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Bulletin of the American Physical Society

A family of spin-S chain representations of SU(2) level k Wess-Zumino-Witten models

Martin Greiter, Ronny Thomale, Stephan Rachel, Peter Schmitteckert

Published Feb 27, 2012• 0 citations
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We investigate a family of spin-SSS chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For S=1/2S=1/2S=1/2, it corresponds to the Haldane-Shastry model. For general spin SSS, we find indication that the low-energy theory of these spin chains is described by the SU(2)k{}_{k}k​ Wess-Zumino-Witten model with coupling k=2Sk=2Sk=2S. In particular, we investigate the S=1S=1S=1 model whose ground state is given by a Pfaffian for even number of sites NNN. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary NNN with trial states obtained by Schwinger projection of two Haldane-Shastry chains.
The authors have identified critical points in the Wess-Zumino-Witten (WZW) SU(2)5{}_{5}5​ universality class in the phase diagram of a spin-5/2 chain with frustrated next-nearest-neighbor and 3-site interactions, providing a concrete example of this universality class in a realistic, nonintegrable model. These points lie on a transition line to a dimerized phase and occur when the transition turns from first order into WZW SU(2)3{}_{3}3​ in agreement with the generic possibilities allowed by field theory arguments.
We study the long-wavelength limit of a spin SSS Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level 2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten (WZW) model. This effective theory is then mapped into a compact U(1)U(1)U(1) boson interacting with Z2S{Z}_{2S}Z2S​ parafermions. The analysis of this effective theory allows us to show that when SSS is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case and the SU(2)2S{\mathrm{SU}(2)}_{2S}SU(2)2S​ WZW model flows towards the SU(2)1{\mathrm{SU}(2)}_{1}SU(2)1​ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin SSS.