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

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published Nov 20, 2012• 1 citation

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.

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

N. M. Bogoliubov, Cyril Leonidovich Malyshev

Published Nov 20, 2012• 1 citation

OpenAlex

Abstract

Related papers

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

N. M. Bogoliubov, C. MALYSHEV

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.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.

arXiv:0810.4816International Conference on Path IntegralsNov 1, 20082 citations

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

N. M. Bogoliubov, C. Malyshev

The $XXZ$ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $\Delta \to 0$ 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.The $XXZ$ Heisenberg chain is considered for two specific limits of the anisotropy parameter: $\Delta \to 0$ 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.

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arXiv:0903.3227Theoretical and Mathematical PhysicsMay 1, 200913 citations

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.

The correlation functions of the

XXZ

Heisenberg chain in the case of zero or infinite anisotropy, and random walks of vicious walkers

The

XXZ

Heisenberg chain is considered for two specific limits of the anisotropy parameter:

\Delta \to 0

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.

We present a review of the method we have elaborated to compute the correlation functions of the

XXZ

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.