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 **N**_{1,2} by *w*_{} to

The *Z*_{2} gauge transformation

where *(**r*,*)=1. This **Z*_{2}gauge 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, *Z*_{2}, 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 *Z*_{2}, we need a Higgs scalar, *, that carries U(1) charge 2, so that **e*^{2i}* *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 **Z*_{2} 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 *Z*_{2} 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 NĂ©el 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 ****S**_{i} obeys equation (6) with **N**_{1,2 }given by equation (7) and the wavevector *K*=(,)+2*. Thus, this state is a coplanar spin-ordered state. The **Z*_{2} gauge transformation in equation (8) is the same as the *Z*_{2 }*= transformation.*

(4) *s*>0, : this state has *z*_{}=0 and . This is the *Z*_{2} spin-liquid (or Higgs) state we are after. Spin-rotation invariance is preserved.

Hi Ross,

ReplyDeleteI can't view the equations in your post -- they seem to be hosted at a URL behind UQ's library proxy.

I can't read them even at UQ!

ReplyDelete