Shawn Cui | Searcharxiv

arXiv2025

Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds

We give invariants of flat bundles over 4-manifolds generalizing a result by Chaidez, Cotler, and Cui (Alg. \& Geo. Topology '22). We utilize a structure called a Hopf $G$-triplet for $G$ a group, which generalizes the notion of a Hopf triplet by Chaidez, Cotler, and Cui. In our construction, we present flat bundles over 4-manifolds using colored trisection diagrams: a direct analogue of colored Heegaard diagrams as described by Virelizier. Our main result is that involutory Hopf $G$-triplets of finite type yield well-defined invariants of $G$-colored trisection diagrams, and that if the monodromy of a flat bundle has image in $G$ we obtain invariants of flat bundles. We also show that a special Hopf $G$-triplet yields the invariant from Hopf $G$-algebras described by Mochida, thus generalizing the construction.

math.GT↗

arXiv2024

Refined Phase Diagram for a Spin-1 System Exhibiting a Haldane Phase

We provide the phase diagram of a 2-parameter spin-1 chain that has a symmetry-protected topological (SPT) Haldane phase using computational algorithms along with tensor-network tools. We improve previous results, showing the existence of a new phase and new triple points. New striking features are the triple end of the Haldane phase and the complexity of phases bordering the Haldane phase in proximity allowing moving to nearby non-SPT phases via small perturbations. These characteristics make the system, which appears in Rydberg excitons, e.g. in Cu$_2$O, a prime candidate for applications.

cond-mat.str-el↗

arXiv2022

Ternary Logic Design in Topological Quantum Computing

A quantum computer can perform exponentially faster than its classical counterpart. It works on the principle of superposition. But due to the decoherence effect, the superposition of a quantum state gets destroyed by the interaction with the environment. It is a real challenge to completely isolate a quantum system to make it free of decoherence. This problem can be circumvented by the use of topological quantum phases of matter. These phases have quasiparticles excitations called anyons. The anyons are charge-flux composites and show exotic fractional statistics. When the order of exchange matters, then the anyons are called non-Abelian anyons. Majorana fermions in topological superconductors and quasiparticles in some quantum Hall states are non-Abelian anyons. Such topological phases of matter have a ground state degeneracy. The fusion of two or more non-Abelian anyons can result in a superposition of several anyons. The topological quantum gates are implemented by braiding and fusion of the non-Abelian anyons. The fault-tolerance is achieved through the topological degrees of freedom of anyons. Such degrees of freedom are non-local, hence inaccessible to the local perturbations. In this paper, the Hilbert space for a topological qubit is discussed. The Ising and Fibonacci anyonic models for binary gates are briefly given. Ternary logic gates are more compact than their binary counterparts and naturally arise in a type of anyonic model called the metaplectic anyons. The mathematical model, for the fusion and braiding matrices of metaplectic anyons, is the quantum deformation of the recoupling theory. We proposed that the existing quantum ternary arithmetic gates can be realized by braiding and topological charge measurement of the metaplectic anyons.

quant-ph↗