Chapter 5 
Quantum rotor models: large N limit

The quantum Ising model studied previously had a discrete Z2 symmetry. An important new ingredient in the rotor models is the presence of a continuous symmetry: the physics is invariant under a uniform, global O(N) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We introduce the important concept of the spin stiffness, which characterizes the rigidity of the ordered state, and determines the dispersion spectrum of the low energy `spin-wave' excitations. Apart from this, much of the technology and the physical ideas introduced earlier for d = 1 Ising chain generalizes straightforwardly, although we are no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high T and the two low T regions on either side of the quantum critical point, continues to be extremely useful, and is again the basis of our discussion. Because we consider models in spatial dimensions d > 1, it is possible to have a thermodynamic phase transition at a non-zero temperature. We are particularly interested in the interplay between the critical singularities of the finite temperature transition and those of the quantum critical point.

The analysis is carried out using a simple and important technical tool: the large N expansion. This chapter largely confines itself to the results obtained at N = ¥. The results so obtained will give an adequate description of gross features of the phase diagram and some static observables, but are quite inadequate for dynamical properties at non-zero temperatures. The latter problems will be addressed in subsequent chapters.