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 Zap. Nauchn. Sem. POMI, 2015, Volume 437, Pages 15–34 (Mi znsl6171)

Combinatorial aspects of correlation functions of the $XXZ$ Heisenberg chain in limiting cases

N. M. Bogoliubovab, C. Malyshevb

a ITMO University, 49 Kronverksky, 197101 St. Petersburg, Russia
b St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191023 St. Petersburg, Russia

Abstract: We discuss the connection between quantum integrable models and some aspects of enumerative combinatorics and the theory of partitions. As a basic example, we consider the spin $XXZ$ Heisenberg chain in the limiting cases of zero and infinite anisotropy. The representation of the Bethe wave functions via Schur functions allows us to apply the theory of symmetric functions to calculating the thermal correlation functions as well as the form factors in the determinantal form. We provide a combinatorial interpretation of the correlation functions in terms of nests of self-avoiding lattice paths. The suggested interpretation is in turn related to the enumeration of boxed plane partitions. The asymptotic behavior of the thermal correlation functions is studied in the limit of small temperature provided that the characteristic parameters of the system are large enough. The leading asymptotics of the correlators are found to be proportional to the squared numbers of boxed plane partitions.

Key words and phrases: $XXZ$ Heisenberg chain, correlation functions, symmetric functions, plane partitions, $q$-binomial determinant.

 Funding Agency Grant Number Russian Science Foundation 14-11-00598 Supported in part by the RSF grant 14-11-00598.

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English version:
Journal of Mathematical Sciences (New York), 2016, 216:1, 8–22

Bibliographic databases:

UDC: 517.958+530.145.86+519.116
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Citation: N. M. Bogoliubov, C. Malyshev, “Combinatorial aspects of correlation functions of the $XXZ$ Heisenberg chain in limiting cases”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXVI. Representation theory, dynamical systems, combinatorial methods, Zap. Nauchn. Sem. POMI, 437, POMI, St. Petersburg, 2015, 15–34; J. Math. Sci. (N. Y.), 216:1 (2016), 8–22

Citation in format AMSBIB
\Bibitem{BogMal15} \by N.~M.~Bogoliubov, C.~Malyshev \paper Combinatorial aspects of correlation functions of the $XXZ$ Heisenberg chain in limiting cases \inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~XXVI. Representation theory, dynamical systems, combinatorial methods \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 437 \pages 15--34 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6171} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3499906} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 216 \issue 1 \pages 8--22 \crossref{https://doi.org/10.1007/s10958-016-2884-0} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84969902733}