Rank–2 $U(1)$ Spin Liquid on the Breathing Pyrochlore Lattice

Higher-rank generalizations of electrodynamics have recently attracted considerable attention because of their ability to host “fracton” excitations, with connections to both fracton topological order and gravity. However, the search for higher-rank gauge theories in experiment has been greatly hindered by the lack of materially relevant microscopic models. Here we show how a spin liquid described by rank-2 $U(1)$ gauge theory can arise in a magnet on the breathing pyrochlore lattice. We identify Yb-based breathing pyrochlores as candidate systems, and make explicit predictions for how the rank-2 $U(1)$ spin liquid would manifest itself in experiment.

Figure 1

Breathing-pyrochlore lattice, and singular correlations characteristic of a rank-2 $U(1)$ gauge theory. (a) BP lattice, with $A$- and $B$-sublattice tetrahedra of unequal size. The vectors associated with Dzyaloshinskii-Moriya interactions on the $A$ sublattice [Eq. (6)] are illustrated with green arrows. (b) Prediction of R2-U1 theory for the correlation function $⟨E_{\mathsf{x}\mathsf{y}}(\mathbf{q})E_{\mathsf{x}\mathsf{y}}(-\mathbf{q})⟩$ [Eq. (17)], showing a twofold pinch point in the [0kl] plane. (c) Perpendicular section, showing a fourfold pinch point in the [hk0] plane. (d) Equivalent results from MC simulation of the breathing pyrochlore model [Eq. (6)].

Figure 2

Finite-temperature phase diagram of the BP model, Eq. (6), as a function of DM interaction $D_A$. The crossover between the R2-U1 spin liquid, and the $U(1)\times U(1)\times U(1)$ spin liquid (HAF) is shown with a dashed line. The thin solid line indicates a continuous transition into all-in all-out order (AIAO), while the thick solid line denotes a first order phase transition into a state with $\mathbf{q}=W$ order. Results are taken from MC simulation with $J_A=J_B=1$, $D_B=0$.

Figure 3

Spin structure factor found in MC simulation of the BP model, Eq. (6), showing fourfold pinch points characteristic of a R2-U1 spin liquid. (a) Correlations in the $[h0k]$ plane, in the spin-flip (SF) channel measured using polarized neutrons. 4FPP are visible at [0,0,2] and points related by symmetry. Results are for parameters Eq. (18), $T=2.5\times {10}^{-3}{J}_A$. (b) Equivalent results for parameters motivated by ${\mathrm{Ba}}_3{\mathrm{Yb}}_2{\mathrm{Zn}}_5{\mathrm{O}}_{11}$, Eq. (19), $T=252\mathrm{mK}$.