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Evaluation of Integrals Representing Correlations in the XXX Heisenberg Spin Chain

Herman E. Boos, V. E. Korepin

Published Jan 1, 2002• 25 citations

Abstract

We study the XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in the thermodynamical limit. We prove that for short strings the probability can be expressed in terms of the Riemann zeta function with odd arguments. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1007/978-1-4612-0087-1_4

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We study the XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in the thermodynamical limit. We prove that for short strings the probability can be expressed in terms of the Riemann zeta function with odd arguments. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the spin-1/2 Heisenberg XXX antiferromagnet for which the spectrum of the Hamiltonian was found by Bethe in 1931. We study the probability of the formation of ferromagnetic string in the antiferromagnetic ground state, which we call emptiness formation probability P ( n ). This is the most fundamental correlation function. We prove that, for short strings, it can be expressed in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and rational coefficients. This adds yet another link between statistical mechanics and number theory. We have obtained an analytical formula for P (5) for the first time. We have also calculated P ( n ) numerically by the density matrix renormalization group. The results agree quite well with the analytical results. Furthermore, we study the asymptotic behaviour of P ( n ) at finite temperature by quantum Monte Carlo simulation. This also agrees with our previous analytical results.

Abstract We study the correlation functions of su (2) invariant spin- s chains in the thermodynamic limit. We derive nonlinear integral equations for an auxiliary correlation function ω for any spin s and finite temperature T . For the spin-3/2 chain for arbitrary temperature and zero magnetic field we obtain algebraic expressions for the reduced density matrix of two-sites. In the zero temperature limit, the density matrix elements are evaluated analytically and appear to be given in terms of Riemann’s zeta function values of even and odd arguments.

Abstract We study the correlation functions of , n > 2, invariant spin chains in the thermodynamic limit. We formulate a consistent framework for the computation of short-range correlation functions via functional equations which hold even at finite temperature. We give the explicit solution for two- and three-site correlations for the case at zero temperature. The correlators do not seem to be of factorizable form. From the two-site result we see that the correlation functions are given in terms of Hurwitz’s zeta function, which differs from the case where the correlations are expressed in terms of Riemann’s zeta function of odd arguments.

The exact expression for the density matrix of the kink ground state of the ferromagnetic XXZ chain is obtained. Utilizing this, we exactly calculate various correlation functions such as the longitudinal and transverse spin-spin correlation functions, and the ferromagnetic and antiferromagnetic string formation probabilities. The asymptotic behaviors of these correlation functions are also analyzed. As a consequence, we find that the spin-spin correlation functions decay exponentially for large distances, while the string formation probabilities exhibit Gaussian decay for large strings. We also evaluate the entanglement entropy, which shows interesting behaviors due to the lack of the translational invariance of the state.