Exact Universal Amplitude Ratios for Two-Dimensional Ising Models and a Quantum Spin Chain

Abstract

Let ${f}_{N}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ lattice, with ${f}_{N}\ensuremath{\rightarrow}{f}_{\ensuremath{\infty}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain analytic expressions for ${a}_{k}$ and ${b}_{k}$ in the expansions $N({f}_{N}{\ensuremath{-}f}_{\ensuremath{\infty}})\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{a}_{k}/{N}^{2k\ensuremath{-}1}$ and ${\ensuremath{\xi}}_{N}^{\ensuremath{-}1}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\Sigma}{k\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}{b}_{k}/{N}^{2k\ensuremath{-}1}$ for square, honeycomb, and plane-triangular lattices, and find that ${b}_{k}/{a}_{k}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({2}^{2k}\ensuremath{-}1)/({2}^{2k\ensuremath{-}1}\ensuremath{-}1)$ for all of these lattices, i.e., the amplitude ratio ${b}_{k}/{a}_{k}$ is universal. We also obtain similar results for a critical quantum spin chain and find that such results could be understood from a perturbated conformal field theory.