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The Kitaev Honeycomb Model

intro - strong spin orbit coupling leads to anisotropic spin exchange (as opposed to isotropic exchange like the Heisenberg model) - geometrical frustration leads to QSL ground state with long range entanglement (not simple paramagnet)

RuCl_3 is the classic QSL candidate material
really follows the Kitaev-Heisenberg model
experimental probes include inelastic neutron scattering, Raman scattering

The Model

Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom u_{jk} = \pm 1 with arrows that point in the direction u_{jk} = +1 (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \mathbb{Z}_2 gauge field u_{ij}. This leavies a single Majorana c_i per site. — Figure 1: (a) The standard Kitaev model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each (b). We represent the antisymmetric gauge degree of freedom \(u_{jk} = \pm 1\) with arrows that point in the direction \(u_{jk} = +1\) (c). The Majorana transformation can be visualised as breaking each spin into four Majoranas which then pair along the bonds. The pairs of x,y and z Majoranas become part of the classical \(\mathbb{Z}_2\) gauge field \(u_{ij}\). This leavies a single Majorana \(c_i\) per site.

strong spin orbit coupling yields spatial anisotropic spin exchange leading to compass models¹
spin model of the Kitaev model is one
has extensively many conserved fluxes

A mapping to Majorana Fermions

Gauge Fields

Anyons, Topology and the Chern number

Phase Diagram

1.

Kugel’, K. I. & Khomskiĭ, D. I. The Jahn-Teller effect and magnetism: transition metal compounds. Sov. Phys. Usp. 25, 231 (1982).