Dissipation-induced Luttinger liquids in the Caldeira-Leggett formalism

Abstract

It is well known that dissipation usually induces decoherence in quantum systems. However, recently it has been employed to engineer zero-energy Majorana modes on the boundary of a one-dimensional fermionic chain. In this thesis, by employing the Caldeira-Leggett theory of quantum dissipation, we study the equivalence between a particle on a one-dimensional lattice coupled to a thermal bath, and the massless Luttinger liquid with an impurity on the boundary in a closed system. In particular, the Luttinger liquid with a single impurity at the boundary of a semi-infinite chain is bosonized into the boundary sine-Gordon theory. Here, a zero-energy Majorana mode appears at the boundary when the Luttinger parameter K=1/2. The equivalence is shown to apply also in the case of helical Luttinger liquids, which play an important role in topological insulators. This equivalence is then generalized to the full sine-Gordon theory, such that massive Luttinger liquids can be mapped to a single-particle one-dimensional tunneling Hamiltonian with a dissipative Ohmic Caldeira-Leggett bath.