  We propose a methodology to construct excited states with a fixed angular
momentum, namely, "yrast excited states" of finite-size one-dimensional bosonic
systems with periodic boundary conditions. The excitation energies such as the
first yrast excited energy are calculated through the system-size asymptotic
expansion and expressed analytically by dressed energy.
  Interestingly, they are grouped into sets of almost degenerate energy levels.
The low-lying excitation spectrum near the yrast state is consistent with the $
U(1)$ conformal field theories if the total angular momentum is given by an
integral multiple of particle number; i.e., if the system is supercurrent.
