Spin-Ice State of the Quantum Heisenberg Antiferromagnet on the Pyrochlore Lattice

We study the low-temperature physics of the SU(2)-symmetric spin-$1/2$ Heisenberg antiferromagnet on a pyrochlore lattice and find “fingerprint” evidence for the thermal spin-ice state in this frustrated quantum magnet. Our conclusions are based on the results of bold diagrammatic Monte Carlo simulations, with good convergence of the skeleton series down to the temperature $T/J=1/6$. The identification of the spin-ice state is done through a remarkably accurate microscopic correspondence for the static structure factor between the quantum Heisenberg, classical Heisenberg, and Ising models at all accessible temperatures, and the characteristic bowtie pattern with pinch points observed at $T/J=1/6$. The dynamic structure factor at real frequencies (obtained by the analytic continuation of numerical data) is consistent with diffusive spinon dynamics at the pinch points.

Figure 1

Sketch of the finite-temperature phase diagram for the $XXZ$ model based on perturbation theory. For $J_{xy}\ll J_{zz}$, the first crossover at $T\sim J_{zz}$ (dotted line) is to the thermal spin-ice state; it is followed by a second crossover at $T\sim J_{xy}^3/J_{zz}^2$ to the low-temperature $U(1)$ spin-liquid ground state. Whether the spin-ice state survives on approach to the isotropic Heisenberg point, $J_{xy}/J_{zz}=1$ is beyond perturbation theory.

Figure 2

Uniform susceptibility ${\chi }_u$ as a function of temperature from the DMC approach (red circles) and from the high temperature expansion (HTE) method [30] truncated at different expansion orders. Inset: ${\chi }_u$ at $T/J=1/2$ as a function of inverse maximal skeleton diagram order $N$. The error bar on the final answer, shown as the blue region, is a combination of statistical Monte Carlo errors for fixed-$N$ points and the systematic error of the extrapolation to the $N\to \infty$ limit.

Figure 3

Structure factor $S(\mathit{Q})$ in the $([hh0][00l])$ plane at $T/J=2$ (left panel) and $T/J=1/6$ (right panel). Note that the color scheme contrast (shown at the bottom) is significantly enhanced for the left panel.

Figure 4

Upper panel: normalized static susceptibilities (by modulus) $|{\chi }_s(\mathit{r})/{\chi }_s(0)|$ in the quantum Heisenberg, classical Heisenberg, and classical Ising models at temperatures $T_{\mathrm{QH}}/J=1/2$, $T_{\mathrm{CH}}/J=0.8340$, $T_{\mathrm{I}}/J=2.5374$ (left panel), and $T_{\mathrm{QH}}/J=1/6$, $T_{\mathrm{CH}}/J=0.4279$, $T_{\mathrm{I}}/J=1.4501$ (right panel). The QCC is satisfied within the error bars at all distances. Lower panel: quantum-to-classical temperature relationship plot of $T_{\mathrm{CH}}$ vs $T_{\mathrm{QH}}$. The straight black line is the high-$T$ relation $T_{\mathrm{CH}}=(4/3)T_{\mathrm{QH}}$. Inset: temperature relationship $T_{\mathrm{CH}}$ vs $T_{\mathrm{I}}$ between the classical Heisenberg and Ising systems. The straight black line is the high-$T$ relation $T_{\mathrm{CH}}=(1/3)T_{\mathrm{I}}$.

Figure 5

Dynamic structure factor as a function of frequency at the pinch point ${\mathit{Q}}_1=(0,0,2\pi /a)$ (left panel) and on the nodal line at ${\mathit{Q}}_2=(0,0,5\pi /4a)$ (right panel).