BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Aspects of universality in the XXZ spin 1/2-chain Kozlowski, Karol


The \(L\)-site XXZ spin-1/2 chain is an exactly solvable quantum model of one dimensional condensed matter physics. As such is provides an excellent playground for testing the various manifestations of universality in one dimensional quantum models. For such model, universality manifests itself in the \(L \rightarrow + \infty \) limit and concerns various observables such as the structure of the low-lying spectrum of the model or the large-distance asymptotic behavior of its correlation functions. The XXZ chain is solvable within the Bethe Ansatz approach; its eigenvectors are parametrized by solutions to the Bethe equations, a set of high degree algebraic equations in a number of variables blowing up with \(L\). The whole description of the thermodynamic limit \(L \rightarrow +\infty\) of the model's observables is based on a conjecture due to Hulten in 1938 which stipulates that the roots of the equation associated with the model's ground are real and form a dense distribution in the \(L \rightarrow +\infty\) limit. Generalizations of Hulten's conjecture exist for excited states as well. Variants of Hulten's conjecture adjoined to other techniques allow one to establish various universality results for the chain. In 2009 Dorlas and Samsonov managed to prove the conjecture for some values of the model's coupling constants where it is possible to build the proof on certain convexity arguments. In this talk, I will present the main ideas of the method that I developed so as to prove condensation properties of Bethe roots corresponding to certain classes of solutions to the Bethe equations. The method works independently of the value taken by the coupling constants and appears to be generalisable to many other Bethe Ansatz solvable models. However, first, I shall discuss in more details the nature and history of the problem as well as its connections to universality results. I will also review how the proof of Hulten's conjecture completes the proof of certain multiple integral representations for the correlations functions. These multiple integrals bare certain structural similarities with those of \(\beta\)-ensembles.

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