Emergent Honeycomb Lattice in ${\mathrm{LiZn}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$

We introduce the idea of emergent lattices, where a simple lattice decouples into two weakly coupled lattices as a way to stabilize spin liquids. In ${\mathrm{LiZn}}_2{\mathrm{Mo}}_3{\mathrm{O}}_8$, the disappearance of $2/3$ of the spins at low temperatures suggests that its triangular lattice decouples into an emergent honeycomb lattice weakly coupled to the remaining spins, and we suggest several ways to test this proposal. We show that these orphan spins act to stabilize the spin liquid in the $J_1$-$J_2$ honeycomb model and also discuss a possible 3D analogue, ${\mathrm{Ba}}_2{\mathrm{MoYO}}_6$ that may form a “depleted fcc lattice.”

Figure 1

(a) $J_1$-$J_2$-$J^{\prime }$ lattice, where $J^{\prime }=J_1$ describes the triangular lattice and $J^{\prime }=0$ describes decoupled honeycomb ($J_1$-$J_2$) and triangular ($J_2$) lattices. The $A$ and $B$ sublattices of the honeycomb lattice and the $C$ sublattice of central spins are labeled. (b) Unit cells: Blue dotted lines show the small initial unit cell, while orange dashed lines show the larger final unit cell. Both have trigonal symmetry—only the lattice vector changes. (c) These rotations convert the triangular lattice into the $J_1$-$J_2$-$J^{\prime }$ lattice: the $A$ and $B$ clusters rotate in opposite directions, while the $C$ clusters do not rotate. Inset shows original configuration. (d) The basic unit of the depleted fcc lattice: strong bonds are shown as red (solid) lines, weak bonds as blue (dashed) lines. The central layer forms the emergent honeycomb lattice.

Figure 2

(a) Rough phase diagram of the $J_1$-$J_2$ honeycomb lattice [10], with Néel, plaquette VBS (pVBS) and staggered VBS (sVBS) states, with a small controversial spin liquid region, thought to be the sublattice pairing state (SPS). (b) Diagram for the second order energy shift, $\Delta E_{\mathrm{SPS}}^{(2)}$ generated by a single central spin impurity in the SPS. Solid lines are fermionic spinons, while the dashed line represents the central spin. (c) Diagrams for the second order energy shift, $\Delta E_{\mathrm{AFM}}^{(2)}$ for the single central impurity in the Néel state. Squiggly lines represent Holstein-Primakoff bosons, ${\alpha }_{\mathbf{k}}^{†}$, not magnons, and the dashed lines are the Holstein-Primakoff bosons, $d^{†}$ representing the central spin. (d) Initial and final spin configurations for calculating $\Delta E_{\mathrm{sVBS}}^{(2)}$, where the red ellipses represent singlet valence bonds. Diagrams for $\Delta E_{\mathrm{pVBS}}^{(2)}$ are similar.