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

TL;DR

This study employs the density-matrix renormalization group method to analyze the spin correlation functions of the antiferromagnetic spin-1/2 Heisenberg chain, confirming theoretical predictions about their asymptotic behavior and scaling properties.

Contribution

It applies DMRG to compute correlation functions for large chains and verifies the logarithmic corrections predicted by field theory.

Findings

Correlation functions obey the proposed scaling relation.

Asymptotic correlation functions show logarithmic corrections.

Magnetic structure factor $S(q=)$ exhibits predicted behavior.

Abstract

We use the density-matrix renormalization group technique developed by White \cite{white} to calculate the spin correlation functions $<{S}_{n+l}^z{S}_n^z>=(-1)^l \omega(l,N)$ for isotropic Heisenberg rings up to $N=70$ sites. The correlation functions for large $l$ and $N$ are found to obey the scaling relation $\omega(l,N)=\omega(l,\infty)f_{XY}^{\alpha} (l/N)$ proposed by Kaplan et al. \cite{horsch} , which is used to determine $\omega(l,\infty)$. The asymptotic correlation function $\omega(l,\infty)$ and the magnetic structure factor $S(q=\pi)$ show logarithmic corrections consistent with $\omega(l,\infty)\sim a\sqrt{\ln{cl}}/l$, 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.