Pith. sign in

REVIEW 1 major objections 4 minor 1 cited by

A note on spontaneous symmetry breaking in the mean-field Bose gas

T0 review · 1 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For the mean-field Bose gas at the critical temperature, this paper proves that U(1) symmetry breaking and Bose-Einstein condensation are equivalent: the Bogoliubov quasi-average of the zero mode equals the square root of the condensate…

desk verdict The main equivalence is plausible, but the proof of Theorem 1(b) has a genuine order-of-limits gap; as written the central claim is not established. read the letter →

arxiv 2501.19402 v3 pith:AP2KGELQ submitted 2025-01-31 math-ph math.APmath.MPquant-ph

Andreas Deuchert , Phan Thanh Nam , Marcin Napiorkowski This is my paper · ORCID
classification math-phmath.APmath.MPquant-ph MSC 81V7082B1082B2681R40
keywords Bose-EinsteincondensationspontaneoussymmetrybreakingU(1)Bogoliubovquasi-averagemean-fieldBosegasgrandcanonicalensemblecriticaltemperatureone-particledensitymatrix
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 studies $N$ bosons on the three-dimensional unit torus with a mean-field interaction of strength $1/N$, at inverse temperatures of order $N^{-2/3}$, the scale of the Bose-Einstein condensation transition. It establishes that, after the particle-number limit is taken, the system shows spontaneous breaking of the U(1) particle-number symmetry if and only if the one-particle density matrix of the Gibbs state has a macroscopic eigenvalue, that is, if and only if there is Bose-Einstein condensation. The quantitative statement is that the Bogoliubov quasi-average of the zero-momentum annihilation operator tends to the square root of the condensate fraction, both governed by the same factor $[1-\kappa^{-3/2}]_+$ with $\kappa$ the ratio of the inverse temperature to its critical value. This matters because it turns the heuristic identification of symmetry breaking with condensation into a theorem for an interacting model, and it gives a symmetry-breaking route to computing the condensate fraction.

What carries the argument

The machinery is the Bogoliubov quasi-average combined with a variational analysis of the grand potential. The quasi-average is defined by adding the symmetry-breaking perturbation $\lambda N(\beta,\mu)^{1/2}(a_0+a_0^*)$ to the Hamiltonian, taking the thermodynamic limit $\eta\to\infty$ first and then letting $\lambda\to 0$. The proof bounds the perturbed grand potential from above with a trial state consisting of a coherent state in the zero-momentum mode tensored with the ideal-gas Gibbs state in the excited modes, and from below with an Onsager lower bound on the interaction, a c-number substitution that replaces the zero-mode operators by a complex integration variable, and a relative-entropy estimate controlling how far an arbitrary state is from the ideal excited gas. A Hellmann-Feynman (Griffith) argument then differentiates the concave perturbed grand potential with respect to the perturbation parameters $\lambda$ and $\delta$ to extract the expectations of $a_0$ and $a_0^* a_0$, while the effective chemical potential $\tilde\mu$, defined by the gap equation (1.23), fixes the particle number and the critical temperature.

What would settle it

Take a repulsive interaction such as $v=1$ on a finite torus with $N=10^5$ and $\beta=2\beta_c(\mu,\eta)$, and compute the grand canonical expectation $|\mathrm{Tr}[a_0 G^\lambda_{\beta,\mu}]|/N^{1/2}$ for $\lambda=10^{-2},10^{-3},10^{-4}$; if the values do not approach $\sqrt{1-2^{-3/2}}\approx 0.804$ as $\lambda$ decreases, the equivalence stated in Theorem 1(b) is wrong.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1. For the homogeneous mean-field Bose gas in the grand canonical ensemble, with inverse temperature $\beta=\kappa \beta_c(\mu,\eta)$ and $\kappa\in[0,\infty)$, the condensate fraction of the unperturbed Gibbs state is $\lim_{\eta\to\infty} \mathrm{Tr}[a_0^* a_0 G_{\beta,\mu}]/N(\beta,\mu) = [1-\kappa^{-3/2}]_+$. For the state perturbed by $\lambda N(\beta,\mu)^{1/2}(a_0+a_0^*)$, the Bogoliubov quasi-average satisfies $\lim_{\lambda\to 0}\lim_{\eta\to\infty} |\mathrm{Tr}[a_0 G^\lambda_{\beta,\mu}]|/N(\beta,\mu)^{1/2} = \sqrt{[1-\kappa^{-3/2}]_+}$. Because both limits are governed by the same nonnegative factor, the U(1) symmetry is broken in the quasi-average sense exactly when the zero mode is macroscopically occupied, and the amount of symmetry breaking equals the square root of the condensate fraction. The same theorem shows that the condensate fraction computed in the perturbed state is continuous at $\lambda=0$.

Load-bearing premise

The final Hellmann-Feynman step needs a free-energy error bound that is sharp enough in the perturbation parameter $\lambda$; with the bound as written and the step $\varepsilon=|\lambda|/2$, the normalized error $C\sqrt{\eta}\,|\lambda|^{1/3}$ diverges for fixed $\lambda$ as $\eta\to\infty$, so a sharper $\lambda$-dependent estimate is required to fully close the argument.

Editorial extensions

If this is right

  • If the theorem is correct, the standard definition of Bose-Einstein condensation through a macroscopic eigenvalue of the one-particle density matrix and the Bogoliubov quasi-average definition of U(1) symmetry breaking are interchangeable for this model; either can be used to detect the phase transition.
  • The condensate fraction in the interacting mean-field gas at temperature ratio $\kappa$ is exactly $[1-\kappa^{-3/2}]_+$, the ideal-gas formula evaluated at an effective chemical potential, so the interaction shifts only the critical chemical potential and not the universal shape of the condensation curve.
  • The quasi-average limit equals the square root of the condensate fraction, so measuring the symmetry-breaking order parameter directly yields the condensate fraction in the thermodynamic limit.
  • Part (c) shows that the symmetry-breaking perturbation does not change the condensate fraction as $\lambda\to 0$, so the order parameter and the density-matrix criterion stay consistent.

Reading between the lines

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

  • Editorial inference: the same equivalence should hold in the dilute-gas scaling regime with short-range, weak interactions, where Bose-Einstein condensation has already been proved at positive temperature; the c-number substitution and relative-entropy steps here are not tied to the $1/N$ coupling, so a parallel theorem is a plausible extension.
  • Editorial inference: for trapped or non-translation-invariant systems, the condensed mode is not fixed to momentum zero, and the quasi-average would have to be taken with the maximizing one-particle orbital; the natural conjecture is that the quasi-average still equals the square root of the condensate fraction for that orbital.
  • Editorial inference: the critical inverse temperature $\beta_c(\mu,\eta)$ defined in the paper is the temperature at which the quasi-average first becomes nonzero, giving an experimentally accessible signature of the transition that does not require resolving the full one-particle density matrix.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper studies the homogeneous Bose gas on the unit torus with mean-field interaction N^{-1}v and grand canonical ensemble at inverse temperatures beta ~ N^{-2/3}, near the condensation scale. It defines an effective chemical potential via a self-consistent equation and proves that the condensate fraction of the unperturbed Gibbs state is [1-kappa^{-3/2}]_+ (part a), that the Bogoliubov quasi-average of the zero-momentum mode in the perturbed Gibbs state has modulus the square root of this fraction, establishing U(1) symmetry breaking iff BEC (part b), and that the condensate fraction in the perturbed state is continuous at lambda=0 (part c). The proofs use variational upper and lower bounds for the (perturbed) grand potential, a c-number substitution with relative-entropy estimates, and a Hellmann-Feynman/concavity argument to extract expectations.

Significance. The criterion connecting BEC to U(1) symmetry breaking in this mean-field, finite-temperature model is conceptually important and extends earlier thermodynamic-limit results to a tractable scaling. The paper's variational bounds are carefully quantified, the effective chemical potential is defined with no fitted parameters, and parts (a) and (c) appear sound after normalizing by the particle number. The main result (b), however, is not established by the given proof; see the major comment. If a correct proof of (b) is supplied, the paper would be a valuable contribution.

major comments (1)
  1. [Section 4, proof of Theorem 1(b), Eqs. (4.7)-(4.8)] The finite-difference step does not prove the claimed inner limit. Applying Proposition 4 with delta=0 to the concavity inequalities (4.7) gives a bound of order C eta (|lambda+epsilon|^{4/3} + |lambda|^{4/3} + eta^{-1/3} ln eta)/epsilon, and with epsilon=|lambda|/2 this yields |sqrt(N) Tr[(a0*+a0)G^lambda] - 2 sqrt(N N0)| of order eta |lambda|^{1/3} up to the eta^{-1/3}ln eta term. Dividing by sqrt(N) ~ sqrt(eta) leaves an error of order sqrt(eta) |lambda|^{1/3} + eta^{1/6} ln eta / |lambda|, which diverges for every fixed nonzero lambda as eta -> infinity. Thus the inner limit lim_{eta->infinity} in the iterated limit (1.30) is not established, and the claimed equivalence 'BEC iff U(1) symmetry breaking' is unproven; the paper's own Remark 1.2 identifies (b) as the main new contribution. Moreover, a merely sharper error in Proposition 6, such as O(eta(|lambda|^2 + ...)), would still leave a term sqrt(eta) |lambda| after the same normalization because epsilon is constrained by |lambda|/2 to keep the secant away from the kink of the reference term -2|lambda| sqrt(N N0); closing the gap requires a genuinely different estimate on the derivative or on the difference quotient.
minor comments (4)
  1. [Section 1.7, first paragraph] The phrase 'apart from the the symmetry breaking perturbation' contains a duplicated article.
  2. [Section 4, proof of Theorem 1(b)] 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.
  3. [Section 1.7, last paragraph] The name 'Griffith (Hellmann-Feynman) argument' is unusual; please clarify whether this refers to Griffiths-type inequalities or simply to the Hellmann-Feynman theorem.
  4. [Equation (1.29)] The equality of the two limits is asserted in the display before being proven; the proof in part (a) does show both equal [1-kappa^{-3/2}]_+, but the wording could state this explicitly to avoid confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the BEC–U(1) equivalence is obtained from self-contained variational bounds; self-citations are contextual only.

full rationale

The central claim does not reduce to its inputs. The effective chemical potential ẽµ is defined for every µ,η by the self-consistency equation (1.23) and is later shown (Appendix A, Lemma A.1) to satisfy the bounds needed; it is not selected to reproduce the condensate fraction. Proposition 1 derives N(β,µ) from the variational upper bound (Prop. 2) and lower bound (Prop. 3), and Theorem 1(a) then shows the 1-pdm condensate fraction equals the reference value [1−κ−3/2]+ via (1.27), which is a standard ideal-gas computation, not an assumption. The symmetry-breaking statement (1.30) is obtained from the concavity inequalities (4.7) together with the perturbed grand-potential bounds (Prop. 4); the latter are proved here (Propositions 5 and 6) using coherent trial states, the Onsager lower bound, and the c-number entropy inequalities of [16] as external published tools. No parameter is fitted to the predicted quasi-average. Citations to the authors' companion paper [14] occur in the introduction and in the motivation for the admissible range of µ, but the actual bound (1.24) used in the proofs is established in Appendix A, so [14] is not load-bearing. The technical concern raised by the reviewer—that the choice ε=|λ|/2 in the passage from (4.7) to (4.8) leaves an error C√η|λ|1/3 that does not vanish for fixed λ—is a gap in the uniformity of the error estimate, not a circularity: the target quantity is not an input of Proposition 4. Hence no circular step is present; the low score reflects only minor, non-load-bearing self-citations.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard tools of quantum statistical mechanics (Gibbs variational principle, c-number substitution, bosonic relative entropy bounds from [16]) and on the specific scaling assumptions on the potential, temperature, and chemical potential. The auxiliary effective chemical potential tilde mu is defined by a self-consistency equation and is not a fitted parameter. No new physical entities are introduced.

free parameters (1)
  • Effective chemical potential tilde mu
    Defined as the unique solution of the self-consistent equation (1.23); not fitted to data, but it is an auxiliary parameter chosen so that the variational trial state matches the ideal-gas condensate number. The proof depends on its bounds in Lemma A.1.
assumptions (6)
  • standard math Gibbs variational principle for the grand canonical potential (1.12)
    Used throughout to convert free-energy bounds into statements about the Gibbs state; stated in Section 1.3.
  • standard math Lemmas 3.1 and 3.2 from [16] (entropy bound and bosonic relative entropy lower bound)
    Imported without proof; they are the key tool for the lower bound on the perturbed grand potential in Section 3.3.
  • standard math Wick's theorem for quasi-free states
    Used in the upper bound Proposition 2 to evaluate expectation values in the trial state.
  • domain assumption Fourier coefficients of v satisfy 0 <= v_hat in ell^1 and v_hat(0) > 0
    Assumed in Theorem 1 and Proposition 1; excludes sign-changing and non-summable potentials.
  • domain assumption Scaling regime beta ~ eta^{-2/3}, -eta^{2/3} less than or similar to mu less than or similar to 1, and existence of kappa = lim beta / beta_c(mu, eta)
    This is the BEC critical temperature window; the proof relies on the bounds in Lemma A.1 for the effective chemical potential.
  • ad hoc to paper The self-consistent ideal-gas reference (1.23) with critical temperature (1.26) defines the comparison model
    This is a modeling choice specific to the paper; it is motivated variationally but is not an external physical law.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A note on spontaneous symmetry breaking in the mean-field Bose gas." pith.science (2026). https://pith.science/paper/AP2KGELQ

@misc{pith2026250119402,
  author       = {Pith},
  title        = {Pith review of: A note on spontaneous symmetry breaking in the mean-field Bose gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AP2KGELQ}},
  note         = {Machine review of arXiv:2501.19402}
}
abstract

We consider the homogeneous Bose gas in the three-dimensional unit torus, where $N$ particles interact via a two-body potential of the form $N^{-1} v(x)$. The system is studied at inverse temperatures of order $N^{-2/3}$, which corresponds to the temperature scale of the Bose--Einstein condensation phase transition. We show that spontaneous $U(1)$ symmetry breaking occurs if and only if the system exhibits Bose--Einstein condensation in the sense that the one-particle density matrix of the Gibbs state has a macroscopic eigenvalue.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Gibbs state of the mean-field Bose gas

    math-ph 2025-01 accept novelty 8.0 of 10

    The Gibbs state of the mean-field Bose gas at critical temperatures is proven to be, in trace norm, a coherent-state weighted average of Bogoliubov states governed by a one-mode Phi^4 condensate distribution.

Reference graph

Works this paper leans on

43 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    Adhikari, C

    A. Adhikari, C. Brennecke andB. Schlein, Bose–Einstein Condensation Beyond the Gross–Pitaevskii Regime,Ann. Henri Poincaré, 22, 1163-1233 (2021)

    work page 2021

  2. [2]

    Boccato, C

    C. Boccato, C. Brennecke, S. Cenatiempo andB. Schlein, Complete Bose–Einstein condensation in the Gross–Pitaevskii regime,Commun. Math. Phys.359 (3), 975-1026 (2018)

    work page 2018

  3. [3]

    Boccato, C

    C. Boccato, C. Brennecke, S. Cenatiempo andB. Schlein, Optimal rate for Bose–Einstein condensation in the Gross–Pitaevskii regime,Commun. Math. Phys.376 (2), 1311-1395 (2020) 23

    work page 2020

  4. [4]

    Boccato, A

    C. Boccato, A. Deuchert andD. Stocker, Upper bound for the grand canonical free energy of the Bose gas in the Gross–Pitaevskii limit,SIAM J. Math. Anal.56, no. 2, 2611-2660 (2024)

    work page 2024

  5. [5]

    N. N. Bogoliubov, On the theory of superfluidity,J. Phys. (USSR), 11, p. 23 (1947)

    work page 1947

  6. [6]

    N. N. Bogoliubov, Quasiaverages in Problems of Statistical Mechanics,Communication JINR, D-781, JINR, Dubna (1961)

    work page 1961

  7. [7]

    N. N. Bogoliubov,Lectures on quantum statistics, volume 2: Quasi-Averages. Gordon and Breach Sci. Publ., (1970)

    work page 1970

  8. [8]

    S. N. Bose, Plancks Gesetz und Lichtquantenhypothese,Zeitschrift für Physik26 (1), 178–181 (in German) (1924)

    work page 1924

Show all 43 references
  1. [9]

    Bossmann, S

    L. Bossmann, S. Petrat andR. Seiringer, Asymptotic expansion of the low-energy excitation spectrum for weakly interacting bosons,Forum Math. Sigma9, 28 (2021)

    2021

  2. [10]

    Brennecke, M

    C. Brennecke, M. Brooks, C. Caraci andJ. Oldenburg, A Short Proof of Bose–Einstein Condensation in the Gross–Pitaevskii Regime and Beyond,Ann. Henri Poincaré, online first (2024)

    2024

  3. [11]

    Caporaletti andA

    M. Caporaletti andA. Deuchert, Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit for general interaction potentials,Ann. Henri Poincaré(Online first, 2024)

    2024

  4. [12]

    Anderson, J.R

    M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman andE.A. Cornell, Observation of Bose– Einstein Condensation in a Dilute Atomic Vapor,Science269, pp. 198–201 (1995)

    1995

  5. [13]

    Derezi´nski andM

    J. Derezi´nski andM. Napi ´orkowski, Excitation spectrum of interacting bosons in the mean-field infinite- volume limit,Ann. Henri Poincaré, 15, pp. 2409–2439 (2014)

    2014

  6. [14]

    Deuchert, P.T

    A. Deuchert, P.T. Nam andM. Napi ´orkowski, The Gibbs state of the mean-field Bose gas, arXiv:2501.19396 [math-ph] (2025)

    2025 arXiv

  7. [15]

    Deuchert, R

    A. Deuchert, R. Seiringer andJ. Yngv ason, Bose–Einstein condensation in a dilute, trapped gas at positive temperature,Commun. Math. Phys.368, 723- 776 (2019)

    2019

  8. [16]

    Deuchert andR

    A. Deuchert andR. Seiringer, Gross–Pitaevskii limit of a homogeneous Bose gas at positive temperature, Arch. Ration. Mech. Anal.236, 1217–1271 (2020)

    2020

  9. [17]

    Deuchert andR

    A. Deuchert andR. Seiringer, Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons,J. Funct. Anal.281, Issue 6, 109096 (2021)

    2021

  10. [18]

    Einstein, Quantentheorie des einatomigen idealen Gases,Königliche Preußische Akademie der Wissenschaften, Sitzungsberichte261–267 (in German) (1924)

    A. Einstein, Quantentheorie des einatomigen idealen Gases,Königliche Preußische Akademie der Wissenschaften, Sitzungsberichte261–267 (in German) (1924)

    1924

  11. [19]

    Einstein, Quantentheorie des einatomigen idealen Gases,Sitzungsberichte der Preussischen Akademie der Wissenschaften1: 3 (in German) (1925)

    A. Einstein, Quantentheorie des einatomigen idealen Gases,Sitzungsberichte der Preussischen Akademie der Wissenschaften1: 3 (in German) (1925)

    1925

  12. [20]

    Fannes, J

    M. Fannes, J. V . Pul`e andA. Verbeure, On Bose condensation,Helv. Phys. Acta55, 391 (1982)

    1982

  13. [21]

    Fournais, Length scales for BEC in the dilute Bose gas

    S. Fournais, Length scales for BEC in the dilute Bose gas. In:Partial Differential Equations, Spectral Theory, and Mathematical Physics. The Ari Laptev Anniversary Volume. EMS Series of Congress Reports, vol. 18, pp. 115-133 (2021)

    2021

  14. [22]

    Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Commun

    J. Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many boson systems, Commun. Math. Phys., 8, pp. 26–51 (1968) 24

    1968

  15. [23]

    Grech andR

    P. Grech andR. Seiringer, The excitation spectrum for weakly interacting bosons in a trap,Commun. Math. Phys., 322, pp. 559–591 (2013)

    2013

  16. [24]

    Hainzl, Another proof of BEC in the GP-limit,J

    C. Hainzl, Another proof of BEC in the GP-limit,J. Math. Phys.62, 051901 (2021)

    2021

  17. [25]

    Da vis, M

    K. Da vis, M. Mewes, M. Andrews, N.v anDruten, D. Durfee, D. Kurn, W. Ketterle, Bose–Einstein condensation in a gas of sodium atoms,Phys. Rev. Lett.75, pp. 3969–3973 (1995)

    1995

  18. [26]

    Lewin, P

    M. Lewin, P. T. Nam andN. Rougerie, Derivation of Hartree’s theory for generic mean–field Bose systems, Adv. Math.254, 570–621 (2014)

    2014

  19. [27]

    Lewin, P

    M. Lewin, P. T. Nam, S. Serfaty andJ. P. Solovej, Bogoliubov spectrum of interacting Bose gases,Comm. Pure Appl. Math., 68, pp. 413–471 (2015)

    2015

  20. [28]

    E.H. Lieb, M. Loss,Analysis, 2nd Ed., AMS, Providence, Rhode Island (2010)

    2010

  21. [29]

    E. H. Lieb andR. Seiringer, Proof of Bose-Einstein Condensation for Dilute Trapped Gases,Phys. Rev. Lett., 88, p. 170409 (2002)

    2002

  22. [30]

    E. H. Lieb, R. Seiringer andJ. Yngv ason, Justification of c-Number Substitutions in Bosonic Hamiltonians, Phys. Rev. Lett., 94, p. 080401 (2005)

    2005

  23. [31]

    E. H. Lieb, R. Seiringer, J.P. Solovej andJ. Yngv ason,Mathematics of Bose gas and its condensation, Oberwolfach Seminar Series, V ol. 34, Birkhaeuser (2005)

    2005

  24. [32]

    P. T. Nam andM. Napi ´orkowski, Two-term expansion of the ground state one-body density matrix of a mean-field Bose gas,Calc. Var. PDE60 (3), 1-30 (2021)

    2021

  25. [33]

    P. T. Nam, M. Napi´orkowski, J. Ricaud andA. Triay, Optimal rate of condensation for trapped bosons in the Gross–Pitaevskii regime,Anal. and PDE15 (6), 1585–1616 (2022)

    2022

  26. [34]

    P. T. Nam, N. Rougerie,andR. Seiringer, Ground states of large bosonic systems: The Gross–Pitaevskii limit revisited,Anal. and PDE, 9, no. 2, 459–485 (2016)

    2016

  27. [35]

    Nam andR

    P.T. Nam andR. Seiringer, Collective excitations of Bose gases in the mean-field regime,Arch. Ration. Mech. Anal.215 (2), 381-417 (2015)

    2015

  28. [36]

    Napi ´orkowski, R

    M. Napi ´orkowski, R. Reuvers andJ.P. Solovej, Bogoliubov free energy functional I. Existence of minimizers and phase diagram,Arch. Ration. Mech. Anal.229 (3), 1037-1090 (2018)

    2018

  29. [37]

    Ohy a andD

    M. Ohy a andD. Petz,Quantum entropy and its use, Texts and Monographs in Physics, SpringerVerlag, Berlin (1993)

    1993

  30. [38]

    Penrose andL

    O. Penrose andL. Onsager, Bose–Einstein Condensation and Liquid Helium,Phys. Rev.104, 576 (1956)

    1956

  31. [39]

    J. V . Pul´e, A. F. Verbeure andV . A. Zagrebnov, On nonhomogeneous Bose condensation,J. Math. Phys. 46, 083301 (2005)

    2005

  32. [40]

    Seiringer, The excitation spectrum for weakly interacting bosons,Commun

    R. Seiringer, The excitation spectrum for weakly interacting bosons,Commun. Math. Phys., 306, pp. 565– 578 (2011)

    2011

  33. [41]

    S ¨ut˝o, Equivalence of Bose-Einstein Condensation and Symmetry Breaking,Phys

    A. S ¨ut˝o, Equivalence of Bose-Einstein Condensation and Symmetry Breaking,Phys. Rev. Lett.94, 080402 (2005)

    2005

  34. [42]

    Thirring,Quantum Mathematical Physics, 2 nd ed., Springer, New York (2002) 25

    W. Thirring,Quantum Mathematical Physics, 2 nd ed., Springer, New York (2002) 25

    2002

  35. [43]

    C. N. Yang, Concept of Off–Diagonal Long–Range Order and the Quantum Phases of Liquid He and of Superconductors,Rev. Mod. Phys.34, 694 (1962) (Andreas Deuchert) Department of Mathematics, Virginia Tech 225 Stanger Street, Blacksburg, V A, 24060-1026, USA E-mail address:andreas...

    1962

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.