The co-operative Jahn-Teller effect is a phase transition which is driven by the interaction between localized orbital electronic states and the crystal lattice. It involves the simultaneous splitting of the electronic states and a symmetry-lowering distortion of the lattice. Recent advances in the understanding of this effect have been spectacular both because of the discovery of a family of transparent rare earth compounds in which it is easy to measure the changes of energy of the electronic states and because of the increasing use which has been made of ultrasonic techniques for investigating lattice distortions. It will be shown that in most materials the coupling is predominantly to a strain (which changes the shape of the crystal) and the ultrasonic method provides a direct observation of the softening of the appropriate elastic constant near the transition temperature. This strain mode coupling provides an effective interaction between the Jahn-Teller active ions which has a very long range. As a result molecular field theory can be used to provide a quantitative description of very many of the properties of these systems.The article discusses in detail the possible origins, symmetries and properties of the electron-lattice interactions and how they lead to possible hamiltonians for the coupled system. The relation of these interactions with quadrupolar interactions and with magnetostriction is included. The methods of solution of the hamiltonians which are discussed lead to an understanding of the electronic states, the phonon spectrum and the mixed normal modes. A very wide variety of experimental techniques has been brought to bear on this problem. These are reviewed in detail and the results are compared with theoretical expressions whenever possible. T h e application of external stress and magnetic field turns out to be of particular significance in the case of the rare earth compounds, because here they can produce effects which are larger than the low-temperature spontaneous effects.

We study the T = 0 frustrated phase of the 1D quantum spin-1 2 system with nearest-neighbour and next-nearest-neighbour isotropic exchange known as the Majumdar-Ghosh Hamiltonian. We first apply the coupled-cluster method of quantum many-body theory based on a spiral model state to obtain the ground state energy and the pitch angle. These results are compared with accurate numerical results using the density matrix renormalisation group method, which also gives the correlation functions. We also investigate the periodicity of the phase using the Marshall sign criterion. We discuss particularly the behaviour close to the phase transitions at each end of the frustrated phase.

We apply a recent adaptation of White's density matrix renormalisation group (DMRG) method to a simple quantum spin model, the dimerised XY chain, in order to assess the applicabilty of the DMRG to quantum systems at non-zero temperature. We find that very reasonable results can be obtained for the thermodynamic functions down to low temperatures using a very small basis set. Low temperature results are found to be most accurate in the case when there is a substantial energy gap.Since it's recent inception, White's denstiy matrix renormalisation group (DMRG) method has been established as the method of choice for determining static, low energy properties of one-dimensional quantum lattice systems [1]- [2]. Extensions to the calculation of dynamical properaties [3] and even to the study of low temperature properties of two dimensional systems [4] have been forthcoming. Moreover, Nishino's formulation of the DMRG for two dimensional classical systems [5] has paved the way for the study of one dimensional quantum systems at non-zero temperature. In this letter we present what is, to the best of our knowledge, the first application of the DMRG to the thermodynamics of a quantum system.The system that we consider is a simple spin chain model-the dimerised, S = 1/2, XY modelwhere S i is a spin-1/2 operator for site i on an (even) chain of N sites, with periodic boundary conditions. 0

The study of changes in the refractive indices (with consequent changes of birefringence) of a transparent magnetic crystal which accompany changes in the magnetic order is becoming more popular. In this article we review why this is. The first reason is that birefringence can be measured very accurately: the different experimental arrangements are reviewed. The second reason is because a birefringence measurement is an integrational spectroscopic technique and therefore it is studied both experimentally and theoretically as a branch of magneto-optics and hence gives information on the detailed energy level structure of the solid. The third reason is that in a number of interesting systems the birefringence is proportional to the magnetic energy over a wide temperature range and it is often a more convenient method of obtaining the magnetic specific heat than direct specific heat measurements; this is particularly true in magnetic crystals which show low dimensional ordering. The last reason is that in all magnetic crystals the birefringence change should vary like one of the thermodynamic critical exponents near to the phase transition. We review in detail the reasons why birefringence studies have become so successful for measuring critical exponents in pure and particularly mixed crystals.The method is restricted to crystals which are transparent in some region of the electromagnetic spectrum in or close to the visible. All the available experimental data are summarised in the review.We also review the extent to which a birefringence measurement is sensitive to critical fluctuations and also the use of this technique to study domains.