where are the 22 Pauli matrices. Note that this mapping from to z is redundant. We can make a space-time-dependent change in the phase of
zby the field (x,)
and leave unchanged. All physical properties must therefore also be invariant under equation (4), and so the quantum field theory for z has a U(1) gauge invariance, much like that found in quantum electrodynamics. The effective action for z therefore requires the introduction of an 'emergent' U(1) gauge field A (where =x, is a three-component space-time index). The field A is unrelated to the electromagnetic field, but is an internal field that conveniently describes the couplings between the spin excitations of the antiferromagnet. Expressing the spin-wave fluctuations in terms of z and A is a matter of choice, but is completely equivalent to the more conventional theory for the vector field .
The action for the quantum field theory for zand A is determined by the constraints of symmetry and gauge invariance to be
Here the condensation of z associated with commensurate antiferromagnetic order qenches the dynamics of the A gauge boson.
A key point is that U(1) gauge fields in 2+1 dimensions are always confining. This is because of quantum tunneling of instantons. The physical consequence of this for commensurate antiferromagnets are that the spinons are always bound together.
It turns out incommensurate antiferromagnets are different. More on that to follow....