UC Riverside

This thesis investigates the intricate relations between entanglement, symmetry factorization, and topological states of matter, advancing our understanding from the theoretical aspects of the emergent behavior of gapped electronic systems to potential applications in quantum information or computing. Our findings span four key areas.

First, we use the entanglement negativity to measure the multipartite entanglement arising from cutting and gluing ($2+1$)-dimensional topological fluids. We focus on the entangling properties of Abelian general Laughlin states and non-Abelian general Moore-Read states, where we analyze the real-space structure of their ground state wavefunctions.

We find that while the conventional disentangling condition is necessary and sufficient for disentangling general Abelian Laughlin states and untwisted Moore-Read states, it fails for twisted Moore-Read states. Even when this condition is met, the twisted Moore-Read states remain entangled due to the presence of the Ising twisted field, which scrambles the associated reduced density matrix. We conjecture that such obstruction to disentanglement prevails among any non-Abelian topological fluids.

Second, we investigate the topological states of matter that emerge from the symmetry factorization of $E_8$ bosonic integer quantum Hall states. These states are built up on the exactly solvable coupled-wire formalism. We factorize the $E_8$ Kac-Moody current into sub-currents that are governed by Lie groups, preserving the total chiral central charge of the $E_8$. This technique circumvent the challenges in mathematically describing strongly interacting bosons, leading to the bosonic analog of fractional quantum Hall states that conserve charge and momentum symmetries. The resulting states support a rich variety of emergent topological orders.

Third, we extend our studies of symmetry factorization to classical Lie groups, $SU(N)$ and $SO(N)$ at level $1$. This provides us with a microscopic understanding of the emergence of quantum Hall states, topological insulators, and quantum spin liquids with non-Abelian excitations like the Majorana, metaplectic, and Fibonacci types. We discover a theorem of branching rule from these classical Lie group symmetry embeddings via the coupled-wire construction. The theorem states that the pairing of various anyons and anti-anyon from the embedded subgroups can be glued together, forming local chain operators that can be expressed as products of their parent state's current operators, which do not appear in the tensor product of current operators of the embedded symmetries.

Finally, we investigate topological defects in quantum Low-Density Parity-Check (qLDPC) codes. We showed how quantum information encoded by defects paves the way to non-local quantum error corrections in qLDPC codes. Our analysis reveals the relation between such qubit-carrying defects and the general topological entanglement entropy, demonstrating that the non-local error correction feature in the qLDPC is validated by entanglement entropy that is both finite and nonzero.