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arXiv:0810.4816 · International Conference on Path Integrals

A PATH-INTEGRATION APPROACH TO THE CORRELATORS OF XY HEISENBERG MAGNET AND RANDOM WALKS

N. M. Bogoliubov, C. MALYSHEV

Published Nov 1, 2008• 2 citations
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Abstract

The path integral approach is used for the calculation of the correlation functions of the $XY$ Heisenberg chain. The obtained answers for the two-point correlators of the $XX$ magnet are of the determinantal form and are interpreted in terms of the generating functions for the random turns vicious walkers.

DOI
10.1142/9789812837271_0074

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The path integral approach is used for the calculation of the correlation functions of the XYXYXY Heisenberg chain. The obtained answers for the two-point correlators of the XXXXXX magnet are of the determinantal form and are interpreted in terms of the generating functions for the random turns vicious walkers.
The correlation functions of the XXZXXZXXZ Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers
The XXZXXZXXZ Heisenberg chain is considered for two specific limits of the anisotropy parameter: Δ→0\Delta \to 0Δ→0 and Δ→−∞\Delta \to -\inftyΔ→−∞. 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 NNN-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.
We present a review of the method we have elaborated to compute the correlation functions of the XXZXXZXXZ spin-1/2 Heisenberg chain.This method is based on the resolution of the quantum inverse scattering problem in the algebraic Bethe Ansatz bamework, and leads to a mul- tiple integral representation of the dynamical correlation functions.We describe in particular some recent advances concerning the two-point functions: in the finite chain, they can be expressed in terms of a single multiple integral.Such a formula provides a direct analytic connection between the previously obtained multiple integral representations and the form factor expansions for the correlation functions.
The XX0 Heisenberg model on a cyclic chain is considered. The representation of the Bethe wave functions via the Schur functions allows to apply the well-developed theory of the symmetric functions to the calculation of the thermal correlation functions. The determinantal expressions of the form-factors and of the thermal correlation functions are obtained. The q-binomial determinants enable the connection of the form-factors with the generating functions both of boxed plane partitions and of self-avoiding lattice paths. The asymptotical behavior of the thermal correlation functions is studied in the limit of low temperature provided that the characteristic parameters of the system are large enough.
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.