Numerical Transfer-Matrix Study of Interfaces in Ising Models
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NUMERICAL TRANSFER-MATRIX STUDY OF INTERFACES IN ISING MODELS. M. A. Novotnyt, H. L. Richardsft,
and P. A. Rikvoldtf*¶
f Supercomputer Computations Research Institute, B-186, Florida State University, Tallahassee, Florida 32306 : Physics Department and Center for Materials Research and Technology, B-159, Florida State University, Tallahassee, Florida 32306 * Tohwa Institute for Science, Tohwa University, Fukuoka 815, Japan ¶ Department of Physics, Kyushu University 33, Fukuoka 812, Japan ABSTRACT Results are reported for the surface tension, the surface free energy, the surface stiffness coefficient, and the single-step free energy for the Ising model in two and three dimensions. These are obtained by numerical transfer-matrix calculations, testing detailed predictions for the scaling of the largest eigenvalues of the transfer-matrix. INTRODUCTION The study of surface and interface phenomena is an active current area of research. The phenomena to which it is relevant range from wetting through nucleation and growth to crystal-faceting, and span areas from materials science to lattice-gauge theory [1]. A great deal of analytical work [1], including low-temperature series [2], has been done to study interfaces in Ising and lattice-gas systems, and it has been supplemented with Monte Carlo simulations [3-9]. Recently there has been a renewed interest in calculating interfacial properties by transfer-matrix (TM) methods [10-14]. However, most of these TM studies have either dealt with exact solutions for the d=2 Ising model, or with theoretical estimates of the associated scaling phenomena. In this paper we apply scaling relations [10,111 for the TM to numerical studies of the Ising model to test whether large-scale TM calculations can penetrate the scaling region and hence obtain information about interfacial properties. For systems such as crystalline solids and theoretical lattice models the interfacial free energy between different phases depends on the orientation of the interface. Here we investigate both d=2 square lattices and d=3 cubic lattices, and we restrict ourselves to interfaces which are nearly parallel to one of the lattice axes. In the TM method lattice strips of size L x oo in d=2 and of size L x M x oo in d=3 are investigated, and for definiteness we assume that there is an infinite number of spins along the x-direction of the lattice. We will consider the dependence on temperature, T, of the angle-dependent surface tension, cr(0, T). When measured per unit projected distance along the n-direction, a has a quadratic minimum [15] cr(O,T)/cos(O) = r(T) + -K(T)82 + 0(04) 2
(1)
with the surface free energy given by r(T) = ur(0, T) > 0
(2)
,.(T) = r(0,T) + c,"(0, T) > 0.
(3)
and the surface stiffness coefficient
The above analysis is appropriate for all temperatures below the critical temperature, Tc, and above the roughening temperature, TR. The later is the temperature at and above which the interface separating two domains becomes rough in the sense that the root-meansquare width of the inter
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