Superselection - Superselection Sectors - Continuous Symmetries

Continuous Symmetries

If a statistical or quantum field has three real valued scalar fields, and the energy or action only depends on combinations which are symmetric under rotations of these components into each other, the contributions with the lowest dimension are (summation convention):

and define the action in a quantum field context or free energy in the statistical context. There are two phases. When t is large, the potential tends to move the average to zero. For t large and negative, the quadratic potential pushes out, but the quartic potential prevents it from becoming infinite. If this is done in a quantum path integral, this is a quantum phase transition, in a classical partition function, a classical phase transition.

So as t moves toward more negative values in either context, the field has to choose some direction to point. Once it does this, it cannot change its mind. The system has ordered. In the ordered phase, there is still a little bit of symmetry--- rotations around the axis of the breaking. The field can point in any direction labelled by all the points on a unit sphere in space, which is the coset space of the unbroken SO(2) subgroup in the full symmetry group SO(3).

In the disordered phase, the superselection sectors are described by the representation of SO(3) under which a given configuration transforms globally. Because the SO(3) is unbroken, different representations will not mix with each other. No local fluctuation will ever bring in nontrivial SO(3) configurations from infinity. A local configuration is entirely defined by its representation.

There is a mass gap, or a correlation length, which separates configurations with a nontrivial SO(3) transformations from the rotationally invariant vacuum. This is true until the critical point in t where the mass gap disappears and the correlation length is infinite. The vanishing gap is a sign that the fluctuations in the SO(3) field are about to condense.

In the ordered region, there are field configurations which can carry topological charge. These are labeled by elements of the second homotopy group . Each of these describe a different field configuration which at large distances from the origin is a winding configuration. Although each such isolated configuration has infinite energy, it labels superselection sectors where the difference in energy between two states is finite. In addition, pairs of winding configurations with opposite topological charge can be produced copiously as the transition is approached from below.

When the winding number is zero, so that the field everywhere points in the same direction, there is an additional infinity of superselection sectors, each labelled by a different value of the unbroken SO(2) charge.

In the ordered state, there is a mass gap for the superselection sectors labeled by a nonzero integer, because the topological solitons are massive even infinitely massive. But there is no mass gap for the all the superselection sectors labeled by zero because there are massless Goldstone bosons describing fluctuations in the direction of the condensate.

If the field values are identified under a Z₂ reflection (corresponding to flipping the sign of all the fields), the superselection sectors are labelled by a nonnegative integer (the absolute value of the topological charge).

It is interesting that O(3) charges only make sense in the disordered phase and not at all in the ordered phase. This is because when the symmetry is broken there is a condensate which is charged, which is not invariant under the symmetry group. Conversely, the topological charge only makes sense in the ordered phase and not at all in the disordered phase, because in some hand-waving way there is a "topological condensate" in the disordered phase which randomizes the field from point to point. The randomizing can be thought of as crossing many condensed topological winding boundaries.

The very question of what charges are meaningful depends very much on the phase. Approaching the phase transition from the disordered side, the mass of the charges particles approaches zero. Approaching it from the ordered side, the mass gap associated with fluctuations of the topological solitions approaches zero.