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Chapter 2
The mapping to classical statistical mechanics: single site models
This chapter discusses the reason for the central importance of the quantum
Ising and rotor models in the theory of quantum phase transitions, quite
apart from any experimental motivations: quantum transitions in these models
in d dimensions are intimately connected to certain wellstudied
finite temperature phase transitions in classical statistical mechanics
models in D = d+1 dimensions. This mapping is described in
some detail in its simplest context of d = 0, D = 1: we will
consider single site quantum Ising and rotor models, and explicitly discuss
their mapping to classical statistical mechanics models in D = 1.
These very simple classical models in D = 1, actually do not have
any phase transitions. Nevertheless, it is quite useful to examine them
thoroughly as they do have regions in which the correlation `length' x
becomes very large: the properties of these regions are very similar to
those in the vicinity of the phase transition points in higher dimensions.
In particular, we will introduce the central ideas of the scaling limit
and
universality in this very simple context. We will then go on to
map the classical models to equivalent zerodimensional quantum models
and demonstrate that this mapping becomes exact in the scaling limit.
Exercises
Corrections

In Eqn. (2.29) a factor of 1/T is missing from the argument of the
cosh

In Eqn. (2.37) the summation in the first term is over n_{i}.
n_{i+1}

In Eqn. (2.52) the right hand side is missing a factor (2p);
this is from the integral over q'(0)
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