Phases of strongly correlated one-dimensional quantum systems

Studying phases and phase transitions is one of the central topics of condensed matter physics. For the one-dimensional quantum system, powerful analytical and numerical tools allow us to gain much understanding of the novel aspects of the phase of matters beyond the Landau-Ginzburg paradigm. In particular, the bosonization techniques can capture the quasi-orders of one-dimensional gapless phases. On the other hand, reliable numerical data can be obtained via optimization methods of matrix product

states, like the density matrix renormalization group (DMRG). Proper data analysis makes this method even more powerful. This thesis combines DMRG and bosonization approach to study strongly correlated

one-dimensional phases, with a focus on the gapless phases. In one of the projects, we use iDMRG to obtain the phase diagram of the mass imbalanced Hubbard

model. We combine the standard bosonization method to reveal one of the phases as a gapless phase with some characters of a symmetry-protected topological phase. In particular, we determine long-range string orders and the "filling anomaly"; the latter refers to the relation among the single-particle gap, inversion symmetry, and filling imbalance for open systems. In addition, we also study the formation of bound states in a one-dimensional, single-component Fermi chain with attractive interactions. The phase diagram, also

computed from DMRG, shows not only a super fluid of paired fermions (pair phase) and a liquid of three-fermion bound states (trion phase), but also a phase with

two gapless modes. We show that the latter phase is described by a 2-component Tomonaga-Luttinger liquid (TLL) theory, consisting of one charged and one neutral

mode. We argue based on our numerical data, that the single, pair, and trion phases are descendants of the 2-component TLL theory.