Non-analytic finite-size corrections in the one-dimensional Bose gas and Heisenberg chain

The energy spectra of the one-dimensional Bose gas and Heisenberg chain in an external field (chemical potential in the case of the Bose gas and magnetic field in the case of the Heisenberg chain) are calculated. It is found that, at general values of the external field, the spectra are not of the form expected on the basis of conformal invariance. The conformal structure can be recovered only if some extra conditions are imposed on the size of the system and the external field. It is argued that these additional conditions must be satisfied when taking the continuum limit in order to arrive at a conformally invariant theory.