Residual entropy and spin fractionalizations in the mixed-spin Kitaev model

We investigate the ground-state and finite-temperature properties of the mixed-spin $(s,S)$ Kitaev model. When one of the spins is a half integer and the other is an integer, we introduce two kinds of local symmetries, which results in a macroscopic degeneracy in each energy level. Applying the exact diagonalization to several clusters with $(s,S)=(1/2,1)$, we confirm the presence of this large degeneracy in the ground states, in contrast to conventional Kitaev models. By means of the thermal pure quantum state technique, we calculate the specific heat, entropy, and spin-spin correlations in the system. We find that in the mixed-spin Kitaev model with $(s,S)=(1/2,1)$, at least, the double-peak structure appears in the specific heat and the plateau in the entropy at intermediate temperatures, indicating the existence of spin fractionalization. Deducing the entropy in the mixed-spin system with $s,S\le 2$ systematically, we clarify that the smaller spin $s$ is responsible for the thermodynamic properties at higher temperatures.

Figure 1

(a) Mixed-spin Kitaev model on a honeycomb lattice. Solid (open) circles represent spin $s$ ($S$). Red, blue, and green lines denote $x$, $y$, and $z$ bonds between nearest-neighbor sites, respectively. (b) Plaquette with sites marked $1–6$ is shown for the corresponding operator $W_p$ defined in Eq. (2).

Figure 2

(a) Specific heat, (b) entropy, and (c) spin-spin correlation as a function of temperatures. Shaded areas stand for the standard deviation of the results obtained from the TPQ states.

Figure 3

(a) $\mathcal{S}-{\mathcal{S}}_{\infty }$ in the generalized $(s,S)$ Kitaev model at higher temperatures. Squares with lines represent data for the $S=1/2$ Kitaev model obtained from the Monte Carlo simulations [24, 25]. Circles, triangles, and diamonds with lines represent the TPQ data for the $S=1$, $3/2$, and 2 cases [28]. (b) $(\mathcal{S}-{\mathcal{S}}_{\infty })/ln(2s+1)$ and $C_s/\left(sS\right)$ as a function of $T/JS$.