Topological Invariants of Gapped Quantum Lattice Systems

Citation

Sopenko, Nikita A. (2023) Topological Invariants of Gapped Quantum Lattice Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/715e-q184. https://resolver.caltech.edu/CaltechTHESIS:06032023-021309617

Abstract

In the first part of the thesis, a systematic way to construct topological invariants of gapped states of quantum lattices systems is proposed. It provides a generalization of the Berry phase and its equivariant analogue to systems with locality in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, it gives a closed (d + 2)-form on the parameter space which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. We prove quantization properties of the invariants for low-dimensional invertible systems.

In the second part, we discuss the properties of the invariant associated with the Hall conductance for 2d lattice systems with U(1)-symmetry. We define anyonic states associated with the flux insertions and relate their statistics to this invariant. We also provide the construction of states realizing chiral topological order with a non-trivial value of this invariant. The construction is based on the data of a unitary regular vertex operator algebra.

Thesis Files