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JHEP, 2014, Issue 1, 070, 28 pp.
(Mi jhep10)
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Spectrum of quantum transfer matrices via classical many-body systems
A. Gorskyab, A. Zabrodinabcd, A. Zotovabe a ITEP,
Bolshaya Cheremushkinskaya str. 25, 117218, Moscow, Russia
b MIPT,
Inststitutskii per. 9, 141700, Dolgoprudny, Moscow region, Russia
c National Research University Higher School of Economics,
Myasnitskaya str. 20, 101000, Moscow, Russia
d Institute of Biochemical Physics,
Kosygina str. 4, 119991, Moscow, Russia
e Steklov Mathematical Institute, RAS,
Gubkina str. 8, 119991, Moscow, Russia
Abstract:
In this paper we clarify the relationship between inhomogeneous quantum
spin chains and classical integrable many-body systems. It provides an alternative (to the
nested Bethe ansatz) method for computation of spectra of the spin chains. Namely, the
spectrum of the quantum transfer matrix for the inhomogeneous $\mathfrak{gl}_n$-invariant XXX spin
chain on $N$ sites with twisted boundary conditions can be found in terms of velocities of
particles in the rational $N$-body Ruijsenaars–Schneider model. The possible values of the
velocities are to be found from intersection points of two Lagrangian submanifolds in the
phase space of the classical model. One of them is the Lagrangian hyperplane corresponding
to fixed coordinates of all $N$ particles and the other one is an $N$-dimensional Lagrangian
submanifold obtained by fixing levels of $N$ classical Hamiltonians in involution. The latter
are determined by eigenvalues of the twist matrix. To support this picture, we give a direct
proof that the eigenvalues of the Lax matrix for the classical Ruijsenaars–Schneider model,
where velocities of particles are substituted by eigenvalues of the spin chain Hamiltonians,
calculated through the Bethe equations, coincide with eigenvalues of the twist matrix, with
certain multiplicities. We also prove a similar statement for the $\mathfrak{gl}_n$ Gaudin model with
$N$ marked points (on the quantum side) and the Calogero–Moser system with $N$ particles
(on the classical side). The realization of the results obtained in terms of branes and
supersymmetric gauge theories is also discussed.
Funding Agency |
Grant Number |
Russian Foundation for Basic Research  |
12-02-00284 12-02-91052 11-02-01220 12-02-91052 12-02-92108 14-01-00860 |
Ministry of Education and Science of the Russian Federation  |
8207 NSh-3349.2012.2 NSh-4724.2014.2 |
Dynasty Foundation  |
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The work of A. G. was supported in part by grants RFBR-12-02-00284 and PICS-12-02-91052. A. G. thanks the organizers of Simons Summer School at Simons Center for Geometry and Physics where the part of this work has been done for the hospitality and support. The work of A.Zabrodin was supported in part by RFBR grant 11-02-01220, by joint RFBR grants 12-02-91052-CNRS, 12-02-92108-JSPS and by Ministry of Science and Education of Russian Federation under contract 8207 and by grant NSh-3349.2012.2 for support of leading scientific schools. The work of A.Zotov was supported in part by RFBR grants 14-01-00860, by grant NSh-4724.2014.2 for support of leading scientific schools and by the D. Zimin's fund "Dynasty". |
DOI:
https://doi.org/10.1007/JHEP01(2014)070
Bibliographic databases:
Received: 13.11.2013 Accepted:23.12.2013
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