Bosonic many-body theory of quantum spin ice

We carry out an analytical study of quantum spin ice, a $\mathrm{U}\left(1\right)$ quantum spin liquid close to the classical spin-ice solution for an effective spin-$\frac{1}{2}$ model with anisotropic exchange couplings $J_{zz}$, $J_{\pm }$, and $J_{z\pm }$ on the pyrochlore lattice. Starting from the quantum rotor model introduced by Savary and Balents [Phys. Rev. Lett. 108, 037202 (2012)], we retain the dynamics of both the spinons and gauge field sectors in our treatment. The spinons are described by a bosonic representation of quantum XY rotors, while the dynamics of the gauge field is captured by a phenomenological Hamiltonian. By calculating the one-loop spinon self-energy, which is proportional to $J_{z\pm }^2$, we determine the stability region of the $\mathrm{U}\left(1\right)$ quantum spin-liquid phase in the $J_{\pm }/J_{zz}$ versus $J_{z\pm }/J_{zz}$ zero-temperature phase diagram. From these results, we estimate the location of the boundaries between the spin-liquid phase and classical long-range ordered phases.

Figure 1

The pyrochlore lattice and its medial lattice, the diamond lattice. The red and blue spheres are the A and B sublattices of the diamond lattice, respectively. $\mathbf{x}$ labels sites on the diamond lattice. $\alpha =0,1,2,3$ labels the four sites of the primitive basis of the pyrochlore lattice. $\widehat{\mu }$ denotes the four vectors connecting site $\mathbf{x}$ on the A sublattice to its nearest neighbors on the diamond lattice. The light-blue shaded region highlights a hexagonal plaquette on the diamond lattice. The lowest-order quantum tunneling process between two spin-ice configurations involves the flip of six alternating spins around the plaquette.

Figure 2

The spinon dispersion ${\omega }_{\mathbf{k}}$ along high-symmetry directions in the fcc Brillouin zone [59] for $j_{\pm }=0.08$.

Figure 3

Photon energy ${\epsilon }_{\mathbf{k}}$ along the high-symmetry directions in the fcc Brillouin zone [59] for $j_{\pm }=0.08$.

Figure 4

The stability region of the U(1) liquid phase (top) and the estimated phase diagram (bottom). The phase boundary between the $\mathrm{U}\left(1\right)$ liquid phase and the XY antiferromagnetic phase (vertical red dotted line), labeled as XY AFM, is determined based on the results of quantum Monte Carlo simulation in Ref. [43]. The blue dashed oval in the bottom panel is the perturbative stability region (shown in the top panel) of the U(1) liquid phase. The phase boundaries between the splayed ferromagnetic [53] (SFM) phase and the XY antiferromagnetic phase have not been determined in this work. Consequently, we do not draw a boundary between these phases in the top panel. See Refs. [8, 54, 61] for estimations of the phase boundaries between the SFM phase and the XY antiferromagnet long-range ordered phases.

Figure 5

The limiting behavior of the stability boundary $j_c\left(j_{\pm }\right)$ for $j_{\pm }\to 0$ (top) and $j_{\pm }\to \frac{1}{12}$ (bottom). ${\overline{j}}_{\pm }\equiv \frac{1}{12}-j_{\pm }$. The numerical data are plotted against fitted forms $0.167j_{\pm }^{1/2}$ and $0.169{\overline{j}}_{\pm }^{1/2}$, respectively.

Figure 6

${\chi }_0\equiv \left|\langle {\psi }_{\mathbf{x}}^{*}{\psi }_{\mathbf{x}+\widehat{0}}\rangle \right|$ is shown for $0\le \theta \le \pi /2$. $j_{\pm }=0.1$ and $j_{z\pm }=0.1$.