Equivalent critical behavior of a helical point contact and a two-channel Luttinger liquid-topological superconductor junction

We demonstrate the equivalence between two distinct Luttinger liquid impurity problems. The first concerns a one-dimensional topological superconductor coupled at one end to the ends of two single-channel Luttinger liquids. The second concerns a point contact in the quantum spin Hall effect, where four helical Luttinger liquids meet at a point. Both of these problems have been studied previously and exhibit several stable phases, depending on the Luttinger parameter K, that can be characterized in terms of simple conformally invariant boundary conditions describing perfect normal (or Andreev) transmission or reflection. In addition, both problems exhibit critical points that are described by intermediate fixed points similar to those found in earlier studies of an impurity in a Luttinger liquid with spin. Though these two models have different symmetries and numbers of modes, we show they are equivalent and are related by a duality transformation, and we show that the nontrivial intermediate critical points are the same. In the noninteracting limit, K = 1, the duality involves two distinct free fermion representations that are related by a nonlocal transformation that derives from the triality of SO(8). Using the explicit translation between the two theories, we translate results from one problem to the other and vice versa. This allows us to make predictions about the topological superconductor-Luttinger liquid junction, including predictions about the global behavior of the critical conductance G* (K), as well predictions for the critical exponents and universal crossover scaling functions. In this paper, we introduce both models from scratch, using a common notation that facilitates their comparison, and we discuss in detail the dualities that relate them, along with their free fermion limits. We close with a discussion of open problems and future directions.