Numerical renormalization group study of the correlation functions of the antiferromagnetic spin-$\frac{1}{2}$ Heisenberg chain

Abstract

We use the density-matrix renormalization group technique developed by White \cite{white} to calculate the spin correlation functions $<{S}_{n+l}^z{S}_n^z>=(-1)^l ω(l,N)$ for isotropic Heisenberg rings up to $N=70$ sites. The correlation functions for large $l$ and $N$ are found to obey the scaling relation $ω(l,N)=ω(l,\infty)f_{XY}^α (l/N)$ proposed by Kaplan et al. \cite{horsch} , which is used to determine $ω(l,\infty)$. The asymptotic correlation function $ω(l,\infty)$ and the magnetic structure factor $S(q=π)$ show logarithmic corrections consistent with $ω(l,\infty)\sim a\sqrt{\ln{cl}}/l$, where $c$ is related to the cut-off dependent coupling constant $g_{eff}(l_0)=1/\ln(cl_0)$, as predicted by field theoretical treatments.