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Algebra i Analiz, 2010, Volume 22, Issue 3, Pages 32–59 (Mi aa1185)  

This article is cited in 11 scientific papers (total in 11 papers)

Research Papers

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

N. M. Bogoliubov, K. Malyshev

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences, St. Petersburg, Russia

Abstract: The $XXZ$ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $\Delta\to0$ 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.

Keywords: $XXZ$ Heisenberg chain, Schur functions, random walks, plane partitions.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:3, 359–377

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Received: 19.02.2010

Citation: N. M. Bogoliubov, K. Malyshev, “The correlation functions of the $XXZ$ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers”, Algebra i Analiz, 22:3 (2010), 32–59; St. Petersburg Math. J., 22:3 (2011), 359–377

Citation in format AMSBIB
\by N.~M.~Bogoliubov, K.~Malyshev
\paper The correlation functions of the $XXZ$ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 3
\pages 32--59
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 3
\pages 359--377

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    This publication is cited in the following articles:
    1. N. M. Bogolyubov, K. L. Malyshev, “Ising limit of a Heisenberg $XXZ$ magnet and some temperature correlation functions”, Theoret. and Math. Phys., 169:2 (2011), 1517–1529  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    2. J. Math. Sci. (N. Y.), 200:6 (2014), 662–670  mathnet  crossref
    3. Perez-Garcia D., Tierz M., “Mapping Between the Heisenberg XX Spin Chain and Low-Energy QCD”, Phys. Rev. X, 4:2 (2014), 021050  crossref  isi  elib  scopus
    4. Bogoliubov N.M., Malyshev C., “Correlation Functions of Xxo Heisenberg Chain, Q-Binomial Determinants, and Random Walks”, Nucl. Phys. B, 879 (2014), 268–291  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. N. M. Bogolyubov, K. L. Malyshev, “Integrable models and combinatorics”, Russian Math. Surveys, 70:5 (2015), 789–856  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. J. Math. Sci. (N. Y.), 216:1 (2016), 8–22  mathnet  crossref  mathscinet
    7. Perez-Garcia D., Tierz M., “Chern–Simons theory encoded on a spin chain”, J. Stat. Mech.-Theory Exp., 2016, 013103  crossref  mathscinet  isi  scopus
    8. Saeedian M., Zahabi A., “Phase Structure of Xx0 Spin Chain and Nonintersecting Brownian Motion”, J. Stat. Mech.-Theory Exp., 2018, 013104  crossref  mathscinet  isi  scopus
    9. J. Math. Sci. (N. Y.), 242:5 (2019), 628–635  mathnet  crossref
    10. Wang N., “Young Diagrams in An N X M Box and the Kp Hierarchy”, Nucl. Phys. B, 937 (2018), 478–501  crossref  mathscinet  zmath  isi  scopus
    11. N. M. Bogoliubov, “Enumerative combinatorics of $XX0$ Heisenberg chain”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 26, Zap. nauchn. sem. POMI, 487, POMI, SPb., 2019, 53–67  mathnet
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