Quantum Liquid with Deconfined Fractional Excitations in Three Dimensions

Excitations which carry “fractional” quantum numbers are known to exist in one dimension in polyacetylene, and in two dimensions, in the fractional quantum Hall effect. Fractional excitations have also been invoked to explain the breakdown of the conventional theory of metals in a wide range of three-dimensional materials. However, the existence of fractional excitations in three dimensions remains highly controversial. In this Letter we report direct numerical evidence for the existence of an extended quantum liquid phase supporting fractional excitations in a concrete, three-dimensional microscopic model—the quantum dimer model on a diamond lattice. We demonstrate explicitly that the energy cost of separating fractional monomer excitations vanishes in this liquid phase, and that its energy spectrum matches that of the Coulomb phase in ($3+1$)-dimensional quantum electrodynamics.

Figure 1

Conjectured form for the ground state phase diagram of the quantum dimer model on a bipartite lattice in 3D, as a function of the ratio $\mu$ of potential to kinetic energy, following 6, 7, 18.

Figure 2

(a) Two adjacent 16-bond cubic unit cells of the diamond lattice, showing a dimer configuration in the maximally-flippable $R$ state. The two flippable hexagons contained within this picture are shaded blue. (b) Nonlocal “string” defect (dashed line) created by separating two monomer excitations.

Figure 3

(a) Order parameter $m_R$ of the $R$ state as a function of $\mu$, calculated using Green’s function Monte Carlo (GFMC) calculations for [100] clusters with 128, 432, 1024 and 2000 bonds (solid and dotted lines serve as a guide for the eye). Results are normalized such that $m_R=1$ for $\mu \to -\infty$. Exact diagonalization (ED) results for 128 bonds are also shown. A sharp jump in $m_R$ can be seen for ${\mu }_c\approx 0.5–0.7$, suggesting a first-order transition out of the $R$ state. (b) This coincides with a collapse in the string tension ${\Delta }_1/L$. The inset in panel (b) shows the finite-size scaling of $m_R$ at the RK point ($\mu =1$). The inset in panel (b) shows the finite-size scaling of ${\mu }_c$. We obtain a value ${\mu }_c=0.77\pm 0.02$.

Figure 4

Energy gap ${\Delta }_{\varphi }$ normalized to the square of the “magnetic” flux $\varphi$ and distance from RK point, $1-\mu$, plotted on a log-linear scale for values of $\mu$ spanning the ordered $R$ state and the proposed $U(1)$ liquid phase. In a $U(1)$ liquid, ${\Delta }_{\varphi }/{\varphi }^2\sim \mathrm{const}$, while deep within in the ordered $R$ state, ${\Delta }_{\varphi }/{\varphi }^2\sim 1/\varphi$. Here, a clear division is observed between $\mu \ge 0.7$ (liquid) and $\mu \le 0.6$ (ordered). Results are from Green’s function Monte Carlo (GFMC) calculations for a cluster of 1024 bonds.