Quantum Theory of Collective Phenomena (Dover Books on Chemistry) by G. L. Sewell

Author:G. L. Sewell [Sewell, G. L.]
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2014-04-05T00:00:00+00:00

5.2. Order parameters, symmetry breakdown, and long-range correlations

In order to formulate the concept of an order parameter, we start by considering the specific case of the two-dimensional Ising model. This model undergoes a phase transition of the second kind at a critical temperature Tc such that, for T < Tc, it has two pure equilibrium phases, with equal and opposite polarizations, while for T > Tc it has just one equilibrium phase, which is unpolarized. The fact that the pure phases are polarized, for T < Tc, means that the spins are preferentially aligned then, and we refer to the system as being ordered by virtue of this alignment. Since the polarization represents a measure of the degree of alignment, we refer to it as the order parameter, η. This parameter evidently represents the strength of an ‘inner field’ (polarization) that acts on its own source, namely the spins, in such a way as to preserve their alignment and thereby to stabilize the ordering.

The concept of an order parameter as a stabilized, symmetry-breaking inner field may be formulated in a general context as follows. Suppose that Σ is an arbitrary system which undergoes a phase transition of the second kind, satisfying the following conditions.

(a) Σ has a space-dependent local observable, , which transforms linearly under a symmetry group, G = {α(g)}, of automorphisms of the algebra of the observables. Thus, if has n components , say

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