Fractional Spin Fluctuations as a Precursor of Quantum Spin Liquids: Majorana Dynamical Mean-Field Study for the Kitaev Model

Experimental identification of quantum spin liquids remains a challenge, as the pristine nature is to be seen in asymptotically low temperatures. We here theoretically show that the precursor of quantum spin liquids appears in the spin dynamics in the paramagnetic state over a wide temperature range. Using the cluster dynamical mean-field theory and the continuous-time quantum Monte Carlo method, which are newly developed in the Majorana fermion representation, we calculate the dynamical spin structure factor, relaxation rate in nuclear magnetic resonance, and magnetic susceptibility for the honeycomb Kitaev model whose ground state is a canonical example of the quantum spin liquid. We find that dynamical spin correlations show peculiar temperature and frequency dependence even below the temperature where static correlations saturate. The results provide the experimentally accessible symptoms of the fluctuating fractionalized spins evincing the quantum spin liquids.

Figure 1

(a) Schematic picture of the Kitaev model on the honeycomb lattice [Eq. (1)] and the mapping to a 26-site cluster used in the Majorana CDMFT. (b) The specific heat $C_v$ and equal-time spin correlations for NN sites $⟨S_j^z{S}_k^z⟩$ obtained by the Majorana CDMFT for the isotropic FM case. QMC data in Ref. [17] are plotted by gray symbols for comparison.

Figure 2

Dynamical spin structure factor $S(\mathbf{q},\omega )$ obtained by the Majorana $\mathrm{CDMFT}+\mathrm{CTQMC}$ calculation for the (a),(c),(e),(g) FM and (b),(d),(f),(h) AFM cases at (a),(b) $T\simeq 0.018$, (c),(d) $T\simeq 0.094$, (e),(f) $T\simeq 0.24$, and (g),(h) $T\simeq 2.4$.

Figure 3

$S(\mathrm{\Gamma },\omega )$ for the (a) FM and (c) AFM cases and (e) $S(K,\omega )$ at several $T$. (e) is common to the FM and AFM cases. The corresponding contour plots in the $T\text{−}\omega$ plane are shown in (b),(d),(f). The arrows indicate the temperatures used for the data in (a),(c),(e). The dashed curves represent the average frequency of $S(\mathbf{q},\omega )$ (see the text for details). In (a),(c),(e), the error bars are shown for every ten points.

Figure 4

$T$ dependences of (a) the NMR relaxation rate $1/T_1$ and (b) the magnetic susceptibility $\chi$. In (b), the dashed curves represent the Curie-Weiss behavior ${\chi }_{\mathrm{CW}}$.