Scaling Relation for Excitation Energy under Hyperbolic Deformation

We introduce a one-parameter deformation for one-dimensional quantum lattice models, the hyperbolic deformation, where the scale of the local energy is proportional to cosh λj at the jth site. From the quantum-classical correspondence with a two-dimensional classical system, it is expected that the deformation does not modify the ground state conspicuously, if there is a finite excitation gap. Under this situation, the excited quasi particle is weakly confined around the center of the system. We derive scaling relations among the mean-square width 〈w2〉 of confinement, energy correction to the excitation gap Δ, and deformation parameter λ. This scaling is useful for the extraction of the bulk excitation gap in the limit λ = 0.