Critical Behavior of the Two-Dimensional Ising Susceptibility (original) (raw)

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Critical Behavior of the Two-Dimensional Ising Susceptibility

Article in Physical Review Letters ⋅\cdot April 2001
DOI: 10.1103/PhysRevLett.86.4120 ⋅\cdot Source: PubMed

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Critical behaviour of the two-dimensional Ising susceptibility

W. P. Orrick, 1,∗{ }^{1, *} B. G. Nickel, 2,†{ }^{2, \dagger} A. J. Guttmann, 1,‡{ }^{1, \ddagger} and J. H. H. Perk 3,§{ }^{3, \S}

1{ }^{1} Department of Mathematics G\mathcal{G} Statistics,
The University of Melbourne, Parkville, Vic. 3010, Australia
2{ }^{2} Department of Physics, University of Guelph,
Guelph, Ontario, Canada N1G 2W1
3{ }^{3} Department of Physics, Oklahoma State University,
Stillwater, Oklahoma 74078-3072, U.S.A.
(Dated: February 1, 2008)

Abstract

We report computations of the short-distance and the long-distance (scaling) contributions to the square-lattice Ising susceptibility in zero field close to TcT_{c}. Both computations rely on the use of nonlinear partial difference equations for the correlation functions. By summing the correlation functions, we give an algorithm of complexity O(N6)\mathrm{O}\left(N^{6}\right) for the determination of the first NN series coefficients. Consequently, we have generated and analysed series of length several hundred terms, generated in about 100 hours on an obsolete workstation. In terms of a temperature variable, τ\tau, linear in T/Tc−1T / T_{c}-1, the short-distance terms are shown to have the form τp(ln⁡∣τ∣)q\tau^{p}(\ln |\tau|)^{q} with p≥q2p \geq q^{2}. To O(τ14)\mathrm{O}\left(\tau^{14}\right) the long-distance part divided by the leading τ−7/4\tau^{-7 / 4} singularity contains only integer powers of τ\tau. The presence of irrelevant variables in the scaling function is clearly evident, with contributions of distinct character at leading orders ∣τ∣9/4|\tau|^{9 / 4} and ∣τ∣17/4|\tau|^{17 / 4} being identified.

[1]

1. *orrick@labri.u-bordeaux.fr
   †{ }^{\dagger} BGN@physics.uoguelph.ca
   ‡{ }^{\ddagger} tonyg@ms.unimelb.edu.au
   §\S perk@okstate.edu ↩︎

I. INTRODUCTION

The two-dimensional Ising model has been extremely useful as a testing ground for new theoretical ideas and methods in the study of phase transitions and critical phenomena. Our present understanding is the result of a series of dramatic developments spanning more than half a century, starting with Onsager’s exact computation of the free energy [1], followed by Yang’s derivation of the spontaneous magnetization [2] and by the work of many researchers on the correlation functions, including Toeplitz determinantal formulae [3], exact expressions for their behaviour at large separation [4], and nonlinear partial difference equations for their efficient computation [5,6,7][5,6,7], to mention only those results which are used in the present work. All results above apply to the zero-field case. While an exact expression for the susceptibility as the sum of two-point correlation functions over all separations [4] exists, a useful closed form expression does not. Moreover, as we discuss, there are strong indications that the susceptibility has a natural boundary in the complex plane [8, 9], a feature which rules out any expression in terms of the “standard” functions of mathematical physics.

Nevertheless, it is desirable to obtain as detailed information about the susceptibility as possible, not only because of its physical importance, but also because of the significant role it plays in ideas about scaling and the renormalization group. In the vicinity of the ferromagnetic critical point at temperature T=TcT=T_{c}, the susceptibility exhibits a singularity of the form

β−1χ±=C0±(2Kc2)7/4∣τ∣−7/4F±(τ)+B(τ)\beta^{-1} \chi_{ \pm}=C_{0_{ \pm}}\left(2 K_{c} \sqrt{2}\right)^{7 / 4}|\tau|^{-7 / 4} F_{ \pm}(\tau)+B(\tau)

Here β=(kBT)−1,τ=12(s−1−s),s=sinh⁡2K\beta=\left(k_{B} T\right)^{-1}, \tau=\frac{1}{2}\left(s^{-1}-s\right), s=\sinh 2 K and 2Kc=ln⁡(1+2)2 K_{c}=\ln (1+\sqrt{2}) with K=βJK=\beta J the conventional Ising model coupling constant. The scaling-amplitude functions F±(τ)F_{ \pm}(\tau) are normalized to unity at τ=0\tau=0. As a consequence of the exact knowledge of the long-range correlations, the coefficients C0pmC_{0_{ \pm}}C0pmwere calculated exactly [10] in terms of the solution of a Painlevé III equation. Additionally, the leading behaviour of both F±(τ)F_{ \pm}(\tau) was computed to be 1+12τ1+\frac{1}{2} \tau. The antiferromagnetic susceptibility, on the other hand, is dominated by the short-distance correlation functions and has leading singularity (const ×τln⁡∣τ∣\times \tau \ln |\tau| ). Such short-distance “background” terms are present as well in the ferromagnetic susceptibility and are denoted by B(τ)B(\tau) in (1). The leading amplitudes of the analytic and singular parts of B(τ)B(\tau) were computed for a general wavevector dependent susceptibility in [11,12][11,12].

An analysis 13 of a 51 term high-temperature series by means of differential approximants yielded two further correction terms in the scaling-amplitude function F+F_{+}, with numerical amplitudes close to rational values, 58τ2+316τ3\frac{5}{8} \tau^{2}+\frac{3}{16} \tau^{3}, and confirmed that the same scalingamplitude function is numerically consistent with the first 11 terms in the low-temperature expansion. These results agreed with the prediction 14 that the corrections to scaling are entirely due to the nonlinearity of the scaling fields and not to the presence of irrelevant operators 14. However, a recent analysis of 115 term high- and low-temperature series 15 showed that this prediction appears to break down in the amplitude of τ4\tau^{4}.

The study reported in this letter substantially improves on all the above results. We extend the methods of 11, 12 to compute both antiferromagnetic and ferromagnetic background amplitudes on the isotropic lattice to O(τ14(ln⁡∣τ∣)3)\mathrm{O}\left(\tau^{14}(\ln |\tau|)^{3}\right). All such terms are seen to be of the form τp(ln⁡∣τ∣)q\tau^{p}(\ln |\tau|)^{q} with p≥q2p \geq q^{2}. We simultaneously compute high-temperature series to order 323 and low-temperature series to order 646 in 123 hours on a 500 MHz DEC Alpha with 21164 processor running Maple TM{ }^{\mathrm{TM}} V version 5.1.

We analyze these series by two independent methods, making use of the computed background amplitudes and the known complex singularity structure 9, 15 to obtain the scalingamplitude functions FpmtotomathrmOleft(tau14right)F_{ \pm}toto \mathrm{O}\left(\tau^{14}\right)FpmtotomathrmOleft(tau14right).

Several important conclusions can be drawn from our results. Firstly, only pure integer powers of τ\tau enter the scaling-amplitude functions and no logarithmic terms are present. Secondly, the high- and low-temperature scaling-amplitude functions are not equal to each other. The amplitudes start to differ at O(τ6)\mathrm{O}\left(\tau^{6}\right). Thirdly, the amplitudes of τ4\tau^{4} and τ5\tau^{5}, which are clearly rational, are not those predicted by simple two-variable scaling 14. We surmise that at least two irrelevant operators must be invoked to account for the above results-one entering at τ4\tau^{4}, the other at τ6\tau^{6}.

Further remarks on the scaling implications of our work can be found in section V while the remainder of the letter will outline the methods by which the ferromagnetic results were obtained. A fuller account, including details of the antiferromagnetic singularity, will appear elsewhere.

II. SINGULARITY STRUCTURE AND NATURAL BOUNDARY

It was argued in [8] that on the anisotropic lattice, the contribution to the susceptibility of the high-temperature graphs with 2N2 N vertical bonds contains more and more poles as NN increases, and that in the limit N→∞N \rightarrow \infty these poles form a dense set in the complex plane. In [9] it was shown that in the expansion of the susceptibility in jj-particle contributions [4]

β−1χ={∑j odd χ(j)T>Tc∑j even χ(j)T<Tc\beta^{-1} \chi= \begin{cases}\sum_{j \text { odd }} \chi^{(j)} & T>T_{c} \\ \sum_{j \text { even }} \chi^{(j)} & T<T_{c}\end{cases}

the higher-particle components give rise to an ever increasing number of singularities that appear to form a dense set on the circle ∣s∣=1|s|=1. In fact, the two phenomena are precisely correlated, with the former being the highly anisotropic limit, and the latter the isotropic limit, of the set of singularities for the generic anisotropic model. These occur at

cosh⁡(2K)cosh⁡(2K′)−sinh⁡(2K)cos⁡2πmj−sinh⁡(2K′)cos⁡2πm′j=0\begin{aligned} & \cosh (2 K) \cosh \left(2 K^{\prime}\right)-\sinh (2 K) \cos \frac{2 \pi m}{j} \\ & \quad-\sinh \left(2 K^{\prime}\right) \cos \frac{2 \pi m^{\prime}}{j}=0 \end{aligned}

with m,m′=1,2,…jm, m^{\prime}=1,2, \ldots j, and K,K′=βJx,βJyK, K^{\prime}=\beta J_{x}, \beta J_{y}. It will be noted that the left-hand side of (3) is the denominator in the Onsager integral for the free-energy and thus we find the (to us) surprising result that the singularity of χ(j)\chi^{(j)}, a property of the Ising model in a magnetic field, is intimately connected with a property in zero field. Barring unexpected cancellation (and we have evidence against this) in the N→∞N \rightarrow \infty limit, we believe that this set forms a natural boundary.

The existence of a natural boundary has strong implications for our series analysis. In the τ\tau plane, the boundary is formed on the imaginary axis by the logarithmic branch cuts coming from the singularities of [9,15][9,15]. Summing the contributions of the discontinuities across these cuts gives Disc⁡(β−1χ)∼exp⁡(−39.76/T2)\operatorname{Disc}\left(\beta^{-1} \chi\right) \sim \exp \left(-39.76 / \mathcal{T}^{2}\right) to leading order, where τ=iT\tau=i \mathcal{T} is a point on the cut. Assuming that the contribution of the physical singularity at τ=0\tau=0 is additive, we expect that the coefficient of τp\tau^{p} in the limit p→∞p \rightarrow \infty in the τ\tau-expansion of the susceptibility will grow as Γ(p/2)/ap/2\Gamma(p / 2) / a^{p / 2} where a=39.76a=39.76. This effect only becomes appreciable at order p∼2ap \sim 2 a and is too small to be seen in the terms we have. However it has the implication that the τ\tau-expansion of χ\chi can be at best asymptotic and not analytic. Whether this non-analyticity comes from the long-distance or the short-distance part, or a combination of the two, has not been determined.

We note that the existence of a dense set of singularities also implies that the susceptibility cannot be a member of the class of functions called D\mathcal{D}-finite, that is, solutions to linear differential equations with polynomial coefficients. Nevertheless, it can be shown from the integral expressions for the χ(j)\chi^{(j)} that they individually are D\mathcal{D}-finite (also called holonomic) [16]. This phenomenon is related to that of perturbative expansions in quantum field theory, where any individual Feynman diagram is holonomic whereas the summed series may not be [17][17].

III. COMPUTATION OF SHORT-DISTANCE AMPLITUDES AND HIGH- AND LOW-TEMPERATURE SERIES

The essential tool for the computation of both the background amplitudes and the highand low-temperature series coefficients is the set of nonlinear partial difference equations for the two-point correlation functions, C(m,n)=⟨σ00σmn⟩C(m, n)=\left\langle\sigma_{00} \sigma_{m n}\right\rangle, given in [6]. These completely determine all the off-diagonal two-point functions once the diagonal ones (m=n)(m=n) are given. The latter can be computed either by means of an independent set of difference equations [7] or, as we have done here, directly from the Toeplitz determinant expressions. The susceptibility, β−1χ=∑C(m,n)−⟨σ00⟩2\beta^{-1} \chi=\sum C(m, n)-\left\langle\sigma_{00}\right\rangle^{2}, is computed by successively adding the contributions of pairs of square shells CN=∑C(m,n)C_{N}=\sum C(m, n) with ∣m∣+∣n∣=2N|m|+|n|=2 N and ∣m∣+∣n∣=|m|+|n|= 2N+12 N+1.

The implementation of the difference equations to obtain high- and low-temperature expansions is straightforward using the multiple precision integer arithmetic capabilities of Maple TM{ }^{\mathrm{TM}} or Mathematica TM{ }^{\mathrm{TM}}, and the time complexity is no worse than O(N6)\mathrm{O}\left(N^{6}\right).

The key to computing the short-distance background amplitudes is to obtain expansions of the partial sums SN=∑n=0NCnS_{N}=\sum_{n=0}^{N} C_{n} in τ\tau directly and to identify which terms in the series contribute to the short-distance part and which to the long-distance part. A combination of analytic work and numerical fitting leads us to a conjecture for the short-distance expansion of the shell sums, namely

sCN=N3/4∑p=0∞(ln⁡∣Nτ∣)p(Nτ)p2AN(p)\sqrt{s} C_{N}=N^{3 / 4} \sum_{p=0}^{\infty}(\ln |N \tau|)^{p}(N \tau)^{p^{2}} A_{N}^{(p)}

where the AN(p)A_{N}^{(p)} are Taylor series in τ\tau with coefficients that are asymptotic Laurent series in

N−1N^{-1}; the highest power of NN multiplying τq\tau^{q} in AN(p)A_{N}^{(p)} is NqN^{q}. The partial sums SNS_{N} are

sSN=∑n=0NCn=∑p=q2∑q=0RN(p,q)τp(ln⁡∣τ∣)q\sqrt{s} S_{N}=\sum_{n=0}^{N} C_{n}=\sum_{p=q^{2}} \sum_{q=0} R_{N}^{(p, q)} \tau^{p}(\ln |\tau|)^{q}

with RN(p,q)R_{N}^{(p, q)} functions of NN only. Asymptotically, for large N,RN(p,q)N, R_{N}^{(p, q)} is a sum of powers N7/4+p′N^{7 / 4+p^{\prime}}, with possible multiplicative ln⁡(N)\ln (N) corrections, plus a constant b(p,q)b^{(p, q)} which arises from the small nn terms in the sum (5) where the asymptotic expressions are not valid and sum and integral are not synonymous. The p′p^{\prime} are integers p′≤pp^{\prime} \leq p.

We must assume that (5) remains valid up to NN of the order 1/τ1 / \tau where it can, in principle, be matched term by term to a large distance expansion that properly describes the roughly exponential exp⁡(−Nτ)\exp (-N \tau) decay of correlations as N→∞N \rightarrow \infty. Explicit matching formed the basis of the previous calculations of terms in the short-distance χ\chi (cf. [11, 12]) but this becomes extremely cumbersome at higher order. Here we argue that the exponential decay implies a cutoff on NN that is proportional to 1/τ1 / \tau and that we can identify the temperature behaviour of terms in SNS_{N} in eqn. (5) by the replacement N→1/τN \rightarrow 1 / \tau. All terms whose variation is as a fractional power of τ\tau, with possibly logarithmic multipliers, are discarded as assumed contributions to the long-distance part of χ\chi. Clearly all that remains is the constant part of RN(p,q)R_{N}^{(p, q)}, namely b(p,q)b^{(p, q)}, and this is extracted by numerical fitting to give

sB=∑p=q2∑q=0b(p,q)τp(ln⁡∣τ∣)q\sqrt{s} B=\sum_{p=q^{2}} \sum_{q=0} b^{(p, q)} \tau^{p}(\ln |\tau|)^{q}

for the short-distance part of χ\chi in eqn. (1). The coefficients b(p,q)b^{(p, q)} must be determined to very high accuracy to be useful for the subtraction procedure described in the next section; the complete list for p<15p<15 will be given elsewhere. The result p≥q2p \geq q^{2} we call the fermionic constraint since it can be traced back to the Toeplitz determinant that led to the correlations of the form in eqn. (4).

IV. SCALING AMPLITUDES

The contribution of the short-distance terms may now be subtracted from the high- and low-temperature series, leaving the long-distance part, from which the scaling amplitudes may be computed using any of a variety of series analysis techniques. Such analysis is vastly simplified by the observation that there are no logarithmic or non-integer power contributions to the scaling-amplitude functions F±F_{ \pm}.

To show this, independently of any fitting procedure, we have noted that any contribution to FpmwhichisnotapositiveintegerpowerofwhichisnotapositiveintegerpoweroftauF_{ \pm}whichisnotapositiveintegerpowerofwhich is not a positive integer power of \tauFpmwhichisnotapositiveintegerpowerofwhichisnotapositiveintegerpoweroftau would manifest itself in the high order series coefficients of the scaled susceptibility, (1−s±4)−1/4χ±\left(1-s^{ \pm 4}\right)^{-1 / 4} \chi_{ \pm}. The 1+τ/21+\tau / 2 terms in F±F_{ \pm}, as poles in the scaled susceptibility, also contribute but as their amplitudes are known to high precision, they can be subtracted. The residual coefficients are comparable in magnitude to those expected from the first neglected short-distance term which enters at τ15\tau^{15}. We may place limits on the size of the amplitudes of any putative non-analytic terms in the scalingamplitude functions. For example, for terms of the form Apτpln⁡∣τ∣A_{p} \tau^{p} \ln |\tau|, the bounds,

∣Ap∣<10−35300p/Γ(p−1),T>Tc∣Ap∣<10−37600p/Γ(p−1),T<Tc\begin{aligned} & \left|A_{p}\right|<10^{-35} 300^{p} / \Gamma(p-1), T>T_{c} \\ & \left|A_{p}\right|<10^{-37} 600^{p} / \Gamma(p-1), T<T_{c} \end{aligned}

reasonably exclude all pp less than about 15 .
On purely numerical grounds, the absence of logarithmic corrections is surprising since it implies the cancellation of the many logarithmic multipliers in the scaling terms we discarded in the previous section. On the other hand, the absence of logarithms appears to be a requirement of the combination of the fermionic constraint p≥q2p \geq q^{2} in (6) and renormalizationgroup scaling as discussed in the next section.

To compute the amplitudes of the integer powers of τ\tau in the scaling-amplitude functions F±F_{ \pm}, we have carried out two independent analyses, one in the ss-plane, the other in the vv-plane, where v=tanh⁡Kv=\tanh K is the conventional high-temperature variable. The natural boundary singularities at ∣s∣=1|s|=1 are mapped to two circles, ∣v±1∣=2|v \pm 1|=\sqrt{2}. As they are farther from the origin than the ferromagnetic and antiferromagnetic singularities at v=±(2−1)v= \pm(\sqrt{2}-1) their amplitudes are exponentially damped and may be neglected in the analysis. The ss plane analysis must take account of these singularities explicitly. The two analyses are in complete agreement.

We find numerically that the scaling-amplitude functions multiplied by s\sqrt{s} appear to be even functions of τ\tau, the amplitudes of the odd terms being comparable in magnitude to the uncertainties in the even amplitudes. The rational amplitudes of τ2\tau^{2} and τ4\tau^{4} below we conjecture to be exact, and these values were fixed in the final fitting. The results, with

uncertainty only in the final digits, are

sF+=1+τ2/2−τ4/12−0.1235292285752086663τ6+0.136610949809095τ8−0.13043897213τ10+0.1215129τ12−0.113τ14+O(τ15)sF−=1+τ2/2−τ4/12−6.321306840495936623067τ6+6.25199747046024329τ8−5.6896599756180τ10+5.142218271τ12−4.67472τ14+O(τ15)\begin{aligned} \sqrt{s} F_{+} & =1+\tau^{2} / 2-\tau^{4} / 12-0.1235292285752086663 \tau^{6} \\ & +0.136610949809095 \tau^{8}-0.13043897213 \tau^{10} \\ & +0.1215129 \tau^{12}-0.113 \tau^{14}+\mathrm{O}\left(\tau^{15}\right) \\ \sqrt{s} F_{-} & =1+\tau^{2} / 2-\tau^{4} / 12 \\ & -6.321306840495936623067 \tau^{6} \\ & +6.25199747046024329 \tau^{8}-5.6896599756180 \tau^{10} \\ & +5.142218271 \tau^{12}-4.67472 \tau^{14}+\mathrm{O}\left(\tau^{15}\right) \end{aligned}

V. COMPARISON WITH SCALING PREDICTIONS

Prior to the analysis of [15], all known amplitudes were in agreement with the hypothesis that corrections to scaling were due to scaling-field nonlinearity, and not to the presence of irrelevant variables. Here for the first time, we quantify the error in this “simple” scaling theory. Ignoring irrelevant operators, the expressions for the free energy, magnetization and susceptibility in zero magnetic field are [14]

f(τ)=−A(a0(τ))2ln⁡∣a0(τ)∣+A0(τ)M(τ<0)=Bb1(τ)∣a0(τ)∣1/8β−1χ±(τ)=C±(b1(τ))2∣a0(τ)∣−7/4−Ea2(τ)a0(τ)ln⁡∣a0(τ)∣+D(τ)\begin{aligned} f(\tau) & =-A\left(a_{0}(\tau)\right)^{2} \ln \left|a_{0}(\tau)\right|+A_{0}(\tau) \\ M(\tau<0) & =B b_{1}(\tau)\left|a_{0}(\tau)\right|^{1 / 8} \\ \beta^{-1} \chi_{ \pm}(\tau) & =C_{ \pm}\left(b_{1}(\tau)\right)^{2}\left|a_{0}(\tau)\right|^{-7 / 4} \\ & -E a_{2}(\tau) a_{0}(\tau) \ln \left|a_{0}(\tau)\right|+D(\tau) \end{aligned}

where A,B,C±A, B, C_{ \pm}, and EE are constants and A0(τ)A_{0}(\tau) and D(τ)D(\tau) are analytic functions of τ\tau. The functions a0(τ)a_{0}(\tau) and b1(τ)b_{1}(\tau) are the leading terms in the expansion of the scaling fields, and can be determined from the free energy and magnetization. The result for chipm\chi_{ \pm}chipmis of the form (1) but with FF replacing FpmF_{ \pm}Fpmwhere

sF=1+τ22−31τ4384+125τ63072+O(τ8)\sqrt{s} F=1+\frac{\tau^{2}}{2}-\frac{31 \tau^{4}}{384}+\frac{125 \tau^{6}}{3072}+\mathrm{O}\left(\tau^{8}\right)

Note that this expression should hold in both temperature regimes.
The difference between (11) and (9) we believe to be due to the effects of one or more irrelevant operators, confluent with the “simple” scaling terms. As there is no free parameter to vary in the model, we can’t identify these operators from the information we have.

However, it is likely that there are at least two mechanisms at work, one entering at O(τ4)\mathrm{O}\left(\tau^{4}\right) which preserves the equality of F+F_{+}F+and F−F_{-}, and a second entering at O(τ6)\mathrm{O}\left(\tau^{6}\right) which breaks this symmetry. In order to probe these effects further, we hope to study the model with anisotropy, and on other periodic lattices.

The corrections to scaling we have found are confluent with expected analytic terms and in the renormalization group picture of scaling this leads to the possibility of logarithmic terms as well (cf. [18]). Logarithmic corrections are not demanded - the issue is whether the scaling fields are coupled and this depends on microscopic details. Barma and Fisher [19] have investigated a model renormalization group flow in detail and conclude that in the case of a confluence, here labelled by integer index mm, one must expect either no coupling between fields or corrections of the form (τmlog⁡∣τ∣)k\left(\tau^{m} \log |\tau|\right)^{k} to all order kk. Since the latter violates the fermionic constraint mk≥k2m k \geq k^{2} we conclude there cannot be any logarithmic terms in the scaling-amplitude function FpmaswehaveverifiedtoaswehaveverifiedtomathrmOleft(tau15right)F_{ \pm}aswehaveverifiedtoas we have verified to \mathrm{O}\left(\tau^{15}\right)FpmaswehaveverifiedtoaswehaveverifiedtomathrmOleft(tau15right).

Acknowledgments

We are pleased to acknowledge M. Bousquet-Mélou, M. E. Fisher, M. L. Glasser, B. M. McCoy, A. Pelissetto and A. D. Sokal for helpful comments and criticisms. AJG and WPO would like to thank the Australian Research Council for financial support, and JHHP thanks the NSF for support in part by Grant PHY 97-22159.
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