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arXiv:2005.06030 · Journal of High Energy Physics

Analytic continuation of Bethe energies and application to the thermodynamic limit of the SL(2, ℂ) non-compact spin chains

Etienne Granet, Jesper Lykke Jacobsen, Hubert Saleur

Published Aug 1, 2020• 3 citations
arXivHTMLPDF

Abstract

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 , $$ \overline{s} $$ 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.

DOI
10.1007/jhep08(2020)069

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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.
Abstract We compute the exact overlaps between integrable boundary states and on-shell Bethe states for integrable spin chains. Our method is based on the coordinate Bethe Ansatz and does not rely on the “rotation trick” of the corresponding lattice model. It leads to a rigorous proof of the factorized overlap formulae in a number of cases, some of which were hitherto inaccessible to earlier methods. As concrete examples, we consider the compact XXX and XXZ Heisenberg spin chains, and the non-compact SL(2, ℝ) spin chain.
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We suggest an approach to the problem of finding integral equations for the excited states of an integrable model, starting from the Thermodynamic Bethe Ansatz equations for its ground state. The idea relies on analytic continuation through complex values of the coupling constant, and an analysis of the monodromies that the equations and their solutions undergo. For the scaling Lee-Yang model, we find equations in this way for the one- and two- particle states in the spin-zero sector, and suggest various generalisations. Numerical results show excellent agreement with the truncated conformal space approach, and we also treat some of the ultraviolet and infrared asymptotics analytically.
It is known that for the Heisenberg XXZ spin - 12 chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.