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arXiv:2112.04733 · Symmetry Integrability and Geometry Methods and Applications

How to Draw a Correlation Function

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published Dec 9, 2021• 0 citations
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Abstract

We discuss connection between the XX0 Heisenberg spin chain and some aspects of enumerative combinatorics. The representation of the Bethe wave functions via the Schur functions allows to apply the theory of symmetric functions to the calculation of the correlation functions. We provide a combinatorial derivation of the dynamical auto-correlation functions and visualise them in terms of nests of self-avoiding lattice paths. Asymptotics of the auto-correlation functions are obtained in the double scaling limit provided that the evolution parameter is large.

DOI
10.3842/sigma.2021.106

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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.
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