Based on a renormalization-group (RG) analysis, we study the bosonization formulas for spin-
S Kitaev-Gamma and Kitaev-Heisenberg-Gamma chains in the parameter regime
K<0,
Γ>0, and
J>0, with
S being a half-odd integer. We find that the effects associated with the breaking of emergent continuous symmetries in the bosonization formulas scale as
1/S in the large-
S limit, in qualitative agreement with density-matrix renormalization-group (DMRG) results for Kitaev-Gamma chains. For Kitaev-Heisenberg-Gamma chains, the low-energy excitations are characterized by incommensurate gapless wave vectors, and symmetry analysis yields ten independent bosonization coefficients, five of which are predicted by the RG analysis to be independent of the Heisenberg coupling up to linear order. Our results provide useful input for determining magnetic ordering tendencies in two-dimensional Kitaev spin models within a quasi-one-dimensional approach.
This brief review introduces the method and application of real-space renormalization group to strongly disordered quantum systems. The focus is on recent applications of the strong disorder renormalization group to the physics of disordered-boson systems and the superfluid–insulator transition in one dimension. The fact that there is also a well-understood weak disorder theory for this problem allows us to illustrate what aspects of the physics change at strong disorder. In particular, the strong disorder RG analysis suggests that the transitions at weak disorder and strong disorder belong to distinct universality classes, but this question remains under debate and is not fully resolved to date. Further applications of the strong disorder renormalization group to higher-dimensional Bose systems and to bosons coupled to dissipation are also briefly reviewed.