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 Comm. Math. Phys., 2013, Volume 322, Issue 3, Pages 807–834 (Mi cmph9)

The geometry of antiferromagnetic spin chains

D. Bykovab

a Steklov Mathematical Institute of Russ. Acad. Sci., Gubkina str. 8, 119991 Moscow, Russia
b Nordita, Roslagstullsbacken 23, 106 91 Stockholm, Sweden

Abstract: It has been a conjecture of F.D.M. Haldane that long-wavelength excitations around antiferromagnetic vacua of certain spin chains are related to relativistic two-dimensional $\sigma$-models. We construct a class of such spin chains and associated $\sigma$-models, using symplectic geometry as a main tool. The target space of the $\sigma$-model can be an arbitrary complex flag manifold, and we find universal expressions for the metric and $\theta$-term. The derivation relies on the fact that the flag manifold is a Lagrangian submanifold in a direct product of Grassmannians associated (via geometric quantization) to the representations of spins at consecutive sites of the spin chain.

 Funding Agency Grant Number Russian Foundation for Basic Research 11-01-00296-a11-01-12037-ofi-m12-01-31298-mol-a Ministry of Education and Science of the Russian Federation NSh-4612.2012.1 My work was supported in part by grants RFBR 11-01-00296-a, 11-01-12037-ofi-m-2011, 12-01-31298-mol-a and in part by grant for the Support of Leading Scientific Schools of Russia NSh-4612.2012.1.

DOI: https://doi.org/10.1007/s00220-013-1702-5

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Accepted:17.10.2012
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