Finite-temperature phase transition to a quantum spin liquid in a three-dimensional Kitaev model on a hyperhoneycomb lattice

The quantum spin liquid is an enigmatic entity that is often hard to characterize within the conventional framework of condensed matter physics. We here present theoretical and numerical evidence for the characterization of a quantum spin liquid phase extending from the exact ground state to a finite critical temperature. We investigate a three-dimensional variant of the Kitaev model on a hyperhoneycomb lattice in the limit of strong anisotropy; the model is mapped onto an effective Ising-type model, where elementary excitations consist of closed loops of flipped Ising-type variables on a diamond lattice. Analyzing this effective model by Monte Carlo simulation, we find a phase transition from the quantum spin liquid to a paramagnet at a finite critical temperature $T_c$, accompanied by a divergent singularity of the specific heat. We also compute the magnetic properties in terms of the original quantum spins. We find that the magnetic susceptibility exhibits a broad hump above $T_c$, while it obeys the Curie law at high temperature and approaches a nonzero Van Vleck–type constant at low temperature. Although the susceptibility changes continuously at $T_c$, its temperature derivative shows a critical divergence at $T_c$. We also clarify that the dynamical spin correlation function is momentum independent but shows quantized peaks corresponding to discretized excitations. Although the phase transition accompanies no apparent symmetry breaking in terms of the Ising-type variables or of the original quantum spins, we characterize it from a topological viewpoint. We find that, by defining the flux density for loops of the Ising-type variables, the transition can be interpreted as one occurring from the zero-flux quantum spin liquid to the nonzero-flux paramagnet; the latter has a Coulombic nature due to the local constraints. The role of global constraints on the Ising-type variables is examined in comparison with the results in the two-dimensional loop model. The correspondence of our model to the Ising model on a diamond lattice is also discussed. A possible relevance of our results to the recently discovered hyperhoneycomb compound $\beta$-${\text{Li}}_2$${\text{IrO}}_3$ is mentioned.

Figure 1

(a) Schematic picture of the hyperhoneycomb lattice structure and interactions in the 3D Kitaev model in Eq. (1). The dotted line represents a ten-site loop on which the conserved quantity $K_p$ in Eq. (2) is defined. (b) Schematic picture of the diamond lattice which is formed by contracting the dimers of $z$ bonds on the hyperhoneycomb lattice. The dotted hexagon represents a six-site loop corresponding to the ten-site loop in (a). The centers of four six-site loops sharing their edges are represented by small circles, which constitute a tetrahedral primitive cell of the pyrochlore lattice (see the main text and Fig. 3). (c) Schematic picture of the pyrochlore lattice on which the effective Hamiltonian in Eq. (3) is defined. The corresponding loop model is defined on the diamond lattice composed of the centers of tetrahedra connected by thick yellow lines.

Figure 2

Four kinds of inequivalent six-site loops on which $B_p$ are defined. Each $B_p$ is given by the product of six ${\tau }_m^l$ which are shown in the figure. See also Eq. (4).

Figure 3

(a) A loop configuration on the diamond lattice shown in Fig. 1. Light (dark) circles represent sites with $B_p=+1$ $(-1)$. (b) The shortest loop excitation consisting of six sites.

Figure 4

MC results for (a) $E$, (b) $C/T$, and (c) $S$ as functions of $T$. Insets in (a) and (c) are the plots in a wide $T$ range. The dotted curve in the inset of (a) shows $-tanh\left(T^{-1}\right)$ for comparison. The inset in (b) shows the peak temperature of $C$ as a function of $1/L$. The dotted curve represents the quadratic fit for the five largest $L$. The dotted horizontal line in the inset of (c) shows the value of $\frac{1}{2}ln2$.

Figure 5

Two kinds of inequivalent eight-site loops of $B_p$ whose $p\in {\mathcal{A}}_m$ (thick lines). These loops contribute the magnetic susceptibility in Eq. (11). See the text for details.

Figure 6

MC results for (a) the magnetic susceptibility ${\chi }^{zz}$ and (b) its narrow-$T$-range plot near $T_c$. In (a), the curve shows the Curie behavior $1/T$, and the horizontal dotted line represents the Van Vleck–type component ${\chi }_0=1/8$. (c) $T$ derivative $d{\chi }^{zz}/dT$. (d) Arrhenius plot for ${\chi }^{zz}-{\chi }_0$. The line shows a linear function whose slope is $-16$.

Figure 7

$T$ dependence of the magnetic susceptibility in the toric code limit of the Kitaev model on a honeycomb lattice. The curve shows the Curie behavior $1/T$.

Figure 8

(a) The dynamical spin correlation function ${\mathcal{S}}^{zz}$ as a function of $\omega$ at several $T$. The $\delta$ functions in Eq. (21) are shown by Lorentzian functions with the width of $0.1$. (b) $T$ dependence of the intensities $g\left(\mathcal{Q}\right)$ of the peaks in ${\mathcal{S}}^{zz}$. The definition of $g\left(\mathcal{Q}\right)$ is given in the text. The system size is $L=30$.

Figure 9

(a) MC results for $\langle {\overline{\varphi }}^2\rangle /L$. The inset in (a) shows $\langle {\overline{\varphi }}^2\rangle /L$ in a wide $T$ range. (b) $T$ dependence of $(\langle {\overline{\varphi }}^2\rangle /L)/L^{-z}$ and (c) scaling plot for $\langle {\overline{\varphi }}^2\rangle /L$. We assume $z=1$ in both (b) and (c). See the text for details.

Figure 10

Schematic pictures of (a) the loop configuration below $T_c$ and (b) that above $T_c$. The thin red (thick blue) lines represent the loops with zero (nonzero) flux.

Figure 11

MC results for (a) $C/T$ and (b) the loop parity parameter $\langle \eta \rangle$ for the effective model in Eq. (3) without the two global constraints. The inset in (a) shows the peak temperature of $C$ as a function of $1/L$. The dotted curve represents the quadratic fit for the five largest $L$.