Tuning topological orders by a conical magnetic field in the Kitaev model

We show that a conical magnetic field $\mathbf{H}=(1,1,1)H$ can be used to tune the topological order and hence anyon excitations of the ${\mathrm{Z}}_2$ quantum spin liquid in the isotropic antiferromagnetic Kitaev model. A novel topological order, featured with Chern number $C=4$ and Abelian anyon excitations, is induced in a narrow range of intermediate fields $H_{c1}\leq H\leq H_{c2}$\mathrm{.}Ontheotherhand,the$C=1$ Ising-topological order with non-Abelian anyon excitations, is previously known to be present at small fields, and interestingly, is found here to survive up to $H_{c1}$, and revive above $H_{c2}$, until the system becomes trivial above a higher field $H_{c3}$. The results are obtained by devoloping and applying a ${\mathrm{Z}}_2$ mean field theory, that works at zero as well as finite fields, and the associated variational quantum Monte Carlo.