Investigations on the Kitaev Model and some of its generalisations.

Hsiang-Hsuan Hung, Congjun Wu, Daniel P. Arovas

Sebastian Halbig, Ulrich Krähmer

The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input of the model are ribbon graphs, that is, the combinatorial data of cellular decompositions of oriented closed surfaces. The other input are certain Hopf bimodules that are closely related to the coefficients in Hopf-cyclic homology. As in previous generalisations of the Kitaev model, a Yetter-Drinfeld module is constructed and shown to be a topological invariant of the surface with boundary that is obtained by "thickening" the graph. The generalisation of the protected space is defined using bitensor products of modules-comodules. Provided that the Hopf bimodule coefficients correspond to pairs in involution, this is shown to depend only on the genus of the graph. As examples, group algebras of finite groups and bosonisations of Nichols algebras are considered.

Paul Watts, J Vala

We describe how the Kitaev honeycomb model with a finite number \nof sites on a torus may be easily generalised to cylindrical and rectangular \ntopologies using a recently developed fermionisation technique. We then look \nfor criteria which determine if zero-energy modes exist for these cases in the \nabsence of an external magnetic field. For the cylindrical case, we determine the \ncombination of spin–spin coupling strengths and vortex configurations which give \nzero modes and construct them explicitly, and for the rectangular case we show \nthat no zero modes can exist.

Zhoushen Huang, Daniel P. Arovas

Itamar Kimchi, Yi‐Zhuang You, Ashvin Vishwanath

Felix Dusel, Tobias Hofmann, Atanu Maity +5 more

We analyze the Kitaev model on the {9,3} hyperbolic lattice. The {9,3} is formed by a regular tricoordinated tiling of nonagons, where the three-color coding of bonds according to the inequivalent Kitaev Ising spin couplings yields the natural generalization of the original Kitaev model for Euclidean regular honeycomb tiling. Upon investigation of the bulk spectrum for large finite size droplets, we identify a gapless chiral Z_{2} spin liquid state featuring spontaneous time-reversal symmetry breaking. Because of its noncommutative translation group structure, such type of hyperbolic spin liquid is conjectured to feature chiral quasiparticles with a potentially non-Abelian Bloch profile.