Corrections to finite-size scaling for quantum chains

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Abstract
The authors compute the leading order correction to the energy gap at the critical point for the anisotropic Ising and the Z3 Potts quantum chains. Periodic, anti-periodic and free boundary conditions are considered. For the Ising case the corrections are independent of the boundary conditions. In the Z3 case the correction is the same for anti-periodic and free boundary conditions but is 2/5 smaller for the periodic case. The authors have no explanation for this phenomenon.