Ground states, excitations, thermodynamics and exact results in various lattice systems

Abstract

This thesis covers problems in a few different aspects of interacting many-particle systems on a lattice. The first problem is of a quantum dynamical system, and the rest are of quantum and classical magnetic systems. We discuss their ground states, excitations, and thermodynamics, and make several exact statements about these properties. The first chapter concerns the quantum-mechanical Toda lattice. The Toda lattice (or Toda chain) is a one-dimensional chain of atoms on equally spaced lattice sites, which can be displaced from their equilibrium positions and whose interactions are exponential, rather than quadratic, in their relative displacements. The classical problem has been much studied and its solution is well understood; the quantum problem's treatment has not been so clean or thorough. On the one hand was the approach by Sutherland [2] who used his technique of the "asymptotic Bethe ansatz,” which is in general approximate but was argued by him to be exact in the thermodynamic limit. He extracted the classical limit successfully from his formulation. On the other hand, an exact formal solution was developed by Gutzwiller [3], and extended by Sklyanin [4], Pasquier and Gaudin [5] and others, but it is cumbersome and they did not use it for actual calculation of physical quantities like dispersions and velocities of excitations. Prior to our work, there has been no attempt to synthesise these two very different approaches, and the limitations of Sutherland's approach have been much exaggerated. In this chapter we show that Gutzwiller's equations in fact reduce to Sutherland's in two limits-large particle number, or large Planck's constant. Moreover, by comparing the asymptotic Bethe ansatz results with the exact diagonalisation results of Matsuyama [6], we show that even for ? = 1 (on the scale of other quantities) and 6 particles, the Bethe ansatz results are very accurate (we reproduce Matsuyama’s figures to his quoted accuracy). Using these equations, then, we calculate dispersion relations and sound velocities; we compare the quantum excitations with the well-known classical ones; we study the extreme quantum limit, and show that the excitations remain qualitatively the same but undergo quantitative changes; using conformal field theory, we find correlation functions and show that the energy calculated by the asymptotic Bethe ansatz method seems to be accurate to order 1/N² in particle number, i.e., much better than it would a priori be expected to be. Sutherland's original argument is that the ABA is expected to be valid for densities of extensive variables only. We wrap things up by discussing the classical problem and solution and its connection with our quantum treatment, the proof of integrability in the quantum case, and other topics which are hard to find in one place in the literature. This chapter is based on work published in 1997 in Physical Review B [7]. The second chapter considers the s = ½ Heisenberg model on an unusual two-dimensional lattice, a sort of stretched version of the well-known tegone lattice. On this lattice the system's ground state can be given exactly in terms of singlet pairs ("dimerisation"), much like the Majumdar-Ghosh model [8] in one dimension. This is not the first two-dimensional system with this property (an early example was due to Shastry and Sutherland [9]) but other examples had non-degenerate ground states; this seems to be the first example with multiple, degenerate ground states and domain-wall-like excitations, and in this respect it is very similar to one-dimensional systems like the Majumdar-Ghosh chain and the sawtooth chain [10]. It is worth studying for this reason, and also because it decouples in the limit of strong (rather than weak) inter-chain exchange constants, into a system of non-interacting (effectively one-dimensional) chains. This chapter is based on work published in 1999 in Physical Review B (Rapid Comm.) [11]. The third chapter is a follow-up to recent experiments on a class of pyrochlores, the rare earth titanates, and based on collaborative work published in 1999 [12,13]. It has been known for some time that an Ising model on the pyrochlore lattice (a three-dimensional structure of corner-sharing tetrahedra) has a macroscopically degenerate ground state (and therefore a nonzero ground-state entropy) quite analogously to ice (frozen water) and with pretty nearly the same mechanism, except that the local ordering rule responsible for this governs the arrangement of protons for ice and the alignment of spins in the pyrochlore. These systems have therefore been called "spin ice”. It was found experimentally that dysprosium titanate seems to be an excellent approximation to this ideal behaviour. We use crystal field calculations to confirm what had already been known experimentally, that dysprosium titanate is extremely well approximated by an Ising model; note that the dominant interactions between spins is really a long-ranged dipole-dipole interaction, rather than a nearest-neighbour-only interaction; and find that a simulation of such a model gives excellent agreement with experimental specific heat data provided the interaction is scaled down somewhat. We attribute the scaling down required here to the presence of another interaction, an antiferromagnetic superexchange. It is difficult to calculate this interaction a priori and it is simplest to assume that it is nearest neighbour only. In the case of dysprosium titanate, that gives very good results, but a uniform scaling down of the entire dipole-dipole interaction gives even better results. We confirm that it is an additional interaction at work here, and not a change in the value of the dipole moment itself, by comparing with experimental data in the presence of different magnetic fields (to which only the dipole moment will couple). Again we get results similar to what one sees experimentally.