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Topological phases of matter in low dimensions with and without disorder


While the essential features of wide classes of phases of matter are well-captured by the conventional frameworks such as the Landau theory of symmetry breaking, there do exist unconventional phases of mater, including topological phases, that provide examples of exotic phases of matter lying beyond the conventional paradigms. Although such phases of matter are hard to understand in general, there are cases where explicit model wave functions can be written. Using the model wave functions, one can extract topological information using the entanglement structures of the wave functions.

In this thesis, we study various kinds of unconventional phases of matter where the quantum effects are essential, using various methodologies. We study not only the ground states of clean systems but also excited states of disordered systems, as the disorder may protects quantum order at highly excited states.

First, we study the crossover between the ground state and the excited state criticality where the quantum criticality is protected by the quenched disorder even at finite energy density. Using the strong-disorder renormalization-group technique, we are able to extract the long distance physics at zero and finite temperature exactly, as well as the crossover between the two.

Second, we consider the fractional quantum effect, which is a strongly interacting quantum system in two dimensions. We first establish the algebraic formulations of the fractional quantum Hall effect by speculating the underlying algebraic structures in the lowest Landau level. We then focus on a model state called the Gaffnian state, which is believed to represent a quantum critical state as opposed to conventional gapped states. Using the Girvin-MacDonald-Platzman mode analysis together with the exact diagonalization, we provide a possible gap closing picture at the Gaffnian critical point.

Next, we consider disordered non-chiral topological phases and the phase transitions between the topological and the trivial phase. With the help of quenched disorder, the ground state topological order may survive at finite energy density, possibly realizing a many-body localization in two dimensions. Using the strong-disorder real-space renormalization group technique, we found a super-universal quantum phase transition between the non-chiral topological phase and the trivial phase, where the quantum critical point is governed by an infinite-randomness fixed point.

Finally, we present how the momentum polarization can be defined in quantum lattice models where the translational symmetries are discrete. We explain how the discrete, non-onsite symmetry is gauged to yield the crystalline gauge field. Using the gauging of the translational symmetry, we present the Berry phase formulation of the momentum polarization, a quantity that is closely related to the Hall viscosity.

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