Quantum Anisotropic Sigma and Lambda Models as Spin Chains

Calan Appadu, Timothy J. Hollowood, Dafydd Price, Daniel C. Thompson

Published:

We consider lambda and anisotropic deformations of the SU(2) principal chiral
model and show how they can be quantized in the Hamiltonian formalism on a
lattice as a suitable spin chain. The spin chain is related to the higher spin
XXZ Heisenberg chain and can be solved by using the Bethe Ansatz. This yields
the spectrum and S-matrix of the excitations. In particular, we find the
S-matrix in the gapped anti-ferromagnetic regime. In this regime, a continuum
limit does not exist and this suggests that the field theories in this regime,
precisely ones with a cyclic RG like the Yang-Baxter deformations, may only
exist as effective theories. In a certain limit, we show that the XXZ type
lambda model gives the symmetric space SU(2)/U(1) lambda model and, hence, we
are able to find its spectrum and S-matrix and show that it gives the S-matrix
of the O(3) sigma model in the appropriate limit. Finally, we show the full
consistency of the S-matrix and the Lagrangian formulations of the lambda
model, by coupling to a conserved charge and computing the way the ground state
energy changes in both pictures.