The spectral gap for the ferromagnetic spin-J XXZ chain

Details The spectral gap for the ferromagnetic spin-J XXZ chain The spectral gap for the ferromagnetic spin-J XXZ chain

2001-10-14

arxiv.org/abs/math-ph/0110017v4

Adv.Theor.Math.Phys. 5 (2001) 1047--1090

Physics

Mathematical Physics

48 pages, 7 figures, 1 table, note-added-in-proof added

Scientific paper

We investigate the spectrum above the kink ground states of the spin J ferromagnetic XXZ chain. Our main theorem is that there is a non-vanishing gap above all ground states of this model for all values of J. Using a variety of methods, we obtain additional information about the magnitude of this gap, about its behavior for large \Delta, about its overall behavior as a function of \Delta and its dependence on the ground state, about the scaling of the gap and the structure of the low-lying spectrum for large J, and about the existence of isolated eigenvalues in the excitation spectrum. By combining information obtained by perturbation theory, numerical, and asymptotic analysis we arrive at a number of interesting conjectures. The proof of the main theorem, as well as some of the numerical results, rely on a comparison result with a Solid-on-Solid (SOS) approximation. This SOS model itself raises interesting questions in combinatorics, and we believe it will prove useful in the study of interfaces in the XXZ model in higher dimensions.

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