Tensor network analysis of the maple-leaf antiferromagnet spangolite

Abstract

Spangolite (Cu$_6$Al(SO$_4$)(OH)$_{12}$Cl$\cdot$3H$_2$O) is a hydroxy-hydrated copper sulfate mineral with a one-seventh depleted triangular lattice of Cu$^{2+}$ ions in each layer. Experimental measurements revealed a non-magnetic ground state at $T \sim 8\, \text{K}$ with magnetic properties dominated by dimerization. We propose a spatially anisotropic Heisenberg model for the Cu$^{2+}$ spin-$1/2$ degrees of freedom on this geometrically frustrated and effectively two-dimensional maple-leaf lattice, featuring five symmetry inequivalent couplings with ferromagnetic bonds on hexagons and antiferromagnetic triangular bonds. The validity of the proposed Hamiltonian is demonstrated by state-of-the-art tensor network calculations, which can assess both the nature of the ground state as well as low-temperature thermodynamics, including the effects of a magnetic field. We provide theoretical support for a picture of a non-trivially correlated dimer ground state, which accounts for the appreciable reduction of the magnetic moment at high temperatures observed in experiment, thereby resolving a long-standing puzzle. We predict the static spin structure factor as well as the emergence of magnetisation plateaus at high values of an external magnetic field, explore the nature of the quantum states in them, and study their melting with increasing temperature.