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MORITA, Tohru

TANAKA, Kazuyuki

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TitleAsymptotic Behavior of Spin-Pair Correlation Function ofIsing Model on Checkerboard Lattice(New Developments inStatistical Physics Similarities in Diversities,YITP Workshop)Author(s)TANAKA, Kazuyuki; MORITA, TohruCitation物性研究 (1993), 60(4): 358-362Issue Date1993-07-20URL http://hdl.handle.net/2433/95120RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityAsymptotic Behavior of Spin-Pair Correlation Functionof Ising Model on Checkerboard LatticeKazuyuki TANAKA and Tohru MORITADepartment of Engineering Science, FacuLty of Engineering,Tohoku University, Sendai 980, JapanVdovichenko 1 ) introduced the diagrammat ical method forthe two-dimensional Ising model with non-crossing interaction.Morita2 ) extended the method by introducing techniques of divisionand integration in the lattice sites for the Ising model.We consider the Ising model on the checkerboard lattice withadditional interaction J y ' as shown in Fig.l(a). The Hamiltonianis given byK LH =kE l LE1[-JySk.L.1Sk.L+l.l+h(Sk.L+l.2,Sk.L,2,Sk+I,L,I,Sk,L,I)]'where h(sl,s2,s3,s4) is the Hamiltonian of a shaded square cluster.For four spins s.J(j=l,2,3,4) in the shaded-square as shownin Fig.2, we assume that(i,j,k=l,2,3,4),- 358-(b)(c)(1,1,1) (1,1.2) (2.l.I) (2,1,2) (3,1.1>Fig. 1. Ising model on the checkerboard lattice with an additional interaction Jy, and its reduction tothe layered free-fermion eight-vertex model on the square lattice. By the techniq~eof division. weobtain (b) from (a). By the technique of integration. we obtain (c) from (b).Fig. 3. Vertex configurations eit.I,1I which are per-mitted in the eight-vertex model.Fig. 2. Cluster consisting of four lattice sites.The spin variable of the spin on a latticesite i (i =1,2,3,4) is denoted by Si. Thehamiltonian of this cluster is denoted byh(SI, S2, S3, s.).- 359-III---.._.,I,l: _1~ - I+ --+--~ :2 ~ =3III•II,~:4where ~T=l/kBT, kB is the Boltzmann constant and T is the absolutetemperature. The model is reduced to the free-fermion eight-vertexmodel consisting of two kinds of sublattice, as shown in Fig.i.The partition function of the equivalent free-fermion eight-vertexmodel is given asK LZ = Tr8V k~l t~lWl(~k,L,l)W2(~k, L,2)'where the symbol T8V means the summation over all the possiblesolid-bold complexions. The possible vertex configurations areden0 ted b y ~ k.. 1 =1 , 2 , . . .. ' 8 assh 0 wn in Fig. 3 ., v, Vthe eight-vertex model are given byThe weights ofWl(~) - sinh(~TJy)' (~= 2,3,6,8),}wl(~) - cosh(~TJy)' (~= 1,4,5,7),w2(1) - (Z2/ 4 ), w2(2) = (Z2/4)<81828384)2'w2 (3) - (Z2/ 4 )<8 182 )2' w2(4) - (Z2/4)<8384>2~w2(5) - (Z2/ 4 )<8283 )2' w2 (6) - (Z2/ 4 )<8 18 4 >2'w2(7) - (Z2/ 4 )<8 183 )2' w2 (8) - (Z2/ 4 )<8284 >2'wv(1)wv(2)+wv(3)wv(4)=wv(5)wv(6)+~v(7)wv(8), (v = 1,2).Next, we consider the spin-pair correlation function of theIsing model- 360-The numerator TrIs l • l . 1sl.l+N, 1exp( -13 TH ) is equal to thepartition function of the free-fermion eight-vertex model whichis obtained by replacing the weight wl (Cl, ~.1) for 2~L~N byWl(C) - cosh(t3TJ y )'w 1 (c) - sinh(t3TJ y )'U : 2.3.6.8).}(c - 1.4,5,7).Hence, the spin-pair correlation function can be obtained byapplying the Vdovichenko r s method to the layered free- fermioneight-vertex model and is expressed in terms of a block'Toeplitzdeterminant whose generating function is a 2x2 matrix function.The detailed calculation was given in Ref.3. And the moregeneralized version for the diagrammatical techniques in reducingfrom two-dimensional Ising models to vertex models was given inRef.4. The asymptotic behavior is obtained by using the theoryof the block Toeplitz determinant reformulated by Tanaka, Moritaand Hiroike. 5 )The model studied here includes the Ising model on thegeneralized Kagome lattice as shown in Fig.4. 3 ,6) The Ising modelwith J 5=J6=J7=!8' J l =J2 and J 3=J4 corresponds to the model treatedby Debauche et al. 7 ,8) In Ref.6, We calculated the asymptoticbehavior of the spin-pair correlation function of the Ising modeland discussed about the nature of the disordered points.- 361-Fig. 4. Ising model on the generalized Kagomelattice.Re:Cerences1 ) N. V. Vdovichenko: Sov. Phys. -JETP 20 (1965)t 477.2) T. Morita: J. Phys. Soc. Jpn. 61 (1992) 470.3) K. Tanaka and 'T. Morita: Prog. Theor. Phys. 88 (1992) 865.4) T. Morita and K. Tanaka: J. Phys. Soc. Jpn. 62 (1993) 873.5) K. Tanaka. T. Morita and K. Hiroike: Physica A171 (1991) 350.6) K. Tanaka and T. Morita: J. Phys. Soc. Jpn. 62 (1993) 868.7) P. Azaria. H. T. Diep and H. Giacomini: Phys. Rev. Lett'. 59(1987) 1629.8) M. Debauche. H. T. Diep. P. Azaria and ~. Giacomini: Phys. Rev.B44 (1991)~2369.- 362-

Asymptotic Behavior of Spin-Pair Correlation Function of Ising Model on Checkerboard Lattice(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

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Asymptotic Behavior of Spin-Pair Correlation Function of Ising Model on Checkerboard Lattice(New Developments in Statistical Physics Similarities in Diversities,YITP Workshop)

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