Thursday, April 15, 2010

Towards a Z2 spin liquid

I spent some of this morning reading a nice Nature Physics review article by Subir Sachdev, Quantum magnetism and criticality.
The key thing I wanted to understand was how gauge fields arise in frustrated spin models and how this can lead to deconfinement of spinons. Here are a few relevant extracts.

One can express the vector field Phi [describing antiferromagnetic spin fluctuations] in terms of an S=1/2 complex spinor field zalpha , where alpha=up arrowdown arrow [these are Schwinger bosons] by


where sigma are the 2times2 Pauli matrices. Note that this mapping from Phi to zalpha is redundant. We can make a space-time-dependent change in the phase of

zalphaby the field theta(x,tau)


and leave Phi unchanged. All physical properties must therefore also be invariant under equation (4), and so the quantum field theory for zalpha has a U(1) gauge invariance, much like that found in quantum electrodynamics. The effective action for zalpha therefore requires the introduction of an 'emergent' U(1) gauge field Amu (where mu=x,tau is a three-component space-time index). The field Amu 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 zalpha and Amu is a matter of choice, but is completely equivalent to the more conventional theory for the vector field Phi.

The action for the quantum field theory for zalphaand Amu is determined by the constraints of symmetry and gauge invariance to be


Here the condensation of zalpha associated with commensurate antiferromagnetic order qenches the dynamics of the Amu 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....

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