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 SIGMA, 2014, Volume 10, 006, 18 pp. (Mi sigma871)

The Master $T$-Operator for Inhomogeneous $XXX$ Spin Chain and mKP Hierarchy

Anton Zabrodinabcd

a Institute of Biochemical Physics, 4 Kosygina, 119334, Moscow, Russia
b ITEP, 25 B. Cheremushkinskaya, 117218, Moscow, Russia
c National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russia
d MIPT, Institutskii per. 9, 141700, Dolgoprudny, Moscow region, Russia

Abstract: Following the approach of [Alexandrov A., Kazakov V., Leurent S., Tsuboi Z., Zabrodin A., J. High Energy Phys. 2013 (2013), no. 9, 064, 65 pages, arXiv:1112.3310], we show how to construct the master $T$-operator for the quantum inhomogeneous $GL(N)$ $XXX$ spin chain with twisted boundary conditions. It satisfies the bilinear identity and Hirota equations for the classical mKP hierarchy. We also characterize the class of solutions to the mKP hierarchy that correspond to eigenvalues of the master $T$-operator and study dynamics of their zeros as functions of the spectral parameter. This implies a remarkable connection between the quantum spin chain and the classical Ruijsenaars–Schneider system of particles.

Keywords: quantum integrable spin chains; classical many-body systems; quantum-classical correspondence; master $T$-operator; tau-function.

DOI: https://doi.org/10.3842/SIGMA.2014.006

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ArXiv: 1310.6988
MSC: 37K10; 81Q80; 05E05
Received: October 18, 2013; in final form January 8, 2014; Published online January 11, 2014
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Citation: Anton Zabrodin, “The Master $T$-Operator for Inhomogeneous $XXX$ Spin Chain and mKP Hierarchy”, SIGMA, 10 (2014), 006, 18 pp.

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Levin A. Olshanetsky M. Zotov A., “Relativistic Classical Integrable Tops and Quantum R-Matrices”, J. High Energy Phys., 2014, no. 7, 012
2. Levin A. Olshanetsky M. Zotov A., “Planck Constant as Spectral Parameter in Integrable Systems and Kzb Equations”, J. High Energy Phys., 2014, no. 10, 109
3. Galleas W., “Off-Shell Scalar Products For the Xxz Spin Chain With Open Boundaries”, Nucl. Phys. B, 893 (2015), 346–375
4. A Zabrodin, “Quantum Spin Chains and Integrable Many-Body Systems of Classical Mechanics”, Springer Proceedings in Physics, 163 (2015), 29–48
5. Tsuboi Z. Zabrodin A. Zotov A., “Supersymmetric Quantum Spin Chains and Classical Integrable Systems”, J. High Energy Phys., 2015, no. 5, 086
6. N. Rozhkovskaya, “Action of Clifford algebra on the space of sequences of transfer operators”, Algebr. Represent. Theory, 21:5 (2018), 1165–1176
7. A. Zabrodin, “Quantum spin chains and classical integrable systems”, Stochastic Processes and Random Matrices, Lecture Notes of the Les Houches Summer School, 104, 2018, 578-613
8. V. Prokofev, A. Zabrodin, “Toda lattice hierarchy and trigonometric ruijsenaars?schneider hierarchy”, J. Phys. A-Math. Theor., 52:49 (2019), 495202
9. V. V. Prokofev, A. V. Zabrodin, “Elliptic solutions of the Toda lattice hierarchy and the elliptic Ruijsenaars–Schneider model”, Theoret. and Math. Phys., 208:2 (2021), 1093–1115
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