Управляющий $T$-оператор для вершинных моделей с тригонометрическими $R$-матрицами как классическая $\tau$-функция

Zabrodin, A.

doi:10.4213/tmf8373

Cited by 4 publications

(1 citation statement)

References 33 publications

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“…Различные аспекты этого замечательного соответствия обсуждаются также в статьях [2]- [7]. В работах [8], [9] оно было обобщено на модели типа XXZ, ассоциированные с квантовыми аффинными алгебрами U q ( sl N ). В настоящей статье мы иллюстрируем это соответствие на простейшем примере неоднородной асимметричной шестивершинной модели, параметризованной тригонометрическими (гиперболическими) функциями, которая оказывается связанной с тригонометрической (гиперболической) системой частиц Рейсенарса-Шнайдера (РШ).…”

Section: Introductionunclassified

Asymmetric six-vertex model and the classical Ruijsenaars - Schneider system of particles

Zabrodin¹,

Zotov²,

Ляшик³

et al. 2017

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Section: Introductionunclassified

Asymmetric six-vertex model and the classical Ruijsenaars - Schneider system of particles

Zabrodin¹,

Zotov²,

Ляшик³

et al. 2017

TMF

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Spectrum of quantum transfer matrices via classical many-body systems

Gorsky

Zabrodin

Zotov

2014

J. High Energ. Phys.

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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 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 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.

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Trigonometric version of quantum–classical duality in integrable systems

et al. 2016

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We extend the quantum-classical duality to the trigonometric (hyperbolic) case. The duality establishes an explicit relationship between the classical N -body trigonometric Ruijsenaars-Schneider model and the inhomogeneous twisted XXZ spin chain on N sites. Similarly to the rational version, the spin chain data fixes a certain Lagrangian submanifold in the phase space of the classical integrable system. The inhomogeneity parameters are equal to the coordinates of particles while the velocities of classical particles are proportional to the eigenvalues of the spin chain Hamiltonians (residues of the properly normalized transfer matrix). In the rational version of the duality, the action variables of the Ruijsenaars-Schneider model are equal to the twist parameters with some multiplicities defined by quantum (occupation) numbers. In contrast to the rational version, in the trigonometric case there is a splitting of the spectrum of action variables (eigenvalues of the classical Lax matrix). The limit corresponding to the classical Calogero-Sutherland system and quantum trigonometric Gaudin model is also described as well as the XX limit to free fermions.

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Управляющий $T$-оператор для вершинных моделей с тригонометрическими $R$-матрицами как классическая $\tau$-функция

Cited by 4 publications

References 33 publications

Asymmetric six-vertex model and the classical Ruijsenaars - Schneider system of particles

Asymmetric six-vertex model and the classical Ruijsenaars - Schneider system of particles

Spectrum of quantum transfer matrices via classical many-body systems

Trigonometric version of quantum–classical duality in integrable systems

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