Inozemtsev's Elliptic Spin Chains Asymptotic Bethe Ansatz and Thermodynamics
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Inozemtsev's elliptic spin chain and its infinite limit are interesting models from many perspectives: both of these models are most likely integrable, but their precise structure is not known yet. They form interpolating models between two prime examples of two very different classes of spin chains, the Heisenberg XXX spin chain and the Haldane-Shastry spin chain. Moreover, the infinite spin chain can be used to study the spectrum of the dilatation operator in N=4 super Yang-Mills theory. Finally, there seems to be a strong relationship between the solvability of these spin chains and their Calogero-Sutherland-Moser counterparts. In this thesis, we derive the eigenfunctions of Inozemtsev's infinite spin chain and use these eigenfunctions to study the thermodynamic behaviour of these models by employing the Asymptotic Bethe Ansatz. Using an approach first proposed by Hulthén, we derive an expression for the antiferromagnetic ground state and we follow a method by Yang and Yang to derive integral equations that govern the thermodynamics at arbitrary density. Finally, we classify all the asymptotic (bound-state) solutions of the Bethe equations of Inozemtsev's elliptic spin chain. This leads to interesting new phenomena and a reason to revisit the derivation of asymptotic bound-state solutions of other models. After identifying the spectrum within the set of solutions of the Bethe equations, we can plot the spectrum of bound states.