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FRG analysis for a relativistic BEC in arbitrary spatial dimensions

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Pith's one-line read FRG flow kills the condensate in one and two spatial dimensions, consistent with Mermin–Wagner.

desk verdict A careful, modest FRG study that gives credible numerical evidence for Mermin-Wagner suppression of relativistic BEC at finite density in d<=2, but the advertised 'analytical confirmation' is conditional on assumptions the authors admit are not generally proven. read the letter →

arxiv 2504.17668 v2 pith:36WO5LQJ submitted 2025-04-24 hep-ph cond-mat.quant-gashep-th

Fumio Terazaki , Kazuya Mameda This is my paper · ORCID
classification hep-phcond-mat.quant-gashep-th MSC 82B2881T1082B27 PACS 05.10.Cc03.75.Hh11.10.Wx
keywords functionalrenormalizationgroupMermin–WagnertheoremrelativisticBose–EinsteincondensatelocalpotentialapproximationfinitechemicalspontaneoussymmetrybreakingcriticalexponentsTaylorexpansionmethod
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 asks whether the functional renormalization group (FRG), in its simplest local-potential approximation, can reproduce a fundamental theorem: continuous symmetries do not break spontaneously at finite temperature in d≤2 spatial dimensions. It studies a relativistic complex scalar field with a chemical potential, i.e., a relativistic Bose–Einstein condensate, and finds numerically that for d≤2 the condensate flows to zero as the infrared scale is removed, for every tested chemical potential. For d>2 the condensate remains finite and actually grows with chemical potential. The authors also give an analytical argument, based on the flow equation for the potential minimum, that directly produces the vanishing of the condensate for d≤2. The work matters because it tests whether a widely used nonperturbative method can be trusted at finite density in low dimensions.

What carries the argument

The load-bearing object is the flow equation for the potential minimum, Eq. (20): k∂ρ0,k/∂k = 4ad T $k^{{d-2}}$ f_k, with f_k = (1+x_k+$x_k^{2}$+y_k/2)/(1+2x_k)^2, where x_k and y_k measure the curvature and cubic coupling at the minimum. Under the assumptions u3,k≥0 and f_k Taylor-expandable near k=0, f_k approaches a positive constant, and the equation integrates to the explicit solutions in Eq. (21): for d<2 a power-law approach to zero and for d=2 a logarithmic approach, with a critical scale kc in Eq. (22) below which ρ0,k=0. This machinery converts the dimensional factor $k^{{d-2}}$ into a decisive suppression for d≤2, while for d>2 the same factor leaves room for a nonzero condensate.

What would settle it

Find a parameter set for which u3,k becomes negative or f_k becomes singular as k→0 (for example, a slightly larger chemical potential at d=2, near √115.1/111, where the paper itself reports numerical instability), solve the full FRG flow without the Taylor expansion approximation, and check whether ρ0,k remains strictly positive down to k=0; a finite condensate in such a case would refute the claim that the FRG, even beyond the Taylor truncation, always enforces Mermin–Wagner suppression.

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Extended reading notes

Core claim

Within the local potential approximation and a Taylor expansion of the effective potential around its flowing minimum, the infrared value of the condensate ρ0,k→0 is zero for d≤2 and strictly positive for d>2, for all studied values of the chemical potential. This dimensional dichotomy is the Mermin–Wagner theorem as seen by the FRG: fluctuations become sufficiently strong in d≤2 to restore the U(1) symmetry, even when a chemical potential tries to favor condensation. The analytical revisit uses a regulator that makes the low-momentum flow equation tractable; the flow of ρ0,k then satisfies k∂ρ0,k/∂k = 4ad T $k^{{d-2}}$ f_k with a positive function f_k, and this forces ρ0,k to hit zero at a finite scale kc for d<2 and a finite scale kc for d=2 (where it falls logarithmically). The paper takes this as confirmation that the FRG is consistent with Mermin–Wagner in the imaginary-time formalism, resolving a subtlety left open by earlier d=3-only studies.

Load-bearing premise

The analytical proof that the condensate must vanish for d≤2 assumes the cubic coefficient u3,k stays non-negative and that the function f_k can be Taylor-expanded at k=0 with a positive constant term; the authors state these conditions hold only for some parameter sets, and if they fail the explicit solution forcing ρ0,k=0 is not guaranteed by the flow equation.

Editorial extensions

If this is right

  • The FRG under the local potential approximation reproduces the Mermin–Wagner theorem for a relativistic Bose–Einstein condensate at finite chemical potential in arbitrary spatial dimension.
  • For d>2, the condensate is enhanced by increasing chemical potential, extending the known d=3 behavior to continuous dimensions down to d=2.
  • The analytical flow equation gives a critical scale kc below which the condensate is exactly zero for d≤2, so the FRG predicts complete symmetry restoration in the deep infrared.
  • The critical exponents extracted at d=3, ν≈0.6672 and β≈0.3670 at T/|m̄|=0.1, together with the high-temperature and zero-temperature values, match known O(2)/XY and mean-field expectations, supporting the method's reliability.
  • The numerical instability in low dimensions is characterized by a divergence of y_k∼k^{-d+2}, which imposes practical limits on FRG calculations but is evadable for certain parameter choices.

Reading between the lines

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

  • If the positivity of the right-hand side of the flow equation holds beyond the local potential approximation, the same analytic argument would suggest that Mermin–Wagner suppression is a robust feature of FRG flows at finite density, not an artifact of the Taylor expansion.
  • The logarithmic vanishing of ρ0,k at d=2 is the expected precursor of Berezinskii–Kosterlitz–Thouless physics, so the same flow equation could be used to study the BKT transition in the relativistic case, an extension the paper mentions only as future work.
  • The analytical solution offers a direct falsifiable criterion: any parameter set for which u3,k becomes negative or f_k is singular near k=0 should invalidate the predicted vanishing, and such sets may already be accessible numerically with a grid method.
  • A regulator that preserves Lorentz symmetry might simultaneously restore the Silver-Blaze property and still show the MW suppression, which would make the FRG a more reliable tool for zero-temperature finite-density systems.
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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 manuscript studies a relativistic complex scalar field with a U(1) chemical potential, using the functional renormalization group under the local potential approximation. For the numerical part, the Litim regulator is combined with a Taylor expansion of the effective potential around its minimum, and the scale-dependent condensate is computed for spatial dimensions from d=3 down to d=1, for a range of chemical potentials. The main numerical finding is that the condensate flows to zero for d≤2 and to a finite value for d>2, consistent with the Mermin-Wagner theorem, and that the d≤2 behavior is insensitive to the chemical potential. The paper also presents an analytical revisit using a frequency-dependent regulator, obtaining an explicit flow equation for the condensate minimum and a closed solution that yields a finite vanishing scale for d≤2, subject to two stated assumptions. Critical exponents at d=3 are computed and compared with the 3D XY and mean-field values, and a numerical instability in low dimensions is discussed.

Significance. If the central claim holds, the paper provides a valuable cross-check that the FRG in the imaginary-time formalism can reproduce the Mermin-Wagner suppression of relativistic Bose-Einstein condensation at finite density, a nonperturbative requirement that is nontrivial at finite chemical potential. The numerical study covers a range of dimensions, verifies the flow with a second regulator, and reports critical exponents consistent with known universality classes, which strengthens confidence in the LPA-based approach. The analytical solution in Eqs. (21)-(22) is an instructive demonstration of how the dimension-dependent power of k in the flow equation can force ρ0,k to vanish for d≤2. The main caveat is that the analytical confirmation is conditional on assumptions that are only partially verified, and the numerical evidence is limited to parameter sets that avoid a documented instability.

major comments (3)
  1. [Section IV, Eq. (20)-(22)] The claimed analytical confirmation of the Mermin-Wagner theorem is conditional on two assumptions that the authors themselves state are not generally established: u3,k≥0 is said to hold only in their numerical computation, and the Taylor-expandability of fk is said to be fulfilled only by some parameter sets. Because the explicit solution (21) is exactly what forces ρ0,k=0 for d≤2, failure of either assumption invalidates the analytical conclusion for general parameters. The abstract's phrase 'analytically confirmed from the flow equation' is therefore stronger than what the derivation establishes. The authors should either prove these assumptions within the LPA truncation for the parameter range of interest or reformulate the claim as a conditional consistency check.
  2. [Section III, Figs. 1-3 and instability discussion] The numerical demonstration is restricted to parameter choices that avoid the instability described in Section III; for example, a slightly larger chemical potential, µ/|mbar|=sqrt(115.1/111), is stated to produce numerical instability at d=2. Thus the conclusion that the condensate vanishes for d≤2 independently of µ is demonstrated only in a selected part of parameter space, not for arbitrary parameters. The domain of validity of the numerical claim should be stated explicitly in the abstract and conclusions, or the stability assessment should be extended.
  3. [Section IV, Eq. (20) and Fig. 4] The analytical argument for all d<2 relies on assumptions whose verification is shown only for d=2: Fig. 4 displays yk→0 for d=2.0, while for d>2 yk behaves as k^{-d+2}. No numerical evidence is presented for d<2 that fk is Taylor-expandable and u3,k≥0. The statement that the MW theorem is confirmed for all d≤2 therefore extrapolates the analytic solution beyond the parameter sets that were actually checked. The manuscript should either supply such checks or explicitly limit the analytical claim to the cases for which the assumptions are verified.
minor comments (5)
  1. [Abstract and Title] The phrase 'arbitrary spatial dimensions' is broader than what is demonstrated: the numerical results cover d=1.0 to d=3.0 in steps of 0.2, and the analytical claim is restricted by unproven assumptions. A more precise wording would help the reader.
  2. [Section III, Fig. 2 and Fig. 6] The figure captions refer to line styles and colors ('blue dotted line') without a full legend; adding an explicit legend would improve reproducibility of the described comparisons.
  3. [Section III, Grid method comparison] The authors state that they confirmed the smooth flow of ρ0,k using the Grid method, but no grid-method results are shown. A brief quantitative statement or a supplementary figure would make this verification checkable.
  4. [Section IV, Eq. (17)] The sentence 'The latter assumption holds at least our numerical computation' should read 'holds at least in our numerical computation.'
  5. [References] Several references lack complete bibliographic data, e.g., Ref. [12] has no volume or article number; the authors should ensure all references are fully specified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central MW-consistency result is a genuine numerical/analytical output checked against an external theorem, not an input or a fitted parameter; admitted technical assumptions are robustness limitations, not circular reductions.

full rationale

The paper's central claim is that the FRG flow of the condensate ρ0,k vanishes for d≤2 and remains finite for d>2, consistently with the Mermin–Wagner theorem. This claim is not built into the inputs: the physical parameters (m̄, μ, T, λ̄) are fixed before the flow, and ρ0,k→0, critical exponents, and μ-dependence are outputs of solving Eq. (7). The comparison with MW is an external consistency check, not a fitted target, and the critical exponents are independently benchmarked against the 3D XY universality class and free-theory expectations. The only self-citation is Ref. [24], the authors' prior d=3 paper, used as a derivation template for the flow equation and as a reference for the d=3 μ-enhancement trend. That is not load-bearing for the new d≤2 result: the arbitrary-d flow equation is written out in the paper (Eq. (7)), and the lower-dimensional extension is not an instance of the cited result. The analytical confirmation in Section IV is explicitly conditional on u3,k≥0 and Taylor-expandability of fk near k=0; the authors concede these are not proven for all parameters. This is a limitation on the force of the analytic derivation, but it is not circularity—the assumptions do not themselves contain the conclusion that ρ0,k=0 for d≤2, and the numerical flows are presented as the primary evidence. Similarly, the discussion of numerical instability is an acknowledged technical caveat, not a circular step. Overall, the derivation chain is self-contained with respect to its inputs and the external MW benchmark.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No parameters are fitted to force the MW result. The mass, coupling, temperature and chemical potential are physical inputs chosen before the flow; the critical exponents are outputs read from the flow. The only hand-set numerical choices are truncation order lmax=7 and regulator forms, which are method choices rather than fitted constants.

assumptions (3)
  • domain assumption Local potential approximation (derivative expansion truncated at potential level, no field renormalization).
    The effective action (4) keeps only a potential Vk(ρ); all derivative corrections are dropped. The paper itself attributes the deviation of high-temperature critical exponents to this truncation.
  • domain assumption Taylor expansion of Uk around the running minimum with lmax=7 and u_{lmax+1}=0.
    Eqs. (10)-(11) solve the flow in a local Taylor basis. The authors check convergence and compare with a grid method for select cases, but systematic truncation error is not quantified.
  • ad hoc to paper For the analytical MW argument, u3,k ≥ 0 and fk can be Taylor-expanded with f0 > 0.
    Section IV assumes these to derive the closed solution (21) and vanishing scale (22). The authors state these hold only for some parameter sets, so the analytical proof is conditional.

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Pith. "Pith review of FRG analysis for a relativistic BEC in arbitrary spatial dimensions." pith.science (2026). https://pith.science/paper/36WO5LQJ

@misc{pith2026250417668,
  author       = {Pith},
  title        = {Pith review of: FRG analysis for a relativistic BEC in arbitrary spatial dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36WO5LQJ}},
  note         = {Machine review of arXiv:2504.17668}
}
read the original abstract

A relativistic Bose-Einstein condensate (BEC) is studied within the complex scalar field theory using the functional renormalization group (FRG) under the local potential approximation. We investigate fluctuation effects on the relativistic BEC through numerical analyses for various spatial dimensions and chemical potentials. Our numerical results are consistent with the Mermin-Wagner theorem, and this consistency is also analytically confirmed from the flow equation. We also discuss a numerical instability of the FRG in lower spatial dimensions, which is evadable for certain parameter choices.

Figures

Figures reproduced from arXiv: 2504.17668 by the authors.

Figure 1
Figure 1. also exhibits the apparent difference between d > 2 (i.e., the upper five lines) and d ≤ 2 (i.e., the lower six lines). For d > 2, the condensate ρ0,k converges to a finite value of ρ0,k→0. The convergence for d = 2.2 is much slower, but our numerical calculation shows that the flow eventually converges to ρ0,k→0/|m¯ | d−1 ≈ 0.049 (left) and ρ0,k→0/|m¯ | d−1 ≈ 0.062 (right) around log(k/Λ) ≈ −25. These trends for d … view at source ↗
Figure 2
Figure 2. FIG. 2. Dependence of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Flows of ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Chemical potential dependence of Taylor coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Critical exponents [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Works this paper leans on

56 extracted references · 23 canonical work pages

  1. [1]

    In general, however, another k-dependent function yk can diverge more rapidly than k−d+1

    A general feature of well-defined theories, u2,k→0 ≥ 0 [34], implies xk = ρ0,ku2,k/k2 ≥ 0, for which (1 + 2xk)−3/2F (k√1 + 2xk) is not singular. In general, however, another k-dependent function yk can diverge more rapidly than k−d+1. As seen from the definition yk = ρ0,ku3,k/u2,k, this divergence occurs when the potential Uk(ρ) becomes highly flat around...

    work page

  2. [2]

    Kapitza, Viscosity of liquid helium below the λ-point, Nature 141, 74 (1938)

    P. Kapitza, Viscosity of liquid helium below the λ-point, Nature 141, 74 (1938)

    work page 1938

  3. [3]

    J. F. Allen and A. D. Misener, Flow of liquid helium ii, Nature 141, 75 (1938)

    work page 1938

  4. [4]

    Hall and W

    H. Hall and W. F. Vinen, The rotation of liquid helium ii ii. the theory of mutual friction in uniformly rotating helium ii, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 238, 215 (1956)

    work page 1956

  5. [5]

    R. F. Sawyer, Condensed pi- phase in neutron star matter, Phys. Rev. Lett. 29, 382 (1972)

    work page 1972

  6. [6]

    Barshay, G

    S. Barshay, G. Vagradov, and G. E. Brown, Possibility of a phase transition to a pion condensate in neutron stars, Phys. Lett. B 43, 359 (1973)

    work page 1973

  7. [7]

    Baym, Pion condensation in nuclear and neutron star matter, Phys

    G. Baym, Pion condensation in nuclear and neutron star matter, Phys. Rev. Lett. 30, 1340 (1973)

    work page 1973

  8. [8]

    S. L. Shapiro and S. A. Teukolsky, Black holes, white dwarfs, and neutron stars: The physics of compact objects (Wiley-VCH, 1983)

    work page 1983

Show all 56 references
  1. [9]

    C. J. Pethick, T. Schaefer, and A. Schwenk, Bose-einstein condensates in neutron stars, in Universal Themes of Bose-Einstein Condensation , edited by N. P. Proukakis, D. W. Snoke, and P. B. Littlewood (Cambridge University Press, Cambridge, UK, 2017) pp. 573–592, arXiv:1507.05...

    2017 arXiv

  2. [10]

    B. Fore, N. Kaiser, S. Reddy, and N. C. Warrington, Mass of charged pions in neutron-star matter, Phys. Rev. C 110, 025803 (2024), arXiv:2301.07226 [nucl-th]

    2024 arXiv

  3. [11]

    Vijayan, N

    V. Vijayan, N. Rahman, A. Bauswein, G. Mart´ ınez-Pinedo, and I. L. Arbina, Impact of pions on binary neutron star mergers, Phys. Rev. D 108, 023020 (2023), arXiv:2302.12055 [astro-ph.HE]

    2023 arXiv

  4. [12]

    Yasuda et al., Extraction of the Landau-Migdal Parameter from the Gamow-Teller Giant Resonance in Sn132, Phys

    J. Yasuda et al., Extraction of the Landau-Migdal Parameter from the Gamow-Teller Giant Resonance in Sn132, Phys. Rev. Lett. 121, 132501 (2018)

    2018

  5. [13]

    L. A. Urena-Lopez, Bose-Einstein condensation of relativistic Scalar Field Dark Matter, JCAP 01, 014, arXiv:0806.3093 [gr-qc]

    arXiv

  6. [14]

    C. G. Boehmer and T. Harko, Can dark matter be a Bose-Einstein condensate?, JCAP 06, 025, arXiv:0705.4158 [astro-ph]

    arXiv

  7. [15]

    Wetterich, Exact evolution equation for the effective potential, Phys

    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301, 90 (1993), arXiv:1710.05815 [hep-th]

    1993 arXiv

  8. [16]

    Schaefer and J

    B.-J. Schaefer and J. Wambach, The Phase diagram of the quark meson model, Nucl. Phys. A 757, 479 (2005), arXiv:nucl-th/0403039

    2005 arXiv

  9. [17]

    Tetradis and C

    N. Tetradis and C. Wetterich, Critical exponents from effective average action, Nucl. Phys. B 422, 541 (1994), arXiv:hep-ph/9308214

    1994 arXiv

  10. [18]

    N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17, 1133 (1966)

    1966

  11. [19]

    P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158, 383 (1967)

    1967

  12. [20]

    Hawashin, J

    B. Hawashin, J. Rong, and M. M. Scherer, Ultraviolet-Complete Local Field Theory of Persistent Symmetry Breaking in 2+1 Dimensions, Phys. Rev. Lett. 134, 041602 (2025), arXiv:2409.10606 [hep- th]

    2025 arXiv

  13. [21]

    Defenu, P

    N. Defenu, P. Mati, I. G. Marian, I. Nandori, and A. Trombettoni, Truncation Effects in the Functional Renormalization Group Study of Spontaneous Symmetry Breaking, JHEP 05, 141, arXiv:1410.7024 [hep-th]

    arXiv

  14. [22]

    Mati, Vanishing beta function curves from the functional renormalization group, Phys

    P. Mati, Vanishing beta function curves from the functional renormalization group, Phys. Rev. D 91, 125038 (2015), arXiv:1501.00211 [hep-th]

    2015 arXiv

  15. [23]

    E. E. Svanes and J. O. Andersen, Functional renormalization group at finite density and Bose conden- sation, Nucl. Phys. A 857, 16 (2011), arXiv:1009.0430 [hep-ph]

    2011 arXiv

  16. [24]

    L. F. Palhares, Exploring the different phase diagrams of Strong Interactions , Ph.D. thesis, Rio de Janeiro Federal U. (2012), arXiv:1208.0574 [hep-ph]

    2012 arXiv

  17. [25]

    Terazaki, K

    F. Terazaki, K. Mameda, and K. Suzuki, Relativistic BEC extracted from a complex FRG flow equation, PTEP 2024, 123B02 (2024), arXiv:2409.04361 [hep-ph]

    2024 arXiv

  18. [26]

    Floerchinger and C

    S. Floerchinger and C. Wetterich, Functional renormalization for Bose-Einstein Condensation, Phys. Rev. A 77, 053603 (2008), arXiv:0801.2910 [cond-mat.supr-con]. 15

    2008 arXiv

  19. [27]

    Drews and W

    M. Drews and W. Weise, Functional renormalization group studies of nuclear and neutron matter, Prog. Part. Nucl. Phys. 93, 69 (2017), arXiv:1610.07568 [nucl-th]

    2017 arXiv

  20. [28]

    K. Otto, M. Oertel, and B.-J. Schaefer, Hybrid and quark star matter based on a nonperturbative equation of state, Phys. Rev. D 101, 103021 (2020), arXiv:1910.11929 [hep-ph]

    2020 arXiv

  21. [29]

    W.-j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D 101, 054032 (2020), arXiv:1909.02991 [hep-ph]

    2020 arXiv

  22. [30]

    Prokof’ev and B

    N. Prokof’ev and B. Svistunov, Worm Algorithms for Classical Statistical Models, Phys. Rev. Lett. 87, 160601 (2001), arXiv:cond-mat/0103146

    2001 arXiv

  23. [31]

    M. G. Endres, Method for simulating O(N) lattice models at finite density, Phys. Rev. D 75, 065012 (2007), arXiv:hep-lat/0610029

    2007 arXiv

  24. [32]

    Gattringer and T

    C. Gattringer and T. Kloiber, Lattice study of the Silver Blaze phenomenon for a charged scalar ϕ4 field, Nucl. Phys. B 869, 56 (2013), arXiv:1206.2954 [hep-lat]

    2013 arXiv

  25. [33]

    Rindlisbacher, Infinite-range correlations in 1D systems with continuous symmetry (2020), arXiv:2012.03396 [hep-lat]

    T. Rindlisbacher, Infinite-range correlations in 1D systems with continuous symmetry (2020), arXiv:2012.03396 [hep-lat]

    2020 arXiv

  26. [34]

    J. I. Kapusta, Bose-Einstein Condensation, Spontaneous Symmetry Breaking, and Gauge Theories, Phys. Rev. D 24, 426 (1981)

    1981

  27. [35]

    Berges, N

    J. Berges, N. Tetradis, and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363, 223 (2002), arXiv:hep-ph/0005122

    2002 arXiv

  28. [36]

    G. R. Golner, Nonperturbative Renormalization Group Calculations for Continuum Spin Systems, Phys. Rev. B 33, 7863 (1986)

    1986

  29. [37]

    J. F. Nicoll, T. S. Chang, and H. E. Stanley, Approximate Renormalization Group Based on the Wegner-Houghton Differential Generator, Phys. Rev. Lett. 33, 540 (1974)

    1974

  30. [38]

    K. G. Wilson and J. B. Kogut, The Renormalization group and the epsilon expansion, Phys. Rept. 12, 75 (1974)

    1974

  31. [39]

    M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory (Addison-Wesley, Reading, USA, 1995)

    1995

  32. [40]

    D. F. Litim, Optimized renormalization group flows, Phys. Rev. D 64, 105007 (2001), arXiv:hep- th/0103195

    2001

  33. [41]

    T. D. Cohen, Functional integrals for QCD at nonzero chemical potential and zero density, Phys. Rev. Lett. 91, 222001 (2003), arXiv:hep-ph/0307089

    2003 arXiv

  34. [42]

    Akerlund, P

    O. Akerlund, P. de Forcrand, A. Georges, and P. Werner, Extended Mean Field study of complex ϕ4- theory at finite density and temperature, Phys. Rev. D 90, 065008 (2014), arXiv:1405.6613 [hep-lat]

    2014 arXiv

  35. [43]

    Mark´ o, U

    G. Mark´ o, U. Reinosa, and Z. Sz´ ep, Bose-Einstein condensation and Silver Blaze property from the two-loop Φ-derivable approximation, Phys. Rev. D 90, 125021 (2014), arXiv:1410.6998 [hep-ph]

    2014 arXiv

  36. [44]

    N. Khan, J. M. Pawlowski, F. Rennecke, and M. M. Scherer, The Phase Diagram of QC 2D from Functional Methods (2015), arXiv:1512.03673 [hep-ph]. 16

    2015 arXiv

  37. [45]

    T¨ opfel, J

    S. T¨ opfel, J. M. Pawlowski, and J. Braun, Phase structure of quark matter and in-medium properties of mesons from Callan-Symanzik flows (2024), arXiv:2412.16059 [hep-ph]

    2024 arXiv

  38. [46]

    O. Bohr, B. J. Schaefer, and J. Wambach, Renormalization group flow equations and the phase tran- sition in O(N) models, Int. J. Mod. Phys. A 16, 3823 (2001), arXiv:hep-ph/0007098

    2001 arXiv

  39. [47]

    Springer and B

    P. Springer and B. Klein, O(2)-scaling in finite and infinite volume, Eur. Phys. J. C 75, 468 (2015), arXiv:1506.00909 [hep-ph]

    2015 arXiv

  40. [48]

    Wang and P

    Z. Wang and P. Zhuang, Critical Behavior and Dimension Crossover of Pion Superfluidity, Phys. Rev. D 94, 056012 (2016), arXiv:1511.05279 [hep-ph]

    2016 arXiv

  41. [49]

    Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, Phys

    M. Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, Phys. Rev. B 100, 224517 (2019), arXiv:1910.05916 [cond-mat.stat-mech]

    2019 arXiv

  42. [50]

    V. M. Kaspi and A. Beloborodov, Magnetars, Ann. Rev. Astron. Astrophys. 55, 261 (2017), arXiv:1703.00068 [astro-ph.HE]

    2017 arXiv

  43. [51]

    Chatterjee, J

    D. Chatterjee, J. Novak, and M. Oertel, Magnetic field distribution in magnetars, Phys. Rev. C 99, 055811 (2019), arXiv:1808.01778 [nucl-th]

    2019 arXiv

  44. [52]

    V. L. Berezinsky, Destruction of long range order in one-dimensional and two-dimensional systems having a continuous symmetry group. I. Classical systems, Sov. Phys. JETP 32, 493 (1971)

    1971

  45. [53]

    V. L. Berezinsky, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems., Sov. Phys. JETP 34, 610 (1972)

    1972

  46. [54]

    J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973)

    1973

  47. [55]

    Jakubczyk, N

    P. Jakubczyk, N. Dupuis, and B. Delamotte, Reexamination of the nonperturbative renormalization- group approach to the Kosterlitz-Thouless transition, Phys. Rev. E 90, 062105 (2014), arXiv:1409.1374 [cond-mat.stat-mech]

    2014 arXiv

  48. [56]

    Ran¸ con and N

    A. Ran¸ con and N. Dupuis, Universal thermodynamics of a two-dimensional bose gas, Phys. Rev. A 85, 063607 (2012)

    2012

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Reviewed August 16, 2026 · model on record in the stance chip above.