Emergence of a two-dimensional macrospin liquid in a highly frustrated three-dimensional quantum magnet

The classical Ising model on the frustrated three-dimensional (3D) swedenborgite lattice has disordered spin liquid ground states for all ratios of inter- and intraplanar couplings. Quantum fluctuations due to a transverse field give rise to several exotic phenomena. In the limit of weakly coupled kagome layers we find a 3D version of disorder by disorder degeneracy lifting. For large out-of-plane couplings one-dimensional macrospins are formed, which realize a disordered macrospin liquid phase on an emerging two-dimensional triangular lattice. We speculate about a possibly exotic version of quantum criticality that connects the polarized phase to the macrospin liquid.

Figure 1

Schematic phase diagram of the 3D swedenborgite TFIM. Dashed line: Classical phase transition line between two different types of spin liquids. Red lines: Disordered phase in kagome and chain limits. Orange line: Ordered phase in the chain limit, encountered on the other end of a second-order phase transition at $\left(0,J_2^c/\mathrm{\Gamma }\right)$. Inset: The structure and spin interactions of the swedenborgite lattice.

Figure 2

The five different plaquettes allowed on the emergent triangular lattice, up to rotation. In green, the underlying macrospin configuration (unique up to global flip of macrospins).

Figure 3

Upper panel: The minimally (left) and maximally zigzagging (right) stripy structures that form the ground states on the emergent triangular lattice, with horizontal monocultural chains of respectively IV.1 and IV.2 plaquettes. Lower panel: Sketch of the finite macrospin configuration used for the ED. Gray (cyan) lines denote the unit cells and indicate the chosen periodic boundary conditions. Each circle represents a triangle in the kagome planes of the swedenborgite lattice so that each finite macrospin configuration contains 24 spins in each kagome plane. These 24-site kagome clusters are considered in the ED.

Figure 4

Critical exponent $z\nu$ as a function of $\alpha$, where $tan\left(\alpha \right)=J_1/J_2$, obtained from various Dlog-Padé extrapolations of the high-field gap $\mathrm{\Delta }(J_1,J_2)$. Solid and dashed horizontal lines indicate known critical exponents for certain universality classes.