The magnetic moment for a system with ordering wavevector K [which is incommensurate with the reciprocal lattice] can be written as
One introduces a spinor w, which parameterizes N1,2 by w to
The Z2 gauge transformation
where (r,)=1. This Z2gauge invariance plays an important role in stabilising a spin liquid. [Compare this to a commensurate antiferromagnetic where there is a U(1) symmetry associated with the Schwinger bosons].
The idea is to use the Higgs mechanism to reduce the unbroken gauge invariance from U(1) to a discrete gauge group, Z2, and so reduce the strength of the gauge-flux fluctuations. [This is key because U(1) gauge theories are confining in 2+1 dimensions, whereas Z2 can have a deconfined phase]. To break U(1) down to Z2, we need a Higgs scalar, , that carries U(1) charge 2, so that e2i under the transformation in equation (4). Then a phase with would break the U(1) symmetry, in the same manner that the superconducting order parameter breaks electromagnetic gauge invariance. However, a gauge transformation with =, while acting non-trivially on the z, would leave invariant: thus, there is a residual Z2 gauge invariance that constrains the structure of the theory in the Higgs phase.
What is the physical interpretation of the field ?
We see below it desribes deviations of the local magnetic order from the commensurate Neel order.
How does its presence characterize the resulting quantum state, that is, what are the features of this Z2 RVB liquid that distinguish it from other quantum states? The effective action for , constrained only by symmetry and gauge invariance, including its couplings to z , where we expand the theory to , is
The crucial term is the last one coupling a and z: it indicates that is a molecular state of a pair of spinons in a spin-singlet state; this pair state has a 'p-wave' structure, as indicated by the spatial gradient a.
The mean-field phase diagram of , as a function of the two 'masses' s and . We have two possible condensates, and hence four possible phases.
(1) s<0,>: this state has and =0. We may ignore the gapped modes, and this is just the Néel state
(2) s>0, : this state has z=0 and =0. Again, we may ignore the gapped modes, and this is a VBS
(3) s<0,>: this state has and . Because of the z condensate, this state breaks spin-rotation invariance, and we determine the spin configuration by finding the lowest-energy z mode in the background of a non-zero in equation (15), which is
with w being a constant spinor. Inserting equation (16) into equation (3), we find that is space dependent so that Si obeys equation (6) with N1,2 given by equation (7) and the wavevector K=(,)+2. Thus, this state is a coplanar spin-ordered state. The Z2 gauge transformation in equation (8) is the same as the Z2 = transformation.
(4) s>0, : this state has z=0 and . This is the Z2 spin-liquid (or Higgs) state we are after. Spin-rotation invariance is preserved.