Specific Heat Exponent for the 3-d Ising Model from a 24-th Order High Temperature Series
a r X i v :h e p -l a t /9312048v 1 10 D e c 1993
IASSNS-93/83WUB-93-40Dec 1993
Speci?c Heat Exponent for the 3-d Ising Model from a 24-th Order High Temperature Series.
by
Gyan Bhanot 12,Michael Creutz 3,Uwe Gl¨a ssner 4,Klaus Schilling 4
ABSTRACT
We compute high temperature expansions of the 3-d Ising model using a recursive transfer-matrix algorithm and extend the expansion of the free energy to 24th https://www.wendangku.net/doc/1a17267032.html,ing ID-Pad′e
and ratio methods,we extract the critical exponent of the speci?c heat to be α=0.104(4).
1INTRODUCTION
High-and low-temperature expansions constitute major tools for the calculation of critical properties in statistical systems.The Ising and Potts model low temperature expansions were recently extended[1,2,3,4]using a technique based on the method of recursive counting[5].In a separate development,Vohwinkel[6]implemented the shadow-lattice technique of Domb[7]in a very clever way and added many new terms to the series. However,the extraction of critical parameters from low temperature series is hampered by the presence of unphysical singularities.This is especially true of the3-d Ising model. For this reason,low temperature analytic methods are very often inferior to Monte-Carlo methods for computing critical exponents.
High-temperature(HT)expansions on the other hand,generally have better analytic behavior and yield more accurate exponents.Very recently,two variants of the recursive counting technique for HT expansions have been pursued.While Enting and Guttmann[4] keep track of spin con?gurations on a set of rectangular?nite lattices,ref.[3]counts HT-graphs on?nite,helical lattices.Such computer based series expansions have very large memory requirements.This makes them ideal candidates for large parallel computers if communication issues can be handled e?ciently.In this paper we will present the results of a HT expansion of the3-d Ising model to24th order,obtained on a32node1GByte Connection Machine CM-5.The implementation is based on a bookkeeping algorithm of binary coded spin con?gurations in helical geometry.
2COMPUTATION OF THE SERIES
We start with a discussion of the HT algorithm to compute the partition functions on?nite 3-d Ising lattices.Starting from the action
E{s}=?
Z= {s}exp(?βE)= {s}
Z=(cosh3β)V {s}
f=?1
β
?
1
recursion step,which requires knowledge only of those spin states that are contained in the exposed two-dimensional surface layer.To minimize ?nite size e?ects,it is best to use helical boundary conditions [2,3].One can visualize helical boundary conditions by imagining all spins in the layer laid out along a straight line.In this picture,the nearest neighbours to a given site in the sequence in the i th direction can be chosen to be h i sites away,with i =x,y,z .It is convenient to assume h x Z =(2cosh 3β)V k s 1,...,s h z p (k ;s 1,...,s h z )t k . (5) The recursion step,which consists of adding another spin s 0to the system,changes the partition function into Z =2V (cosh 3 β) V +1 s 0 k s 1,...,s h z p (k ;s 1,...,s h z )t k (6) ×(1+s 0s h x t )(1+s 0s h y t )(1+s 0s h z t ). s h x ,s h y and s h z are the backward nearest neighbours of the site s 0.The site s 0will displace its backward z neighbour site s h z after the counting of the added spin is completed.Since s h z will not be referred to in the subsequent steps of the algorithm,the summation over s h z can be carried out: Z =2V (cosh 3β)V +1 s 0 k s 1,...,s h z ?1 ×[ +p (k ;s 1,...,s h z ?1,s 0)t k (1+s 0s h x t )(1 +s 0s h y t )(1+t )(7) +p (k ;s 1,...,s h z ?1,ˉs 0)t k (1+s 0s h x t )(1 +s 0s h y t )(1?t )]. The contribution in the second (third)line of this equation contains the part with s h z being parallel (antiparallel,denoted by ˉs 0)to s https://www.wendangku.net/doc/1a17267032.html,paring this expression with the HT series (5)for the new system yields the recursion relation induced for the coe?cients p : 2p ′(k ;s 0,s 1,...,s h z ?1)=p (k ?0;s 1,...,s h z ?1,s 0) +p (k ?0;s 1,...,s h z ?1,ˉs 0) +p (k ?1;s 1,...,s h z ?1,s 0)(s 0s h x +s 0s h y +1)+p (k ?1;s 1,...,s h z ?1,ˉs 0)(s 0s h x +s 0s h y ?1)+p (k ?2;s 1,...,s h z ?1,s 0)(s h x s h y +s 0s h x +s 0s h y )(8) +p (k ?2;s 1,...,s h z ?1,ˉs 0)(s h x s h y ?s 0s h x ?s 0s h y )+p (k ?3;s 1,...,s h z ?1,s 0)(s h x s h y )+p (k ?3;s 1,...,s h z ?1,ˉs 0)(?s h x s h y ) It is crucial to remove ?nite-size errors by combining the results of di?erent lattice structures as described in refs.[2,3].We use the set of lattices listed in table 1and obtain the free energy coe?cients up to 24th order as given in table 2.In order to eliminate the contribution from (unphysical)loops with an odd number of links in any direction,we use h x1510141495101 h y121513151616191918 h z141617171720202122 -3-3333-1-1-222 Table1:Structures and weights w of the lattices used order k 4 22 8 1980 12 319170 16 201010408/3 20 16397040750 24 the cancellation technique of ref.[3].This amounts to inserting additional signature factors into eq.6for each of the three link-factors (1+s0s h i t)→σi(1+s0s h i t),i=x,y,z(9) with{σx,σy,σz}={±,±,±}.By performing8separate runs corresponding to all possible values of σand adding the results,one achieves a complete elimination of the unwanted loops.Possible contributions of higher-order?nite-size-loops are at least of order25for this set of lattices.Since we use open boundary conditions,the coe?cients p are invariant under the global transformation s i→?s i.This Z(2)symmetry enables us to reduce memory requirements by a factor of two.Unlike refs.[2,3,4]we use multiple-word arithmetic to account for the size of the coe?cients.This implementation needs about100%more memory but leads to a doubling in performance.Since the number of words can be adjusted separately for every order,the computational e?ort can be reduced accordingly.On the32 node CM-5the total time for all computations was about50hours. Compared to the?nite-lattice approach of Enting and Guttmann[4],our method ap-pears to require more CPU-time since we need to cancel unphysical loops.It should be noted,however,that helical lattices are very naturally implemented in data parallel soft-ware environments and thus lead to better performance.In the usual?nite lattice method [4],the HT expansion can only be extended in fairly coarse steps,using lattices with(4×5) cross-section for22nd order and(5×5)cross-section for26th order,respectively.For this reason,a24th order computation would not have been feasible using that method with our computer resources. 3CRITICAL EXPONENT The speci?c heat is de?ned as c|h=0=β2 ?2 Figure 1:Critical exponent αas a function of t 2C . -0.25-0.2-0.15-0.1-0.0500.050.10.15 0.20.250.0471 0.04720.0473 0.04740.04750.04760.04770.0478 C r i t i c a l e x p o n e n t a l p h a Critical temperature tc^2 J=0 J=1J=2 J=3 IDPs can also be used to predict the most signi?cant digits of the next term in the speci?c heat series [4].The estimate of the 24th order term as obtained in ref.[4]agrees perfectly with our exact https://www.wendangku.net/doc/1a17267032.html,ing the same method we can estimate the 26th order term in the expansion to be f 26=443762(4)×107,(13)where the errors quoted are two standard deviations. 3.2Ratio-Test The main problem in the determination of critical exponents in the low-temperature case is the presence of unphysical singularities nearer to the origin than the physical one.Since the expansion coe?cients c n are dominated by these unphysical singularities,ratio-methods cannot be applied. In the HT-expansion,the physical singularity dominates the asymptotic behaviour,so that the ratio r n =c n /c n ?1of successive coe?cients of the series is expected to behave as [9]r n = 1 n +c n 1+2θ +O 1n 1/2 + d n 3/2 .(15) A plot of this sequence against n is shown in ?gure 2.Obviously the ?rst four values are 0.050.10.150.20.25 0.30.350.40.45246 8101214 s (n ) n Figure 2:Plot of the sequence s n against n .The error of s 13,obtained from the ID-Pad′e extrapolation,is too small to be visible. dominated by higher order corrections.To obtain estimates for αwe therefore use only the values {s 6,...,s 13}.A 3-parameter least-square-?t using the ansatz of eq.(15)yields the values shown as diamonds in ?gure 3.The value of α=0.113obtained by the ?t to the points {s 6,...,s 11}is in perfect agreement with the result of ref.[4].Their estimate of α=0.110using the extrapolated term s 12appears to be slightly above our value of α=0.108using the exact term.Including our value for s 13of the ID-Pad′e extrapolation eq.(13)we obtain α=0.105(2).The error represents the uncertainty of the extrapolation.However,from ?g.3it it quite suggestive that the α-values might converge to a value below 0.105. To get an estimate of the uncertainties of our results,we investigate the stability of the ?ts.For this purpose,we repeat the analysis after eliminating the point s 6from the data.As a result we obtain sizeable changes for α.The new data are shown as crosses in ?g.3. In ?gure 4we present the results for the ?rst correction-to-scaling coe?cient c from our 3-parameter ?ts.In contrast to ref.[4],our values suggest that c changes sign with increasing n max .Because of the sensitivity of the ?ts to the number of terms we keep,it is di?cult to determine the value of c very precisely.Our best estimate is c =0.01(4).Since c vanishes within error,it seems reasonable to also try a 2-parameter-ansatz with c =0to ?t the data.The results of these ?ts are shown in ?gure 5.We now ?nd that the ?ts are much more stable and the αestimates show much more of a convergence to their asymptotic values.The best value (from the largest n max )is α=0.1045(3).This value supports the impression of the 3-parameter ?ts,which suggested that αwas slightly below 0.105.Taking into account the fact that neglecting c causes a systematic error,our ?nal 0.095 0.10.1050.110.1150.12 0.1250.13 8910 11121314 e s t i m a t e f o r a l p h a nmax nmin=6nmin=7 Figure 3:Estimates of alpha using a 3-parameter-?t.Each point represents the results of a ?t to the set of values {s n min ,...,s n max }.The error bars of the rightmost values represent the uncertainty of the extrapolated 13th term. -0.15 -0.1 -0.05 0.05 0.1 8910 11121314 c o r r e c t i o n t o s c a l i n g c o e f f . c nmax nmin=6nmin=7 Figure 4:Estimates of c using a 3-parameter-?t.Each point represents the results of a ?t to the set of values {s n min ,...,s n max }.The error bars of the rightmost values represent the uncertainty of the extrapolated 13th term. 0.104 0.10420.10440.10460.10480.1050.10520.10548910 11121314 e s t i m a t e f o r a l p h a nmax nmin=6nmin=7 Figure 5:Estimates of αusing a 2-parameter-?t with c =0.Each point represents the results of a ?t to the set of values {s n min ,...,s n max }.The error bars of the rightmost values represent the uncertainty of the extrapolated 13th term.estimate for the critical exponent is, α=0.104(4). (16) 4DISCUSSION AND OUTLOOK The crucial element in the estimate of the error in α(eq.16)is our neglect of the correction-to-scaling coe?cient c .The resulting systematic error is rather large.From ?g.4one might speculate that the estimates for c begin to exhibit asymptotic behaviour at the 26th order.Therefore an exact calculation of the 26-th term of the expansion might reduce the uncertainty of c signi?cantly.If the magnitude of c turns out to be really negligible,one could adopt the errors of the linear ?ts,and αwould be obtained accurate to the fourth signi?cant digit. 5ACKNOWLEDGEMENTS This work was partly funded under Contracts No.DE-AC02-76CH00016and DE-FG02-90ER40542of the U.S.Department of Energy.Accordingly,the https://www.wendangku.net/doc/1a17267032.html,ernment retains a nonexclusive,royalty-free license to publish or reproduce the published form of this contribution,or allow others to do so,for https://www.wendangku.net/doc/1a17267032.html,ernment purposes.The work of GB was also partly supported by a grant from the Ambrose Monell Foundation.UG and KS are grateful to Deutsche Forschungsgemeinschaft for its support to the Wuppertal CM-Project. 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