We study a correlation function for the one-dimensional isotropic XY model (XX0 model), which is called the Emptiness Formation Probability (EFP). It is the probability of the formation of a ferromagnetic string in the anti-ferromagnetic ground state. Using the expression of the EFP as a Toeplitz determinant, we discuss its asymptotic behaviors. We also compare the analytical results with numerical calculations as the density-matrix renormalization group and the quantum Monte-Carlo method.

We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the Hamiltonian was found by Hans Bethe in 1931, [1]. 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 analytical results.

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Recently a new integral equation describing the thermodynamics of the 1D Heisenberg model was discovered by Takahashi. Using the integral equation we have succeeded in obtaining the high-temperature expansion of the specific heat and the magnetic susceptibility up to O[(J/T)(100)]. This is much higher than those obtained so far by the standard methods such as the linked-cluster algorithm. Our results will be useful to examine various approximation methods to extrapolate the high-temperature expansion to the low temperature region.