; not derived.
domain assumption The maximum possible number of elementary quasiparticles in an N-boson system is N, and wavefunctions (11) and (12) form a complete description of excited states.
Taken from the author's prior works [18, 29]; no independent derivation appears in this paper.
ad hoc to paper For a dilute Bose gas, the zero-order wavefunctions (15)-(19) are valid, including for states with N quasiparticles, and interaction effects only blur the condensate further.
Introduced in Section II as an approximation and extrapolated to dense gas and liquid without proof.
domain assumption At ultrahigh temperatures the system is a gas described by the Bogoliubov dispersion at large momenta.
Based on experimental observation and stated in Section II; not proven for arbitrary interaction strength.

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Cite this review

Pith. "Pith review of Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures." pith.science (2026). https://pith.science/paper/IL54ZQ63

@misc{pith2026250103029,
  author       = {Pith},
  title        = {Pith review of: Why a Bose-Einstein condensate cannot exist in a system of interacting bosons at ultrahigh temperatures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IL54ZQ63}},
  note         = {Machine review of arXiv:2501.03029}
}

abstract

It is well known that a Bose-Einstein (BE) condensate of atoms exists in a system of interacting Bose atoms at $T\lesssim T^{(i)}_{c}$, where $T^{(i)}_{c}$ is the BE condensation temperature of an ideal gas. It is also generally accepted that BE condensation is impossible at ``ultrahigh'' temperatures $T\gg T^{(i)}_{c}$. While the latter property has been theoretically proven for an ideal gas, no such proof exists for an interacting system, to our knowledge. In this paper, we propose an approximate mathematical proof for a finite, nonrelativistic, periodic system of $N$ spinless interacting bosons. The key point is that, at $T\gg T^{(i)}_{c}$, the main contribution to the occupation number $N_{0}=\frac{1}{Z}\sum_{\wp}e^{-E_{\wp}/k_{B}T}\langle \Psi_{\wp}|\hat{a}^{+}_{\mathbf{0}}\hat{a}_{\mathbf{0}}|\Psi_{\wp}\rangle$, corresponding to atoms with zero momentum, originates from the states containing $N$ elementary quasiparticles. These states do not contain the BE condensate of zero-momentum atoms, implying that an ultrahigh temperature should ``blur'' such a condensate.

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Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

Novel Superﬂuids , K. H. Bennemann and J. B. Ketterson (eds.), Oxford University Press, Oxford (2014), Vol. 2. https://doi.org/10.1093/acprof:oso/9780198719267.001.0001

arXiv 2014

[2] [3]

blurring

and are similar to free atoms; the number of the lat- ter is equal to N , hence the maximum possible number of elementary quasiparticles with large |p| is also equal to N . The treatment in [18, 29] shows that this conclu- sion holds for elementary quasiparticles with arbitrary momenta. Interestingly, the number of rotons in 4He at T =Tλ is of the order o...

work page

[3] [4]

Mannarelli, Particles 2, 411 (2019)

M. Mannarelli, Particles 2, 411 (2019). https://doi.org/10.3390/particles2030025

doi:10.3390/particles2030025 2019

[4] [5]

S. O. Demokritov, V. E. Demidov, O. Dzyapko, G. A. Melkov, A. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430 (2006). https://doi.org/10.1038/nature05117

doi:10.1038/nature05117 2006

[5] [6]

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doi:10.1016/0370-1573(90)90132-l 1990

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Kogar, M

A. Kogar, M. S. Rak, S. Vig, et al., Science 358, 1314 (2017). https://doi.org/10.1126/science.aam6432

work page doi:10.1126/science.aam6432 2017

[7] [8]

D. Wang, N. Luo, W. Duan, and X. Zou, J. Phys. Chem. Lett. 12, 5479 (2021). https://doi.org/10.1021/acs.jpclett.1c01370

work page doi:10.1021/acs.jpclett.1c01370 2021

[8] [9]

(see also review [11] and books [1, 2]). Furthermore, while BE condensation in liquid 4He and trapped gases occurs at low ( ∼ 1K) and ultralow ( < ∼ 10−5K) temper- atures, respectively, BE condensation of quasiparticles and mesons occurs at high temperatures, ∼ 100K and ∼ 1012K respectively. Despite the signiﬁcant variation in temperature, in all these fo...

work page