The phase diagram of the one-dimensional quantum sine-Gordon system (β = 4π) with a linear spatial modulation

The one-dimensional quantum sine-Gordon system with a linear spatial modulation is investigated in a special case, β2 =4π. The model is tranformed into a massive Thirring model and then is exactly diagonalized, the energy spetrum of the model is obtained. Our result clearly demonstrates that cancelling the cosine term without any considering is unadvisable. The one-dimensional (1D) quantum sine-Gordon model [1] may probably be the most useful quantum model since it can be used to describe the most of the oneand two-dimensional (2D) models of either fermi or bose system, [2], [3], [4] this fact attaches particular importance to the quantum sine-Gordon model. Many works have been done about this model both in field theory and in condensed matter physics. [1], [5], [6], [7], [8] It is exactly solvable by quantum inverse scattering method. [1] By a variational method Coleman [5] first discovered that the energy density of the system is unbounded below when the coupling constant β exceeds 8π, so there is a phase transition as the coupling constant varies. This corresponds to the Kosterlitz-Thouless (K-T) phase transition by the equivalence of the 2D Coulomb gas and sine-Gordon model. The soliton mode of the sine-Gordon model corresponds to a one-fermion excitation in the fermi picture, which was clarified soon later by Mandelstam by introducing a Fermi-Bose relation. [7] As there are other spatial variations in the cosine potential, the low energy properties of the 1D quantum sine-Gordon system are more difficult to be analysized. Here we shall discuss a simple case that there are a linear spatial modulation in the cosine potential. The Hamlitonian reads as H = ∫ 