Existence of Low-Temperature Critical Regime in 1D Luttinger Liquid with a Weak Link

The exact solution of the boundary sine-Gordon model is studied in the region where the scaling dimension of the boundary field $1 < \Delta < 1$. It is shown that at $\Delta > 2/3$ the infrared fixed point belongs to the universality class of the underscreened anisotropic Kondo model. The boundary contribution to the specific heat scales as $C \sim T^{2\Delta^{-1} - 2}$ at small temperatures.