Z. Phys. B - Condensed Matter 46, 245-250 (1982) Conder d Zeitschrift Matter for Physik B 9 Springer-Verlag 1982 The Influence of a Surface on Ising and Conjugate Model Spin Correlations at T= T c J. Kroemer and W. Pesch Physikalisches Institut der Universit~it, Bayreuth, Federal Republic of Germany Received January 20, 1982 We calculate rigorously the two-point correlation function in the 2-d Ising model on a quasi-cylindrical lattice at the critical temperature T~. The scaling predictions for the surface critical exponents r/• = 5/8, t/II = 1 are confirmed and the change from surface to bulk behavior is studied. By a correspondence relation the results are mapped to the conjugate model, a subcase of the 8-vertex model. We find in this model t/• = 5/4 and t/ll =2. I. Introduction and Summary In this paper the spin-spin correlation function in the Ising model and the vertical-arrow correlation function of the so-called conjugate model [1, 2] - a special case of the 8-vertex model - are determined at the critical temperature T~ in the presence of a free boundary. Both models are considered in two dimensions. We are mainly interested in the influ- ence of the surface on the critical exponent t/, which describes the spatial decay of the correlation func- tion at T c. This exponent is determined for the two spins arranged perpendicular (t/• and parallel (/711) to the boundary. It is well known that neglecting boundary effects the conjugate model corresponds to an Ising model [3]. The correspondence remains valid for mixed free and cyclic boundary conditions and leads to a shear- ed Ising model (2) with somewhat unusual cou- plings between the spins. We calculate first the correlation function of the Ising model with the help of the combinatorial method [4] and then translate the results to the conjugate model. Within our calculational scheme the spin correlation function including boundary ef- fects is evaluated directly at T~. In contrast to for- mer work [5, 6] one needs not introduce an arti- ficial four spin correlation function, which factorizes for T+ To. It is not clear a priori, that this factori- zation can be extended to To, where the corre- lations do not decay exponentially. The critical exponents ~ll, /7• can be read off from (13), (19). There is agreement with results obtained previously [-5, 7]. The scaling law 2t/• I [-8, 9] is fulfilled. In our case also the relation 711=v-1 [-9] is fulfilled, which has been shown to hold not in general [10]. From our result (13) the change from bulk to surface behavior can be followed directly. No additional crossover exponent is involved. Furthermore, in the extreme anisotropic limit (15), the critical exponents are shown to be not universal at T~. Our results for the conjugate model are given in (21a), (21b). Some calculational details are deferred to the appendices. II. The Correspondence Between the Conjugate and the Ising Model The conjugate model is an eight vertex model with the vertex weights COl=Coz=a, e)3=co~=a -1, 095 =co6=b , co7=6o8=b -1, where the vertices are la- beled as in [2]. We consider a vertex lattice wrap- ped on a cylinder leading to free boundaries in the vertical direction and cyclic ones in the horizontal direction. Neglecting boundary effects, the conjugate model can be mapped on two interpenetrating un- coupled square Ising models 11, I2, oriented at an angle of 45 ~ with respect to the original lattice (cf. Chap. III in [-11]). Both Ising models have the same horizontal and vertical coupling constants Jx and J2, 0722-3277/82/0046/0245/$01.20