Dynamics of Topological Excitations in a Model Quantum Spin Ice

We study the quantum spin dynamics of a frustrated $XXZ$ model on a pyrochlore lattice by using large-scale quantum Monte Carlo simulation and stochastic analytic continuation. In the low-temperature quantum spin ice regime, we observe signatures of coherent photon and spinon excitations in the dynamic spin structure factor. As the temperature rises to the classical spin ice regime, the photon disappears from the dynamic spin structure factor, whereas the dynamics of the spinon remain coherent in a broad temperature window. Our results provide experimentally relevant, quantitative information for the ongoing pursuit of quantum spin ice materials.

Figure 1

(a) $XXZ$ model on pyrochlore lattice. Gray box shows a cubic unit cell along with the orientation of the cubic axes $a$, $b$, $c$. Small red and blue spheres denote $S^z=1/2$ and $-1/2$ states respectively. Starting from a spin configuration in the ice manifold ($a_1$), one may flip a spin and create a pair of spinons with charges $Q=1$ (gold sphere) and $Q=-1$ (light green sphere) residing on neighboring tetrahedra ($a_2$). The spinons may propagate in the lattice by flipping a string of spins (red solid line) ($a_3$). (b) Thermal entropy $\mathcal{S}$ (orange open circles, right vertical axis) and specific heat $C$ (red open squares, left vertical axis) as a function of temperature $T$. The regions corresponding to the trivial paramagnetic regime, the classical spin ice regime, and the quantum spin ice regime are shaded in red, green, and blue, respectively. Bright yellow arrows mark the temperatures at which we carry out QMC study.

Figure 2

Left panel: (a),(b) Dynamic spin structure factor $S^{zz}(\mathbf{q},\omega )\equiv \sum _{\alpha }{S}_{\alpha \alpha }^{zz}(\mathbf{q},\omega )$ obtained from QMC-SAC at temperature $T_1$ along high symmetry cubic directions (010) and (111). The photon appears as a gapless branch of excitation with dispersion starting from Brillouin Zone center. White dots mark the position of spectral peaks. Pink open circles show the integrated spectral weight at each momentum point with maximal spectral weight rescaled to 1. (a’),(b’) Photon spectra calculated from a Gaussian QED model. Right panel: (c),(d) Dynamic spin structure factor $S^{+-}(\mathbf{q},\omega )\equiv \sum _{\alpha }{S}_{\alpha \alpha }^{+-}(\mathbf{q},\omega )$ obtained from QMC-SAC at $T_1$. The spectra show a dispersive continuum of two-spinon excitations. (c’),(d’) The results from a tight-binding model calculation, where the spinons are modeled as free particles. The calculated spectra are then broadened with a Lorentzian to mimic interaction effects. The spinon continuum boundaries calculated from the tight-binding model are marked as white dots in both QMC-SAC spectra (c),(d) and the theoretical spectra (c’),(d’).

Figure 3

Left panel: Dynamic spin structure factor $S^{zz}(\mathbf{q},\omega )$ obtained from QMC-SAC at temperature $T_2$ (a),(b) and $T_3$ (a’),(b’). White dots mark the position of spectral peaks. Both $T_2$ and $T_3$ are inside the classical spin ice regime. The photon disappears, and the spectra are diffusive. The peak positions in (a’),(b’) are slightly above the horizontal ($\omega =0$) axis. This is likely an artifact due to the uncertainties in the SAC method. Note the $\omega$ axis scale of (a’),(b’) is different from (a),(b). Right panel: Dynamic spin structure factor $S^{+-}(\mathbf{q},\omega )$ obtained from QMC-SAC at temperature $T_2$ (c),(d) and $T_3$ (c’),(d’). Comparing to the spectra at $T_1$, the spinon continuum is still present but with narrower and flatter dispersion at $T_2$. At $T_3$, the continuum becomes dispersionless.