Quantum correlations and number theory

Abstract

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.