Residual Entropy and Spin Fractionalizations in the Mixed-Spin Kitaev Model

TL;DR

This study explores the unique ground-state and thermal properties of the mixed-spin Kitaev model, revealing macroscopic degeneracy, spin fractionalization, and the influence of smaller spins on thermodynamics.

Contribution

It introduces local symmetries in the mixed-spin Kitaev model, demonstrating macroscopic degeneracy and spin fractionalization, which differ from conventional models.

Findings

Presence of macroscopic degeneracy in ground states

Double peak structure in specific heat at finite temperatures

Entropy plateau indicating spin fractionalization

Abstract

We investigate ground-state and finite temperature properties of the mixed-spin $(s, S)$ Kitaev model. When one of spins is half-integer and the other is 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 the 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 the 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.

Tables

Table 1. Table 1: Ground state energy E g subscript 𝐸 𝑔 E_{g} and its degeneracy N d subscript 𝑁 𝑑 N_{d} in the mixed-spin ( s , S ) 𝑠 𝑆 (s,S) Kitaev models with the twelve-site clusters.

$s$	$S$	$N=12a$		$N=12b$
		$E_g/JN$	$N_d$	$E_g/JN$	$N_d$
1/2	1/2	-0.20417	4	-0.21102	1
1/2	1	-0.33533	8	-0.34235	8
1/2	3/2	-0.47208	4	-0.47389	1
1/2	2	-0.60214	8	-0.60260	8
1	1	-0.66487	1	-0.67421	1
1	3/2	-0.92855	8	-0.93437	8
1	2	-1.19271	1	-1.19567	1
3/2	3/2	-1.37169	4	-1.38840	1
3/2	2	-1.76901	8	-1.77691	8
2	2	-2.33449	1	-2.35306	1

Table 2. Table 2: Ground state profile for several clusters in the Kitaev model with ( s , S ) = ( 1 / 2 , 1 ) 𝑠 𝑆 1 2 1 (s,S)=(1/2,1) . N p subscript 𝑁 𝑝 N_{p} is the number of plaquettes, where the local operator W p subscript 𝑊 𝑝 W_{p} is diagonal in the basis set. N d subscript 𝑁 𝑑 N_{d} is the degeneracy in the ground state.

$N$	$N_p$	$E_g/JN$	$\Delta /J$	$N_d$	$N$	$N_p$	$E_g/JN$	$\Delta /J$	$N_d$
12a	1	-0.33981	0.0071	8	20a	2	-0.33550	0.0013	20
12b	2	-0.34235	0.0024	8	20b	3	-0.34210	0.0041	32
16a	2	-0.33543	0.0002	20	22	2	-0.33531	0.0016	20
16b	2	-0.33895	0.0019	16	24a	4	-0.33525	0.0031	64
18a	3	-0.33533	0.0018	8	24b	4	-0.33511	0.0010	64
18b	2	-0.33537	0.0015	40

Equations

$$\mathcal{H}$$

$$W_p$$

Full text

Kitaev model Kitaev (2006) and its related models have attracted much interest in condensed matter physics since the possibility of the direction-dependent Ising interactions has been proposed in the realistic materials Jackeli and Khaliullin (2009). Among them, low temperature properties in the candidate materials such as $A_{2}{\rm IrO_{3}}$ ($A={\rm Na,K}$) Singh and Gegenwart (2010); Singh et al. (2012); Choi et al. (2012); Comin et al. (2012); Modic et al. (2014); Takayama et al. (2015); Kitagawa et al. (2018) and $\alpha$-${\rm RuCl_{3}}$ Plumb et al. (2014); Kubota et al. (2015); Sears et al. (2015); Majumder et al. (2015); Kasahara et al. (2018) have been examined extensively. To clarify the experimental results, the roles of the Heisenberg interactions Chaloupka et al. (2010); Jiang et al. (2011); Singh and Oitmaa (2017), off-diagonal interactions Katukuri et al. (2014); Suzuki et al. (2015), interlayer coupling Tomishige et al. (2018); Seifert et al. (2018); Tomishige et al. , and the spin-orbit couplings Koga et al. (2018a) have been theoretically investigated for both ground state and finite temperature properties. One of the important issues characteristic of the Kitaev models is the fractionalization of the spin degree of freedom. In the Kitaev model with $S=1/2$ spins, the spins are exactly shown to be fractionalized into itinerant Majorana fermions and localized fluxes, which manifest themselves in the ground state and thermodynamic properties Nasu et al. (2015, 2017). It has been observed as the half-quantized thermal quantum Hall effects, which is a clear evidence of the Majorana quasiparticles fractionalized from quantum spins Kasahara et al. (2018). Recently, the Kitaev model with larger spins has theoretically been examined Baskaran et al. (2008); Suzuki and Yamaji (2018); Koga et al. (2018b); Oitmaa et al. (2018); Peter Stavropoulos et al. . In the spin-$S$ Kitaev model, the specific heat exhibits double peak structure, and plateau appears in the temperature dependence of the entropy Koga et al. (2018b). This suggests the existence of the fractionalization even in this generalized Kitaev model. However, it is still hard to explain how the spin degree of freedom is divided in the generalized Kitaev models beyond the exactly solvable $S=1/2$ case Kitaev (2006); Nasu et al. (2015, 2017).

The key to understand the “fractionalization” in the spin-$S$ Kitaev model should be the multiple entropy release phenomenon. The half of spin entropy $\sim\frac{1}{2}\ln(2S+1)$ in higher temperatures emerges with a broad peak in the specific heat. Then, a question arises how the plateau structure appears in the entropy in the Kitaev model composed of multiple kinds of spins (the mixed-spin Kitaev model). In other words, how is the many-body state realized in the system, with decreasing temperatures? The extension to the mixed-spin models should be a potential to exhibit an intriguing nature of the ground states. In fact, the mixed-spin quantum Heisenberg model has been examined Fujii et al. (1996); Pati et al. (1997); Fukui and Kawakami (1997); Tonegawa et al. (1998); Koga et al. (1998, 1999); Kolezhuk et al. (1997); Takushima et al. (2000), and the topological nature of spins and lattice plays an important role in stabilizing the non-magnetic ground states. Moreover, mixed-spin Kitaev model can be realized by replacing transition metal ions to other ions in the Kitaev candidate materials. Therefore, it is desired to study this model to discuss the nature of the spin fractionalization in the Kitaev system.

In this manuscript, we investigate the mixed-spin Kitaev model, where two distinct spins $(s,S)\;[s<S]$ are periodically arranged on the honeycomb lattice (see Fig. 1). First, we show the existence of the $Z_{2}$ symmetry in each plaquette in the system. In addition, by considering another local symmetry, we show that the macroscopic degeneracy exists in each energy level when one of the spins is half-integer and the other integer. The exact diagonalization (ED) in the system with $(s,S)=(\frac{1}{2},1)$ reveals that the ground state has a macroscopic degeneracy, which is consistent with the presence of the two kinds of local symmetries. Using thermal pure quantum (TPQ) state methods Sugiura and Shimizu (2012, 2013), we find that, at least, the double peak structure appears in the specific heat and the plateau appears at intermediate temperatures in the entropy, which are similar to those in the spin-$S$ Kitaev models Koga et al. (2018b). From systematic calculations for the mixed-spin systems with $s,S\leq 2$, we clarify that the smaller spin-$s$ is responsible for the high-temperature properties. The deconfinement picture to explain the “spin fractionalization” in the Kitaev model is addressed.

We consider the Kitaev model on a honeycomb lattice, which is given by the following Hamiltonian as

[TABLE]

where $s_{i}^{\alpha}(S_{i}^{\alpha})$ is the $\alpha(=x,y,z)$ component of a spin-$s(S)$ operator at the $i$th site. $J$ is the exchange constant between the nearest neighbor spin pairs $\langle i,j\rangle_{\gamma}$. The model is schematically shown in Fig. 1(a). We consider here the following local Hermite operator defined on each plaquette $p$ as,

[TABLE]

where $\eta=[3(s+S)]$ is a phase factor. By using the following relation for the spin operators, $e^{i\pi S^{\alpha}}S^{\beta}e^{-i\pi S^{\alpha}}=(2\delta_{\alpha\beta}-1)S^{\beta},$ we find $[{\cal H},W_{p}]=0$ for each plaquette and $W_{p}^{2}=1$. Therefore, the mixed-spin Kitaev system has a $Z_{2}$ local symmetry.

It is known that this local $Z_{2}$ symmetry is important to understand ground state properties in the Kitaev model. We wish to note that the local operator $W_{p}$ on a plaquette $p$ commutes with those on all other plaquettes in the spin-$S$ Kitaev models, while this commutation relation is not always satisfied in the present mixed-spin Kitaev model. In fact, we obtain $[W_{p},W_{q}]\propto[e^{i\pi(s_{i}^{x}+S_{j}^{y})},e^{i\pi(s_{i}^{y}+S_{j}^{x})}]\propto\sin[\pi(s_{i}^{y}-S_{j}^{x})]$ when the plaquettes $p$ and $q$ share the same $z$ bond $\langle ij\rangle_{z}$. This means that the local operator does not commute with the adjacent ones in the mixed-spin Kitaev model with one of spins being half-integer and the other integer. Instead, we introduce another local symmetry specific in this case. When either $s$ or $S$ is half-integer and the other is integer, the Hilbert space is divided into subspaces specified by the set of the eigenvalues $w_{p}(=\pm 1)$ of the $N_{p}(\leq N/6)$ local operators $W_{p}$ defined on the plaquettes $p\in{\cal P}$, where ${\cal P}$ is a set of the plaquettes whose corners are not shared with each other. Now, we assume the presence of the local operator $R_{p}$ on a plaquette $p(\in{\cal P})$ so as to satisfy the conditions, $R_{p}^{2}=1$ and the following commutation relations $[{\cal H},R_{p}]=0$, $[W_{p},R_{q}]=0\;(p\neq q)$, and $\{W_{p},R_{p}\}=0$. In the case that half-integer and integer spins are mixed, such an operator can be introduced so that the spins located on its corners are inverted as ${\bf S}_{2i-1}\rightarrow-{\bf S}_{2i-1},{\bf s}_{2i}\rightarrow-{\bf s}_{2i}\;(i=1,2,3)$ and the signs of the six exchange constants are changed on the bonds connecting with a corner site belonging to the plaquette, shown as the dashed lines in Fig. 1(b). When a wavefunction for the energy level $E$ is given by the set of $\{w_{p}\}$ as $|\psi\rangle=|\psi;\{w_{1},w_{2},\cdots,w_{p},\cdots\}\rangle$, we obtain ${\cal H}|\psi^{\prime}\rangle=E|\psi^{\prime}\rangle$ with the wave function $|\psi^{\prime}\rangle=R_{p}|\psi\rangle=|\psi;\{w_{1},w_{2},\cdots,-w_{p},\cdots\}\rangle$. Since the operators $R_{p}$ for arbitrary plaquettes in ${\cal P}$ generates degenerate states, the presence of $R_{p}$ results in, at least, $2^{N/6}$-fold degenerate ground states.

This qualitative difference in the spin magnitudes $s$ and $S$ can be confirmed in the small clusters. By using the ED method, we obtain ground state properties in the twelve-site systems, as shown in Table 1. We clearly find that, as for the ground-state degeneracy, the mixed-spin systems can be divided into three groups. When both spins $s$ and $S$ are integer, the ground state is always singlet. In the half-integer case, the four-fold degenerate ground state is realized in the $N=12a$ system, while the singlet ground state is realized in the $N=12b$ system. This feature is essentially the same as ground state properties in the $S=1/2$ Kitaev model, where the ground-state degeneracy depends on the topology in the boundary condition. By contrast, the eight-fold degenerate state is realized in the system with one of spins being half-integer and the other integer, which suggests the macroscopic degeneracy in the thermodynamic limit.

To confirm this, we focus on the mixed-spin system with $(s,S)=(1/2,1)$. By using the ED method, we obtain the ground-state energies for several clusters up to 24 sites [see Fig. 1(a)]. The obtained results are shown in Table 2. It is clarified that a finite size effect slightly appears in the ground state energy, and its value is deduced as $E_{g}/JN=-0.335$. We also find that the ground state is $N_{\cal S}(=N_{d}/2^{N_{p}})$-fold degenerate in each subspace and its energy is identical in all subspaces ${\cal S}[\{w_{p}\}]$ except for the $N=18a$ system N (18). The large ground-state degeneracy $N_{d}\geq 2^{N/6}$ is consistent with the above conclusion. We also find that the first excitation energy $\Delta$ is much smaller than the exchange constant $J$, as shown in Table 2. These imply the existence of multiple low-energy states in the system.

Next, we consider thermodynamic properties in the Kitaev model. It is known that there exist two energy scales in the $S=1/2$ Kitaev model Kitaev (2006), which clearly appear as double peak structure in the specific heat and a plateau in the entropy Nasu et al. (2015, 2017). Similar behavior has been reported in the spin-$S$ Kitaev model Koga et al. (2018b). These suggest the existence of the fractionalization in the generalized spin-$S$ Kitaev model. An important point is that the degrees of freedom for the high energy part depend on the magnitude of spins $\sim(2S+1)^{N/2}$. On the other hand, in the mixed-spin case, it is unclear which spin is responsible for the high-temperature properties.

Here, we calculate thermodynamic quantities for twelve-site clusters, by diagonalizing the corresponding Hamiltonian. Furthermore, we apply the TPQ state method Sugiura and Shimizu (2012, 2013) to larger clusters. In this calculation, the thermodynamic quantities are deduced by the statistical average of the results obtained from, at least, 25 independent TPQ states. Here, we calculate specific heat $C(T)=dE(T)/dT$, entropy $S(T)=S_{\infty}-\int_{T}^{\infty}C(T^{\prime})/T^{\prime}dT^{\prime}$, and the nearest-neighbor spin-spin correlation $C_{S}(T)=\langle s_{i}^{\alpha}S_{j}^{\alpha}\rangle_{\alpha}=-2E(T)/(3J)$, where $S_{\infty}=\frac{1}{2}\ln(2s+1)(2S+1)$ and $E(T)$ is the internal energy per site.

The results for the mixed-spin systems with $(s,S)=(1/2,1)$ are shown in Fig. 2. We clearly find the multiple-peak structure in the specific heat. Note that finite size effects appear only at low temperatures. Therefore, our TPQ results for the 24 sites appropriately capture the high temperature properties ($T\gtrsim 0.01J$) in the thermodynamic limit. Then, we find the broad peak around $T_{H}\sim 0.6J$, which is clearly separated by the structure at low temperatures ($T<0.01J$). Now, we focus on the corresponding entropy, which is shown in Fig. 2(b). This indicates that, decreasing temperature, the entropy monotonically decreases and the plateau structure is found around $T/J\sim 0.1$. The released entropy is $\sim\frac{1}{2}\ln 2$, which is related to the smaller spin $(s=1/2)$. Therefore, multiple temperature scales do not appear at high temperatures although the system is composed of two kinds of spins $(s,S)$. However, it does not imply that only smaller spins are frozen and larger spins remain paramagnetic at the temperature since the spin-spin correlations develop around $T\sim T_{H}$ and a quantum many-body spin state is formed, as shown in Fig. 2(c). We have also confirmed that local magnetic moments do not appear even in the wavefunction constructed by the superposition of the ground states with different configurations of $\{w_{p}\}$.

By contrast, the value $\frac{1}{2}\ln 2$ reminds us of the high-temperature feature for itinerant Majorana fermions in spin-1/2 Kitaev model Nasu et al. (2015, 2017). Then, one expects that, in the mixed-spin $(s,S)$ Kitaev model, higher temperature properties are described by the smaller spin-$s$ Kitaev model, where degrees of freedom $\sim(2s+1)^{1/2}$ are frozen at each site Koga et al. (2018b). In the case, a peak structure appears in the specific heat and the plateau structure at $\sim S_{\infty}-\ln(2s+1)/2$ in the entropy. These interesting properties at higher temperatures will be examined systematically. Further decrease of temperatures decreases the entropy and finally $S\sim S_{\infty}-\ln 2$ at lower temperatures, as shown in Fig. 2(b). This may suggest that thermodynamic properties in this mixed-spin Kitaev model with $(s,S)=(1/2,1)$ are governed by two kinds of fractional quasiparticles originating from the smaller $s=1/2$ spin by analogy with the spin fractionalization in the spin-1/2 Kitaev model. In the case, the existence of the remaining entropy $S\sim S_{\infty}-\ln 2$ should be consistent with macroscopic degeneracy in the ground state as discussed before. However, our TPQ data have a large system size dependence at low temperatures, and conclusive results could not be obtained. Therefore, a systematic analysis is desired to clarify the nature of low temperature properties.

To clarify the role of the smaller spins in the mixed-spin Kitaev models, we calculate the entropy in the systems with $s,S\leq 2$ and $N=12a$ by means of the TPQ state methods. The results are shown in Fig. 3.

The plateau structure is clearly observed in the curve of the entropy in the mixed-spin Kitaev models. In addition, we find that the plateau is located around $S=S_{\infty}-\frac{1}{2}\ln(2s+1)$, as expected above. Therefore, we can say that, decreasing temperatures, the half of the degree of freedom in the smaller spin-$s$ are released.

This may be explained by the deconfined-spin picture in the Kitaev model. In the picture, each spin $S$ is divided into two kinds of quasiparticles with distinct energy scales: $2S$ $L$-quasiparticles and $2S$ $H$-quasiparticles, which are dominant at lower and higher temperatures, respectively. In the exactly solvable $S=1/2$ Kitaev model, $H$- ($L$-)quasiparticles are identical to itinerant Majorana fermions (localized fluxes). In addition, this should explain the double peak structure in the specific heat of the spin-$S$ Kitaev model, each of which corresponds to the half entropy release Koga et al. (2018b). In our mixed-spin $(s,S)$ system, the entropy release at higher temperatures can be interpreted as follows: $2s$ fractional $H$-quasiparticles are present with the energy scale of $\sim J$. On the other hand, remaining $H$-quasiparticles originating from larger spin-$S$ posses the energy that is much smaller than $J$ due to the absense of the two-dimensional network. Therefore, only $2s$ $H$-quasiparticles form the many-body state at high temperatures, resulting in the plateau structure in the entropy.

Interestingly, the temperature $T^{*}$ characteristic of the plateau in the entropy, which may be defined such that $S(T^{*})=S_{\infty}-\frac{1}{2}\ln(2s+1)$, depends on the magnitude of the larger spin. In fact, we find that $T^{*}$ should be scaled by the larger spin $T^{*}\sim JS$, which is shown in Fig. 3(b). This is in contrast to the conventional temperature scale $T^{**}\sim J\sqrt{s(s+1)S(S+1)}$, which is derived from the high-temperature expansion. This discrepancy is common to the spin-$S$ Kitaev model Koga et al. (2018b), implying that quantum fluctuations are essential even in this temperature range in the mixed-spin Kitaev models. As for the spin-spin correlation, decreasing temperatures, it develops around $T/JS\sim 1$ and is almost saturated around $T^{*}$, as shown in Fig. 3(b). This means that the many-body spin state is indeed realized at the temperature. We also find that at low temperatures, the normalized spin-spin correlation $C_{s}/(sS)\sim 0.4$ is less than unity when $s$ and $S$ are large. This suggests that the quantum spin liquid state is, in general, realized in the generalized mixed-spin Kitaev model, which is consistent with the presence of magnetic fluctuations even in the classical limit Suzuki and Yamaji (2018).

In summary, we have studied the mixed-spin Kitaev model. First, we have clarified the existence of the local $Z_{2}$ symmetry at each plaquette. We could introduce an operator $R_{p}$ on the plaquette $p$ so as to (anti)commute with the Hamiltonian ($W_{p}$), which leads to the macroscopic degeneracy for each energy level in the mixed-spin system with one of spins being half-integer and the other integer. Using the TPQ state methods for several clusters, we have found the double peak structure in the specific heat and plateau in the entropy, which suggests the existence of the fractionalization in the mixed-spin system. Deducing the entropy in the mixed-spin system with $s,S\leq 2$ systematically, we have clarified that the smaller spin plays a crucial role in the thermodynamic properties at higher temperatures. We expect that the present mixed-spin Kitaev systems are realizable in the real materials by substituting the magnetic ions in the Kitaev candidate materials to other magnetic ions with larger spins, and therefore, the present work should stimulate material researches for mixed-spin Kitaev systems.