Crystalline Topological Invariants in Invertible Fermionic States and Fractional Chern Insulators

Abstract

Topological phases of matter have long been a central theme in condensed matter physics. When subject to certain symmetries, these phases are classified by topological invariants that often take fractional, quantized values. Despite significant theoretical advancements, major open questions remain regarding the computation of these invariants in Chern insulators or fractional Chern insulators with crystalline symmetry. The first part of this dissertation establishes the necessary theoretical background, defining symmetry operators, geometrical measures, and topological actions used to determine crystalline topological invariants. The second part focuses on extracting these invariants numerically for invertible fermionic states in multiple ways. Specifically, we calculate the discrete shift $\mathscr{S}{\text{o}}$, the electric polarization $\vec{\mathscr{P}}{\text{o}}$, and the invariants $\Theta_{\text{o}}^{\pm}$. We discuss the properties of these invariants, such as how $\mathscr{S}{\text{o}}$ and $\vec{\mathscr{P}}{\text{o}}$ determines the universal charge response of crystalline defects and boundaries, and how they depend on an origin $\text{o}$ in real space. Concrete numerical methods are demonstrated using the Hofstadter model, which exhibits non-zero Chern number and magnetic field. These invariants, together with several established invariants in the literature, gives a complete classification of ground states and reveal novel colorings of the Hofstadter butterfly. The third part of this dissertation extends the calculation of crystalline topological invariants to fractional quantum Hall states, such as the 1/2-Laughlin state constructed by projecting parton states onto the same position. We show that the numerically extracted values of these invariants align with theoretical predictions from conformal field theory and $G$-crossed braided tensor categories.