*Harvard* *Condensed Matter Theory Seminars*

**Gauge fields, quantized fluxes, and monopole confinement of the honeycomb lattice**-
Electron hoppings on the honeycomb lattice are studied. The lattice consists of two triangular sublattices which are not equivalent, and thus it is non-Bravais. The dual space has non-trivial topology, being a 2$d$ M\"{o}bius.

Gauge fields of Bloch electrons have $U(1)$ symmetry, and thus represent superconducting states on the 2$d$ M\"{o}bius. Two Abrikosov quantized fluxes exist at the Dirac points and have fluxes $2\pi$ and $-2\pi$ respectively.

We define a non-Abelian $SO(3)$ gauge theory in the extended 3$d$ dual space and it is shown that the monopole and anti-monopole solution is stable. The $SO(3)$ gauge symmetry is broken into $U(1)$ at the 2$d$ M\"{o}bius boundary.

The quantized fluxes are related to quantized Hall conductance by the topological expression. Based on this, monopole confinement and deconfinement are discussed in relation to the Quantized Hall Effect.

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