On quantum separation of variables
| Autor: | J. M. Maillet, Giuliano Niccoli |
|---|---|
| Přispěvatelé: | Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
High Energy Physics - Theory
Pure mathematics fusion model: lattice Integrable system [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] Separation of variables FOS: Physical sciences integrability 01 natural sciences symbols.namesake algebra: Yang-Baxter Lattice (order) 0103 physical sciences charge: conservation law [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI] transfer matrix Quantum inverse scattering method 010306 general physics Quantum Mathematical Physics Mathematics Nonlinear Sciences - Exactly Solvable and Integrable Systems [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] 010308 nuclear & particles physics Hilbert space deformation Statistical and Nonlinear Physics Mathematical Physics (math-ph) algebra: Yangian Transfer matrix R-matrix inverse scattering method High Energy Physics - Theory (hep-th) flow symbols Exactly Solvable and Integrable Systems (nlin.SI) Trigonometry spin: chain |
| Zdroj: | J.Math.Phys. J.Math.Phys., 2018, 59 (9), pp.091417. ⟨10.1063/1.5050989⟩ Journal of Mathematical Physics Journal of Mathematical Physics, American Institute of Physics (AIP), 2018, 59 (9), pp.091417. ⟨10.1063/1.5050989⟩ |
| ISSN: | 0022-2488 |
| Popis: | We present a new approach to construct the separate variables basis leading to the full characterization of the transfer matrix spectrum of quantum integrable lattice models. The basis is generated by the repeated action of the transfer matrix itself on a generically chosen state of the Hilbert space. The fusion relations for the transfer matrix, stemming from the Yang-Baxter algebra properties, provide the necessary closure relations to define the action of the transfer matrix on such a basis in terms of elementary local shifts, leading to a separate transfer matrix spectral problem. Hence our scheme extends to the quantum case a key feature of Liouville-Arnold classical integrability framework where the complete set of conserved charges defines both the level manifold and the flows on it leading to the construction of action-angle variables. We work in the framework of the quantum inverse scattering method. As a first example of our approach, we give the construction of such a basis for models associated to Y(gln) and argue how it extends to their trigonometric and elliptic versions. Then we show how our general scheme applies concretely to fundamental models associated to the Y(gl2) and Y(gl3) R-matrices leading to the full characterization of their spectrum. For Y(gl2) and its trigonometric deformation a particular case of our method reproduces Sklyanin's construction of separate variables. For Y(gl3) it gives new results, in particular through the proper identification of the shifts acting on the separate basis. We stress that our method also leads to the full characterization of the spectrum of other known quantum integrable lattice models, including in particular trigonometric and elliptic spin chains, open chains with general integrable boundaries, and further higher rank cases that we will describe in forthcoming publications. 61 pages, v2: small typos corrected and some references added |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|