Friday, April 30, 2010
A circumstantial discovery
Thursday, April 29, 2010
Realisation of a Z2 spin liquid
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 gaugeflux 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 nontrivially 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 spinsinglet state; this pair state has a 'pwave' structure, as indicated by the spatial gradient _{a}.
The meanfield 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 spinrotation invariance, and we determine the spin configuration by finding the lowestenergy z_{} mode in the background of a nonzero 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 spinordered 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} spinliquid (or Higgs) state we are after. Spinrotation invariance is preserved.
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