Anyon polarons as a window into the competing phases of the Kitaev-Gamma-Gamma' model

Abstract: We investigate the dispersions of anyon quasi-particles in the Kitaev honeycomb spin-liquid perturbed by $\Gamma$ and $\Gamma&#39;$ couplings in order to understand phase transitions into competing states through anyon gap-closing instabilities. We demonstrate how anyon gap closings allow to understand phase transitions into a plethora of previously identified competing phases -- including zigzag, stripy, $120<sup>\circ$, and incommensurate spiral phases -- and are in agreement with numerical studies not only on the nature of the phases, but also on the specific critical values of $\Gamma$ and $\Gamma&#39;$ couplings. Remarkably, when the anti-ferromagnetic Kitaev model is perturbed by an ferromagnetic $\Gamma$ interaction, we find that the single-vison and fermion gaps remain open while the gap of a magnon-like local boson vanishes, implying that the resulting state has coexistence of a spontaneous broken symmetry and the fractionalization pattern of the Kitaev spin liquid. The magnetic long-range order could be either a stripy antiferromagnet or an incommensurate spiral, depending on the sign of $\Gamma&#39;$.