Motivated by previous observations that geometrizing statistical mechanics
offers an interesting alternative to more standard approaches,we have recently
calculated the curvature (the fundamental object in this approach) of the
information geometry metric for the Ising model on an ensemble of planar random
graphs. The standard critical exponents for this model are alpha=-1, beta=1/2,
gamma=2 and we found that the scalar curvature, R, behaves as
epsilon^(-2),where epsilon = beta_c - beta is the distance from criticality.
This contrasts with the naively expected R ~ epsilon^(-3) and the apparent
discrepancy was traced back to the effect of a negative alpha on the scaling of
R.
  Oddly,the set of standard critical exponents is shared with the 3D spherical
model. In this paper we calculate the scaling behaviour of R for the 3D
spherical model, again finding that R ~ epsilon^(-2), coinciding with the
scaling behaviour of the Ising model on planar random graphs. We also discuss
briefly the scaling of R in higher dimensions, where mean-field behaviour sets
in.Comment: 7 pages, no figure

A.C. Irving

B.P. Dolan

C.R. Rao

D. A. Johnston

D. Brody

D. O’Connor

D.V. Boulatov

E.J. Brody

G. Ruppeiner

H. Janyszek

J. Gaite

L. Onsager

R. Kenna

R.A. Fisher

T. Berlin

V.A. Kazakov

W. Janke

English

arXiv

Crossref

Information geometry of the spherical model

Motivated by the observation that geometrizing statistical mechanics offers an interesting alternative to more standard approaches, we calculate the scaling behavior of the curvature R of the information geometry metric for the spherical model. We find that R∼ε−2, where ε=βc−β is the distance from criticality. The discrepancy from the naively expected scaling R∼ε−3 is explained and compared with that for the Ising model on planar random graphs, which shares the same critical exponents

Janke, W.

Johnston, D. A.

Kenna, Ralph

Coventry University Pure Portal

CURVE is the Institutional Repository for Coventry University   Information geometry of the spherical model  Janke, W. , Johnston, D. A. and Kenna, R.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Janke, W. , Johnston, D. A. and Kenna, R. (2003) Information geometry of the spherical model. Physical Review E, volume 67 : 046106 http://dx.doi.org/10.1103/PhysRevE.67.046106   DOI 10.1103/PhysRevE.67.046106 ISSN 1539-3755 ESSN 1550-2376  Publisher: American Physical Society  Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.   This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.        arXiv:cond-mat/0210571v1  [cond-mat.stat-mech]  25 Oct 2002 The Information Geometryof the Spherical ModelW. JankeInstitut fu¨r Theoretische PhysikUniversita¨t LeipzigAugustusplatz 10/11D-04109 Leipzig, GermanyD.A. JohnstonDepartment of MathematicsHeriot-Watt UniversityRiccartonEdinburgh, EH14 4AS, ScotlandandR. KennaSchool of Mathematical and Information SciencesCoventry UniversityCoventry, CV1 5FB, EnglandApril 19, 2016AbstractMotivated by previous observations that geometrizing statistical mechanics offersan interesting alternative to more standard approaches, we have recently calculatedthe curvature (the fundamental object in this approach) of the information geometrymetric for the Ising model on an ensemble of planar random graphs. The standardcritical exponents for this model are α = −1, β = 1/2, γ = 2 and we found that thescalar curvature, R, behaves as ǫ−2,where ǫ = βc−β is the distance from criticality.This contrasts with the naively expected R ∼ ǫ−3 and the apparent discrepancywas traced back to the effect of a negative α on the scaling of R.Oddly, the set of standard critical exponents is shared with the 3D sphericalmodel. In this paper we calculate the scaling behaviour of R for the 3D sphericalmodel, again finding that R ∼ ǫ−2, coinciding with the scaling behaviour of theIsing model on planar random graphs. We also discuss briefly the scaling of R inhigher dimensions, where mean-field behaviour sets in.1 The Information Geometry of Spin ModelsThe idea of endowing the space of parameters with a metric and geometrical structure hasbeen borrowed from parametric statistics [1] and employed to some eﬀect in statisticalmechanics [2, 3, 4, 5, 6, 7, 8]. The approach seems to be particularly fruitful for a spinmodel in ﬁeld where the parameters are β, the inverse temperature, and h, the externalﬁeld. In this case the (Fisher-Rao) metric is simply given byGij = ∂i∂jf , (1)where f is the reduced free energy per site and ∂i = (∂/∂β, ∂/∂h).For such a metric the scalar curvature may be calculated asR = − 12G2∣∣∣∣∣∣∣∂2βf ∂β∂hf ∂2hf∂3βf ∂2β∂hf ∂β∂2hf∂2β∂hf ∂β∂2hf ∂3hf∣∣∣∣∣∣∣ , (2)where G = det(Gij) is the determinant of the metric itself.The work in [2, 3, 4, 5] has made it clear that, as one might expect, the scalar curvatureplays a central role in any attempt to look at statistical mechanics from a geometricalperspective. In particular, for all the models that have been considered so far, the curva-ture diverges only at a phase transition point for physical ranges of the parameter values.We assume a standard scaling form for the free energy per sitef(ǫ, h) = λ−1f(ǫλaǫ , hλah) , (3)where ǫ = βc−β (with βc marking the critical point) and aǫ, ah are the scaling dimensionsfor the energy and spin operators.The spherical model has a one-sided critical point [9] and we are interested in thehigh-temperature domain (where ǫ > 0). There, standard scaling assumptions allow usto write (3) asf(ǫ, h) = ǫ1/aǫψ+(hǫ−ah/aǫ) , (4)where we have introduced the scaling function ψ+. Consideration of the behaviour of thecomponents of R in equ. (2) then leads to the scalingR ∼ ǫα−2 , (5)for the curvature itself.Equation (5) shows that the scalar curvature not only characterises phase transitionpoints as divergences, as do the more standard statistical mechanical quantities such asthe speciﬁc heat C and susceptibility χ, it displays a scaling behaviour that allows theextraction of the critical exponent α. Indeed, with hyperscaling, equ. (5) can be recast asR ∼ ξd , (6)where ξ is the correlation length and d is the dimensionality of the system. So thecurvature is proportional to the correlation volume, an intuitively reasonable result ondimensional grounds.1At ﬁrst sight, our calculation of the scaling behaviour of R in [10] for the Ising modelon planar random graphs puts a spanner in the works. There, α = −1 but R ∼ ǫ−2 ratherthan the expected R ∼ ǫ−3. However, returning to the detailed scaling of the individualterms in equ. (2) showed that a negative α aﬀected some of the scaling of the componentsand the end result could be traced back to these modiﬁcations. In principle this modiﬁedbehaviour should apply to any model with negative α, so an interesting test would be tocalculate R, or at least its scaling limit, for another model with this property.The spherical model provides just such a test case. It was solved (in ﬁeld) in the classicBerlin and Kac paper [9] and the critical exponents in 3D found to be α = −1, β =1/2, γ = 2 which are identical to those of the Ising model on 2D planar random graphs[11]. This is remarkable, because there are no obvious physical similarities between thetwo models. An additional motivation for any such calculation is the general paucity ofspin models that have been solved in ﬁeld. Indeed, R for spin models has been obtainedexplicitly only for the 1D Ising [3], Bethe lattice Ising [6] and 1D Potts [12] models andthe scaling form calculated for the Ising model on planar random graphs [10]. This formsa rather small sample on which to formulate hypotheses about its general behaviour andits scaling properties. In the sequel, we extend this short list incrementally by lookingat the scaling behaviour of R in the spherical model. We concentrate on the 3D case,but also discuss other dimensions, including the mean-ﬁeld like behaviour which sets inat d = 4.2 The Spherical ModelBerlin and Kac [9] introduced the spherical model (and the Gaussian model) in an attemptto understand how generic some of the features of Onsager’s solution [13] of the 2DIsing model are for ferromagnetic spin models, particularly for other dimensions. In thespherical model, the±1 condition on the value of the Ising spins is relaxed, whilst retaininga global constraint on the total spin magnitude. With si denoting the value of a spin ata site i of a hypercubic lattice, the partition function is [9]Z =∫ds1 . . . dsN expβ∑〈ij〉sisj + h∑isi δ(∑is2i −N), (7)where N is the total number of sites. This can be evaluated by exponentiating theconstraint and using steepest descent, resulting in the following expression for the reducedfree energy per site in the thermodynamic limit, N →∞:f =12log(πβ)+ βz − 12g(z) +h24β(z − d) , (8)whereg(z) =1(2π)d∫ 2π0dω1 . . . dωd log(z −d∑k=1cos(ωk)). (9)The saddle point value of z which appears in the expression for the free energy in equ. (8)is determined fromg′(z) = 2β − h22β(z − d)2 . (10)2The solution reveals no transition for d = 1 and 2, and a transition with the exponentsα = −1, β = 1/2, γ = 2 for d = 3. For d ≥ 4, on the other hand, mean-ﬁeld behaviourwith α = 0, β = 1/2, γ = 1 sets in (modiﬁed by multiplicative logarithmic corrections inthe d = 4 case [14]). For the following formulae we conﬁne our attention to the d = 3case.It is useful to note that, with h = 0, equ. (10) givesdzdβ=2g′′(z), (11)and henced2zdβ2= −4g(3)(z)[g′′(z)]3. (12)The critical point is given by z = d = 3 and h = 0 [9], and the behaviour of g(z) in thisregion is determined by diﬀerentiating equ. (9) twice and then expanding for the smallωk values which give the dominant contribution. One ﬁndsg′′(z) ∼ − 12√2π(z − 3)−1/2 . (13)A further diﬀerentiation givesg(3)(z) ∼ 14√2π(z − 3)−3/2 , (14)and an integration yieldsg′(z) =1√2π(z − 3)1/2 + g′(3) , (15)where g′(3) = (18 + 12√2− 10√3− 7√6)[K(2√3 +√6− 2√2− 3)]2 ≈ 0.505 462 019 . . .is the exactly known massless 3D lattice propagator at the origin, with K(k2) denotingthe standard elliptic integral. This latter expression can be combined with equ. (10) withh = 0 to give(z − 3) ∼ 8π2(βc − β)2 ∼ ǫ2 , (16)in which βc = g′(3)/2 ≈ 0.252 731 009 . . .. Equations (13) and (14) may then be substi-tuted in equs. (11) and (12) to give the scaling of dz/dβ and d2z/dβ2,limz→3dzdβ= limz→3{−4√2π(z − 3)1/2} = 0 ,limz→3d2zdβ2= 16π2 , (17)which we shall employ below in the calculation of the scalar curvature.33 The Scalar CurvatureWe recapitulate the general considerations of [10] before going on to discuss in detail theresults for the spherical model. In (4), we deﬁne A = 1/aǫ and C = −ah/aǫ, which, interms of the standard exponents, gives A = 2−α and A+C = β. If A > 2 (which is thecase for α < 0) the speciﬁc heat, rather than diverging, will be a constant at the criticalpoint, which we denote by A(A − 1)φ(0). With this in mind, substitution of the scalingfunction into equ. (2) givesR = − 12G2×∣∣∣∣∣∣∣A(A− 1)φ(0) 0 ǫA+2Cψ′′+(0)−A(A− 1)(A− 2)ǫA−3ψ+(0) 0 −(A + 2C)ǫA+2C−1ψ′′+(0)0 −(A + 2C)ǫA+2C−1ψ′′+(0) 0∣∣∣∣∣∣∣ .(18)In equ. (18) terms with an odd number of h derivatives have been set to zero. Thesedo not appear because of the h → −h symmetry of the free energy. The scaling for thedeterminant of the metric G is given byG = A(A− 1)ǫA+2Cφ(0)ψ′′+(0) , (19)so the leading term in R (for A > 2) isR = (A + 2C)22A(A− 1)φ(0)ǫ−2 , (20)or, translating back to the standard critical exponents,R = γ22(2− α)(1− α)φ(0)ǫ−2 . (21)In summary, if α < 0 the expected scaling of R is R ∼ ǫ−2 rather than the R ∼ ǫα−2,which is seen for positive α.We now move on to examine the scaling of the various terms contributing to R inequ. (2) for the spherical model itself. As we have remarked, the h→ −h symmetry in thefree energy per site, f , of the spherical model means that any terms with an odd numberof h derivatives will automatically be zero when h = 0, hence fβh = fββh = fhhh = 0.This leaves the non-zero terms (again when h = 0)fββ =∂z∂β+12β2,fhh =12β(z − 3) ,fβββ =∂2z∂β2− 1β3,fhhβ = − 12β(z − 3)2∂z∂β− 12β2(z − 3) , (22)4which, using equs. (16) and (17) have the following behaviour in the scaling region:fββ ∼ 12β2c,fhh ∼ 116π2βc(βc − β)2 ∼ ǫ−2 ,fβββ ∼ 16π2 − 1β3c,fhhβ ∼ 18π2βc(βc − β)3 −116π2β2c (βc − β)2∼ ǫ−3 . (23)Comparing with the individual terms in equ. (18) we see that the expected general scalingof each term (for α < 0) does indeed apply and that overall we have, as in equ. (21),R ∼ ǫ−2 . (24)We thus see that calculating the scaling of R for the 3D spherical model for which α = −1gives results in accordance with expectations from general scaling arguments which takeinto account the negative α, similarly to the Ising model on planar random graphs.In d ≥ 4 dimensions the exponents of the spherical model attain their mean-ﬁeldvalues, α = 0, β = 1/2, γ = 1. In this case g′(z) ∼ z − d + g′(d) near criticality and theresulting scaling of the components of the expression for R isR = − 12G2∣∣∣∣∣∣∣const 0 ǫ−1const 0 ǫ−20 ǫ−2 0∣∣∣∣∣∣∣ , (25)where the determinant of the metric scales as G = ǫ−1. Thus R itself scales as ǫ−2, in themean-ﬁeld case. This is in agreement with the expected scaling of ǫα−2 when α ≥ 0 since,in that case [4],R = − 12G2×∣∣∣∣∣∣∣A(A− 1)ǫA−2ψ+(0) 0 ǫA+2Cψ′′+(0)−A(A− 1)(A− 2)ǫA−3ψ+(0) 0 −(A+ 2C)ǫA+2C−1ψ′′+(0)0 −(A + 2C)ǫA+2C−1ψ′′+(0) 0∣∣∣∣∣∣∣ ,(26)where the scaling of the metric determinant isG = A(A− 1)ǫ2A+2C−2ψ+(0)ψ′′+(0) . (27)The diﬀerence in scaling between the α ≥ 0 and the α < 0 cases originates in the fββterm which contributes to both R and G in equ. (26) and equ. (27). In the negative αcase, the contribution of this term comes from the regular part of the free energy, and isnon-diverging.Note that for α = 0, A = 2, and although the displayed scaling term A(A − 1)(A −2)ǫA−3ψ+(0) in equ. (26) would be expected to vanish, a regular term could still contribute.5This is indeed what is seen in the explicit calculation for the spherical model. Two furthercaveats arise. Firstly, for the upper critical dimension d = 4 itself, there are multiplicativelogarithmic corrections [14] which require a careful handling. Secondly, for d > 4, wherethe scaling is described by mean-ﬁeld theory, the hyperscaling relation fails. So one shouldbe quite cautious in comparing the explicit calculation in terms of ǫ with expressionsinvolving the correlation length ξ.4 ConclusionsWe have calculated the scaling behaviour of the scalar curvature, R, of the informationgeometry metric for the 3D spherical model, ﬁnding, as for the Ising model on planarrandom graphs, thatR ∼ ǫ−2. Careful considerations of the scaling of the various elementswhich contribute to R show that this is in accordance with expectations. We also brieﬂydiscussed the scaling behaviour of R for d ≥ 4, corresponding to mean-ﬁeld exponents.The calculation reported here provides another example of a statistical mechanicalmodel in which the curvature of the information geometry metric diverges at the criticalpoint, with a clearly quantiﬁable scaling behaviour. Once again it is curious that therelevant exponent is shared with the Ising model on planar random graphs. All criticalexponents now coincide in the two models though there is no obvious physical relationbetween them. It seems that the suggestion in the original Berlin and Kac paper [9]that the 3D spherical model might provide a hint to the behaviour of the 3D Ising modelapplies rather to the Ising model on 2D planar random graphs.5 AcknowledgementsW.J. and D.J. were partially supported by EC IHP network “Discrete Random Geome-tries: From Solid State Physics to Quantum Gravity” HPRN-CT-1999-000161. D.J. andR.K. were also partially supported by an Enterprise Ireland/British Council ResearchVisits Scheme grant.References[1] R.A. Fisher, Phil. Trans. R. Soc. Lond. 222 (1922) 309;C.R. Rao, Bull. Calcutta Math. Soc. 37 (1945) 81.[2] G. Ruppeiner, Rev. Mod. Phys. 67 (1995) 605; Phys. Rev. A 20 (1979) 1608; ibid.A 24 (1980) 488.[3] H. Janyszek, Rep. Math. Phys. 24 (1986) 1; ibid. 11;H. Janyszek and R. Mruga la, Phys. Rev. A 39 (1989) 6515.[4] H. Janyszek, J. Phys. A 23 (1990) 477.[5] E.J. Brody, Phys. Rev. Lett. 58 (1987) 179;D. Brody and N. Rivier, Phys. Rev. E 51 (1995) 1006;6D. Brody and A. Ritz, Nucl. Phys. B 522 (1998) 588;D. Brody and L. Hughston, Proc. Roy. Soc. London A 455 (1999) 1683.[6] B.P. Dolan, Proc. Roy. Soc. London A 454 (1998) 2655.[7] B.P. Dolan, Int. J. Mod. Phys. A 12 (1997) 2413.[8] D. O’Connor and C.R. Stephens, Prog. Theor. Phys. 90 (1993) 747;B.P. Dolan, Int. J. Mod. Phys. A 10 (1995) 2439; ibid. 2703; Nucl. Phys. B 528(1998) 553;J. Gaite and D. O’Connor, Phys. 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Kenna, R.

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