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

The correlation functions of the $ XXZ$ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers


Authors: N. M. Bogoliubov and C. Malyshev
Translated by: the authors
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 359-377
MSC (2010): Primary 81U40
Published electronically: March 17, 2011
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Abstract: The $ XXZ$ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $ \Delta\to 0$ and $ \Delta\to -\infty$. The corresponding wave functions are expressed in terms of symmetric Schur functions. Certain expectation values and thermal correlation functions of the ferromagnetic string operators are calculated over the basis of $ N$-particle Bethe states. The thermal correlator of the ferromagnetic string is expressed through the generating function of the lattice paths of random walks of vicious walkers. A relationship between the expectation values obtained and the generating functions of strict plane partitions in a box is discussed. An asymptotic estimate of the thermal correlator of the ferromagnetic string is obtained in the zero temperature limit. It is shown that its amplitude is related to the number of plane partitions.


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Additional Information

N. M. Bogoliubov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
Email: bogoliub@pdmi.ras.ru

C. Malyshev
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
Email: malyshev@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01146-X
PII: S 1061-0022(2011)01146-X
Keywords: XXZ Heisenberg chain, Schur functions, random walks, plane partitions
Received by editor(s): February 19, 2010
Published electronically: March 17, 2011
Additional Notes: Extended talk at the Conference “Conformal Field Theory, Integrable Systems, and Liouville Gravity” (Chernogolovka, June 30–July 2, 2009).
Partially supported by RFBR (No. 07-01-00358) and by the Russian Academy of Sciences program “Mathematical Methods in Nonlinear Dynamics”
Dedicated: Dedicated to L. D. Faddeev
Article copyright: © Copyright 2011 American Mathematical Society