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REVIEW 3 major objections 5 minor 45 references

On the anomalous density of a dilute homogeneous Bose gas

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The sign of the anomalous density in a dilute Bose gas is an unobservable phase convention, while its absolute value can be extracted from sound-velocity and condensed-fraction measurements.

desk verdict Interesting HFB phase argument, but the extraction formula for |σ| is undercut by a sign inconsistency in the regularization of the divergent anomalous-density integral. read the letter →

arxiv 2411.15816 v1 pith:J5GN4IVX submitted 2024-11-24 cond-mat.quant-gas

Abdulla Rakhimov , Mukhtorali Nishonov This is my paper · ORCID
classification cond-mat.quant-gas PACS 67.85.-d67.40.Mj67.57.Bc
keywords anomalousdensityBose-EinsteincondensationHartree-Fock-BogoliubovtheoryHugenholtz-PinesrelationsoundvelocitycondensedfractionuniformboxpotentialBogoliubovtransformation
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

This paper argues that the anomalous density $\sigma$ in a dilute Bose-Einstein condensate is a phase-dependent object whose sign is not observable. In a conserving Hartree-Fock-Bogoliubov theory that keeps the condensate phase arbitrary, the anomalous density satisfies $\sigma=-\xi^2\bar{\sigma}$, so its sign simply records whether the discrete condensate phase satisfies $\xi^2=+1$ or $\xi^2=-1$; the phase is constrained to these two values by the extended Hugenholtz-Pines relation plus spectrum stability. What is physical is the modulus $|\sigma|$, which the paper shows is linked to two measurable quantities, the sound velocity and the condensed fraction. The result turns a long-puzzling many-body quantity into a concrete experimental target, and the paper provides numerical predictions for $|\sigma|$ in a uniform-box $^{87}$Rb condensate.

What carries the argument

The central objects are the anomalous density $\sigma$ (the density of binary correlated, non-condensed pairs) and the condensate phase $\xi=e^{i\theta}$. The argument is carried by three pieces: a two-chemical-potential Hartree-Fock-Bogoliubov scheme that keeps the spectrum gapless while remaining self-consistent; the phase-generalized Hugenholtz-Pines relation $\Sigma_n(0,0)-\bar{\xi}^2\Sigma_{an}(0,0)=\mu_1$, which together with stability forces $\xi^4=1$; and the identity $\sigma=-\xi^2\bar{\sigma}$ with the extraction formula $|\sigma|/\rho=n_0-ms^2/(g\rho)$, which converts measurements of sound velocity $s$ and condensed fraction $n_0$ into a value for the anomalous density.

What would settle it

Measure the sound velocity and condensed fraction of a single uniform-box $^{87}$Rb condensate over a range of temperatures, insert them into $m_1 = n_0 - ms^2/(g\rho)$, and compare with the paper's predicted curve for $|\sigma|$; a systematic mismatch, or a dependence of the result on which regularization is used to compute $\sigma_0$, would refute the central claim.

Watch

Extended reading notes

Core claim

The authors establish that the phase of the condensate wave function in a homogeneous dilute Bose gas is not arbitrary after all in this theory: combining the phase-generalized Hugenholtz-Pines relation with the demand that a stable equilibrium have positive, real self-energies forces $\xi^4=1$, i.e. $\xi^2=\pm1$. With this constraint the anomalous density is completely determined by the phase through $\sigma=-\xi^2\bar{\sigma}$, where $\bar{\sigma}\ge 0$ is built from momentum integrals over the bogolon dispersion, so its sign is a gauge-degree-of-freedom artifact. The magnitude, however, is tied to the gap parameter $\Delta=ms^2$ via $\Delta=g(\rho_0+\sigma)$, giving the extraction formula $|\sigma|/\rho = n_0 - ms^2/(g\rho)$. Solving the coupled equations numerically for $^{87}$Rb in a uniform box, the paper reproduces measured condensate-fraction data and predicts that $|\sigma|$ rises with temperature, peaks near $t\approx 0.7$, and vanishes at $T_c$.

Load-bearing premise

The predicted value and low-temperature sign of $\sigma$ depend on the regularization chosen for the divergent zero-temperature momentum integral, and the paper does not show that this choice is fixed by a physical condition.

Editorial extensions

If this is right

  • The sign of the anomalous density is not a physical observable; whether $\sigma$ appears positive or negative at low temperature is a phase convention, not a fact about the system.
  • The absolute value $|\sigma|$ can be obtained from measurements of the sound velocity and condensed fraction in a uniform-box condensate using the paper's formula.
  • For dilute $^{87}$Rb in a uniform box, $|\sigma|$ is predicted to grow with temperature, peak around $t\approx 0.7$, and fall to zero at $T_c$, while the condensed fraction and sound velocity remain monotonic.
  • The Hartree-Fock-Popov approximation, which neglects $\sigma$, is adequate for small gas parameter $\gamma$ but becomes unjustified for $\gamma \gtrsim 10^{-2}$.
  • The theory reproduces available condensate-fraction data for $^{87}$Rb in a uniform box without fitted parameters.

Reading between the lines

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

  • If the extracted $|\sigma|$ from Eq. (63) is ever compared with a direct measurement of pair correlations, the scheme-dependence noted in the paper for $\sigma_0$ would make the theory falsifiable in a sharper way than the authors spell out.
  • The discrete-phase constraint $\xi^4=1$ suggests a four-fold phase degeneracy for a homogeneous BEC; an experiment that appears to resolve the condensate phase would actually be probing a non-observable convention, much as the paper argues for interference experiments.
  • The same two-chemical-potential Hartree-Fock-Bogoliubov machinery can be applied to quantum magnets, where the condensate phase is tied to the magnetization direction; there the constraint could select only certain magnetic phases, which the paper mentions but does not pursue.
  • A natural extension would be to derive the regularization condition for $\sigma_0$ from a physical requirement, such as matching the zero-temperature Bogoliubov sound velocity, which would remove the main scheme ambiguity in the predicted $|\sigma|$.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops a Hartree-Fock-Bogoliubov theory for a homogeneous dilute Bose gas with an arbitrary condensate phase. It argues that the Hugenholtz-Pines-Watabe relation plus stability forces the condensate phase to satisfy ξ² = ±1, that the sign of the anomalous density σ is tied to this phase and is therefore not observable, but that |σ| can be extracted from measurements of the sound velocity s and the condensate fraction n0 through the relation |σ|/ρ = n0 - ms²/(gρ). Numerical predictions for |σ| are presented for a uniform 87Rb gas, and the paper also discusses a nonmonotonic temperature dependence of σ.

Significance. If correct, the paper would resolve a long-standing question about the anomal density in Bose-Einstein condensates and provide a practical route to its experimental determination. The phase-invariance argument and the constraint ξ² = ±1 are interesting and potentially useful for the HFB formalism. However, the central quantitative claim—that |σ| can be uniquely extracted from n0 and s—is undermined by an internal sign inconsistency and by a renormalization-scheme dependence in the evaluation of the zero-temperature anomalous density. The paper does include a self-contained derivation with no fitted parameters, which is a strength, but the numerical predictions for |σ| are not robust as presented.

major comments (3)
  1. [Sec. 5, Eqs. (66)-(67); Appendix A, Eq. (A.4)] The extraction formula Eq. (63), m1 = n0 - Δ/(gρ), is derived under the assumption that σ is negative, so that Δ = g(ρ0 - |σ|). However, Eq. (67) and Eq. (A.4) give the zero-temperature anomalous density as positive: σ0 = (mΔ)^{3/2}/π² > 0. For the positive branch, the relation between Δ, n0, and |σ| would instead be |σ|/ρ = Δ/(gρ) - n0. These two branches give different predictions (for example, at T=0 and small γ, the negative branch yields a negative right-hand side for Eq. (63) while the positive branch gives |σ|/ρ ≈ n1). The paper does not specify for which temperatures Eq. (63) applies, and it is presented as a general strategy for estimating |σ|. This sign inconsistency directly affects the central measurement claim.
  2. [Sec. 5, Eqs. (66)-(67)] The regularization of the divergent momentum integral in the zero-temperature anomalous density is not fixed by a physical condition. The subtraction of 1/ε_k removes the leading divergence, but any finite modification of the counterterm (e.g., adding a term that does not reintroduce a UV divergence but changes the finite part) changes σ0 by an amount of order (mΔ)^{3/2}. Since the only energy scale in the problem is Δ, any regularization scheme that respects the symmetries will produce a finite part of this order, but the coefficient is scheme-dependent unless a physical renormalization condition is imposed. The paper labels the procedure as dimensional regularization and obtains σ0 = (mΔ)^{3/2}/π², but no argument is given for why this particular finite part is the physical one. Because the predicted |σ| and the relation between Δ, n0, and |σ| depend on this choice, the central claim of a unique extraction is not established.
  3. [Sec. 5, Figs. 1 and 3] The paper presents predictions for |σ| and states that these can be extracted from future measurements. However, the numerical results in Fig. 1(b) and Fig. 3 are obtained by solving the full coupled HFB equations, while the proposed extraction formula Eq. (63) is a different relation. The paper does not demonstrate that the values obtained from solving the equations coincide with those that an experimenter would obtain from Eq. (63) using the same n0 and s. Given the sign issue in the previous comment, consistency between the two routes is not automatic and should be checked explicitly.
minor comments (5)
  1. [Abstract] The abstract states that "the sign of σ directly related to the phase, and, hence is not observable." This is plausible, but the phrase "not observable" should be qualified: it is the sign relative to the condensate phase that is not observable; the absolute value is claimed to be observable, yet the paper later shows that the magnitude itself depends on the renormalization scheme. The wording should be tightened to avoid overstatement.
  2. [Sec. 2] The definition of the Green function D_{ij} in Eq. (17) uses ω_n in the off-diagonal entries, but the conventional Matsubara Green function has a different sign convention. The authors should verify that this convention is consistent with the subsequent derivation of the dispersion relation.
  3. [Sec. 3] The expression for the total energy per volume in Eq. (46) contains dimensionally inhomogeneous-looking terms (e.g., g(σ² + ρ0²)/2 vs. gρ1ρ0). While these may follow from the preceding definitions, the manuscript would benefit from an explicit statement of units and a check of dimensional consistency.
  4. [Fig. 1] The caption of Fig. 1(a) says "Experimental points are taken from fitting experimental data in [14]", which is unclear. If the points are from the published data, this should be stated as such; if they are from a fit, the fitting procedure should be described.
  5. [References] Reference [27] is listed as "V. I. Yukalov and H. KLeinert"; the correct spelling is "Kleinert". Also, reference [36] is from the "Brazilian Journal of Physics" and the author list is incomplete; please verify the citation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the anomalous-density extraction is an algebraic consequence of the mean-field equations, not a fitted or self-cited input.

full rationale

The paper's derivation chain is self-contained. Its inputs are the two-chemical-potential HFB Hamiltonian (following Yukalov's framework, cited but not load-bearing) and the Watabe-extended Hugenholtz-Pines relation (Eq. 20). The phase constraint ξ²=±1 is derived algebraically from Eq. (21) together with the stability requirement X1,X2≥0 and the Goldstone condition, not assumed from a self-citation. The central relation σ=−ξ²σ̄ (Eq. 50) follows by solving the self-consistency equation (49), and the extraction formula (63), m1=n0−Δ/(gρ), is a rearrangement of the mean-field equation Δ=g(ρ0+σ) with σ=−|σ|; no free parameter is fitted to |σ|. The numerical |σ| curves are obtained by solving the coupled equations with fixed physical inputs (γ and t) and require no optimization to describe the 2013 Cambridge condensate-fraction data. Self-citations occur, for example [38] for dimensional regularization, but the regularization subtraction 1/ε_k and the finite result (Δm)^{3/2}/π² are stated explicitly in the paper, so the derivation does not reduce to an unverified self-cited premise. The internal sign tension between Eq. (50) (negative σ for ξ²=1) and Eq. (67) (positive σ0 at low T), and the associated scheme-dependence of the divergent integral, are robustness/correctness concerns rather than circularity: they affect the uniqueness of the claimed extraction but do not make the prediction equivalent to its inputs by construction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the HPW relation, the two-chemical-potential scheme, a stability requirement, and a specific renormalization counterterm. No fitted free parameters are introduced; inputs are measured density and scattering length plus the ideal-gas critical temperature formula. The counterterm choice is the least anchored ingredient and directly affects the predicted absolute anomalous density.

assumptions (6)
  • domain assumption Hugenholtz-Pines-Watabe relation for arbitrary condensate phase, Eq. (4): Sigma_n(0,0) - exp(-2i theta) Sigma_an(0,0) = mu.
    Adopted from Watabe (ref. 17) without re-derivation; it is the starting point for the phase constraint proof in Sec. 2.
  • domain assumption Stability of the uniform BEC requires real, positive self-energies X1 and X2.
    Used in Sec. 2.2 to reject the solution X1 + X2 = 0 and to force sin 2 theta = 0. The positivity assumption is asserted, not derived.
  • domain assumption Two chemical potentials mu0 and mu1 are needed below the critical temperature (Yukalov representative ensembles).
    Borrowed from Yukalov (ref. 19) to resolve the Hohenberg-Martin dilemma; the whole HFB construction in Sec. 3 depends on this scheme.
  • domain assumption Wick decoupling of the cubic and quartic terms in Eq. (29) retains only normal and anomalous one-body averages.
    This defines the Hartree-Fock-Bogoliubov approximation level and neglects higher-order correlations.
  • ad hoc to paper The divergent momentum integral for the zero-temperature anomalous density is regularized by subtracting 1/epsilon_k (dimensional regularization), Eq. (67).
    The sign and magnitude of the finite anomalous density depend on this counterterm choice, and the paper does not demonstrate scheme independence.
  • domain assumption Contact interaction with coupling g = 4 pi a_s / m, Eq. (26), and ideal-gas critical temperature formula, Eq. (64).
    Standard model for dilute atomic Bose gases; the input scattering length and density are taken from experiments.

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Pith. "Pith review of On the anomalous density of a dilute homogeneous Bose gas." pith.science (2026). https://pith.science/paper/J5GN4IVX

@misc{pith2026241115816,
  author       = {Pith},
  title        = {Pith review of: On the anomalous density of a dilute homogeneous Bose gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J5GN4IVX}},
  note         = {Machine review of arXiv:2411.15816}
}
abstract

Measurement of numerical values of the anomalous density, $\sigma$, which plays important role in Bose -- Einstein condensation, and, especially, determination of its sign, has been a long standing problem. We develop Hartree -- Fock -- Bogoliubov theory taking account arbitrary phase of the condensate wave function. We show that, the sign of $\sigma$ directly related to the phase, and, hence is not observable. Despite this, its absolute value can be extracted from measurements of the sound velocity and condensed fraction. We present theoretical prediction for $\vert \sigma \vert$ for a BEC in a uniform box.

Figures

Figures reproduced from arXiv: 2411.15816 by the authors.

Figure 1
Figure 1. (a) Fraction of uncondensed particles, n1 = 1 − n0 vs temperature. Note that, at T = 0 non-condensed fraction is very small: n1(T = 0) = 0.18 × 10−2 . Experimental points are taken from fitting experimental data in [14]. (b) Our prediction for the absolute value of the anomalous density, |m1| = |σ|/ρ (solid line) and sound velocity (dashed line) for 87Rb condensate in uniform box potential. 5. Numerical results and … view at source ↗
Figure 2
Figure 2. σ/ρ (a) and |σ|/ρ (b) vs reduced temperature t = T /Tc for three values of the gas parameter:γ = 10−4 , γ = 10−3 , γ = 10−2 . The question arises, whether such behavior of σ will affect the monotony of functions n0(γ, t) or s(γ, t) vs temperature. In Figs.3 we plot the condensed fraction n0 = ρ0/ρ (solid curve), the fraction of anomalous density |m1| = |σ|/ρ (dashed curve) and the sound velocity (dotted curve) vs re… view at source ↗
Figure 3
Figure 3. The condensed fraction n0 = ρ0/ρ (solid curve), the fraction of anomalous density |m1| = |σ|/ρ (dashed curve) and the sound velocity (dotted curve) vs reduced temperature. Figs. (a), (b) and (c) correspond to γ = 10−4 , γ = 10−3 and γ = 10−2 , respectively. present theory, HFB, with respect of the gas parameter γ = ρa3 s ?’. Actually, comparison of HFB calculations with Monte – Carlo ones have revealed that, HFB the… view at source ↗

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