By expressing thermodynamic functions in terms of the edge and density of
Lee--Yang zeroes, we relate the scaling behaviour of the specific heat to that
of the zero field magnetic susceptibility in the thermodynamic limit of the
$XY$--model in two dimensions. Assuming that finite--size scaling holds, we
show that the conventional Kosterlitz--Thouless scaling predictions for these
thermodynamic functions are not mutually compatable unless they are modified by
multiplicative logarithmic corrections. We identify these logarithmic
corrections analytically in the case of the specific heat and numerically in
the case of the susceptibility. The techniques presented here are general and
can be used to check the compatibility of scaling behaviour of odd and even
thermodynamic functions in other models too.Comment: 11 pages, latex, 4 figure

A.C. Irving

Allton

Baillie

Biferale

Butera

Dunlop

Edwards

Fisher

Gupta

Hamer

Hasenbusch

Hulsebos

Irving

Itzykson

Kajantie

Kenna

Kosterlitz

Luck

Nightingale

R. Kenna

Roomany

Salmhofer

Suzuki

Wolff

Yang

Physics Letters B

English

arXiv

Crossref

Logarithmic corrections to scaling in the two dimensional XY-model

By expressing thermodynamic functions in terms of the edge and density of Lee-Yang zeroes, we relate the scaling behaviour of the specific heat to that of the zero field magnetic susceptibility in the thermodynamic limit of the XY-model in two dimensions. Assuming that finite-size scaling holds, we show that the conventional Kosterlitz-Thouless scaling predictions for these thermodynamic functions are not mutually compatible unless they are modified by multiplicative logarithmic corrections. We identify these logarithmic corrections analytically in the case of the specific heat and numerically in the case of the susceptibility. The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too

Kenna, Ralph

Irving, A. C.

Coventry University Pure Portal

CURVE is the Institutional Repository for Coventry University   Logarithmic corrections to scaling in the two dimensional XY-model  Kenna, R. and Irving, A. C.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Kenna, R. and Irving, A. C. (1995) Logarithmic corrections to scaling in the two dimensional XY-model. Physics Letters B, volume 351 (1-3): 273–278 http://dx.doi.org/10.1016/0370-2693(95)00316-D  DOI 10.1016/0370-2693(95)00316-D ISSN 0370-2693 ESSN 1873-2445  Publisher: Elsevier  NOTICE: This is the author’s version of a work that was accepted for publication in Physics Letters B. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published Physics Letters B, [351, 1-3, 1995] DOI 10.1016/0370-2693(95)00316-D  © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/  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:hep-lat/9501008v1  12 Jan 1995Liverpool Preprint: LTH 343Logarithmic Corrections to Scaling in the TwoDimensional XY –ModelR. Kenna1 and A.C. IrvingDAMTP, University of Liverpool L69 3BX, EnglandDecember 1994AbstractBy expressing thermodynamic functions in terms of the edge and density of Lee–Yang ze-roes, we relate the scaling behaviour of the specific heat to that of the zero field magneticsusceptibility in the thermodynamic limit of the XY –model in two dimensions. Assuming thatfinite–size scaling holds, we show that the conventional Kosterlitz–Thouless scaling predictionsfor these thermodynamic functions are not mutually compatable unless they are modified bymultiplicative logarithmic corrections. We identify these logarithmic corrections analytically inthe case of the specific heat and numerically in the case of the susceptibility. The techniquespresented here are general and can be used to check the compatibility of scaling behaviour ofodd and even thermodynamic functions in other models too.1Supported by EU Human Capital and Mobility Scheme Project No. CHBI–CT94–1125The Kosterlitz–Thouless Scenario The partition function for the O(n) non–linear σ–modeldefined on a regular d–dimensional lattice Λ with periodic boundary conditions in the presence ofan external magnetic field can be written asZL(β, h) =1N∑{~σx}eβS+h~n·~M , (1)where ~n is a unit vector defining the direction of the external magnetic field and h is a scalarparameter representing its strength. The summation is over all configurations open to the system.The factor N ensures that ZL(0, 0) = 1 andS =∑x∈Λd∑µ=1~σx · ~σx+µ , ~M =∑x∈Λ~σx .Here L represents the linear extent of the system, β = 1/kT and ~σx is an n–component unit lengthspin at site x ∈ Λ. The case d = 2, n = 2 is the two dimensional XY –model or plane rotatormodel and is unusual in that it exhibits an exponentional singularity. Using an approximaterenormalization group approach, Kosterlitz and Thouless [1] demonstrated the existence of a phasetransition driven by the condensation of vortices. The model remains critical (thermodynamicfunctions diverge) for all β > βc and the critical exponents are dependent on temperature. Interms of the reduced temperaturet = 1 −ββc,the scaling behaviour of the correlation length, susceptibility and the specific heat is given in [1] asξ∞(t) ∼ eat−ν , (2)χ∞(t) ∼ ξ2−η∞ , (3)C∞(t) ∼ ξα̃∞ + constant , (4)where for t → 0+, ν = 1/2, η = 1/4 and α̃ = −d = −2. The purpose of this work is to argue that thelatter two scaling formulae are incompatible as they stand. To this end, a method is presented bywhich odd and even thermodynamic functions (the susceptibility and specific heat) can be relatedand expressed in terms of partition function zeroes. Using certain reasonable assumptions regardingfinite–size scaling, it is shown that there have to exist multiplicative logarithmic corrections 2 to(3) and (4). This method can be applied to any model.Partition Function Zeroes Lee and Yang [2] showed the connection between critical behaviourand partition function zeroes. For a finite system these are strictly complex (non–real). As L → ∞one expects the zeroes to condense onto smooth curves. Zeroes in the plane of complex externalmagnetic field h are refered to as Lee–Yang zeroes. Lee and Yang further showed that for certainsystems these zeroes are in fact restricted to the imaginary h axis (the Lee–Yang theorem) [2]. Inthe symmetric phase, t > 0, they lie away from the real h-axis, pinching it only as t → 0+ (in the2The work of [1] contains implicitly a prediction for logarithmic corrections but this seems not to have beenfollowed up quantitatively.1thermodynamic limit). For systems obeying the Lee–Yang theorem, such as the XY Model [3], thispinching occurs at h = 0 prohibiting analytic continuation from Re(h) < 0 to Re(h) > 0. Thismeans that (for the finite–size system or in the thermodynamic limit) the partition function is anentire function of h all of whose zeroes are purely imaginary. The zeroes in the complex fugacityplane (z ≡ exp (h)) are distributed on the unit circle. Writing these zeroes aszj(β) = eiθj(β) ,the partition function isZL(β, h) = ρL(β, h)∏j(z − eiθj(β)), (5)where ρL is a non-vanishing function of h related to the spectral density (ignored in what followssince it contributes only to the regular part of the free energy in the thermodynamic limit). Thefree energy is thenfL(β, h) = L−d∑jln (z − eiθj(β)) . (6)Define the density of zeroes to be [4, 5]gL(β, θ) = L−d∑jδ(θ − θj(β)) =∂GL(β, θ)∂θ. (7)Here GL(β, θ) is the cumulative density of zeroes and monotonically increases from GL(β, 0) = 0to GL(β, π) = 1/2. The distribution of zeroes is symmetric about the real h– (or z–) axis and sogL(β,−θ) = gL(β, θ) . (8)In terms of the density of zeroes, (6) becomesfL(β, h) =12(h + ln 2) +∫ π0ln (cosh h − cos θ)dGL(β, θ) , (9)where (8) has been used.In the high temperature phase there exists a region around θ = 0 which is free from Lee–Yangzeroes [2]. We define the corresponding Yang–Lee edge θYL(β) byg(β, θ) = 0 for − θYL(β) < θ < θYL(β) .The integral in (9) can therefore be taken to begin at θYL(β). Salmhofer has proved the existenceof a unique density of zeroes in the thermodynamic limit [5]. In this limit, integrating (9) by partsand expanding the trigonometric functions gives for the singular part of the free energy [4, 6, 7]f∞(β, h) = −2∫ πθYL(β)θh2 + θ2G∞(β, θ)dθ . (10)The magnetic susceptibility χ∞ is given by the second derivative of the free energy with respect toh. Following [4, 6, 7], this leads toG∞(β, θ) = χ∞(β)(θYL(β))2Φ(θθYL(β)), (11)2Φ(x) being some function of x such that Φ(| x |≤ 1) = 0. From (10) and the fact that G (θYL, β) = 0,one gets the specific heatC∞(β) =∂2f∞(β, h)∂β2∣∣∣∣∣h=0= −2∫ πθYL(β)θ−1d2G∞ (θ, β)dβ2dθ . (12)The above formulae are quite general and hold for any model provided it obeys the Lee–Yangtheorem.To proceed further we insert the (model–specific) critical behaviour. Instead of (3) and (4), assumenow the following modified critical behaviour for the singular parts of the zero field susceptibilityand the specific heat for t >∼ 0.χ∞ ∼ ξ2−η∞ tr , C∞ ∼ ξα̃tq . (13)Similarly, assume that the leading scaling behaviour of the Yang–Lee edge in terms of the reducedtemperature isθYL(t) ∼ ξλtp . (14)For t >∼ 0 the critical indices are independent of t [1]. The scaling behaviour of C∞ can be relatedto that of χ∞ and θYL through (11) and (12). These giveξ(t)α̃tq ∝ ξ(t)2−η+2λt2p+r−2−2ν .Therefore,λ =12(α̃ − 2 + η) , (15)p =12(q − r) + 1 + ν . (16)Finite–Size Scaling The finite–size scaling (FSS) hypothesis, first formulated in 1972 by Fisher[8], is a relationship between the scaling behaviour of thermodynamic quantities in the infinitevolume limit and the size dependence of their finite volume counterparts. The general statement ofFSS, which is expected to hold in all dimensions including the upper critical one [9] is that if PL(t)is the value of some thermodynamic quantity P at reduced temperature t measured on a system oflinear extent L, thenPL(0)P∞(t)= FP(ξL(0)ξ∞(t)), (17)where ξL(t) is the correlation length of the finite–size system. Luck has shown that, for the XY –model in two dimensions, ξL(0) is proportional to L [10]. Fixing the scaling variable, one hasξ∞(t) ∼ x−1L ,and from (2)t ∼ (ln L)−1ν ,for large enough L. Therefore FSS for the susceptibility and the Yang–Lee edge isχL(0) ∼ L2−η(ln L)−rν , (18)θ1(0) ∼ Lλ(ln L)−pν . (19)3For the finite size system it is convenient to consider the zeroes in the complex h–plane. Thesingular part of the free energy corresponds tofL(t, h) = L−d∑jln (h − iθj(t)) (20)where θ1 = θY L. The (reduced) magnetic susceptibility is thereforeχL(t) =∂2fL∂h2∣∣∣∣∣h=0= −Ld∑j1θ2j. (21)The lowest lying zeroes are expected to scale in the same way [11] so thatχL(t) ∝ −L−dθ1(t)−2 . (22)From (18) and (19) we haveL2−η(ln L)−rν ∼ L−d−2λ(ln L)2pν , (23)and concludeα̃ = −d , q = −2(1 + ν) . (24)Thus the scaling behaviour of the singular part of the specific heat indeed exhibits multiplicativelogarithmic corrections. Accepting the KT predictions ν = 1/2, η = 1/4 for t >∼ 0, the leadingcritical exponent for the Yang–Lee edge λ and the correction exponents q and p areλ = −158, q = −3 , p = −r2. (25)The above analytic considerations have yielded no information on the odd correction exponentr. The original renormalisation group analysis of Kosterlitz and Thouless [1], in fact, implicitlycontained the predictionr = −1/16 (26)as noted by [12]. Subsequent analysis have concentrated on the r = 0 form of the scaling behaviour(18) and the verification that η(βc) = 1/4. Allton and Hamer [13] have conjectured that thedeviation of their determination of η from 1/4 might be due to logarithmic corrections.The study of the full scaling form and the numerical determination of r is the subject of whatfollows.Numerical Procedures Numerical methods were used to test the scaling picture of the last sec-tion. Specifically, we analysed the FSS behaviour of the Yang–Lee edge so as to test the prediction(19), (25)θ1(βc) ∼ Lλ(ln L)r . (27)In particular, we sought to confirm λ = 15/8 for the leading behaviour and to check if r 6= 0.An algorithm based on that of Wolff [14] was used to simulate the XY –model at zero magneticfield (h = 0) on square lattices of sizes L = 32, 64, 128 and 256 . The values of β at which thesimulations took place (βo) were evenly spaced in steps of 0.02 between 1.00 and 1.20. At each4simulation point, 50, 000 measurements were made from each of two separate starts. The secondruns were used only to verify equilibration and to improve our understanding of the statisticalerrors. The latter were estimated using bootstrap as described below. The autocorrelation time ofthe susceptibility was also monitored. Histogram reweighting [15] was used to enable extrapolationbetween βo values.The determination of βc is inextricably bound up with the assumed critical scaling behaviour. Forthe present purposes, we adopted a two-stage strategy: we used estimates of βc from our ownand other analyses which had assumed no logarithmic corrections then tested the stability of theseconclusions with respect both to uncertainties in βc and to the form of the scaling asumption.One simple estimate was based on assumed conventional (r = 0) scaling of the susceptibility (3). Ageneralised [16] Roomany-Wyld approximant [17] was used to give estimates of the correspondingrenormalisation group beta function for each pair of lattice sizes related by factor of two scaling:βRWL = (2β2)η − 2 + 2r ln( ln 2Lln L )/ ln 2 + ln(χ2L/χL)/ ln 2∂∂β [χ2LχL]. (28)A preliminary estimate, using η = 1/4 and r = 0, leads to to βc = 1.11(1) which is at the lowerend of the range of published estimates [12], [18], [19]. These span 1.11 to 1.13. The most recenthigh precision result is 1.1197(5) [19].Determination of Lee–Yang Zeroes When the external field is complex (h = hr + ihi), thepartition function (1) can be rewrittenZL(β, hr + ihi) = ReZL(β, hr + ihi) + iImZL(β, hr + ihi) , (29)whereReZL(β, hr + ihi) =1N∑{~σx}eβS+hrM cos (hiM) ,= ZL(β, hr)〈cos (hiM)〉β,hr , (30)andImZL(β, hr + ihi) =1N∑{~σx}eβS+hrM sin (hiM) ,= ZL(β, hr)〈sin (hiM)〉β,hr . (31)According to the Lee–Yang theorem the zeroes are on the imaginary h–axis (hr = 0) where〈sin (hiM)〉β,hr vanishes and so the Lee–Yang zeroes are simply the zeroes ofReZL(β, ihi) ∝ 〈cos (hiM)〉β,0 . (32)Thus the Lee–Yang zeroes are easily found. Moreover, one can find them at any β using reweight-ing techniques [15]. For the lowest lying zeroes at β = 1.11 we found θ1(L) = 0.0023353(7),0.0006350(2), 0.00017278(5) and 0.000047062(13) for L = 32, 64, 128 and 256 respectively. Er-rors were calculated using the bootstrap method where the data for each βo are resampled 1005times (with replacement) leading to 100 estimates for θ1, from which the variance and bias can becalculated.In figure 1 we plot the logarithm of the position of the first Lee–Yang zero against the logarithmof the lattice size L, at β = 1.11. In the absence of any corrections, the slope should give theleading power–law exponent λ. In fact the slope is -1.8776(2), the deviation from the KT value of−15/8 = −1.875 being attributed to the presence of logarithmic corrections. To identify these, andthe correction exponent r in (27), ln (θ1L15/8) is plotted against ln ln L in figure 2. A straight line isidentified. Its slope is −0.012(1) giving evidence for a non–zero value of r, albeit not in agreementwith the RG predictions of −1/16 = −0.0625 from [1].At this point, one must investigate the extent to which these conclusions regarding the existence ofmultiplicative logarithmic corrections depend upon the measurement of βc. We have attempted todo this systematically by studying the above ln ln L fits for r as a function of the assumed criticalbeta. Fits with a range of values of r (including zero) are possible but not all with acceptableχ2/degree of freedom. To be conservative, we take ‘acceptable’ to mean less than 2. This corre-sponds to a minimum confidence level of 14%. In figure 3 we show that acceptable values of χ2are possible only for 1.110 <∼ βc<∼ 1.120. The r = 0 solution would correspond to βc ≈ 1.105and χ2/dof ≈ 4. Similary, we find that a fit with r = −1/16 would correspond to βc ≈ 1.138 andχ2/dof ≈ 16.In summmary, assumming the Kosteritz-Thouless value η = 1/4, we find non-zero logarithmiccorrections to scaling and a corresponding estimate of the critical temperature:r = −0.023 ± 0.010 , βc = 1.115 ± .005 . (33)As a cross check, one may use the above values for r and η to find the Roomany-Wyld approximant(28) and estimate βc from its zeroes. We find βc = 1.115±0.004 in good agreement with the above.However, we emphasize the much higher precision available to an analysis based on Lee-Yang zeroesrather than on the spin susceptibility.Conclusions Theoretical arguments concerning the consistency of the scaling behaviour of oddand even thermodynamic functions at a KT phase transition have been presented. The generallyused scaling formulae have to be modified by multiplicative corrections. These are identified ana-lytically for the specific heat and numerically for the susceptibility. This numerical identificationcomes via an analysis of Lee–Yang zeroes, the FSS of which is linked to that of the susceptibility.We would like to thank P. Lacock for assistance with multihistogram reweighting techniques.6References[1] J. Kosterlitz and D. Thouless, J. Phys. C 6 (1973) 1181; J. Kosterlitz, J. Phys. C 7(1974) 1046.[2] C. N. Yang and T. D. Lee, Phys. Rev. 87 404 (1952); ibid. 410.[3] F. Dunlop and C.M. Newman, Commun. Math. Phys. 44 (1975) 223.[4] R. Kenna and C.B. Lang, Phys. Rev. E 49 (1994) 5012.[5] M. Salmhofer, UBC–Preprint (unpublished); Nucl. Phys. B (Proc. Suppl.) 30, (1993),81.[6] R. Abe, Prog. Theor. Phys. 37 (1967) 1070; Prog. Theor. Phys. 38 (1967) 72; ibid. 322;ibid. 568.[7] M. Suzuki, Prog. Theor. Phys. 38 (1967) 289; ibid. 1225; ibid. 1243; Prog. Theor. Phys.39, (1968) 349.[8] M.E. Fisher, in Critical Phenomena, Proc. 51th Enrico Fermi Summer School, Varena,ed. M.S. Green (Academic Press, NY, 1972).[9] R. Kenna and C.B. Lang, Phys. Lett. B 264 (1991) 396; Nucl. Phys. B (Proc. Suppl.)30 (1993) 697; Nucl. Phys. B 393 (1993) 461; Err. ibid. B 411 (1994) 340.[10] J.M. Luck, J. Phys. A 15 (1982) L169.[11] C. Itzykson, R. B. Pearson, and J. B. Zuber, Nucl. Phys. B 220[FS8] (1983) 415;C. Itzykson and J. M. Luck, Progress in Physics, Critical Phenomena (1983 BrasovConference) Birkhäuser, Boston Inc. Vol 11, 45 (1985).[12] P. Butera and M. Comi, Phys. Rev. B 47 (1993) 11969.[13] C. Allton and C. Hamer, J. Phys. A 21 (1988) 2417.[14] U. Wolff, Phys. Rev. Lett. 62 (1989) 361.[15] K. Kajantie, L. Kärkkäinen and K. Rummukainen, Nucl. Phys. B 357 (1991) 693.[16] A.C. Irving and C. Hamer, Nucl. Phys. B 230[FS10] (1984) 361; C. Hamer and A.C.Irving, J. Phys. A 17 (1984) 1649.[17] M.P. Nightingale, Physica A 83 (1976) 561; H. Roomany and H.W. Wyld, Phys. Rev.D 21 (1980) 3341.[18] R. Gupta, J. De Lapp, G. Batrouni, G.C. Fox, C.F. Baillie and J. Apostolakis, Phys.Rev. Lett. 61 (1988) 1996; L. Biferale and R. Petronzio, Nucl. Phys. B 328 (1989) 677;C.F. Baillie and R. Gupta, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 669; A. Hulsebos,J. smit and J.C. Vink, Nucl. Phys. B 356 (1991) 775; R.G. Edwards, J. Goodman andA.D. Sokal, Nucl. Phys. B 354 (1991) 289.7[19] M. Hasenbusch, M. Marcu and K. Pinn, Physica A 208 (1994) 124.8-10-9-8-7-63 3.5 4 4.5 5 5.5 6Ln (Z1)Ln (L)Figure 1: Leading FSS of Lee–Yang Zeroes at β = 1.11.0.4320.4340.4360.4380.441.2 1.3 1.4 1.5 1.6 1.7Ln(Z1) + (15/8)Ln(L)LnLn(L)Figure 2: Corrections to FSS of Lee–Yang Zeroes at β = 1.11.9-0.06-0.04-0.0200.02r  (a)04812161.1 1.12 1.14Chisq/dofBETA (b)Figure 3: (a) Correction Exponent r Measured over a Range of β and (b) corresponding χ2 perDegree of Freedom (Goodness of Fit).10

CURVE is the Institutional Repository for Coventry University   Logarithmic corrections to scaling in the two dimensional XY-model  Kenna, R. and Irving, A. C.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Kenna, R. and Irving, A. C. (1995) Logarithmic corrections to scaling in the two dimensional XY-model. Physics Letters B, volume 351 (1-3): 273–278 http://dx.doi.org/10.1016/0370-2693(95)00316-D  DOI 10.1016/0370-2693(95)00316-D ISSN 0370-2693 ESSN 1873-2445  Publisher: Elsevier  NOTICE: This is the author’s version of a work that was accepted for publication in Physics Letters B. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published Physics Letters B, [351, 1-3, 1995] DOI 10.1016/0370-2693(95)00316-D  © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/  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:hep-lat/9501008v1  12 Jan 1995Liverpool Preprint: LTH 343Logarithmic Corrections to Scaling in the TwoDimensional XY –ModelR. Kenna1 and A.C. IrvingDAMTP, University of Liverpool L69 3BX, EnglandDecember 1994AbstractBy expressing thermodynamic functions in terms of the edge and density of Lee–Yang ze-roes, we relate the scaling behaviour of the specific heat to that of the zero field magneticsusceptibility in the thermodynamic limit of the XY –model in two dimensions. Assuming thatfinite–size scaling holds, we show that the conventional Kosterlitz–Thouless scaling predictionsfor these thermodynamic functions are not mutually compatable unless they are modified bymultiplicative logarithmic corrections. We identify these logarithmic corrections analytically inthe case of the specific heat and numerically in the case of the susceptibility. The techniquespresented here are general and can be used to check the compatibility of scaling behaviour ofodd and even thermodynamic functions in other models too.1Supported by EU Human Capital and Mobility Scheme Project No. CHBI–CT94–1125The Kosterlitz–Thouless Scenario The partition function for the O(n) non–linear σ–modeldefined on a regular d–dimensional lattice Λ with periodic boundary conditions in the presence ofan external magnetic field can be written asZL(β, h) =1N∑{~σx}eβS+h~n·~M , (1)where ~n is a unit vector defining the direction of the external magnetic field and h is a scalarparameter representing its strength. The summation is over all configurations open to the system.The factor N ensures that ZL(0, 0) = 1 andS =∑x∈Λd∑µ=1~σx · ~σx+µ , ~M =∑x∈Λ~σx .Here L represents the linear extent of the system, β = 1/kT and ~σx is an n–component unit lengthspin at site x ∈ Λ. The case d = 2, n = 2 is the two dimensional XY –model or plane rotatormodel and is unusual in that it exhibits an exponentional singularity. Using an approximaterenormalization group approach, Kosterlitz and Thouless [1] demonstrated the existence of a phasetransition driven by the condensation of vortices. The model remains critical (thermodynamicfunctions diverge) for all β > βc and the critical exponents are dependent on temperature. Interms of the reduced temperaturet = 1−ββc,the scaling behaviour of the correlation length, susceptibility and the specific heat is given in [1] asξ∞(t) ∼ eat−ν , (2)χ∞(t) ∼ ξ2−η∞ , (3)C∞(t) ∼ ξα˜∞ + constant , (4)where for t→ 0+, ν = 1/2, η = 1/4 and α˜ = −d = −2. The purpose of this work is to argue that thelatter two scaling formulae are incompatible as they stand. To this end, a method is presented bywhich odd and even thermodynamic functions (the susceptibility and specific heat) can be relatedand expressed in terms of partition function zeroes. Using certain reasonable assumptions regardingfinite–size scaling, it is shown that there have to exist multiplicative logarithmic corrections 2 to(3) and (4). This method can be applied to any model.Partition Function Zeroes Lee and Yang [2] showed the connection between critical behaviourand partition function zeroes. For a finite system these are strictly complex (non–real). As L→∞one expects the zeroes to condense onto smooth curves. Zeroes in the plane of complex externalmagnetic field h are refered to as Lee–Yang zeroes. Lee and Yang further showed that for certainsystems these zeroes are in fact restricted to the imaginary h axis (the Lee–Yang theorem) [2]. Inthe symmetric phase, t > 0, they lie away from the real h-axis, pinching it only as t→ 0+ (in the2The work of [1] contains implicitly a prediction for logarithmic corrections but this seems not to have beenfollowed up quantitatively.1thermodynamic limit). For systems obeying the Lee–Yang theorem, such as the XY Model [3], thispinching occurs at h = 0 prohibiting analytic continuation from Re(h) < 0 to Re(h) > 0. Thismeans that (for the finite–size system or in the thermodynamic limit) the partition function is anentire function of h all of whose zeroes are purely imaginary. The zeroes in the complex fugacityplane (z ≡ exp (h)) are distributed on the unit circle. Writing these zeroes aszj(β) = eiθj(β) ,the partition function isZL(β, h) = ρL(β, h)∏j(z − eiθj(β)), (5)where ρL is a non-vanishing function of h related to the spectral density (ignored in what followssince it contributes only to the regular part of the free energy in the thermodynamic limit). Thefree energy is thenfL(β, h) = L−d∑jln (z − eiθj(β)) . (6)Define the density of zeroes to be [4, 5]gL(β, θ) = L−d∑jδ(θ − θj(β)) =∂GL(β, θ)∂θ. (7)Here GL(β, θ) is the cumulative density of zeroes and monotonically increases from GL(β, 0) = 0to GL(β, π) = 1/2. The distribution of zeroes is symmetric about the real h– (or z–) axis and sogL(β,−θ) = gL(β, θ) . (8)In terms of the density of zeroes, (6) becomesfL(β, h) =12(h+ ln 2) +∫ π0ln (coshh− cos θ)dGL(β, θ) , (9)where (8) has been used.In the high temperature phase there exists a region around θ = 0 which is free from Lee–Yangzeroes [2]. We define the corresponding Yang–Lee edge θYL(β) byg(β, θ) = 0 for − θYL(β) < θ < θYL(β) .The integral in (9) can therefore be taken to begin at θYL(β). Salmhofer has proved the existenceof a unique density of zeroes in the thermodynamic limit [5]. In this limit, integrating (9) by partsand expanding the trigonometric functions gives for the singular part of the free energy [4, 6, 7]f∞(β, h) = −2∫ πθYL(β)θh2 + θ2G∞(β, θ)dθ . (10)The magnetic susceptibility χ∞ is given by the second derivative of the free energy with respect toh. Following [4, 6, 7], this leads toG∞(β, θ) = χ∞(β)(θYL(β))2Φ(θθYL(β)), (11)2Φ(x) being some function of x such that Φ(| x |≤ 1) = 0. From (10) and the fact thatG (θYL, β) = 0,one gets the specific heatC∞(β) =∂2f∞(β, h)∂β2∣∣∣∣∣h=0= −2∫ πθYL(β)θ−1d2G∞ (θ, β)dβ2dθ . (12)The above formulae are quite general and hold for any model provided it obeys the Lee–Yangtheorem.To proceed further we insert the (model–specific) critical behaviour. Instead of (3) and (4), assumenow the following modified critical behaviour for the singular parts of the zero field susceptibilityand the specific heat for t >∼ 0.χ∞ ∼ ξ2−η∞ tr , C∞ ∼ ξα˜tq . (13)Similarly, assume that the leading scaling behaviour of the Yang–Lee edge in terms of the reducedtemperature isθYL(t) ∼ ξλtp . (14)For t >∼ 0 the critical indices are independent of t [1]. The scaling behaviour of C∞ can be relatedto that of χ∞ and θYL through (11) and (12). These giveξ(t)α˜tq ∝ ξ(t)2−η+2λt2p+r−2−2ν .Therefore,λ =12(α˜− 2 + η) , (15)p =12(q − r) + 1 + ν . (16)Finite–Size Scaling The finite–size scaling (FSS) hypothesis, first formulated in 1972 by Fisher[8], is a relationship between the scaling behaviour of thermodynamic quantities in the infinitevolume limit and the size dependence of their finite volume counterparts. The general statement ofFSS, which is expected to hold in all dimensions including the upper critical one [9] is that if PL(t)is the value of some thermodynamic quantity P at reduced temperature t measured on a system oflinear extent L, thenPL(0)P∞(t)= FP(ξL(0)ξ∞(t)), (17)where ξL(t) is the correlation length of the finite–size system. Luck has shown that, for the XY –model in two dimensions, ξL(0) is proportional to L [10]. Fixing the scaling variable, one hasξ∞(t) ∼ x−1L ,and from (2)t ∼ (lnL)−1ν ,for large enough L. Therefore FSS for the susceptibility and the Yang–Lee edge isχL(0) ∼ L2−η(lnL)−rν , (18)θ1(0) ∼ Lλ(lnL)−pν . (19)3For the finite size system it is convenient to consider the zeroes in the complex h–plane. Thesingular part of the free energy corresponds tofL(t, h) = L−d∑jln (h− iθj(t)) (20)where θ1 = θY L. The (reduced) magnetic susceptibility is thereforeχL(t) =∂2fL∂h2∣∣∣∣∣h=0= −Ld∑j1θ2j. (21)The lowest lying zeroes are expected to scale in the same way [11] so thatχL(t) ∝ −L−dθ1(t)−2 . (22)From (18) and (19) we haveL2−η(lnL)−rν ∼ L−d−2λ(lnL)2pν , (23)and concludeα˜ = −d , q = −2(1 + ν) . (24)Thus the scaling behaviour of the singular part of the specific heat indeed exhibits multiplicativelogarithmic corrections. Accepting the KT predictions ν = 1/2, η = 1/4 for t >∼ 0, the leadingcritical exponent for the Yang–Lee edge λ and the correction exponents q and p areλ = −158, q = −3 , p = −r2. (25)The above analytic considerations have yielded no information on the odd correction exponentr. The original renormalisation group analysis of Kosterlitz and Thouless [1], in fact, implicitlycontained the predictionr = −1/16 (26)as noted by [12]. Subsequent analysis have concentrated on the r = 0 form of the scaling behaviour(18) and the verification that η(βc) = 1/4. Allton and Hamer [13] have conjectured that thedeviation of their determination of η from 1/4 might be due to logarithmic corrections.The study of the full scaling form and the numerical determination of r is the subject of whatfollows.Numerical Procedures Numerical methods were used to test the scaling picture of the last sec-tion. Specifically, we analysed the FSS behaviour of the Yang–Lee edge so as to test the prediction(19), (25)θ1(βc) ∼ Lλ(lnL)r . (27)In particular, we sought to confirm λ = 15/8 for the leading behaviour and to check if r 6= 0.An algorithm based on that of Wolff [14] was used to simulate the XY –model at zero magneticfield (h = 0) on square lattices of sizes L = 32, 64, 128 and 256 . The values of β at which thesimulations took place (βo) were evenly spaced in steps of 0.02 between 1.00 and 1.20. At each4simulation point, 50, 000 measurements were made from each of two separate starts. The secondruns were used only to verify equilibration and to improve our understanding of the statisticalerrors. The latter were estimated using bootstrap as described below. The autocorrelation time ofthe susceptibility was also monitored. Histogram reweighting [15] was used to enable extrapolationbetween βo values.The determination of βc is inextricably bound up with the assumed critical scaling behaviour. Forthe present purposes, we adopted a two-stage strategy: we used estimates of βc from our ownand other analyses which had assumed no logarithmic corrections then tested the stability of theseconclusions with respect both to uncertainties in βc and to the form of the scaling asumption.One simple estimate was based on assumed conventional (r = 0) scaling of the susceptibility (3). Ageneralised [16] Roomany-Wyld approximant [17] was used to give estimates of the correspondingrenormalisation group beta function for each pair of lattice sizes related by factor of two scaling:βRWL = (2β2)η − 2 + 2r ln( ln 2LlnL )/ ln 2 + ln(χ2L/χL)/ ln 2∂∂β [χ2LχL]. (28)A preliminary estimate, using η = 1/4 and r = 0, leads to to βc = 1.11(1) which is at the lowerend of the range of published estimates [12], [18], [19]. These span 1.11 to 1.13. The most recenthigh precision result is 1.1197(5) [19].Determination of Lee–Yang Zeroes When the external field is complex (h = hr + ihi), thepartition function (1) can be rewrittenZL(β, hr + ihi) = ReZL(β, hr + ihi) + iImZL(β, hr + ihi) , (29)whereReZL(β, hr + ihi) =1N∑{~σx}eβS+hrM cos (hiM) ,= ZL(β, hr)〈cos (hiM)〉β,hr , (30)andImZL(β, hr + ihi) =1N∑{~σx}eβS+hrM sin (hiM) ,= ZL(β, hr)〈sin (hiM)〉β,hr . (31)According to the Lee–Yang theorem the zeroes are on the imaginary h–axis (hr = 0) where〈sin (hiM)〉β,hr vanishes and so the Lee–Yang zeroes are simply the zeroes ofReZL(β, ihi) ∝ 〈cos (hiM)〉β,0 . (32)Thus the Lee–Yang zeroes are easily found. Moreover, one can find them at any β using reweight-ing techniques [15]. For the lowest lying zeroes at β = 1.11 we found θ1(L) = 0.0023353(7),0.0006350(2), 0.00017278(5) and 0.000047062(13) for L = 32, 64, 128 and 256 respectively. Er-rors were calculated using the bootstrap method where the data for each βo are resampled 1005times (with replacement) leading to 100 estimates for θ1, from which the variance and bias can becalculated.In figure 1 we plot the logarithm of the position of the first Lee–Yang zero against the logarithmof the lattice size L, at β = 1.11. In the absence of any corrections, the slope should give theleading power–law exponent λ. In fact the slope is -1.8776(2), the deviation from the KT value of−15/8 = −1.875 being attributed to the presence of logarithmic corrections. To identify these, andthe correction exponent r in (27), ln (θ1L15/8) is plotted against ln lnL in figure 2. A straight line isidentified. Its slope is −0.012(1) giving evidence for a non–zero value of r, albeit not in agreementwith the RG predictions of −1/16 = −0.0625 from [1].At this point, one must investigate the extent to which these conclusions regarding the existence ofmultiplicative logarithmic corrections depend upon the measurement of βc. We have attempted todo this systematically by studying the above ln lnL fits for r as a function of the assumed criticalbeta. Fits with a range of values of r (including zero) are possible but not all with acceptableχ2/degree of freedom. To be conservative, we take ‘acceptable’ to mean less than 2. This corre-sponds to a minimum confidence level of 14%. In figure 3 we show that acceptable values of χ2are possible only for 1.110 <∼ βc<∼ 1.120. The r = 0 solution would correspond to βc ≈ 1.105and χ2/dof ≈ 4. Similary, we find that a fit with r = −1/16 would correspond to βc ≈ 1.138 andχ2/dof ≈ 16.In summmary, assumming the Kosteritz-Thouless value η = 1/4, we find non-zero logarithmiccorrections to scaling and a corresponding estimate of the critical temperature:r = −0.023 ± 0.010 , βc = 1.115 ± .005 . (33)As a cross check, one may use the above values for r and η to find the Roomany-Wyld approximant(28) and estimate βc from its zeroes. We find βc = 1.115±0.004 in good agreement with the above.However, we emphasize the much higher precision available to an analysis based on Lee-Yang zeroesrather than on the spin susceptibility.Conclusions Theoretical arguments concerning the consistency of the scaling behaviour of oddand even thermodynamic functions at a KT phase transition have been presented. The generallyused scaling formulae have to be modified by multiplicative corrections. These are identified ana-lytically for the specific heat and numerically for the susceptibility. This numerical identificationcomes via an analysis of Lee–Yang zeroes, the FSS of which is linked to that of the susceptibility.We would like to thank P. Lacock for assistance with multihistogram reweighting techniques.6References[1] J. Kosterlitz and D. Thouless, J. Phys. C 6 (1973) 1181; J. Kosterlitz, J. Phys. C 7(1974) 1046.[2] C. N. Yang and T. D. Lee, Phys. Rev. 87 404 (1952); ibid. 410.[3] F. Dunlop and C.M. Newman, Commun. Math. Phys. 44 (1975) 223.[4] R. Kenna and C.B. Lang, Phys. Rev. E 49 (1994) 5012.[5] M. Salmhofer, UBC–Preprint (unpublished); Nucl. Phys. B (Proc. Suppl.) 30, (1993),81.[6] R. Abe, Prog. Theor. Phys. 37 (1967) 1070; Prog. Theor. Phys. 38 (1967) 72; ibid. 322;ibid. 568.[7] M. Suzuki, Prog. Theor. Phys. 38 (1967) 289; ibid. 1225; ibid. 1243; Prog. Theor. Phys.39, (1968) 349.[8] M.E. Fisher, in Critical Phenomena, Proc. 51th Enrico Fermi Summer School, Varena,ed. M.S. Green (Academic Press, NY, 1972).[9] R. Kenna and C.B. Lang, Phys. Lett. B 264 (1991) 396; Nucl. Phys. B (Proc. Suppl.)30 (1993) 697; Nucl. Phys. B 393 (1993) 461; Err. ibid. B 411 (1994) 340.[10] J.M. Luck, J. Phys. A 15 (1982) L169.[11] C. Itzykson, R. B. Pearson, and J. B. Zuber, Nucl. Phys. B 220[FS8] (1983) 415;C. Itzykson and J. M. Luck, Progress in Physics, Critical Phenomena (1983 BrasovConference) Birkha¨user, Boston Inc. Vol 11, 45 (1985).[12] P. Butera and M. Comi, Phys. Rev. B 47 (1993) 11969.[13] C. Allton and C. Hamer, J. Phys. A 21 (1988) 2417.[14] U. Wolff, Phys. Rev. Lett. 62 (1989) 361.[15] K. Kajantie, L. Ka¨rkka¨inen and K. Rummukainen, Nucl. Phys. B 357 (1991) 693.[16] A.C. Irving and C. Hamer, Nucl. Phys. B 230[FS10] (1984) 361; C. Hamer and A.C.Irving, J. Phys. A 17 (1984) 1649.[17] M.P. Nightingale, Physica A 83 (1976) 561; H. Roomany and H.W. Wyld, Phys. Rev.D 21 (1980) 3341.[18] R. Gupta, J. De Lapp, G. Batrouni, G.C. Fox, C.F. Baillie and J. Apostolakis, Phys.Rev. Lett. 61 (1988) 1996; L. Biferale and R. Petronzio, Nucl. Phys. B 328 (1989) 677;C.F. Baillie and R. Gupta, Nucl. Phys. B (Proc. Suppl.) 20 (1991) 669; A. Hulsebos,J. smit and J.C. Vink, Nucl. Phys. B 356 (1991) 775; R.G. Edwards, J. Goodman andA.D. Sokal, Nucl. Phys. B 354 (1991) 289.7[19] M. Hasenbusch, M. Marcu and K. Pinn, Physica A 208 (1994) 124.8-10-9-8-7-63 3.5 4 4.5 5 5.5 6Ln (Z1)Ln (L)Figure 1: Leading FSS of Lee–Yang Zeroes at β = 1.11.0.4320.4340.4360.4380.441.2 1.3 1.4 1.5 1.6 1.7Ln(Z1) + (15/8)Ln(L)LnLn(L)Figure 2: Corrections to FSS of Lee–Yang Zeroes at β = 1.11.9-0.06-0.04-0.0200.02r  (a)04812161.1 1.12 1.14Chisq/dofBETA (b)Figure 3: (a) Correction Exponent r Measured over a Range of β and (b) corresponding χ2 perDegree of Freedom (Goodness of Fit).10

Logarithmic Corrections to Scaling in the Two Dimensional $XY$--Model

Logarithmic Corrections to Scaling in the Two Dimensional $XY$ --Model

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Logarithmic Corrections to Scaling in the Two Dimensional XYXYXY--Model

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Logarithmic Corrections to Scaling in the Two Dimensional $XY$ --Model