Spin structures and phase diagrams of the spin-$\frac{5}{2}$ triangular-lattice antiferromagnet Na$_2$BaMn(PO$_4$)$_2$ under magnetic field

We combine single-crystal neutron diffraction studies and Monte Carlo simulations to determine the spin structures and finite-temperature phase diagram of the spin-5/2 triangular-lattice antiferromagnet Na$_2$BaMn(PO$_4$)$_2$ in magnetic field. With the application of a magnetic field in two different directions, namely along the $c$-axis and in the $ab$-plane of the trigonal symmetry, we track the evolution of the spin structure through changes of the magnetic propagation vector. We account for these results with a minimal Heisenberg Hamiltonian that includes easy-axis anisotropy and weak, frustrated interlayer couplings in addition to intralayer exchange. Guided by representation analysis, we refine symmetry-allowed modes to the measured intensities and obtain the spin structures for all field-induced phases, which we compare quantitatively with simulated configurations. Taken together, our measurements and simulations show that frustrated interlayer exchange -- rather than purely 2D physics -- organizes the unexpectedly rich field-induced phases of Na$_2$BaMn(PO$_4$)$_2$.