Anisotropic excitonic magnetism from discrete $C_4$ symmetry in ${\mathrm{CeRhIn}}_5$

Anisotropy in strongly correlated materials is a central parameter in determining the electronic ground state and is tuned through the local crystalline electric field. This is notably the case in the ${\mathrm{CeCo}}_x{\mathrm{Rh}}_{1-x}{\mathrm{In}}_5$ system where the ground-state wave function can provide the basis for antiferromagnetism and/or unconventional superconductivity. We develop a methodology to understand the local magnetic anisotropy and experimentally investigate with neutron spectroscopy applied to antiferromagnetic ($T_N=3.8K$) ${\mathrm{CeRhIn}}_5$, which is isostructural to $d$-wave superconducting ($T_c=2.3K$) ${\mathrm{CeCoIn}}_5$. Through diagonalizing the local crystal field Hamiltonian with discrete tetragonal $C_4$ point group symmetry and coupling these states with the random phase approximation, we find two distinct modes polarized along the crystallographic $c$ and $a-b$ planes, agreeing with experiment. The anisotropy and bandwidth, underlying the energy scale of these modes, are tuneable with a magnetic field which we use experimentally to separate in energy single and multiparticle excitations thereby demonstrating the instability of excitations polarized within the crystallographic $a-b$ plane in ${\mathrm{CeRhIn}}_5$. We compare this approach to a $S_{\mathrm{eff}}=\frac{1}{2}$ parametrizations and argue for the need to extend conventional SU(2) theories of magnetic excitations to utilize the multilevel nature of the underlying crystal-field basis states constrained by the local point-group $C_4$ symmetry.