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

Let $f_N$ and ${\xi }_N^{-1\mathrm{}}$ represent, respectively, the free energy per spin and the inverse spin-spin correlation length of the critical Ising model on a $N\times \infty$ lattice, with $f_N\to f_{\infty }$ as $N\to \infty$. We obtain analytic expressions for $a_k$ and $b_k$ in the expansions $N\left(f_N{-f}_{\infty }\right)=\Sigma {k=1\mathrm{}}^{\infty }{a}_k/N^{2k-1}$ and ${\xi }_N^{-1\mathrm{}}=\Sigma {k=1\mathrm{}}^{\infty }{b}_k/N^{2k-1}$ for square, honeycomb, and plane-triangular lattices, and find that $b_k/a_k=(2^{2k}-1\mathrm{})/(2^{2k-1}-1\mathrm{})$ 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.