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arxiv: 2512.07473 · v3 · submitted 2025-12-08 · ⚛️ nucl-th · cond-mat.quant-gas· hep-th

Absence of charged pion condensation in a magnetic field with parallel rotation

Puyuan Bai , Lianyi He This is my paper

Pith reviewed 2026-05-17 00:53 UTC · model grok-4.3

classification ⚛️ nucl-th cond-mat.quant-gashep-th
keywords charged pion condensationmagnetic fieldrotationBose-Einstein condensatequasi-one-dimensionalColeman-Mermin-Wagner-Hohenberg theoremrelativistic bosonsorder parameter
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The pith

The order parameter for charged pion condensation vanishes at any nonzero temperature because the system is quasi-one-dimensional.

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

The paper examines a relativistic system of charged bosons placed in a magnetic field with rotation parallel to the field direction. For non-interacting bosons the critical temperature is zero because the Landau-level structure plus rotation reduces the problem to quasi-one dimension. When a quartic self-interaction is added, the order parameter still vanishes and off-diagonal long-range order is forbidden at any finite temperature by the Coleman-Mermin-Wagner-Hohenberg theorem. A sympathetic reader cares because earlier work proposed that such a condensate might form in rotating nuclear or quark matter under strong magnetic fields.

Core claim

For non-interacting charged bosons the critical temperature vanishes because the system is quasi-one-dimensional. For interacting bosons with quartic self-interaction the order parameter vanishes and off-diagonal long-range order is absent at any nonzero temperature because of the quasi-one-dimensional feature, in accordance with the Coleman-Mermin-Wagner-Hohenberg theorem.

What carries the argument

The quasi-one-dimensional feature of the charged-boson system in a magnetic field with parallel rotation, which directly invokes the Coleman-Mermin-Wagner-Hohenberg theorem to forbid long-range order.

If this is right

  • Non-interacting charged bosons cannot undergo Bose-Einstein condensation in this geometry at any temperature.
  • The condensate order parameter remains strictly zero for interacting bosons at any finite temperature.
  • Off-diagonal long-range order cannot develop in the rotating charged-boson system.
  • The result applies to charged pion condensation in strong magnetic fields combined with rotation.

Where Pith is reading between the lines

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

  • Similar suppression may occur in other relativistic systems that are effectively low-dimensional because of external fields.
  • Numerical checks could test whether small deviations from exact quasi-one-dimensionality allow condensation to reappear.
  • The finding indicates that effective one-dimensional models are enough to rule out certain condensate phases in rotating nuclear matter.

Load-bearing premise

The relativistic system with quartic self-interaction remains strictly quasi-one-dimensional, so that neither relativistic kinematics nor the interaction can restore higher-dimensional behavior or permit long-range order.

What would settle it

A lattice simulation or effective-model calculation that finds a nonzero order parameter or finite critical temperature for the interacting charged bosons at any nonzero temperature would falsify the claim.

Figures

Figures reproduced from arXiv: 2512.07473 by Lianyi He, Puyuan Bai.

Figure 1
Figure 1. Figure 1: FIG. 1. The angular speed [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

We investigate the critical temperature of a relativistic Bose-Einstein condensate of charged bosons driven by rotation in a parallel magnetic field [Y. Liu and I. Zahed, Phys. Rev. Lett. 120, 032001 (2018)]. For non-interacting bosons, the critical temperature can only be determined for a system with fixed angular momentum. We find that the critical temperature of the non-interacting system vanishes due to the fact that the system is quasi-one-dimensional, indicating that non-interacting bosons cannot undergo Bose-Einstein condensation. For interacting bosons, we investigate a system with quartic self-interaction. We show that the order parameter vanishes and the off-diagonal long-range order is absent at any nonzero temperature because of the quasi-one-dimensional feature, in accordance with the Coleman-Mermin-Wagner-Hohenberg theorem.

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

2 major / 1 minor

Summary. The paper investigates charged pion condensation for relativistic charged bosons in a magnetic field with parallel rotation. For the non-interacting case it concludes that the critical temperature vanishes because the system is quasi-one-dimensional, so that Bose-Einstein condensation cannot occur. For the interacting case with a quartic self-interaction it shows that the order parameter vanishes at any T>0 due to the same quasi-one-dimensional reduction, in accordance with the Coleman-Mermin-Wagner-Hohenberg theorem, implying the absence of off-diagonal long-range order.

Significance. If the central claim is correct, the work would establish that charged pion condensation is forbidden at finite temperature in this setup, with direct relevance to models of rotating magnetized nuclear matter and heavy-ion collisions. The explicit invocation of the CMWH theorem to rule out order in the interacting quasi-1D theory is a clear strength when the effective dimensionality is rigorously controlled.

major comments (2)
  1. [interacting bosons / quartic self-interaction analysis] The interacting-bosons analysis applies the Coleman-Mermin-Wagner-Hohenberg theorem directly after projecting onto the lowest Landau level. However, the relativistic quartic self-interaction can generate level-mixing vertices or loop-induced operators carrying transverse momentum; these would violate the strict quasi-1D assumption with short-range interactions that the theorem requires. An explicit demonstration that such corrections are irrelevant in the infrared (or a controlled effective-action derivation) is needed before the no-order conclusion follows.
  2. [non-interacting bosons section] The non-interacting claim that the critical temperature vanishes for fixed angular momentum rests on the quasi-1D density of states. The manuscript should provide the explicit integral or sum over the Landau-level spectrum that shows why the particle-number equation cannot be satisfied at any finite T, rather than invoking dimensionality reduction alone.
minor comments (1)
  1. [Abstract] The abstract states that the non-interacting critical temperature 'vanishes' but does not immediately clarify that this holds only for fixed angular momentum; a parenthetical remark would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate explicit derivations where requested.

read point-by-point responses

  1. Referee: The interacting-bosons analysis applies the Coleman-Mermin-Wagner-Hohenberg theorem directly after projecting onto the lowest Landau level. However, the relativistic quartic self-interaction can generate level-mixing vertices or loop-induced operators carrying transverse momentum; these would violate the strict quasi-1D assumption with short-range interactions that the theorem requires. An explicit demonstration that such corrections are irrelevant in the infrared (or a controlled effective-action derivation) is needed before the no-order conclusion follows.

    Authors: We thank the referee for highlighting this subtlety in the effective theory. In the strong-magnetic-field regime relevant to our setup, the Landau level spacing is parametrically large. After integrating out higher Landau levels, the leading effective quartic interaction projected onto the lowest Landau level remains local in the longitudinal (rotation-axis) coordinate and short-ranged. Any loop-generated operators carrying transverse momentum are suppressed by inverse powers of the cyclotron frequency and are irrelevant in the infrared according to power-counting in the quasi-1D effective theory. We have added a new appendix deriving the effective action and demonstrating the irrelevance of such corrections, thereby justifying the direct application of the Coleman-Mermin-Wagner-Hohenberg theorem. revision: yes

  2. Referee: The non-interacting claim that the critical temperature vanishes for fixed angular momentum rests on the quasi-1D density of states. The manuscript should provide the explicit integral or sum over the Landau-level spectrum that shows why the particle-number equation cannot be satisfied at any finite T, rather than invoking dimensionality reduction alone.

    Authors: We agree that an explicit calculation clarifies the argument and have revised the manuscript accordingly. In the updated Section on non-interacting bosons we now present the explicit sum over Landau levels n together with the integral over longitudinal momentum p_z for the total particle number at fixed angular momentum. For the lowest Landau level the resulting expression reduces to an integral of the Bose distribution whose infrared behavior prevents satisfaction of the fixed-particle-number constraint at any finite temperature; the effective density of states diverges in a manner that forces the critical temperature to vanish. This explicit form replaces the previous dimensionality-reduction statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity: external theorem applied to derived effective model

full rationale

The paper first establishes the quasi-one-dimensional character of the charged boson system from the parallel magnetic field and rotation (via Landau level projection and dominance of the lowest level), for both the non-interacting and quartic-interacting cases. It then directly invokes the external Coleman-Mermin-Wagner-Hohenberg theorem to conclude that off-diagonal long-range order is absent at any T > 0. This theorem is a pre-existing mathematical result from the 1960s-1970s literature, not derived or cited from the authors' prior work. No equations reduce to self-definition, no fitted parameters are relabeled as predictions, and no load-bearing step relies on a self-citation chain. The derivation chain remains self-contained against external benchmarks, with the only potential issue being applicability of the theorem to the interacting relativistic case (a correctness concern, not circularity).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the applicability of the Coleman-Mermin-Wagner-Hohenberg theorem to the quasi-one-dimensional relativistic interacting boson system. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The Coleman-Mermin-Wagner-Hohenberg theorem applies to this quasi-one-dimensional relativistic system with quartic self-interaction.
    Invoked to conclude that off-diagonal long-range order is absent at any nonzero temperature.

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Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [1]

    + 1 2 (∇ϕ1)2 + (∇ϕ2)2 + m2 2 (ϕ2 1 +ϕ 2 2) +V int ϕ2 1 +ϕ 2 2 2 + q2B2ρ2 8 (ϕ2 1 +ϕ 2 2)− iqB 2 (ϕ1ˆlzϕ2 −ϕ 2ˆlzϕ1) +iΩ(π 1ˆlzϕ1 +π 2ˆlzϕ2)(8) At finite temperature, the partition function of the system is given by Z=Tre −β( ˆH−µ ˆQ),(9) whereβ= 1/T, ˆNis the conserved charge associated with the U(1)symmetry, ˆQ= Z dr(π1ϕ2 −π 2ϕ1),(10) andµis the correspo...

    work page

  2. [2]

    NON-INTERACTING BOSONS IN PRM In this section, we consider the non-interacting system,Vint = 0

    + iqB 2 (ϕ1ˆlzϕ2 −ϕ 2ˆlzϕ1).(17) Integrating by parts and using the complex fieldΦ, we can express it as Leff = Φ∗ ∂τ −µ−Ω ˆlz 2 + ˆK2D +∂ 2 z −m 2 Φ−V(Φ ∗Φ)(18) where the operatorK2D is defined as: ˆK2D = ∂2 ∂ρ2 + 1 ρ ∂ ∂ρ − ˆl2 z ρ2 − 1 4 q2B2ρ2 +qB ˆlz.(19) III. NON-INTERACTING BOSONS IN PRM In this section, we consider the non-interacting system,Vint ...

    work page

  3. [3]

    Becattini, F

    F. Becattini, F. Piccinini, and J. Rizzo, Phys. Rev. C77, 024906 (2008)

    work page 2008

  4. [4]

    Becattini, G

    F. Becattini, G. Inghirami, V. Rolando, A. Beraudo, L. Del Zanna, A. De Pace, M. Nardi, G. Pagliara, and V. Chandra, Eur. Phys. J. C75, 406 (2015)

    work page 2015

  5. [5]

    Deng and X

    W. Deng and X. Huang, Phys. Rev. C93, 064907 (2016)

    work page 2016

  6. [6]

    Jiang, Z

    Y. Jiang, Z. Lin, and J. Liao, Phys. Rev. C94, 044910 (2016)

    work page 2016

  7. [7]

    Adamczyket al.(STAR Collaboration), Nature548, 62 (2017)

    L. Adamczyket al.(STAR Collaboration), Nature548, 62 (2017)

    work page 2017

  8. [8]

    Skokov, A

    V. Skokov, A. Y. Illarionov and V. Toneev, Int. J. Mod. Phys. A24, 5925 (2009)

    work page 2009

  9. [9]

    W. T. Deng and X. G. Huang, Phys. Rev. C85, 044907 (2012)

    work page 2012

  10. [10]

    D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Prog. Part. Nucl. Phys.88, 1 (2016)

    work page 2016

  11. [11]

    D. T. Son and M. A. Stephanov, Phys. Rev. Lett.86, 592 (2001)

    work page 2001

  12. [12]

    J. B. Kogut and D. K. Sinclair, Phys. Rev. D66, 034505 (2002)

    work page 2002

  13. [13]

    B. B. Brandt, G. Endrodi, and S. Schmalzbauer, Phys. Rev. D97, 054514 (2018)

    work page 2018

  14. [14]

    R. F. Sawyer, Phys. Rev. Lett.29, 382 (1972)

    work page 1972

  15. [15]

    D. J. Scalapino, Phys. Rev. Lett.29, 386 (1972)

    work page 1972

  16. [16]

    Baym, Phys

    G. Baym, Phys. Rev. Lett.30, 1340 (1973)

    work page 1973

  17. [17]

    A. B. Migdal, Phys. Rev. Lett.31, 257 (1973). 16

    work page 1973

  18. [18]

    B. B. Brandt, G. Endrodi, E. S. Fraga, M. Hippert, J. Schaffner-Bielich and S. Schmalzbauer, Phys. Rev. D98, 094510 (2018)

    work page 2018

  19. [19]

    V. Ruck, M. Gyulassy, and W. Greiner, Z. Phys. A277, 391 (1976)

    work page 1976

  20. [20]

    Zimanyi, G

    J. Zimanyi, G. Fai, and B. Jakobsson, Phys. Rev. Lett.43, 1705 (1979)

    work page 1979

  21. [21]

    Greiner, C

    C. Greiner, C. Gong, and B. Mueller, Phys. Lett. B316, 226 (1993)

    work page 1993

  22. [22]

    M. M. Wygas, I. M. Oldengott, D. Boedeker, and D. J. Schwarz, Phys. Rev. Lett.121, 201302 (2018)

    work page 2018

  23. [23]

    Vovchenko, B

    V. Vovchenko, B. B. Brandt, F. Cuteri, G. Endrodi, F. Hajkarim, and J. Schaffner-Bielich, Phys. Rev. Lett.126, 012701 (2021)

    work page 2021

  24. [24]

    G. Cao, L. He, and P. Zhang, Phys. Rev. D104, 054007 (2021)

    work page 2021

  25. [25]

    Liu and I

    Y. Liu and I. Zahed, Phys. Rev. Lett.120, 032001 (2018)

    work page 2018

  26. [26]

    T. Guo, J. Li, C. Mu, and L. He, Phys. Rev. D106, 094010 (2022)

    work page 2022

  27. [27]

    Blaauwgeers, V

    R. Blaauwgeers, V. B. Eltsov, M. Krusius, J. J. Ruohio, R. Schanen, and G. E. Volovik, Nature404, 471 (2000)

    work page 2000

  28. [28]

    Kanda, B

    A. Kanda, B. J. Baelus, F. M. Peeters, K. Kadowaki, and Y. Ootuka, Phys. Rev. Lett.93, 257002 (2004)

    work page 2004

  29. [29]

    I. V. Grigorieva, W. Escoffier, V. R. Misko, B. J. Baelus, F. M. Peeters, L. Y. Vinnikov, and S. V. Dubonos, Phys. Rev. Lett.99, 147003 (2007)

    work page 2007

  30. [30]

    R. B. G. Kramer, A. V. Silhanek, J. Van de Vondel, B. Raes, and V. V. Moshchalkov, Phys. Rev. Lett.103, 067007 (2009)

    work page 2009

  31. [31]

    T. Cren, L. Serrier-Garcia, F. Debontridder, and D. Roditchev, Phys. Rev. Lett.107, 097202 (2011)

    work page 2011

  32. [32]

    Kasamatsu, M

    K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev. A66, 053606 (2002)

    work page 2002

  33. [33]

    U. R. Fischer and G. Baym, Phys. Rev. Lett.90, 140402 (2003)

    work page 2003

  34. [34]

    Aftalion and I

    A. Aftalion and I. Danaila, Phys. Rev. A69, 033608 (2004)

    work page 2004

  35. [35]

    A. L. Fetter, B. Jackson, and S. Stringari, Phys. Rev. A71, 013605 (2005)

    work page 2005

  36. [36]

    Ding, S.-T

    H.-T. Ding, S.-T. Li, A. Tomiya, and X.-D. Wang, and Y. Zhang, Phys. Rev. D104, 014504 (2021)

    work page 2021

  37. [37]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett.17, 616 (1966)

    work page 1966

  38. [38]

    C. J. Pethick and H. Smith,Bose–Einstein Condensation in Dilute Gases, Cambridge University Press (2002)

    work page 2002

  39. [39]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari,Bose-Einstein Condensation and Superfluidity, Oxford University Press (2016)

    work page 2016

  40. [40]

    Schaefer, D

    T. Schaefer, D. T. Son, M. A. Stephanov, D. Toublan, and J. J. M. Verbaarschot, Phys. Lett. B522, 67 (2001)

    work page 2001

  41. [41]

    V. A. Miransky and I. A. Shovkovy, Phys. Rev. Lett.88, 111601 (2002)

    work page 2002

  42. [42]

    Brauner, Symmetry2, 609 (2010)

    T. Brauner, Symmetry2, 609 (2010)

    work page 2010

  43. [43]

    Watanabe and H

    H. Watanabe and H. Murayama, Phys. Rev. Lett.108, 251602 (2012)

    work page 2012

  44. [44]

    Hidaka, Phys

    Y. Hidaka, Phys. Rev. Lett.110, 091601 (2013)

    work page 2013

  45. [45]

    Watanabe and H

    H. Watanabe and H. Murayama, Phys. Rev. X4, 031057 (2014)

    work page 2014

  46. [46]

    Adamovaet al.(CERES Collaboration), Nucl

    D. Adamovaet al.(CERES Collaboration), Nucl. Phys. A714, 124 (2003)

    work page 2003

  47. [47]

    S. S. Adleret al.(PHENIX Collaboration), Phys. Rev. Lett.93, 152302 (2004)

    work page 2004

  48. [48]

    Aamodtet al.(ALICE Collaboration), Phys

    K. Aamodtet al.(ALICE Collaboration), Phys. Lett. B696, 328 (2011)

    work page 2011

  49. [49]

    Adamet al.(ALICE Collaboration), Phys

    J. Adamet al.(ALICE Collaboration), Phys. Rev. C93, 054908 (2016)

    work page 2016

  50. [50]

    O. V. Teryaev and V. I. Zakharov, Phys. Rev. D96, 096023 (2017)

    work page 2017

  51. [51]

    J. I. A. Li, T. Taniguchi, K. Watanabe, J. Hone, and C. R. Dean, Nat. Phys.13, 751 (2017)

    work page 2017

  52. [52]

    Di Mauro Villari, I

    L. Di Mauro Villari, I. Galbraith, and F. Biancalana, Phys. Rev. A102, 033321 (2020)

    work page 2020

  53. [53]

    Wang, X.-C

    Z.-Y. Wang, X.-C. Cheng, B.-Z. Wang, J.-Y., Zhang, Y.-H. Lu, C.-R. Yi, S. Niu, Y. Deng, X.-J. Liu, S. Chen, and J.-W. Pan, Science372, 271 (2021). [52]NIST handbook of mathematical functions, Frank W. J. Olver,et al., eds. Cambridge University Press, 2010

    work page 2021

  54. [54]

    Mermin, H

    N.D. Mermin, H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Model, Phys. Rev. Lett. 17 (1966) 1133-1136; P.C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158 (1967)383-386; S. Coleman, There are no Goldstone Bosons in Two Dimensions, Comm. Math. Phys. 31 (1973) 259-264

    work page 1966