| This dissertation studies three examples of low-dimensional quantum systems. The contents of the dissertation have been published on Phys. Rev. B [1, 2] and Phys. Rev. Lett. [3, 4].; In the first chapter, we analyze ground-state properties of strictly one-dimensional molecular matter comprised of identical particles. Selecting the pair interaction potential in the Morse form we analytically compute the properties of the Luttinger liquid and its range of existence. We find that as the strength of quantum zero-point motion increases, the system first undergoes a discontinuous evaporation transition into a diatomic gas followed by a continuous dissociation transition into a monoatomic gas. We also investigate the effect of finite pressure on the properties of the liquid and monoatomic gas phases. In particular we estimate a pressure at which molecular hydrogen undergoes an inverse Peierls transition into a metallic state.; In the second chapter, we investigate the Coulomb blockade of a large gated quantum dot coupled via a quantum point contact to a reservoir of one-dimensional spinless electrons whose interactions are characterized by the Luttinger liquid parameter g. We find that the classical step-like dependence of the average number of electrons on the dot n as a function of the gate voltage nx is preserved under certain conditions. The system can belong to either of two universality classes. (i) In the Kondo/Ising class, the n(nx) dependence is always continuous for g ≥ 1, while for g < 1 the n(nx) dependence takes the form of a modified staircase for dots sufficiently isolated from the reservoir. (ii) In the tricritical class we find in addition an intermediate regime where the dot population jumps from near integer value to a half-integer “plateau”. In particular, this tricritical behaviour is realized for non-interacting electrons, g = 1.; In the third chapter, we study macroscopic wave function of low-dimensional Bose liquids beyong Gross-Pitaevskii approximation. We point out that for short-ranged repulsive interactions Gross-Pitaevskii approximation fails in dimensions d ≤ 2, and we propose the appropriate low-dimensional modifications. For d = 1 we analyze density profiles in confining potentials, superfluid properties, solitons, and self-similar solutions. |
