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Differential equation for a correlation function of the spin-\(\frac12\) Heisenberg chain. (English) Zbl 1049.82508

Summary: We consider the probability to find a string of \(x\) adjacent parallel spins in the antiferromagnetic ground state of the model (in a magnetic field). We derive a system of integro-difference equations which define this probability. This system is completely integrable, it has Lax representation and a corresponding Riemann-Hilbert problem. The quantum correlation function is a \(\tau\)-function of this system.

MSC:

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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[1] Bethe, H., Z. Phys., 71, 205 (1931)
[2] Griffiths, R. B., Phys. Rev., 133, A768 (1964)
[3] Korepin, V. E.; Izergin, A. G.; Bogoliubov, N. M., Quantum inverse scattering method, correlation functions and algebraic Bethe ansatz (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0787.47006
[4] Faddeev, L. D.; Takhtajan, L. A., Hamiltonian methods in the theory of solitons (1987), Springer: Springer Berlin · Zbl 1327.39013
[5] Ablowitz, M. J.; Segur, H., Solitons and the inverse scattering method (1981), SIAM: SIAM Philadelphia · Zbl 0299.35076
[6] Deift, P. A.; Its, A. R.; Zhou, X., Long-time asymptotics for integrable nonlinear wave equations, (Fokas, A. S.; Zakharov, V. E., Important developments in soliton theory (1993), Springer: Springer Berlin) · Zbl 0926.35132
[7] Jimbo, M.; Miwa, T.; Mori, Y.; Sato, M., Physica D, 1, 80 (1980)
[8] Korepin, V. E.; Izergin, A. G.; Essler, F. H.L.; Uglov, D. B., Phys. Lett. A, 190, 182 (1994)
[9] Barouch, E.; McCoy, B. M.; Wu, T. T., Phys. Rev. Lett., 31, 1409 (1973)
[10] Tracy, C. A.; McCoy, B. M., Phys. Rev. Lett., 31, 1500 (1973)
[11] Its, A. R.; Izergin, A. G.; Korepin, V. E.; Slavnov, N. A., Int. J. Mod. Phys. B, 4, 1003 (1990)
[12] Jimbo, M.; Miki, K.; Miwa, T.; Nakayashiki, A., Phys. Lett. A, 168, 256 (1992)
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