Abstract
The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified theory G is a continuous Lie group. The ground state symmetry described by G can change spontaneously from G to one of its subgroups H as the external parameters of the system are modified. Such a macroscopic change of the ground state symmetry of a system from G to H correspond to a “phase transition”. Such phase transitions have been extensively studied within a framework due to Landau. A vast range of systems can be described using Landau’s approach, however there are also systems where the framework does not work. Recently there has been growing interest in looking at such non-Landau type of phase transitions. For instance there are several “quantum phase transitions” that are not of the Landau type. In this short review we first describe a refined version of Landau’s approach in which topological ideas are used together with group theory. The combined use of group theory and topological arguments allows us to determine selection rule which forbid transitions from G to certain of its subgroups. We end by making a few brief remarks about non-Landau type of phase transition.
Highlights
In many branches of physics the macroscopic symmetry properties of the ground state of a system can be different from that of its Hamiltonian
A very useful phenomenological approach for tackling this problem and for understanding how spontaneous symmetry breaking transitions can occur which we will use was proposed by Landau [1]
We show how Landau theory constructed for this system allows us to understand why an abrupt change of the symmetry of the density function can happen
Summary
In many branches of physics the macroscopic symmetry properties of the ground state (vacuum state) of a system can be different from that of its Hamiltonian. This is done by combining group theory methods with topological ideas Such an approach does not give the exact functional form for the order parameter but gives the allowed symmetry breaking patterns the system can have. We turn to the general situation using these ideas and proceed to show how to describe the order parameter of a crystal with symmetry group G and describe how to symmetry breaking can be formulated This means constructing a procedure for understanding how the model approach allows the group G to abruptly change to one of its subgroupsHi. Let us look at a crystal which has a certain symmetry group G, where G is a finite group of order n(G). The use of Morse theory in Landau theory is rather natural as there is a natural real valued function, namely the Free Energy, whose critical points, if minima, represent possible equilibrium configurations of the system [9].
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