Coupling quantum spin ice to matter on the centered pyrochlore lattice

Abstract

The low-energy physics of quantum spin ice is known to support an emergent form of quantum electrodynamics (QED), where magnetic monopoles exist and the fine structure constant is material dependent. In this article, we show how this QED is modified via a coupling to dynamical matter on the centered pyrochlore lattice, a structure which has recently been synthesized using metal-organic frameworks. Specifically, we study the low-energy properties of the $S = 1/2$ quantum XXZ model on the centered pyrochlore lattice, with a focus on the sign-problem free region. At fourth order in degenerate perturbation theory this model hosts a quantum spin liquid distinct from the well-known U(1) quantum spin ice on the pyrochlore due to the presence of dynamical matter in the ground state. Exact diagonalization results are consistent with this quantum spin liquid over an extended region of the ground state phase diagram although potential quantum critical points within this region could indicate a richer phase structure. Our work thus expands the physics of quantum spin ice in an experimentally motivated geometry, showing how the emergent QED can be coupled to dynamical matter at zero temperature.

Figures (20)

• Figure 1: The centered pyrochlore lattice with its 6 sublattices: $\{1,2,3,4\}$ for vertex sites, and $\{a,b\}$ for center sites. The former make up a pyrochlore lattice while the latter occupy a diamond lattice.
• Figure 2: Ground-state phase diagram of the classical Ising model with $\gamma=J_1^z/J_2^z$ and $J_1^\perp=J_2^\perp=0$. Reproduced from nutakki2023.
• Figure 3: The eight allowed single-tetrahedron configurations of the Ising model ground state for $1<\gamma<3$. The degenerate set of states which can be constructed from these configurations is a $\mathbb{Z}_2$ classical spin liquid nutakki2023. We refer to this ensemble of ground states as 3:1 states and $1<\gamma<3$ as the 3:1 regime. Unlike 2:2 single-tetrahedron configurations one can obtain a valid 3:1 configuration from another by flipping a pair of majority vertex spins and the center spin, which is how the lower row is obtained from the upper row.
• Figure 4: Illustration of virtual processes contributing to the constant terms of Eq. (\ref{['eq:Hp2']}) in second-order perturbation theory, reading from left (initial) to right (final). The initial and final states are always the same, while the intermediate (virtual) states bears two excited tetrahedra: (a) the one displayed here and the neighboring one connected to spin 2 (not shown), or (b) the two neighboring tetrahedra connected to spins 1 and 2. These virtual excitations are either 2:2 or 4:0 states, and are degenerate for $\gamma=2$, with energy $H_t^z=\frac{J_2^z}{2} \Rightarrow H^z=J_2^z$.
• Figure 5: Processes contributing to third-order perturbation theory for $\gamma = 2$, with the single-tetrahedron energies of the intermediate (virtual) states shown below: $t_0$ is the displayed tetrahedron, $t_{i=\{1,2,3\}}$ are the neighboring tetrahedra connected via site $i$. (a) and (b) are triangular-hoping processes, respectively mediated by a center spin [Eq. (\ref{['eq:Hp3CPy1']})], or not [Eq. (\ref{['eq:Hp3triang']})]. (c) is the off-diagonal ring-exchange term, with clockwise convention to label the sites and tetrahedra around an hexagonal loop. Only the edges of the tetrahedra making up the loop are shown.