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## "Topological order with a twist: Ising anyons from an Abelian model", Hector Bombin, 2010

*Reviewed December 31, 2023*

*Citation:* Bombín, Héctor. "Topological order with a twist: Ising anyons from an Abelian model." Physical review letters 105.3 (2010): 030403.

*Web:* https://arxiv.org/abs/1004.1838

*Tags:* SPT/SETs, Error-correcting-code, Ising-computer, Defects/boundaries

This paper gives a lovely introduction and implementation of the idea of realizing
Ising anyons as twist defects in the toric code. In this approach, the
lattice Hamiltonian from

> Wen, Xiao-Gang. "Quantum orders in an exact soluble model." Physical review letters 90.1 (2003): 016803.

and the twist defects are non-square plaquettes. The idea of using lattice dislocations to realize Majorana fermions in the toric code goes
back to the "odds and ends" section of Kitaev's seminal work

> Kitaev, Alexei. "Anyons in an exactly solved model and beyond." Annals of Physics 321.1 (2006): 2-111.

where he asks the followng: "The weak translational symmetry breaking in the Abelian phase has some interesting
consequences. A particularly unusual phenomenon takes place when the lattice has a
dislocation. A particle winding around the dislocation changes its type: e ↔ m. Since
m = e x ε, the fermionic parity appears not to be conserved. To restore the conservation
law, we must assume that the dislocation carries an unpaired Majorana mode. Therefore,
Abelian phases can also be used for the implementation of quantum memory."

Bombin was studying this question at the same time as Kitaev and Kong, who released their
(more general) approach two years later:

> Kitaev, Alexei, and Liang Kong. "Models for gapped boundaries and domain walls." Communications in Mathematical Physics 313.2 (2012): 351-373.