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.
Using bosonization and path-integral methods, we study general low-temperature behavior of nonmagnetic and magnetic impurity scattering in Tomonaga-Luttinger liquid, and calculate the electron Green function for a general backward scattering potential. We demonstrate that the electron density of state near the impurity site is suppressed by backward scattering, but it mainly remains invariant far away from the impurity, and that at zero temperature the electrons are completely reflected on the impurity site; the system breaks into two subsystems, but right- and left-moving electron fields have a twisted boundary condition. We also show that a testing charge can only be partially screened by conduction electrons, and in a strong interaction region the impurity susceptibility has a
1/T-type low-temperature behavior.
The exact Green's functions of a Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. It is also shown that if the resistances are equal to the characteristic impedance of the Luttinger liquid, then the finite Luttinger liquid connected to resistors is equivalent to an infinite Luttinger liquid. Impedance boundary conditions (IBC) include also as a special limit the case of open boundary conditions, which are explicitly recovered. Finally bosonization for a LL with IBC is proven to hold.