Spectrum of Itinerant Fractional Excitations in Quantum Spin Ice

We study the quantum dynamics of fractional excitations in quantum spin ice. We focus on the density of states in the two-monopole sector, $\rho (\omega )$, as this can be connected to the wave-vector-integrated dynamical structure factor accessible in neutron scattering experiments. We find that $\rho (\omega )$ exhibits a strikingly characteristic singular and asymmetric structure that provides a useful fingerprint for comparison to experiment. $\rho (\omega )$ obtained from the exact diagonalization of a finite cluster agrees well with that, from the analytical solution of a hopping problem on a Husimi cactus representing configuration space, but not with the corresponding result on a face-centered cubic lattice, on which the monopoles move in real space. The main difference between the latter two lies in the inclusion of the emergent gauge field degrees of freedom, under which the monopoles are charged. This underlines the importance of treating both sets of degrees of freedom together, and it presents a novel instance of dimensional transmutation.

Figure 1

(a) Pyrochlore lattice. $\{{\overrightarrow{a}}_1,{\overrightarrow{a}}_2,{\overrightarrow{a}}_3\}$ are the lattice vectors of the fcc lattice of upward tetrahedra. A monopole on an upward tetrahedron, $n^{\prime }$, hops to the neighboring upward tetrahedron, $n$, by the process $a_n^{†}{\sigma }_j^x{\sigma }_{j^{\prime }}^x{a}_{n^{\prime }}$ in Eq. (2), by flipping two intervening spins $j$ and $j^{\prime }$. (b) Schematic picture of the Husimi cactus. (c), (d) Pyrochlore lattice seen along the $[111]$ direction. (c) A monopole hops twice following the solid arrows back to the initial tetrahedron. Note that the spins also return to their initial configuration, Similar three-step motions are possible for the other two choices of initial hopping directions (dashed lines). These hopping processes imply a mapping to the Husimi cactus, (b). (d) If a monopole comes back to the initial tetrahedron along a larger loop, it goes along with flipping the six spins marked by dashed circles.

Figure 2

(a) ${\chi }_{ii}(\omega )$ and two-monopole density of states, $\rho (\omega )$, from exact diagonalization of a 32-site cluster. $\rho (\omega )$ of the tight-binding model on Husimi cactus and fcc lattice are shown for comparison. (b) One-particle density of states on Husimi cactus and fcc lattice. (c) Schematic picture of the wave function at the lower spectral edge $\omega =-6$, depicted on the graph of many-body states.

Figure 3

(a) Two-monopole density of states, $\rho (\omega )$, ${\rho }_{\mathrm{c}}(\omega )$, and ${\rho }_{\mathrm{nc}}(\omega )$. The inset shows the comparison between rescaled $\rho (\omega )$ and ${\tilde{\rho }}_{\mathrm{nc}}(\omega )$. (b)–(c) Schematic picture of (b) contractible and (c) noncontractible monopole pairs. These two configurations are transformed to each other by encircling one monopole around the hexagonal ring. (d) A noncontractible pair on a loopless Husimi cactus. In this case, the pair cannot be deformed to a contractible pair.