The spectrum of two-particle bound states of transfer matrices of Gibbs fields. Part 3: Fields on the three-dimensional lattice

Lakshtanov, E.; Minlos, R.

doi:10.1090/S0077-1554-08-00171-4

The spectrum of two-particle bound states of transfer matrices of Gibbs fields. Part 3: Fields on the three-dimensional lattice
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by E. L. Lakshtanov and R. A. Minlos
Translated by: E. Khukhro

Trans. Moscow Math. Soc. 2008, 255-288

DOI: https://doi.org/10.1090/S0077-1554-08-00171-4

Published electronically: November 19, 2008

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Abstract:

In this paper, which continues our earlier papers, we investigate so-called “adjacent” bound states (that is, bound states which only appear in a neighbourhood of special values of the total quasimomentum of the system) of the transfer matrix of the general spin model on the 3-dimensional lattice in its two-particle subspace for high temperatures $T=1/ \beta$. The case of double non-degenerate extrema of the “symbol” $\omega _\Lambda (k)$, $\Lambda \in \mathbb T^2$, $k \in \mathbb T^2$, is studied. The corresponding points $\Lambda$ are situated on certain “double” curves on the torus $\mathbb T^2$. We also study the case of degenerate extrema $\omega _\Lambda (k)$ situated on caustic curves on the torus. In the first case, conditions under which adjacent levels appear are indicated and the size of a neighbourhood of “double” curves where these levels “live” is estimated. In the second case, it is shown that for a degenerate extremum of $\omega _\Lambda (k)$ “with general position” there are no adjacent levels in a neighbourhood of caustics.

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