Field-induced quantum spin liquid in the Kitaev-Heisenberg model and its relation to $\alpha \text{−}{\mathrm{RuCl}}_3$

Recently considerable excitement has arisen due to the experimental observation of a field-induced spin liquid phase in the compound $\alpha \text{−}{\mathrm{RuCl}}_3$. However, the nature of this putative spin liquid phase and the relevant microscopic model Hamiltonian remain still unclear. In this work, we address these questions by performing large-scale numerical simulations of a generalized Kitaev-Heisenberg model proposed to describe the physics of $\alpha \text{−}{\mathrm{RuCl}}_3$. While an intermediate phase does not appear for in-plane magnetic fields, our results strongly suggest that a stable intermediate spin liquid phase, sandwiched between a magnetically ordered phase at low fields and a high-field polarized phase, can be induced by out-of-plane magnetic fields. Moreover, we show that this field-induced spin liquid phase can be smoothly connected to a spin liquid possessing a spinon Fermi surface as proposed recently for the Kitaev model. The relevance of our results to $\alpha \text{−}{\mathrm{RuCl}}_3$ is also discussed.

Figure 1

(a) Honeycomb cylinder with periodic and open boundary conditions along the directions specified by the lattice basis vectors, $e_2$ and $e_1$, respectively. $L_x$ and $L_y$ are the number of unit cells in the $e_1$ and $e_2$ directions. Bond directions $\gamma =x,y,z$ are labeled by different colors. (b) The first and second Brillouin zone and high symmetry points. (c) Directions of the external magnetic field in spin space.

Figure 2

Ground state phase diagram of the Kitaev-Heisenberg model on $L_y=4$ cylinder under magnetic field ${\mathbf{h}}_{{\mathbf{c}}^{*}}$. $Z_2$ and $U\left(1\right)$ SL denote the $Z_2$ gapped spin liquid and $U\left(1\right)$ gapless spin liquid with spinon Fermi surface, respectively. Insets: zoomed-in phase diagram with small $J_1$ (left) and derivative of magnetization at $h_{c^{*}}=0.343$ (right). Dashed line denotes the set of parameter extracted from neutron scattering experiments [37]. Here $K_1=1$ is the energy unit.

Figure 3

Zigzag order parameter $Z\left(M\right)=\frac{1}{3}{\sum }_{i=1}^3\sqrt{S\left({\mathbf{M}}_i\right)/N}$ along the $J_1=-0.786$ line in Fig. 2 as a function of $h_{c^{*}}$. Phase transitions between distinct phases are labeled by shaded regions. Inset: examples of the spin structure factor $S\left(\mathbf{q}\right)$ in the zigzag and spin liquid phases on $L_y=4$ cylinder of length $L_x=6$.

Figure 4

von Neumann entanglement entropy $S$ on $L_y=4$ cylinder of length $L_x=30$ for the $J_1=-1.0$ model, where $x^{\prime }=\frac{L_x}{\pi }sin\left(\frac{\pi x}{L_x}\right)$. The extracted central charge $c$ is shown in the inset where the dashed line is a guide for eyes.

Figure 5

Ground state phase diagram of the Kitaev-Heisenberg model on $L_y=4$ cylinder under magnetic field ${\mathbf{h}}_{\mathbf{b}}$. Here $K_1=1$ is the energy unit.