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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

Non-Hermitian spin chains with inhomogeneous coupling


Author: A. G. Bytsko
Translated by: the author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 393-410
MSC (2010): Primary 81T10
Published electronically: March 17, 2011
MathSciNet review: 2729941
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Abstract: An open $ U_q(sl_2)$-invariant spin chain of spin $ S$ and length $ N$ with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter $ \gamma$ are determined for which the spectrum of the model is real. For a certain range of $ \gamma$, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous $ U_q(sl_2)$-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.


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Additional Information

A. G. Bytsko
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
Email: bytsko@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01148-3
PII: S 1061-0022(2011)01148-3
Keywords: Quasi-Hermitian Hamiltonians, quantum algebras, spin chains
Received by editor(s): December 18, 2009
Published electronically: March 17, 2011
Dedicated: To Ludwig Dmitrievich Faddeev on his 75th birthday
Article copyright: © Copyright 2011 American Mathematical Society