Seeing the light: Experimental signatures of emergent electromagnetism in a quantum spin ice

The “spin-ice” state found in the rare-earth pyrochlore magnets Ho${}_2$Ti${}_2$O${}_7$ and Dy${}_2$Ti${}_2$O${}_7$ offers a beautiful realization of classical magnetostatics, complete with magnetic monopole excitations. It has been suggested that in “quantum spin-ice” materials, quantum-mechanical tunneling between different ice configurations, could convert the magnetostatics of spin ice into a quantum spin liquid that realizes a fully dynamical, lattice analogue of quantum electromagnetism. Here, we explore how such a state might manifest itself in experiment, within the minimal microscopic model of a such a quantum spin ice. We develop a lattice field theory for this model, and use this to make explicit predictions for the dynamical structure factor that would be observed in neutron scattering experiments on a quantum spin ice. We find that “pinch points,” which are the signal feature of a classical spin ice, fade away as a quantum ice is cooled to its zero-temperature ground state. We also make explicit predictions for the ghostly, linearly dispersing magnetic excitations which are the “photons” of this emergent electromagnetism. The predictions of this field theory are shown to be in quantitative agreement with quantum Monte Carlo simulations at zero temperature.

Figure 1

Spin correlations in a spin ice, as measured by quasielastic neutron scattering. (a) Correlations within the classical spin ice configurations, showing the characteristic pinch-point singularities at wave vector (1,1,1), etc. (b) Correlations in a quantum ice at $T=0$, showing the suppression of pinch points by quantum fluctuations. (c) Correlations in a quantum ice at an intermediate temperature $T=c{a}_0^{-1}$, showing how pinch points are progressively restored by the thermal excitation of magnetic photons. In all cases, results are shown for equal-time structure factors in the $(h,h,0)$ plane, for a polarized neutron scattering experiment in the spin-flip channel considered by Fennell et al.[9] Temperature is measured in units where $c$ is the speed of light associated with magnetic “photon” excitations, $a_0$ the lattice constant, and $\hslash =k_B=1$.

Figure 2

An illustration of the simplest tunneling process between different spin-ice configurations. The ice rules dictate that each tetrahedron within the lattice has two spins that point “in” and two that point “out.” Where these spins form a closed loop on a hexagonal plaquette—here shaded red—the sense of each spin within the loop can be reversed to give a new configuration that also obeys the ice rules.

Figure 3

Ghostly magnetic “photon” excitation as it might appear in an inelastic neutron scattering experiment on a quantum spin ice realising a quantum ice ground state. The photon dispersion $\omega \left(\mathbf{k}\right)$ is taken from lattice gauge theory developed in Sec. 2c of this paper, convoluted with a Gaussian representing the finite energy resolution of the instrument. The intensity of scattering vanishes as $I\propto \omega \left(\mathbf{k}\right)$ at low energies.

Figure 4

Zero-temperature phase diagram of the model of tunneling between ice states ${\mathcal{H}}_{\mu }$ [see Eq. (16)], as determined by quantum Monte Carlo simulation in Ref. 58. The “quantum ice” point, $\mu =0$, lies deep within a quantum liquid phase with low-energy excitations described by a lattice analog of quantum electromagnetism This extends from a “squiggle” ordered phase, found for $\mu <-0.5g$, to the exactly-soluble RK point $\mu =g$. (Here, $g$ is the strength of tunneling between ice states.)

Figure 5

Structure of the pyrochlore lattice realized by the magnetic ions in spin-ice materials. (a) The lattice is built of corner sharing tetrahedra, and can be decomposed into a set of A-sublattice tetrahedra (here coloured red) and B-sublattice tetrahedra (here coloured black), each of which forms an fcc lattice in its own right. The primitive unit of the pyrchlore lattice consists of a single tetrahedron with four lattice sites. However, it is also possible to define a cubic unit cell, of side $a_0$, containing 16 lattice sites. (b) Bipartite, diamond lattice formed by the centers of tetrahedra that make up the pyrochlore lattice. The bonds of this diamond lattice define the easy axes for spins in a spin ice and play an important role in the lattice gauge theory of its excitations.

Figure 6

The different fields used in Sec. 2b to construct lattice gauge theory of the spin liquid state, and the different lattices on which they are defined. (a) The “magnetic” field ${\mathcal{B}}_{\mathbf{r}{\mathbf{r}}^{\prime }}$ [see Eq. (26)], and its conjugate field ${\mathcal{G}}_{\mathbf{r}{\mathbf{r}}^{\prime }}$ [see Eq. (27)] are defined on the links of the diamond lattice, shown here in red. Each link of this diamond lattice corresponds to a site of the original pyrochlore lattice, and ${\mathcal{B}}_{\mathbf{r}{\mathbf{r}}^{\prime }}$ encodes the orientation of the spin on this site. (b) The compact $U\left(1\right)$ gauge field ${\mathcal{A}}_{\mathbf{s}{\mathbf{s}}^{\prime }}$ and the conjugate “electric” field ${\mathcal{E}}_{\mathbf{s}{\mathbf{s}}^{\prime }}$ are defined on the links of a second, dual, diamond lattice, shown here in blue. The midpoints of these bonds also form a second, dual, pyrochlore lattice, corresponding to the centers of hexagonal plaquettes in the original pyrochlore lattice. (c) An illustration of taking the lattice curl on the hexagonal plaquettes of the diamond lattice. The resulting vector lives on the links of the dual, diamond lattice.

Figure 7

Dispersion $\omega \left(\mathbf{k}\right)$ of magnetic photon excitations, calculated for the lattice field theory Eq. (40) in the “quantum-ice” limit $\mathcal{W}/\mathcal{K}\to 0$. The dispersion is plotted in the $(h,h,l)$ plane, following Eq. (67). The dispersion is linear in $\left|\mathbf{k}\right|$ in the long-wavelength limit, with a speed of light $c=\sqrt{\mathcal{U}\mathcal{K}}{a}_0$.

Figure 8

Dispersion $\omega \left(\mathbf{k}\right)$ of magnetic photon excitations, calculated for the lattice field theory Eq. (40) in the limit $\mathcal{U}/\mathcal{K}\to 0$. The dispersion is plotted in the $(h,h,l)$ plane, following Eq. (67). The dispersion is quadratic in $\mathbf{k}$ the long wavelength limit. This situation is realised in the microscopic “quantum ice” model ${\mathcal{H}}_{\mu }$ [Eq. (16)], at the RK point, $\mu =g$.

Figure 9

Comparison between the predictions of the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] and quantum Monte Carlo simulation of the microscopic model ${\mathcal{H}}_{\mu }$ [see Eq. (16)], for a quantum charge ice at $T=0$. (First column) Equal-time structure factor $S_{\mathsf{charge}}(\mathbf{k},t=0)$ calculated using Green's function Monte Carlo (GFMC) simulation of a 2000-site cubic cluster, for a range of $\mu$ ranging from $\mu /g=1$ (RK point) to $\mu /g=0$ (quantum ice). (Second column) Best fit of the finite-size (FS) prediction of the lattice field theory to simulation, following Eq. (83). There is excellent, quantitative, agreement between theory and simulation for all values of $\mu /g$. (Third column) Prediction of lattice field theory in the thermodynamic limit, for parameters obtained from fits to simulation.

Figure 10

Parameters of lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] as a function of $\mu /g$, from comparison with quantum Monte Carlo simulation of spin correlations in the microscopic model ${\mathcal{H}}_{\mu }$ [see Eq. (16)] at $T=0$. The dimensionless combinations of parameters $\overline{\mathcal{U}}$ and $\overline{\mathcal{W}}$ [see Eq. (85)] were obtained by fitting the predictions of the lattice field theory $S_{\mathsf{charge}}{(\mathbf{k},t=0)}_{T=0}$ [see Eq. (83)] to simulation results at fixed $\mu /g$ (see Fig 9).

Figure 11

Comparison between the predictions of the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] and quantum Monte Carlo simulation of the microscopic model ${\mathcal{H}}_{\mu }$ [see Eq. (16)], for a quantum spin ice at $T=0$. (First column) Equal-time structure factor $S_{\mathsf{spin}}^{yy}(\mathbf{k},t=0)$, as measured in neutron scattering by Fennell et al.,[9] calculated using Green's function Monte Carlo (GFMC) simulation of a 2000-site cubic cluster for parameters ranging from $\mu /g=1$ (RK point) to $\mu /g=0$ (quantum ice). (Second column) Best fit of the finite-size (FS) prediction of the lattice field theory to simulation, following Eq. (90). There is excellent, quantitative, agreement between theory and simulation for all values of $\mu /g$. (Third column) Prediction of lattice field theory in the thermodynamic limit, for parameters obtained from fits to simulation.

Figure 12

Relationship between the dispersion of the magnetic photon excitation $\omega \left(\mathbf{k}\right)$ [see Eq. (67)], and the equal time structure factor $S_{\mathsf{spin}}^{yy}(\mathbf{k},t=0)$ [see Eq. (90)] in a quantum spin ice. The photon dispersion $\omega \left(\mathbf{k}\right)$ in the $(h,h,l)$ plane is plotted above the corresponding equal-time structure factor, demonstrating how the photon disperses out of the (suppressed) pinch points at reciprocal lattice vectors. Note that the intensity of the scattering $S_{\mathsf{spin}}^{yy}(\mathbf{k},t=0)\to 0$ where $\omega \left(\mathbf{k}\right)\to 0$ [see Eq. (105)]. Results were calculated within the lattice field theory [see Eq. (40)] for $\mathcal{W}=0$, with energy measured in units such that $\hslash =1$.

Figure 13

Ghostly magnetic “photon” excitation as it might appear in an inelastic neutron scattering experiment on a quantum spin ice realising a quantum ice ground state, for a series of cuts along high symmetry directions in reciprocal space. The prediction of the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] for inelastic scattering by unpolarized neutrons, $I(\mathbf{k},\omega )$ [see Eq. (91)] has been convoluted with a Gaussian of variance $0.3c{a}_0^{-1}$ to represent the finite energy resolution of the instrument. The intensity of scattering vanishes as $\omega \to 0$ and is strongest at high energies. Energy is measured in units such that $\hslash =1$ and the photon dispersion calculated for $\mathcal{W}=0$.

Figure 14

Prediction of the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] for quasielastic neutron scattering performed using unpolarised neutrons, for comparison with Fig. 13. Results for $I\left(\mathbf{k}\right)$ are taken from Eq. (92), and calculated for $\mathcal{W}=0$. The path within the $[h,h,l]$ plane used for plotting the photon dispersion in Fig. 13 is shown using unbroken black arrows, with Brillouin zone boundaries marked as dashed white lines.

Figure 15

Finite-size scaling of the finite-size correction to the ground-state energy per site $\delta E_0/N$ found in quantum Monte Carlo simulations of the quantum ice model ${\mathcal{H}}_{\mu }$ [see Eq. (16)]. Results are shown for cubic clusters of $N=432$, $1024,$ and 2000 sites, for parameters $\mu /g=0$, $0.25$, $0.5$, $0.75$, as function of the linear dimension of the system $L=(a_0/2){(N/2)}^{\frac{1}{3}}$. The fact that $\delta E_0/N\sim 1/L^4$ implies the existence of a linearly dispersing excitation—the photon of the underlying lattice gauge theory.

Figure 16

Slow, cold death of pinch points in a quantum ice. Equal-time structure factor $S_{\mathsf{spin}}^{yy}(\mathbf{k},t=0)$ [see Eq. (90)] in the spin-flip channel measured by Fennell et al.,[9] calculated from the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)], for comparison with neutron scattering experiments on a quantum spin ice. Results are plotted for temperatures ranging from $T=0$ to $4.0c{a}_0^{-1}$, where $c$ is the speed of light and $a_0$ the linear dimension of the cubic unit cell, with temperature measured in units such that $\hslash =k_B=1$. The pinch-point structure observed at finite temperature is progressively “hollowed out” as the system is cooled towards its zero-temperature ground state.

Figure 17

Angle-integrated scattering intensity $I(k\approx 0,T)$ [see Eq. (124)] calculated from the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)], for comparison with neutron scattering experiments on a powder sample of a quantum spin ice. Results are plotted for temperatures ranging from $T=0$ to $1.0c{a}_0^{-1}$, where $c$ is the speed of light and $a_0$ the linear dimension of the cubic unit cell, with temperature measured in units such that $\hslash =k_B=1$. The progressive elimination of pinch points as the sample is cooled manifests itself as a steady loss of scattering for $\left|\mathbf{k}\right|\to 0$.

Figure 18

Comparison of the predictions of the lattice field theory ${\mathcal{H}}_{\mathsf{U}\left(1\right)}^{\prime }$ [see Eq. (40)] with the results of finite-temperature quantum Monte Carlo simulations of a quantum charge ice described by ${\mathcal{H}}_{\mathsf{t}-\mathsf{V}}$ [see Eq. (17)]. Results are shown for the equal time, on-sublattice structure factor $S_{00}\left(\mathbf{k}\right)=\langle n_0\left(-\mathbf{k}\right)n_0\left(\mathbf{k}\right)\rangle$ Simulations are taken from Banerjee et al.[59] and were performed at temperatures $T=1.05g$ and $1.57g$, where $g=12t^3/V^2$ is the size of the leading tunneling matrix element between different charge ice configurations. The temperature dependence of the spin correlations makes it possible to estimate the speed of light $c\approx 1.8g{a}_0{\hslash }^{-1}$.