Z. Phys.B - CondensedMatter 75, 11%125 (1989) Ze,sohri. M a t t e r f~r Physik B 9 Springer-Verlag 1989 Diagonalisation of finite-size corner transfer matrices and related spin chains T.T. Truong and I. Peschel Fachbereich Physik, Freie Universit~it Berlin, Germany Received October 4, 1988 We study Baxter's corner transfer matrices for anisotropic Ising models of finite size. They are related to spin one-half chains with coefficients which increase linearly along the chain. The operators are diagonalised with the help of special polynomials and the eigenvalue spectrum is discussed. The relation to the infinite lattice limit is outlined. 1. Introduction The corner transfer matrix (CTM) of Baxter [1] is a somewhat unusual quantity. It relates, for two- dimensional lattice systems, two rows of variables which meet at an angle 0, normally n/4 or n/6, depen- ding on the lattice. Therefore one might expect it to be quite a complicated object. For the exactly solvable models, however, almost the opposite is true. Writing the CTM as A=exp(-H), the eigenvalues of H were found to be equidistant in the infinite systems, with a splitting dependent on the temperature. Furthermore, order parameters can be calculated quite easily with the CTM and are obtained in the form of simple infinite products. In the usual calculations [1-4], the infinite lattice limit is taken at an early stage and the peculiar spec- trum is a consequence of the star-triangle relations. Numerical calculations in the Ising model show, how- ever, that in general not all but only the low-lying eigenvalues of H are equidistant [5]. Thus certain features are lost in the usual treatment and it is clearly desirable to solve the finite-size problem, too. This has been done so far for the Ising model at the critical point, both numerically [5, 6] and analytically [7], and the lowest eigenvalues of H were in agreement with the predictions of conformal invariance [5]. In the present work we extend the analytical treatment to arbitrary temperatures.+ To simplify matters we consider Ising systems in the anisotropic (Hamiltonian) limit. In this case, the operator H can be written down by inspection. It describes a finite Ising chain in a transverse field where both the field and the couplings increase linearly along the chain. A mathematically similar problem occurs when one treats the motion of electrons in a homo- geneous electric field [8] or the equivalent spin chain [9]. In our case, however, the linear dependence of the coefficients is a geometrical effect: it reflects the wedge- shaped geometry in which the CTM operates. Since the Hamiltonian H is a quadratic form when written in fermion operators, it can be diagonalised by the standard procedure [10]. This is described in Sect. 2. The solution is expressed in terms of the so-called Carlitz polynomials which are closely related to ellip- tic functions. In Sect. 3 we study the zeros of these polynomials which determine the eigenvalues of H. We also treat, in Sect. 4, the problem of two inter- penetrating Ising lattices (the decoupled eight-vertex model). In this case H describes an anisotropic X-Y chain with linearly increasing couplings. Although this looks more symmetric, the solution is slightly more involved. Finally, in the two appendices we review the Carlitz polynomials and make contact with the in- finite lattice formulation of Baxter [2]. 2. The simple lsing model We consider a square lattice with a 90~ as shown in Fig. la. The CTM connects the primed and unprimed spin variables, the central spin a o and the