Robust non-Abelian spin liquid and a possible intermediate phase in the antiferromagnetic Kitaev model with magnetic field

We investigate the non-Abelian topological chiral spin-liquid phase in the two-dimensional Kitaev honeycomb model subject to a magnetic field. By combining density matrix renormalization group and exact diagonalization we study the energy spectra, entanglement, topological degeneracy, and expectation values of Wilson loop operators, allowing for a robust characterization. While the ferromagnetic Kitaev spin liquid is already destroyed by a weak magnetic field with Zeeman energy $H_{*}^{\text{FM}}\approx 0.02$, the antiferromagnetic (AFM) spin liquid remains robust up to a magnetic field that is an order of magnitude larger, $H_{*}^{\text{AFM}}\approx 0.2$. Interestingly, for larger fields $H_{*}^{\text{AFM}}<H<H_{**}^{\text{AFM}}$, an intermediate gapless phase is observed, before a second transition to the high-field partially polarized paramagnet. We attribute this rich phase diagram, and the remarkable stability of the chiral topological phase in the AFM Kitaev model, to the interplay of strong spin-orbit coupling and frustration enhanced by the magnetic field. Our findings suggest relevance to recent experiments on ${\mathrm{RuCl}}_3$ under magnetic fields.

Figure 1

The honeycomb lattice and Wilson loop operators. The honeycomb lattice is spanned by unit vectors (1,0) and $(1/2,\sqrt{3}/2)$ with a lattice size $N=L_x\times L_y\times 2$. Green and blue loops denote Wilson loop operators along the vertical and horizontal periodic boundary conditions on the torus, respectively.

Figure 2

(a), (b) For the (a) AFM and (b) FM Kitaev models in a magnetic field, the pair of topological ground states (approximately degenerate on this $N=4\times 3\times 2$ torus) are separated from higher-energy states by an energy gap, within the topological phase $H<H_{*}$, where $H_{*}^{\text{AFM}}\approx 0.2$ and $H_{*}^{\text{FM}}\approx 0.02$. (c), (d) Wilson loop operators $W_y$ (main panel) and $W_x$ (inset) for the two lowest-energy states, for (c) AFM and (d) FM models. The two states have $W_y=-1$ but are distinguished by $W_x=\pm 1$. (e) The magnetization curves show the transitions and the stark difference between the AFM and FM models. (f) The second-order derivative of ground-state energy with respect to field, or equivalently the magnetic susceptibility: Note the difference in magnitudes between AFM and FM models. In the AFM case, the transition to the polarized high-field phase is achieved only at a second peak with $H_{**}\approx 0.36$.

Figure 3

(a) The DMRG calculation of the three lowest-energy states as a function of $L_x$ at $H_{111}=0.1$ on the torus. The energy difference between the two lowest-energy states becomes vanishingly small with increasing system size while they are separated from the higher-energy sectors by a finite gap. (b) The DMRG results of the Wilson loop operator $W_y$ on the torus for the antiferromagnetic (AFM) Kitaev model with $L_x=6,8$.

Figure 4

(a) Majorana dispersions on infinite ladders ($L_y=3$) for the $W_y=-1$ sector at $H_{111}=0$, which show a finite gap and a quantized value of Wilson loop operators. These are consistent with the numerics for finite fields (see main text). (b) The energy density of two degenerate states on cylinders at $H_{111}=0.1$; the inset shows the von Neumann entanglement entropy for long cylinders, where the flat feature indicates the existence of the finite gap.