Phases of strongly correlated one-dimensional quantum systems

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