Finite lattice methods in statistical mechanics

Abstract

This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.