arXiv:hep-th/0104008 · Journal of Physics A Mathematical and General

Quantum spin chains and Riemann zeta function with odd arguments

Herman E. Boos, V. E. Korepin

Published Jun 22, 2001• 109 citations

Abstract

The Riemann zeta function is an important object of number theory. We argue that it is related to the Heisenberg spin-1/2 anti-ferromagnet. In the XXX spin chain we study the probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in the thermodynamics limit. We prove that for short strings the probability can be expressed in terms of the Riemann zeta function with odd arguments.

DOI: 10.1088/0305-4470/34/26/301

Related papers

Evaluation of Integrals Representing Correlations in XXX Heisenberg Spin Chain

Herman E. Boos, V. E. Korepin

We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.

arXiv:hep-th/0105144May 15, 20010 citations

Evaluation of Integrals Representing Correlations in the XXX Heisenberg Spin Chain

Herman E. Boos, V. E. Korepin

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.

Birkhäuser Boston eBooksJan 1, 200225 citations

Quantum correlations and number theory

Herman E. Boos, V. E. Korepin, Yoshihiro Nishiyama +1 more

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.

arXiv:cond-mat/0202346Journal of Physics A Mathematical and GeneralMay 10, 200247 citations

Correlation functions of the integrable S U ( n ) spin chain

G A P Ribeiro, Andreas Klümper

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.

arXiv:1804.10169Journal of Statistical Mechanics Theory and ExperimentJan 1, 201920 citations

Correlation functions of the integrable spin- s chain ∗

G A P Ribeiro, Andreas Klümper

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.

arXiv:1603.00504Journal of Physics A Mathematical and TheoreticalMay 12, 201611 citations

Correlation functions of the integrable isotropic spin-1 chain: algebraic expressions for arbitrary temperature

Andreas Klümper, Dominic Nawrath, Junji Suzuki

We derive algebraic formulae for the density matrices of finite segments of the integrable s u (2) isotropic spin-1 chain in the thermodynamic limit. We give explicit results for the 2- and 3-site cases for arbitrary temperature T and zero field. In the zero-temperature limit the correlation functions are given in elementary form in terms of Riemann's zeta function at even integer arguments.

arXiv:1304.5512Journal of Statistical Mechanics Theory and ExperimentAug 27, 201317 citations