Exact universal amplitude ratios for the planar Ising model and a quantum spin chain

Let fN and ξ −1 N represent, respectively, the free energy per spin and the inverse correlation length of the critical Ising model on N × ∞ lattice, with fN → f∞ as N → ∞. We obtain analytic expressions for ak and bk in the expansions: N(fN − f∞) = ∑ ∞ k=1 ak/N 2k−1 and ξ N = ∑ ∞ k=1 bk/N 2k−1 for square, honeycomb, and planetriangular lattices, and find that bk/ak = (2 2k−1)/(22k−1−1) for all of these lattices, i.e. the amplitude ratio bk/ak is universal. We also obtain similar expansions for the ground state energy and the first energy gap of a critical quantum spin chain and find that the amplitude ratios have the same values.