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H.E.Boos, V.E.Korepin, Y.Nishiyama, M.Shiroishi
Quantum Correlations and Number Theory
ABSTRACT. We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the
Hamiltonian was found by Hans Bethe in 1931. We study the probability
of 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 the 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 ones.
Furthermore we study asymptotic behavior of P(n) at finite temperature
by Quantum Monte-Carlo simulation. It also agrees with our previous