Transfer matrix, finite-size scaling

This system was modelled in terms of the lattice gas with interactions shown in Fig. Ib. The phase diagram was first calculated by the transfer matrix finite size scaling technique for various choices of the interaction parameters [Pg.122]

Fig. 18. Phase diagram of the centered rectangular lattice gas model with ==0, 3/4 2 = V3> vJ

Fig. 30. Phase diagram of a model for Si/W(110) in the temperature versus 9 plane. Experimentally determined interactions J Jj,are used. Full dots are from Monte Carlo calculations, while triangles are based on transfer matrix finite size scaling using strip widths of 8 and 12. The point labelled L indicates approximate location of Lifshitz point. The dotted line indicates the transition region between the (5 x l)and(6 x 1) phases. (From...

With the availabihty of computers, the transfer matrix method [14] emerged as an alternative and powerful technique for the study of cooperative phenomena of adsorbates resulting from interactions [15-17]. Quantities are calculated exactly on a semi-infinite lattice. Coupled with finite-size scaling towards the infinite lattice, the technique has proved popular for the determination of phase diagrams and critical-point properties of adsorbates [18-23] and magnetic spin systems [24—26], and further references therein. Application to other aspects of adsorbates, e.g., the calculation of desorption rates and heats of adsorption, has been more recent [27-30]. Sufficient accuracy can usually be obtained for the latter without scaling and essentially exact results are possible. In the following, we summarize the elementary but important aspects of the method to emphasize the ease of application. Further details can be found in the above references. 

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

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