Thermodynamics of Chiral Spin Liquids with Abelian and Non-Abelian Anyons

Thermodynamic properties of chiral spin liquids are investigated for a variant of the Kitaev model defined on a decorated honeycomb lattice. Using the quantum Monte Carlo simulation, we find that the model exhibits a finite-temperature phase transition associated with the time reversal symmetry breaking, in both topologically trivial and nontrivial regions. Numerical results for the Chern number and the thermal Hall conductivity indicate that the phase transition changes from a continuous to a discontinuous transition as we vary the coupling constants to reach the non-Abelian phase coming from the Abelian phase of the model. In addition, we find as a diagnostic of the chiral spin liquids, successive crossovers with multistage entropy release above the critical temperature, which indicates that the hierarchical fractionalization of a quantum spin occurs differently between the two regions.

Figure 1

Finite-$T$ phase diagram of the Kitaev model on a decorated honeycomb lattice. The ground-state phase diagram is also presented in the bottom of the figure. The lattice structure is depicted in the inset. The circles represent the phase transition temperature $T_c$. The deduced location of the tricritical point is also shown; for larger (smaller) $\alpha$, the transition is continuous (discontinuous). The triangles, squares, diamonds, and inverted triangles represent the crossover temperatures, $T^{*}$, $T^{**}$, $T_L^{*}$, and $T_H^{*}$, respectively. The solid and dotted lines in the large $\alpha$ region represent $T_c$ and $T^{**}$, respectively, determined by the MC simulation for the effective model for $J^{\prime }/J\gg 1$. The solid, dashed, and dashed-dotted lines in the small $\alpha$ region represent $T_c$, $T_L^{*}$, and $T_H^{*}$, respectively, obtained from the effective model for $J^{\prime }/J\ll 1$. See the text for details.

Figure 2

$T$ dependences of (a) the specific heat, (b) the mean square of the chirality $\kappa$, (c) the mean of $Z_2$ variables $W_h$ per dodecagon, ${\overline{W}}_h=(1/L^2)\sum _h{W}_h$, at $\alpha /\pi =0.3$. The corresponding data at $\alpha =0.4$ are shown in (d)–(f). The vertical dashed (dashed-dotted) line indicates $T_c$ ($T^{**}$). (g) Energy histograms at several $T$ in the vicinity of $T_c$ at $\alpha /\pi =0.3$. (h) Scaling collapse for $⟨{\kappa }^2⟩$ with $1/\nu =1.09$ and $\eta =0.18$ at $\alpha /\pi =0.4$.

Figure 3

$T$ dependences of the absolute values of (a) the Chern number $\nu$ and (b) the thermal Hall conductivity ${\kappa }^{ab}$ at $\alpha /\pi =0.3$. The corresponding data at $\alpha /\pi =0.4$ are shown in (c) and (d). The vertical dashed lines indicate $T_c$, and the dotted line in (b) represents $\pi T/12$.

Figure 4

$T$ dependences of (a) the specific heat and (b) the entropy per site at $\alpha /\pi =0.05$ and 0.4 in the $L=6$ cluster.