Exact asymptotic correlation functions of bilinear spin operators of the Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chain

TL;DR

This paper derives exact asymptotic expressions for correlation functions of bilinear spin operators in the Heisenberg antiferromagnetic chain, identifying logarithmic corrections and confirming numerical predictions.

Contribution

It provides the first exact asymptotic formulas including logarithmic corrections for these correlation functions in the spin-1/2 chain.

Findings

Logarithmic contributions to correlation functions are identified and quantified.

Numerical confirmation of multiplicative logarithmic corrections in staggered parts.

Estimated numerical prefactor for the logarithmic correction as approximately 0.067.

Abstract

Exact asymptotic expressions of the uniform parts of the two-point correlation functions of bilinear spin operators in the Heisenberg antiferromagnetic spin-$\frac{1}{2}$ chain are obtained. Apart from the algebraic decay, the logarithmic contribution is identified, and the numerical prefactor is determined. We also confirm numerically the multiplicative logarithmic correction of the staggered part of the bilinear spin operators $\langle\langle S^{a}_0S^{a}_{1}S^{b}_{r}S^{b}_{r+1} \rangle\rangle=(-1)^rd/(r \ln^{\frac{3}{2}}r) +(3\delta_{a,b}-1) \ln^2r /(12 \pi^4 r^4)$, and estimate the numerical prefactor as $d\simeq 0.067$. The relevance of our results for ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems is discussed at the end of our work.