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Chapter 5

Quantum rotor models: large *N* limit

The quantum Ising model studied previously had a discrete *Z*_{2}
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.

*Exercises*

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