Variational solution of the one-dimensional backward-scattering model with generalized coupling constant

Abstract

We present a variational solution of the sine-Gordon theory, and at the same time the Luther-Emery backward-scattering model, for the range of coupling constants $0<\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}}\ensuremath{\le}1$ or $\ensuremath{-}1<\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}}<~\frac{\ensuremath{-}3}{5}$, respectively. This yields a renormalized mass for the fundamental fermion of the massive Thirring model ${m}_{f}=(\frac{\ensuremath{\pi}}{\sqrt{c}})(\frac{1}{{\ensuremath{\beta}}^{2}}){(\frac{{\ensuremath{\alpha}}_{0}}{{\ensuremath{\Lambda}}^{\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}}}})}^{\ensuremath{\nu}}$ and an energy gap $\ensuremath{\Delta}={v}_{F}{c}^{\ensuremath{\nu}\ensuremath{-}1}{[4+{(\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}})}^{2}]}^{\ensuremath{\nu}\ensuremath{-}1}(\frac{1}{{\ensuremath{\alpha}}_{L}}){(\frac{{U}_{\ensuremath{\perp}}}{2\ensuremath{\pi}{v}_{F}})}^{\ensuremath{\nu}}$ in the fermion spectrum for the backward-scattering model where $\ensuremath{\nu}=\frac{1}{(2\ensuremath{-}\frac{{\ensuremath{\beta}}^{2}}{4\ensuremath{\pi}})}$ and $\frac{{\ensuremath{\beta}}^{2}}{8\ensuremath{\pi}}={[\frac{(1+\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}})}{(1\ensuremath{-}\frac{{U}_{\ensuremath{\parallel}}}{2\ensuremath{\pi}{v}_{F}})}]}^{\frac{1}{2}}$.