Finite-size scaling of string order parameters characterizing the Haldane phase

We have developed a numerical procedure to clarify the critical behavior near a quantum phase transition by analyzing a multi- point correlation function characterizing the ground state. The procedure focuses the gradient of the inversed-system-size dependence of the correlation function on a logarithmic plot. It requires only the correlation functions of several finite sizes under the same condition as a candidate for the long-range order. We apply the analysis to the string order parameter of the $S=1$ $XXZ$ chain with uniaxial single-ion anisotropy obtained by the density matrix renormalization group method. The present analysis gives precise estimates of transition points and critical exponents, $\nu$ and $\eta$, in Ising transitions, Gaussian transitions, and Berezinskii- Kosterlitz- Thouless transitions are consistent with results obtained from the analysis of the energy-level structure. This method will contributes much for a direct observation of quantum phase transitions.