Quantum Excitations in Quantum Spin Ice

Recent work has highlighted remarkable effects of classical thermal fluctuations in the dipolar spin ice compounds, such as “artificial magnetostatics,” manifesting as Coulombic power-law spin correlations and particles behaving as diffusive “magnetic monopoles.” In this paper, we address quantum spin ice, giving a unifying framework for the study of magnetism of a large class of magnetic compounds with the pyrochlore structure, and, in particular, discuss ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, and extract its full set of Hamiltonian parameters from high-field inelastic neutron scattering experiments. We show that fluctuations in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are strong, and that the Hamiltonian may support a Coulombic “quantum spin liquid” ground state in low magnetic fields and host an unusual quantum critical point at larger fields. This appears consistent with puzzling features seen in prior experiments on ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. Thus, ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ is the first quantum spin liquid candidate for which the Hamiltonian is quantitatively known.

Popular Summary

A quantum spin liquid (QSL) is a seemingly paradoxical “magnet without magnetism.” Manifesting a purely quantum effect, microscopic spins in a QSL fluctuate even at absolute zero temperature, but they do so in a highly coordinated fashion. Given the theoretical predictions that QSLs can host exotic excitations with fractional quantum numbers and artificial gauge fields, how to find QSLs has been a hotly debated question since Anderson proposed them in 1973. In this experimental and theoretical paper, we identify a new set of candidate QSL materials that fall in the category of “quantum spin ice” compounds, and in particular, ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$.

We start with a theoretical model that describes how the microscopic (electronic) spins interact with each other. We then experimentally verify the validity of our model on the specific material ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ by showing its striking agreement with the results of neutron scattering. Investigating the model, we conclude that quantum spin fluctuations in ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ are strong and may support a QSL ground state where spin correlation follows a Coulombic-like power law. Going further, we also show that in these materials the QSL state behaves as a type of quantum electrodynamics—the theory that describes the interaction of light and matter in the real world.

Our work therefore paves the way for future experimental searches for QSLs and for new experiments manipulating analogs of quantum electric and magnetic monopoles and photons inside these unique substances.

Figure 1

The measured $S(\mathbf{Q},\omega )$ at $T=30\mathrm{mK}$, sliced along various directions in the HHL plane, for both $H=5\mathrm{T}$ (first row) and $H=2\mathrm{T}$ (third row). The second and fourth rows show the calculated spectrum for these two field strengths, based on an anisotropic exchange model with five free parameters (see text) that were extracted by fitting to the 5 T data set. For a realistic comparison to the data, the calculated $S(\mathbf{Q},\omega )$ is convoluted with a Gaussian of full-width 0.09 meV. Both the 2 T and 5 T data sets, composed of spin wave dispersions along five different directions, are described extremely well by the same parameters. (Note that r.l.u. stands for reciprocal lattice units.)

Figure 2

Representations of the HHL scattering plane, showing the FCC Brillouin-zone boundaries and the corresponding zone centers (labeled in terms of the conventional simple-cubic unit cell). Blue lines indicate the directions of the five different cuts shown in Fig. 1.

Figure 3

Schematic phase diagram in the temperature ($T$) magnetic-field ($H$) plane, for a material in the QSL phase of Eq. (1) at $T=H=0$. At a low field and temperature, the QSL state supports exotic excitations: “magnetic” (red sphere) and “electric” (yellow sphere) monopoles, and an emergent photon (wavy line). The field $H_c$ marks a quantum critical point: the confinement phase transition. For $H>H_c$, the ground state is a simple field-polarized ferromagnet (FM), and the elementary excitations are conventional magnetic dipoles. The gradations in gray scale indicate crossovers to the quantum critical region between them, governed by the $T=0$ confinement phase transition.

Figure 4

Dispersions (white curves) obtained from the fitting procedure over-plotted on the data at $H=5\mathrm{T}$.

Figure 5

Field versus temperature phase diagram obtained from a mean-field analysis of the Hamiltonian of Eq. (1), for a field $\mathbf{H}$ parallel to the [110] direction and with the exchange constant values obtained with our fits, Eq. (3). The system displays a net magnetic moment throughout the $(H,T)$ plane. The blue region denotes a region where the total magnetization lies in the $xy$ plane, and the green region is the paramagnetic phase; the two zones are separated by a continuous transition. In zero field, this transition takes place at $T_c^{\mathrm{MF}}=3.2\mathrm{K}$. At zero temperature, the amplitude of the transition field is $H_c^{\mathrm{MF}}=1.1\mathrm{T}$. The dark blue arrow shows the experimentally reported transition temperature of 240 mK: The actual transition occurs at a much lower temperature than that predicted by mean-field theory $T_c^{\mathrm{MF}}$.

Figure 6

One of the six ground states of $H_{3\mathrm{rdIsing}}^{\mathrm{eff}}=-\frac{3J_{z\pm }^2}{J_{zz}}\sum _{⟨⟨⟨i,j⟩⟩⟩}{\mathsf{S}}_i^z{\mathsf{S}}_j^z$. A blue sphere represents an “in” spin while a black sphere represents an “out” one. Because two of the four chain types contain nonalternating spins, there are no flippable hexagons in any of the six (equivalent) ground states.