Thursday, April 29, 2010

Realisation of a Z2 spin liquid

This follows an earlier post about how spinons and gauge fields emerge in quantum antiferromagnets. It closely follows a review article by Subir Sachdev. Red font highlights important comments.

The magnetic moment for a system with ordering wavevector K [which is incommensurate with the reciprocal lattice] can be written as

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

One introduces a spinor walpha, which parameterizes N1,2 by walpha to


Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The Z2 gauge transformation

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where eta(r,tau)=plusminus1. 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, Lambda, that carries U(1) charge 2, so that Lambdaright arrowe2ithetaLambda under the transformation in equation (4). Then a phase with Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com would break the U(1) symmetry, in the same manner that the superconducting order parameter breaks electromagnetic gauge invariance. However, a gauge transformation with theta=pi, while acting non-trivially on the zalpha, would leave Lambdainvariant: 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 Lambda?

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 Lambda, constrained only by symmetry and gauge invariance, including its couplings to zalpha , where we expand the theory Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com to Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, is

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The crucial term is the last one coupling Lambdaa and zalpha: it indicates that Lambda 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 parta.

The mean-field phase diagram of Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com, as a function of the two 'masses' s and Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. We have two possible condensates, and hence four possible phases.

(1) s<0,>Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com: this state has Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com and left fenceLambdaright fence=0. We may ignore the gapped Lambda modes, and this is just the NĂ©el state

(2) s>0, Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com: this state has left fencezalpharight fence=0 and left fenceLambdaright fence=0. Again, we may ignore the gapped Lambda modes, and this is a VBS

(3) s<0,>Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com: this state has Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.comand Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. Because of the zalpha condensate, this state breaks spin-rotation invariance, and we determine the spin configuration by finding the lowest-energy zalpha mode in the background of a non-zero left fenceLambdaright fence in equation (15), which is

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

with walpha being a constant spinor. Inserting equation (16) into equation (3), we find that Phi is space dependent so that left fenceSiright fence obeys equation (6) with N1,2 given by equation (7) and the wavevector K=(pi,pi)+2left fenceLambdaright fence. Thus, this state is a coplanar spin-ordered state. The Z2 gauge transformation in equation (8) is the same as the Z2 theta=pi transformation.

(4) s>0, Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com: this state has left fencezalpharight fence=0 and Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com. This is the Z2 spin-liquid (or Higgs) state we are after. Spin-rotation invariance is preserved.

2 comments:

  1. Hi Ross,

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

    ReplyDelete

Emergence and protein folding

Proteins are a distinct state of matter. Globular proteins are tightly packed with a density comparable to a crystal but without the spatia...