Purdue University Graduate School
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posted on 2023-04-17, 17:45 authored by Sheng TanSheng Tan

This dissertation consists of two parts. In the first part, we discuss the boundary and domain wall theories of the generalized quantum double lattice realization of the two-dimensional topological orders based on Hopf algebras. The boundary Hamiltonian and domain wall Hamiltonian are constructed by using Hopf algebra pairings and generalized quantum double. The algebraic data behind the gapped boundary and domain wall are comodule algebras and bicomodule algebras, respectively. The topological excitations in the boundary and domain wall are classified by bimodules over these algebras. Finally, via the Hopf tensor network representation of the quantum many-body states, we solve the ground state of the model in the presence of the boundary and domain wall.

In the second part, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems, and we establish weak Hopf symmetry breaking theory based on the fusion closed set of anyons. We present a thorough investigation of the quantum double model based on weak Hopf algebras, including the topological excitations and ribbon operators, and show that the vacuum sector of the model has weak Hopf symmetry. The gapped boundary and domain wall theories are also established. We show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra, and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. We also introduce the weak Hopf tensor network states, by which we solve the weak Hopf quantum double models on closed and open surfaces. Lastly, we discuss the duality of the quantum double phases.


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Uli Walther

Additional Committee Member 2

Shawn X. Cui

Additional Committee Member 3

Ralph Kaufmann

Additional Committee Member 4

Eric Samperton