Pith. sign in

REVIEW 3 major objections 6 minor 40 references

Exact limits in dimension and spin number can be interpolated to predict 3D critical exponents and couplings without free parameters.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-14 08:41 UTC pith:VSKQZGLT

load-bearing objection Clean two-axis interpolation toolkit: spatial Kc and RG polynomials are genuinely parameter-free and useful; spin-axis 1/n^{2} is a two-point fit, so Heisenberg/O(2.5) numbers are not independent forecasts. the 3 major comments →

arxiv 2607.10865 v1 pith:VSKQZGLT submitted 2026-07-12 cond-mat.stat-mech math-phmath.MPphysics.chem-ph

Dimensional and Spin Interpolation for the O(n) Model: From Exact Anchors to RG-Improved Critical Exponents

Kumar Ghosh This is my paper
classification cond-mat.stat-mech math-phmath.MPphysics.chem-ph PACS 05.70.Jk64.60.F-11.10.Hi
keywords O(n) modelcritical exponentsdimensional interpolationWilson–Fisherspherical modelcompatibility criterion1/n expansionIsing universality
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that the O(n) family of critical models can be treated by treating both spatial dimension D and spin-component number n as continuous variables that connect exact solvable limits. On the dimension axis, the exact two-dimensional Onsager solution and infinite-dimensional mean-field theory, combined with a simple geometric weight and Wilson–Fisher slope constraints, produce a closed-form three-dimensional Ising critical coupling and renormalization-group-improved exponents that match Monte Carlo and bootstrap numbers to a few percent. On the spin axis the same idea works only for observables that change monotonically between the Ising and spherical-model anchors; the correlation-length exponent does, while the critical coupling does not. A short 1/n^{2} series fixed at n = 1 and n = 2 then predicts the Heisenberg exponents, and the same formulas extend immediately to non-integer n such as 2.5. The practical payoff is a unified, parameter-light route to critical exponents that also states when the method is allowed to succeed.

Core claim

Two-axis interpolation between exact anchors—Onsager at D = 2, mean-field as D → ∞, and the spherical model as n → ∞—together with Wilson–Fisher slope constraints and a monotonicity compatibility criterion, yields closed-form 3D Ising K_c ≈ 0.2204 and RG-improved exponents ν = 2/3, β = 31/96, η = 35/864, plus a 1/n^{2} prediction ν(3) = 0.7493 that propagates through exact scaling relations to β(3) and γ(3), all without additional free parameters.

What carries the argument

The compatibility criterion: two-anchor interpolation is valid only for observables that vary monotonically between the anchors, so the target lies inside the bracket. It is used together with the geometric weight δ_D = 1/(D − 1) and Wilson–Fisher-constrained polynomials in ε = 4 − D.

Load-bearing premise

A low-order polynomial fixed only by the two exact anchors and the leading Wilson–Fisher slope is already accurate enough at the physical three-dimensional point.

What would settle it

A high-precision Monte Carlo or conformal-bootstrap determination of the correlation-length exponent for a non-integer O(n) model (for example n = 2.5) that lies outside the few-percent window predicted by the 1/n^{2} formula would refute the spin-axis claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proposes a two-axis interpolation scheme for the O(n) family, treating D and n as continuous parameters anchored at exact limits (Onsager, mean-field, spherical/Mermin–Wagner). On the spatial axis, the weight δ_D=1/(D−1) places D=3 at the midpoint and yields a parameter-free K_c≈0.2204, while Wilson–Fisher-constrained polynomials give closed-form ν=2/3, β=31/96, η=35/864 at D=3 and track conformal-bootstrap trends for 3≤D<4. On the spin axis it introduces a monotonicity compatibility criterion, shows that K_c(n) fails it while ν(n) passes, fits a 1/n² series for ν(n) at n=1,2, and propagates the result through exact scaling relations to β(n), γ(n) and a forecast for n=2.5. Extensions to quantum TFIM via the quantum-classical map and a discussion of systematic improvements are included.

Significance. If the claims hold after clarification, the work offers a transparent, systematically improvable interpolation language that unifies several exact anchors and makes the conditions for success explicit (monotonicity; coupling rescaling only when the upper anchor has K_c=0). The spatial K_c result is genuinely parameter-free arithmetic from the midpoint weight and is close to Monte Carlo; the RG polynomials are uniquely fixed by anchors plus leading WF slopes and reproduce bootstrap trends near D=4. The compatibility criterion and the clear separation of the n→∞ and D→∞ limits are useful conceptual contributions. The non-integer-n and quantum-TFIM extensions are falsifiable forecasts. These strengths are real even if the spin-axis quantitative claims need to be restated more carefully as calibrated extrapolations rather than parameter-free predictions.

major comments (3)
  1. Abstract and §V H–I (and the parallel claim in the Conclusion): the statement that the spin-axis results for ν(3), β(3), γ(3) are obtained “without introducing additional parameters” is not accurate. Equations (46)–(48) fix the two free coefficients c1, c2 of the 1/n² series by the accepted Monte Carlo values ν(1)=0.6299 and ν(2)=0.6717; the same pattern appears for K_c in Eqs. (40)–(41), where the XY benchmark is used as a constraint. The Heisenberg and O(2.5) numbers are therefore two-point extrapolations of an empirical interpolant, not a priori predictions. The residual 5.4% error at n=3 measures how well a quadratic 1/n polynomial happens to match the true function, not an independent test of the framework. The abstract, §V H, Table V footnotes, and Conclusion should state explicitly that two coefficients are calibrated at n=1,2 and that only n≥3 (and non-integer n between the fit p
  2. §V I, Eq. (53): the ansatz η(n)=η_RG(D=3)/n = 35/(864 n) is load-bearing for the derived β(n) and γ(n), yet it is inconsistent with the accepted n-dependence. Accepted values are nearly flat (η≈0.0362, 0.0381, 0.0375 for n=1,2,3), whereas Eq. (53) forces a 1/n decay (0.0405, 0.0203, 0.0135), i.e. 47% and 64% errors already at n=2,3. The paper notes that η is a small multiplicative correction, but the wrong trend means the n-dependence of β and γ is not controlled by the same physics that governs the accepted data. Either replace Eq. (53) by a form compatible with the known near-constancy of η (or by literature large-N coefficients), or clearly demote β(n), γ(n) to “illustrative propagation under a provisional η ansatz” and report sensitivity to η.
  3. §V E–F and Remark 5: the bivariate free-energy formula (33) fails at the spherical corner by a factor of two because the multiplicative coupling rescaling was designed for an anchor with K_c=0. The manuscript correctly diagnoses this and switches to affine interpolation for K_c, but then still presents (33) as the unifying free-energy construction. Either restrict the free-energy claim to the spatial axis (where the construction works) or supply a corrected rescaling rule that recovers K_c^(sph) when n→∞ at D=3; otherwise the “two-axis free energy” is not a working object on the spin axis.
minor comments (6)
  1. Throughout: the reduced-coupling convention is stated as K≡J/(2k_B T) in the Onsager section and later as K=βJ with J=1; a single consistent definition should be fixed early and used uniformly when quoting K_c benchmarks.
  2. Table IV and Table V: the dagger footnotes that mark fit constraints are easy to miss; promote them into the table captions so that “prediction” vs “constraint” is unambiguous at a glance.
  3. §V H: the coefficients c1=−0.9431, c2=0.5730 are empirical. A short comparison to the analytic large-N / nonlinear σ-model coefficients at d=3 (even if only at leading 1/n) would clarify how much of the series is universal field theory versus fit.
  4. Fig. 3 caption and text: K^(sph)_c is quoted both as 0.12636 and (in the figure label) 0.1253; use one value consistently.
  5. §IV D / Table III: bootstrap comparison for ν is clear; adding the parallel η column (already computed in the text for D=3.5 and 3.75) into the table would make the non-integer-D benchmark self-contained.
  6. §VI quantum extension: the polynomials (65) are formally identical to the classical ones by the d↔D=d+1 map; state explicitly that no new numerical content is claimed beyond covariance of the framework, to avoid the impression of an independent quantum calculation.

Circularity Check

3 steps flagged

Spin-axis 1/n^{2} coefficients are fixed by the n=1,2 Monte Carlo benchmarks, so Heisenberg and O(2.5) forecasts are two-point extrapolations, not parameter-free first-principles predictions; spatial-axis results remain clean.

specific steps
  1. fitted input called prediction [Sec. V H, Eqs. (45)–(50); Table V; Abstract]
    "The two coefficients are determined by the Ising boundary at n=1 and the XY value at n=2 as constraints. … c1 + c2 = Δν(1) = −0.3701, … c1/2 + c2/4 = Δν(2) = −0.3283 … ν(1)(3) = 1 − 0.3144 + 0.0637 = 0.7493 (benchmark: 0.7112, error 5.4%). … A perturbative 1/n2 expansion yields ν(3) = 0.7493 … without introducing additional parameters."

    c1 and c2 are free parameters fixed by the accepted MC values ν(1)=0.6299 and ν(2)=0.6717. The formula is forced through those two points by construction (Table V marks them † as fit constraints). Evaluating the same quadratic at n=3 (and n=2.5) is therefore a two-point extrapolation, not a parameter-free derivation. Calling the result a prediction “without introducing additional parameters” renames the fitted coefficients as first-principles content.

  2. fitted input called prediction [Sec. V F, Eqs. (39)–(42); Table IV]
    "Going to first order, one writes the 1/n series Kc(n)=K(sph)c[1+A1/n+A2/n2+O(n−3)] and imposes the spatial interpolation result at n=1 and the XY benchmark at n=2 as constraints. … The first-order Heisenberg prediction then follows: K(1)c(3)=0.1409 (benchmark: 0.12130, error 16.2%). … †XY value used as a fit constraint; not a free prediction at this order."

    A1 and A2 are solved from the n=1 and n=2 Kc values (one of which is itself an external MC benchmark). The Heisenberg number is the evaluation of that two-parameter fit at n=3. The table correctly flags the XY point as a constraint, but the abstract still folds the spin-axis Kc discussion into the same “no adjustable parameters” framing used for the genuinely parameter-free spatial Kc.

  3. fitted input called prediction [Sec. V I–J, Eqs. (54)–(63); Table VI–VII; Abstract]
    "propagation through exact scaling relations gives β(3)=0.3797 (benchmark: 0.3689) and γ(3)=1.489 (benchmark: 1.396), without introducing additional parameters. … Evaluating at n=2.5: ν(1)(2.5)=…=0.7143. … The prediction ν(2.5)=0.7143 is therefore a genuine parameter-free forecast for this non-integer universality class."

    β(3), γ(3), and the entire O(2.5) suite are obtained by substituting the already-fitted ν(1)(n) into exact scaling relations. They inherit the two free coefficients fixed at n=1,2; no new independent information enters. Labeling ν(2.5) a “genuine parameter-free forecast” is therefore the same fitted expansion evaluated at a third point.

full rationale

The spatial-axis results are not circular: K_c^interp = K_c^(2D)/2 follows from exact Onsager and mean-field anchors with δ_D(3)=1/2 and no 3D input; the RG polynomials for ν, β, η are fixed by exact anchors at ε=0 and ε=2 plus literature Wilson–Fisher leading slopes, then compared to independent 3D and non-integer-D bootstrap benchmarks. Circularity is confined to the spin axis. There the paper solves a two-coefficient 1/n^{2} (or 1/n^{2} for K_c) system using the accepted Monte Carlo values at n=1 and n=2 as constraints, then evaluates at n=3 and n=2.5 and presents those values as predictions “without introducing additional parameters.” The coefficients are free parameters of an empirical fit; n=1 and n=2 are reproduced by construction (tables mark them †), and residual error at n=3 measures how well a quadratic 1/n polynomial happens to track the true curve, not an independent test of the framework. The paper itself calls the formula an “empirical realization” with coefficients “determined from… benchmarks rather than from σ-model loop integrals,” yet the abstract and conclusion still advertise parameter-free spin-axis forecasts. β(3), γ(3), and the O(2.5) suite inherit the same calibration. No load-bearing self-citation uniqueness theorem is involved; the GKH citations supply only the interpolation philosophy. Overall: partial circularity on one of the two axes, score 5.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 2 invented entities

The framework rests on exact classical solutions (Onsager, mean-field, spherical, Mermin–Wagner), standard RG scaling relations, and the leading Wilson–Fisher ε-expansion coefficients taken from the literature. The only free parameters introduced by the paper are the two empirical coefficients of the 1/n² series (and the analogous pair for K_c). The interpolation weight δ_D = 1/(D−1) and the compatibility criterion are modeling choices of the paper.

free parameters (2)
  • c1, c2 of Δν(n) = c1/n + c2/n² = c1 = −0.9431, c2 = 0.5730
    Fixed by imposing accepted Monte Carlo values ν(1) = 0.6299 and ν(2) = 0.6717 (Eqs. 46–48); used to generate all subsequent spin-axis predictions.
  • A1, A2 of Kc(n) 1/n series = A1 = 0.1469, A2 = 0.5974
    Fixed by the spatial-interpolation Kc at n = 1 and the XY Monte Carlo value at n = 2; used only for the non-monotone Kc discussion.
axioms (7)
  • domain assumption Exact Onsager free energy and critical coupling at D = 2, n = 1
    Used as lower spatial anchor throughout Sections II–IV.
  • domain assumption Mean-field theory exact as D → ∞ with Kc = 1/(2D)
    Upper spatial anchor; standard textbook result.
  • domain assumption Berlin–Kac spherical model exact as n → ∞ at fixed D = 3
    Upper spin-axis anchor for both Kc and exponents.
  • domain assumption Leading Wilson–Fisher slopes ν′(0) = 1/12, β′(0) = −1/6, η ∼ ε²/54
    Taken from the classic ε-expansion literature and imposed as polynomial constraints (Section IV B).
  • standard math Fisher and hyperscaling relations γ = (2 − η)ν, β = ν(1 + η)/2 at D = 3
    Derived in Appendix A and used to obtain β(n), γ(n) from ν(n) and η(n).
  • ad hoc to paper Compatibility criterion: two-anchor interpolation succeeds only for monotone observables
    Formalized as Proposition 1; used to discard Kc(n) and retain ν(n).
  • ad hoc to paper Interpolation weight δ_D = 1/(D − 1) (and δ_n = 1/n)
    Chosen so that D = 3 sits at the midpoint; motivated by lattice geometry but not derived from a uniqueness theorem.
invented entities (2)
  • Bivariate interpolated free energy (Eq. 33) no independent evidence
    purpose: Unifies the four exact corner solutions in the (n, D) plane into a single expression.
    Constructed by weighted sum of the four anchors; suffers a known factor-of-two shift at the spherical corner (Remark 5).
  • RG-improved quadratic/cubic polynomials for ν(ε), β(ε), η(ε) no independent evidence
    purpose: Provide continuous D-dependence of Ising exponents between D = 2 and D = 4.
    Uniquely fixed by anchors plus leading WF coefficients; no independent microscopic derivation.

pith-pipeline@v1.1.0-grok45 · 25823 in / 3639 out tokens · 41563 ms · 2026-07-14T08:41:10.102296+00:00 · methodology

0 comments
read the original abstract

We develop a two-axis interpolation framework for the O$(n)$ universality family, treating the spatial dimension $D$ and the spin-component number $n$ as independent continuous parameters connecting exact limiting solutions. On the spatial axis, anchoring between the Onsager solution at $D=2$ and mean-field theory at $D\to\infty$ yields a closed-form prediction for the 3D Ising critical coupling that agrees well with Monte Carlo benchmarks $K_c = 0.2204$ (benchmark: $0.22165$) with no adjustable parameters. Wilson--Fisher-constrained polynomial interpolation gives $\nu=2/3$, $\beta=31/96$, and $\eta=35/864$ at $D=3$ (benchmarks: $0.6299$, $0.3265$, $0.0362$), and reproduces conformal-bootstrap results across $3 \le D < 4$. On the spin axis, we establish a necessary compatibility criterion: two-anchor interpolation succeeds only for observables that vary monotonically between the anchor values. The critical coupling $K_c(n)$ violates this criterion because the Heisenberg value falls below the spherical limit, whereas the correlation-length exponent $\nu(n)$ satisfies it. A perturbative $1/n^2$ expansion yields $\nu(3) = 0.7493$ (benchmark: $0.7112$), and propagation through exact scaling relations gives $\beta(3) = 0.3797$ (benchmark: $0.3689$) and $\gamma(3) = 1.489$ (benchmark: $1.396$), without introducing additional parameters. The framework naturally extends to non-integer spin, producing the prediction $\nu(2.5) = 0.7143$ for the O$(2.5)$ universality class. These results establish dimensional and spin interpolation as a unified and predictive approach to critical phenomena, while clarifying the structural conditions under which interpolation succeeds.

Figures

Figures reproduced from arXiv: 2607.10865 by Kumar Ghosh.

Figure 1
Figure 1. Figure 1: FIG. 1. RG-improved spatial interpolation for [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The ( [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Critical coupling [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Correlation-length exponent [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

40 extracted references · 5 linked inside Pith

  1. [1]

    (n= 1, D= 2): the Onsager solution,K (1,2) c = 1 2 ln(1 + √ 2)≈0.44069 [7]

  2. [2]

    (n= 1, D→ ∞): mean-field theory,K (1,∞) c = 1/(2D)→0 [14]

  3. [3]

    (n→ ∞, D= 2): the Mermin–Wagner theorem [21] prohibits spontaneous breaking of continuous symmetries inD≤2, soT c = 0 and the free energy is zero for all finiteK[22]

  4. [4]

    Remark 3.The spatial axis (Dat fixedn= 1) and the spin axis (nat fixedD= 3) are geometrically orthogonal in the(n, D)plane

    (n→ ∞, D= 3): the Berlin–Kac spherical model withK (∞,3) c =W 3/4≈0.12636 [11, 20]. Remark 3.The spatial axis (Dat fixedn= 1) and the spin axis (nat fixedD= 3) are geometrically orthogonal in the(n, D)plane. In particular, the spherical model is the upper anchor for the spin axis only; it cannot serve as the spatial upper anchor (Section V A). 9 5 10 15 2...

  5. [5]

    D. R. Herschbach, J. G. Loeser, and W. L. Virgo, The Journal of Physical Chemistry A121, 6336 (2017)

    2017

  6. [6]

    K. J. B. Ghosh, S. Kais, and D. R. Herschbach, Frontiers in Physics8, 331 (2020)

    2020

  7. [7]

    K. J. B. Ghosh, S. Kais, and D. R. Herschbach, Physical Chemistry Chemical Physics23, 7841 (2021). 17

    2021

  8. [8]

    K. J. B. Ghosh, S. Kais, and D. R. Herschbach, The Journal of Physical Chemistry A125, 7581 (2021)

    2021

  9. [9]

    K. J. B. Ghosh, S. Kais, and D. R. Herschbach, Physical Chemistry Chemical Physics24, 9298 (2022)

    2022

  10. [10]

    Ising, Zeitschrift f¨ ur Physik31, 253 (1925)

    E. Ising, Zeitschrift f¨ ur Physik31, 253 (1925)

    1925

  11. [11]

    Onsager, Physical Review65, 117 (1944)

    L. Onsager, Physical Review65, 117 (1944)

    1944

  12. [12]

    Istrail, inProceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000)(ACM, New York,

    S. Istrail, inProceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000)(ACM, New York,

    2000

  13. [13]

    Hasenbusch, Physical Review B82, 174433 (2010)

    M. Hasenbusch, Physical Review B82, 174433 (2010)

    2010

  14. [14]

    F. Kos, D. Poland, D. Simmons-Duffin, and A. Vichi, Journal of High Energy Physics2016, 36 (2016), arXiv:1603.04436 [hep-th]

    Pith/arXiv arXiv 2016

  15. [15]

    T. H. Berlin and M. Kac, Physical Review86, 821 (1952)

    1952

  16. [16]

    H. E. Stanley, Physical Review176, 718 (1968)

    1968

  17. [17]

    Yang, Physical Review85, 808 (1952)

    C.-N. Yang, Physical Review85, 808 (1952)

    1952

  18. [18]

    W. L. Bragg and E. J. Williams, Proceedings of the Royal Society of London. Series A145, 699 (1934)

    1934

  19. [19]

    Pelissetto and E

    A. Pelissetto and E. Vicari, Physics Reports368, 549 (2002), arXiv:cond-mat/0012164

    Pith/arXiv arXiv 2002

  20. [20]

    K. G. Wilson and M. E. Fisher, Physical Review Letters28, 240 (1972)

    1972

  21. [21]

    Ma, Reviews of Modern Physics45, 589 (1973)

    S.-k. Ma, Reviews of Modern Physics45, 589 (1973)

    1973

  22. [22]

    Bonanno, A

    C. Bonanno, A. Cappelli, M. Kompaniets, S. Okuda, and K. J. Wiese, SciPost Physics14, 135 (2023), arXiv:2210.03051 [hep-th]

    Pith/arXiv arXiv 2023

  23. [23]

    Yoshida and A

    B. Yoshida and A. Kubica, arXiv preprint (2014), arXiv:1404.6311 [cond-mat.stat-mech], arXiv:1404.6311 [cond-mat.stat- mech]

    Pith/arXiv arXiv 2014

  24. [24]

    G. N. Watson, The Quarterly Journal of Mathematics10, 266 (1939)

    1939

  25. [25]

    N. D. Mermin and H. Wagner, Physical Review Letters17, 1133 (1966)

    1966

  26. [26]

    G. S. Joyce, inPhase Transitions and Critical Phenomena, Vol. 2, edited by C. Domb and M. S. Green (Academic Press, London, 1972) pp. 375–442

    1972

  27. [27]

    Campostrini, M

    M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Physical Review B63, 214503 (2001), arXiv:cond- mat/0010360

    arXiv 2001

  28. [28]

    Campostrini, M

    M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi, and E. Vicari, Physical Review B65, 144520 (2002), arXiv:cond- mat/0110336

    arXiv 2002

  29. [29]

    Br´ ezin and J

    E. Br´ ezin and J. Zinn-Justin, Physical Review B14, 3110 (1976)

    1976

  30. [30]

    Codello, N

    A. Codello, N. Defenu, and G. D’Odorico, Physical Review D91, 105003 (2015), arXiv:1410.3308 [hep-th]

    Pith/arXiv arXiv 2015

  31. [31]

    Pfeuty, Annals of Physics57, 79 (1970)

    P. Pfeuty, Annals of Physics57, 79 (1970)

    1970

  32. [32]

    Aharony, Y

    A. Aharony, Y. Imry, and S.-k. Ma, Physical Review Letters37, 1364 (1976)

    1976

  33. [33]

    Imry and S.-k

    Y. Imry and S.-k. Ma, Physical Review Letters35, 1399 (1975)

    1975

  34. [34]

    M. E. Fisher, Journal of Mathematical Physics5, 944 (1964)

    1964

  35. [35]

    Widom, The Journal of Chemical Physics43, 3898 (1965)

    B. Widom, The Journal of Chemical Physics43, 3898 (1965)

    1965

  36. [36]

    L. P. Kadanoff, Physics Physique Fizika2, 263 (1966)

    1966

  37. [37]

    J. L. Cardy,Scaling and Renormalization in Statistical Physics(Cambridge University Press, Cambridge, 1996)

    1996

  38. [38]

    Zinn-Justin,Quantum Field Theory and Critical Phenomena, 4th ed

    J. Zinn-Justin,Quantum Field Theory and Critical Phenomena, 4th ed. (Oxford University Press, Oxford, 2002)

    2002

  39. [39]

    M. E. Fisher, Reports on Progress in Physics30, 615 (1967)

    1967

  40. [40]

    Kardar,Statistical Physics of Fields(Cambridge University Press, Cambridge, 2007)

    M. Kardar,Statistical Physics of Fields(Cambridge University Press, Cambridge, 2007). Appendix A: Derivation of the Fisher and Hyperscaling Relations We derive the two exact scaling relations γ= (2−η)νandβ= ν 2 (d−2 +η),(A1) which are used throughout the main text. The first is Fisher’s relation [30]; the second follows from the Widom– Kadanoff scaling hy...

    2007