Quantum Ice: A Quantum Monte Carlo Study

Ice states, in which frustrated interactions lead to a macroscopic ground-state degeneracy, occur in water ice, in problems of frustrated charge order on the pyrochlore lattice, and in the family of rare-earth magnets collectively known as spin ice. Of particular interest at the moment are “quantum spin-ice” materials, where large quantum fluctuations may permit tunnelling between a macroscopic number of different classical ground states. Here we use zero-temperature quantum Monte Carlo simulations to show how such tunnelling can lift the degeneracy of a spin or charge ice, stabilizing a unique “quantum-ice” ground state—a quantum liquid with excitations described by the Maxwell action of $(3+1)$-dimensional quantum electrodynamics. We further identify a competing ordered squiggle state, and show how both squiggle and quantum-ice states might be distinguished in neutron scattering experiments on a spin-ice material.

Figure 1

Quantum mechanics enters ice physics through the tunnelling of the system from one ice configuration to another. The leading tunnelling matrix element for spin ice is the reversal of a set of Ising spins with closed circulation on a 6-link hexagonal plaquette.

Figure 2

Ground-state phase diagram of the quantum-ice model [Eq. (2)] as a function of the ratio $\mu$ of kinetic to potential energy. The 3D quantum-ice point $\mu =0$ exists deep within an extended quantum liquid phase with deconfined fractional excitations and algebraic decay of correlation functions.

Figure 3

The ice configuration possessing the most flippable plaquettes is the squiggle state, shown here within a 40-site tetragonal cell. Arrows show the displacement of protons within $Ic$ water ice or, equivalently, the orientation of spins in spin ice. The squiggle state possesses a net magnetic flux, $\overrightarrow{\varphi }={\overrightarrow{\varphi }}_{\mathrm{squiggle}}$, orientated along a [100] axis of the crystal. In a spin-ice material this would correspond to a net magnetization of $1/5$ the value at saturation, directed along a [100] axis.

Figure 4

Evidence for a phase transition out of the squiggle state. (a) Melting of squiggle order within the squiggle-flux sector $\overrightarrow{\varphi }={\overrightarrow{\varphi }}_{\mathrm{squiggle}}$. The relative number of flippable plaquettes $\nu =N_f(\mu )/N_f(-\infty )$ is calculated using quantum Monte Carlo simulations for clusters of 160 to 1080 sites. The solid lines show different Padé approximants to the series expansion about a perfectly ordered squiggle state. (b) Ground-state level crossing between the flux sector associated with the squiggle state, and the zero-flux sector associated with the quantum U(1) liquid. The ground-state energy per site is calculated for clusters of 432, 1024, and 2000 sites (squiggle-flux sector) and 160, 540, and 1280 sites (zero-flux sector), using quantum Monte Carlo simulation. A clear crossing is observed for $\mu =-0.50\pm 0.03$.

Figure 5

Evidence for the existence of a quantum U(1) liquid. (a) Flux dependence of finite-size energy gaps $E_{\varphi }-E_0$ for parameters bordering the RK point, $\mu =1$. Results are plotted for quantum Monte Carlo simulation of a 1280-site cluster (solid points) and perturbation theory about the RK point (open black circles). The dashed line indicates the scaling expected for a quantum U(1) liquid, following Eq. (1). (b) Flux dependence of finite-size energy gaps at the quantum-ice point, $\mu =0$. Results are obtained using quantum Monte Carlo simulation for 320-, 640-, and 1280-site clusters, and in exact diagonalization for an 80-site cluster. The dashed line indicates the scaling expected for a quantum U(1) liquid.

Figure 6

(a) Static structure factor $S(\mathbf{q})$ in the spin-flip channel for a classical spin ice, as measured by Fennel et al. [18], calculated here from the microscopic model Eq. (2), with $\mu =1$. Results are plotted in the ($h$, $h$, $l$) plane, and show the pinch-point structure associated with $1/r^3$ dipolar correlations in 3 dimensions. (b) Equivalent static structure factor $S(\mathbf{q})$ for a quantum spin ice described by Eq. (2) with $\mu =0$. In this case, correlations between spins show the $1/r^4$ behavior characteristic of dipoles in $(3+1)$ dimensions, and the pinch points are eliminated. All simulations were performed for a cubic cluster with 2000 lattice sites.