Chapter 8 
Physics close to and above the upper-critical dimension

is not too large; the perturbation theory has to be combined with a renormalization group analysis in this case.

The physics described by this perturbative method can, in most cases, also be elucidated by the large N expansion we have developed in the previous chapters. However, there are a number of instances where the underlying principles are most transparently illustrated by studies close to and above d = 3. Our specific reasons for undertaking such an analysis are:

• As we have noted earlier, the quantum critical point at T = 0, g = g_c, extends out to a line of finite temperature phase transitions for the cases d = 2, N = 1,2. The e expansion offers a controlled method obtaining the structure of the crossovers in the vicinity of this line.


We have not yet found a successful description of the low frequency dynamics of the order parameter (n or s^z) in the high T regime in d = 2, although we did succeed in d = 1 in Chapters 4 and 6. We show that the e-expansion leads to an appealing quasi-classical wave description of this dynamics.

For the quantum rotor models being studied here, the crossovers above the upper-critical dimension, with d > 3, are obviously in a physically inaccessible dimension. However the basic structure that will emerge is quite generic to quantum critical points above their upper-critical dimension. The results will therefore be useful in Part 3 where we will consider other models with a lower upper-critical dimension, so that dimensions above the upper-critical can be experimentally studied.

The following chapter studies transport properties of the quantum rotor models in the high T and quantum-paramagnetic low T regions in d = 2. While it is possible to do this within the 1/N expansion, the computation is simplest, and most physically transparent, using the e expansion we shall develop here.

• In Eqn (8.13), there should be a + sign before 3l²/2u