Mathematics Colloquia and Seminars

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Description

I will quickly define a spin system and quantum spin system, using the tools of representation theory for SU(2). The 1d XXZ was solved by Koma and Nachtergaele using the quantum group SU_q(2), but this fails in higher dimensions. So analysis is required.

In a recent paper (by Bolina, Contucci, Nachtergaele and Starr) we considered ground states with a simple form: specifically symmetric simple tensors (we call this the grand canonical ensemble). These exhibit interfaces parametized by CP^1. We considered a linear space of perturbations, analogous to the the 1-magnon pertubations of the homogeneous ground state. The Hamiltonian, restricted and projected to this space, upon proper scaling, is simply the negative Laplacian. Using the variational principle, we determined an upper-bound for the first excited eigenvalue of the Hamiltonian, proportional to 1/R^2 where R is the linear size of the pertubation. This is far better than the closest bounds in the theoretical physics literature.

If time permits, I'll describe a similar result for interface states of the canonical ensemble which is the subject of a second, more difficult paper. There the key step is to demonstrate equivalence with the grand canonical ensemble. We elicited helpful advice on the local central limit theorem (unpon which equivalence of ensembles relies) from a local probabilist (Prof. Gravner).

As with all my talks, everything will be easily understood by even the most unprepared graduate student. In fact I may even understand the material, myself.