Abstract:
The XXZ Heisenberg chain is considered for two specific limits of the anisotropy parameter: Δ→0 and Δ→−∞. 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.
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
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\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
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\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 3
\pages 359--377
\crossref{https://doi.org/10.1090/S1061-0022-2011-01146-X}
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Linking options:
https://www.mathnet.ru/eng/aa1185
https://www.mathnet.ru/eng/aa/v22/i3/p32
This publication is cited in the following 16 articles:
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David Pérez-García, Leonardo Santilli, Miguel Tierz, “Dynamical quantum phase transitions from random matrix theory”, Quantum, 8 (2024), 1271
C Malyshev, N M Bogoliubov, “Spin correlation functions, Ramus-like identities, and enumeration of constrained lattice walks and plane partitions”, J. Phys. A: Math. Theor., 55:22 (2022), 225002
N. M. Bogoliubov, “Enumerative Combinatorics of XX0 Heisenberg Chain”, J Math Sci, 257:4 (2021), 459
Santilli L., Tierz M., “Phase Transition in Complex-Time Loschmidt Echo of Short and Long Range Spin Chain”, J. Stat. Mech.-Theory Exp., 2020:6 (2020), 063102
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
Saeedian M., Zahabi A., “Phase Structure of Xx0 Spin Chain and Nonintersecting Brownian Motion”, J. Stat. Mech.-Theory Exp., 2018, 013104
J. Math. Sci. (N. Y.), 242:5 (2019), 628–635
Wang N., “Young Diagrams in An N X M Box and the Kp Hierarchy”, Nucl. Phys. B, 937 (2018), 478–501
Perez-Garcia D., Tierz M., “Chern–Simons theory encoded on a spin chain”, J. Stat. Mech.-Theory Exp., 2016, 013103
N. M. Bogolyubov, K. L. Malyshev, “Integrable models and combinatorics”, Russian Math. Surveys, 70:5 (2015), 789–856
J. Math. Sci. (N. Y.), 216:1 (2016), 8–22
J. Math. Sci. (N. Y.), 200:6 (2014), 662–670
Perez-Garcia D., Tierz M., “Mapping Between the Heisenberg XX Spin Chain and Low-Energy QCD”, Phys. Rev. X, 4:2 (2014), 021050
Bogoliubov N.M., Malyshev C., “Correlation Functions of Xxo Heisenberg Chain, Q-Binomial Determinants, and Random Walks”, Nucl. Phys. B, 879 (2014), 268–291
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
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