Spin-Liquid–to–Spin-Liquid Transition in Kitaev Magnets Driven by Fractionalization

While phase transitions between magnetic analogs of the three states of matter—a long-range ordered state, paramagnet, and spin liquid—are extensively studied, the possibility of “liquid-liquid” transitions, namely, between different spin liquids, remains elusive. By introducing the additional Ising coupling into the honeycomb Kitaev model with bond asymmetry, we discover that the Kitaev spin liquid turns into a spin-nematic quantum paramagnet before a magnetic order is established by the Ising coupling. The quantum phase transition between the two liquid states accompanies a topological change driven by fractionalized excitations, the $Z_2$ gauge fluxes, and is of first order. At finite temperatures, this yields a persisting first-order transition line that terminates at a critical point located deep inside the regime where quantum spins are fractionalized. It is suggested that similar transitions may occur in other perturbed Kitaev magnets with bond asymmetry.

Figure 1

(a) Twenty-four-site cluster of the honeycomb lattice with periodic boundary conditions, where blue, green, and red lines represent $x$, $y$, and $z$ bonds, respectively. The six-site loop with the dashed line represents $W_p={\sigma }_1^z{\sigma }_2^y{\sigma }_3^x{\sigma }_4^z{\sigma }_5^y{\sigma }_6^x$. (b) Ground-state phase diagram of the pure Kitaev model on the basal plane of $J_x+J_y+J_z=1$. The $\alpha$ and the vertical axes parametrize the bond asymmetry (see the text) and $J_I$, respectively. (c) Ground-state phase diagram of the Kitaev-Ising model obtained by the ED at $T=0$. The circles represent the phase boundary determined by the peaks of $-J_I{d}^{2}E/d{J}_I^2$ (see Fig. 2). The dashed-dotted line shows the asymptotic phase boundary, $J_I=J_y^2{J}_z^2/(16J_x^3)$, between the Kitaev and the spin-nematic liquids in the large-$J_x$ limit. The small arrow on the vertical axis indicates the Ising transition point $J_I=0.3258(1)$ for $\alpha =0$ (see the text). The solid (dashed) line shows the first (second)-order phase boundary in the MF approximation. (d) Finite-$T$ phase diagram at $\alpha =0.7$ obtained by the TPQ state approach in the 24-site cluster in (a). The intensity map shows $-J_I{d}^{2}F/d{J}_I^2$, the peaks of which imply the phase boundaries. The circles represent the peaks of the specific heat (see Fig. 3) and the hatched area corresponds to the fractionalized paramagnetic regime.

Figure 2

(a)–(d) Ground state properties at $\alpha =0.7$ obtained with the ED in the 24-site cluster shown in Fig. 1: (a) the first and (b) the second derivatives of the ground state energy in terms of $J_I$, (c) mean square of the magnetization $⟨m^2⟩$ and the flux density $W$, and (d) the dimensionless measures of kinetic energy of the Majorana fermions $\overline{c}$, $⟨{\overline{K}}_x⟩$, $⟨{\overline{K}}_y⟩$, $⟨\overline{K}⟩=⟨{\overline{K}}_x⟩+⟨{\overline{K}}_y⟩$, and the squared fidelity $F_N^{\mathrm{sq}}$ (see the text). (e)–(h) Majorana fermion MF results for the same or related observables (except for $F_N^{\mathrm{sq}}$) for $\alpha =0.5$: (e) $-dE/d{J}_I$, (f) $-J_I{d}^{2}E/d{J}_I^2$, (g) $⟨m⟩=(1/N)\sum _i⟨{\sigma }_i^z⟩$ and $⟨\eta ⟩=(2/N)\sum _{⟨ij{⟩}_z}⟨{\overline{c}}_j{\overline{c}}_k⟩$, and (h) $⟨{\overline{K}}_x⟩$, $⟨{\overline{K}}_y⟩$, and $⟨\overline{K}⟩$. We note $W\approx ⟨\eta {⟩}^2$ in the MF approximation because of the relation $W_p={\eta }_r{\eta }_{r^{\prime }}$ for each plaquette $p$ [47, 48, 49], where $r$, $r^{\prime }$ represent two $z$ bonds of the plaquette $p$.

Figure 3

(a)–(c) $T$ dependences of (a) the specific heat, (b) the normalized entropy, and (c) the dimensionless measures of kinetic energy $⟨K⟩$ and $⟨\overline{K}⟩$ of $c$ and $\overline{c}$, respectively, at $J_I=0.001$ and $\alpha =0.7$, where the ground state is the Kitaev QSL; $K=(2/N)\sum _{\gamma =x,y}\sum _{⟨jk{⟩}_{\gamma }}{\sigma }_j^{\gamma }{\sigma }_k^{\gamma }=(2/N)\sum _{\gamma =x,y}\sum _{⟨jk{⟩}_{\gamma }}i{c}_j{c}_k$. (d)–(f) Corresponding data for $J_I=0.03$ and $\alpha =0.7$, where the ground state is spin nematic. The shaded broadening represents errors associated with the TPQ state approach.