On the spectral properties of inhomogeneous Luttinger liquids

The physics of one-dimensional systems of interacting fermions strongly differs from the picture offered by Fermi liquid theory. The low-energy properties of metallic, paramagnetic 1D Fermi systems are well described by the Luttinger liquid (LL) theory. Among other properties, the homogeneous LL is known to exhibit power-law behaviour for various correlation functions. For spin rotational invariant models, the exponents can be expressed in terms of a single parameter Kρ, which is known to be non-universal. The LL concept emerged in the study of homogeneous systems with periodic boundary condition (PBC). In these models the local spectral function shows a power-law suppression at energies asymptotically close to the chemical potential µ. Theoretically, this is very well understood and the exact LL parameter Kρ is known for a set of integrable models. In the last decades however, the interest shifted towards the investigation of LLs with PBC including impurities. The fact that these systems have been shown to scale to chains with open ends led to the study of several models where open boundary condition (OBC) are introduced. As the translational invariance is broken, the exponent of the power-law suppression close to the boundary differs from the one in the bulk. The present work is devoted to the study of the local spectral function of two lattice models of LLs with open boundaries, namely the spinless fermions model with nearest neighbour interaction and the 1D Hubbard model, with the methods of perturbation theory. It is demonstrated that many aspects of the spectral properties can be understood within the HF theory. Despite what one usually reads about one-dimensional interacting Fermi systems, in the presence of the boundary perturbation theory is already capable of providing meaningful results. Concurrently, this method also allows us to develop an understanding of the dip appearing in the local spectral function at an energy different from the chemical potential.