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Journal of Physics Conference Series

Jordan blocks and the Bethe ansatz: The eclectic spin chain as a limit

Juan Miguel Nieto García

Published Dec 1, 2023• 0 citations
Publisher

Abstract

Abstract We present a procedure to extract generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation and computing the non-diagonalisable limit of its eigenvectors. As an example, we show how to obtain a subset of the spectrum of the eclectic spin chain from the spectrum of a twisted 𝔰𝔲(3) spin chain.

DOI
10.1088/1742-6596/2667/1/012059

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Abstract We present a procedure to extract generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation and computing the non-diagonalisable limit of its eigenvectors. As an example, we show how to obtain a subset of the spectrum of the eclectic spin chain from the spectrum of a twisted 𝔰𝔲(3) spin chain.
We present a procedure to extract the generalised eigenvectors of a non-diagonalisable matrix by considering a diagonalisable perturbation of it and computing the non-diagonalisable limit of its eigenvectors. As an example of this process, we compute a subset of the spectrum of the eclectic spin chain by means of the Nested Coordinate Bethe Ansatz. This allows us to show that the Bethe Ansatz of the finitely twisted spin chain contains enough information to reconstruct the generalised eigenvectors of the eclectic spin chain.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
A bstract We consider the problem of analytically continuing energies computed with the Bethe ansatz, as posed by the study of non-compact integrable spin chains. By introducing an imaginary extensive twist in the Bethe equations, we show that one can expand the analytic continuation of energies in the scaling limit around another ‘pseudo-vacuum’ sitting at a negative number of Bethe roots, in the same way as around the usual pseudo-vacuum. We show that this method can be used to compute the energy levels of some states of the SL(2, ℂ) integrable spin chain in the infinite-volume limit, and as a proof of principle recover the ground-state value previously obtained in [1] (for the case of spins s = 0 , s‾ \overline{s} s s ¯ = − 1) by extrapolating results in small sizes. These results represent, as far as we know, the first (partial) description of the spectrum of SL(2, ℂ) non-compact spin chains in the thermodynamic limit.