Phase separation as an instability of the Tomonaga-Luttinger liquid

Asymptotic behavior of the Tomonaga-Luttinger liquid in the vicinity of the phase-separated region is investigated in the one-dimensional (Formula presented) model to study the universal property of the (Formula presented) conformal field theory with U(1) symmetry near the (Formula presented) instability. By analogy to the spinless fermion, we argue that the compressibility behaves as (Formula presented) and that the Drude weight is constant and changes to zero discontinuously at the phase boundary. This speculation is confirmed by analyzing the finite-size effect from the result of the exact diagonalization.