Compass Correlations in a Square-Lattice Magnet

The square-lattice compass model is a paradigm of bond-dependent frustration, yet experimental evidence for its characteristic spin correlations has remained elusive. Here we identify compass correlations in the square-lattice intermetallic ErRu$_2$Si$_2$ using single-crystal neutron diffuse scattering. Comparative measurements on the isostructural compounds NdRu$_2$Si$_2$ and ErRu$_2$Si$_2$ reveal a pronounced contrast between nearly uniform diffuse rings in the former and strong intensity modulation between Brillouin zones in the latter. Through the magnetic neutron polarization factor, this modulation reveals compass correlations that couple moment orientation to the correlation wavevector. Crystal-field spectroscopy identifies four preferred in-plane orientations for the Er$^{3+}$ moments. A minimal compass-Heisenberg model incorporating this anisotropy quantitatively reproduces the diffuse scattering and accounts for the double-$Q$ vortex-antivortex ground state. Fits with the compass couplings fixed at values set by the dipolar energy scale retain good agreement with the data, supporting a substantial dipolar contribution to the bond-dependent interactions. Our results establish a square-lattice platform for compass physics and suggest that dipolar interactions, acting together with local crystal-field anisotropy, can shape directional magnetic correlations alongside conduction-electron-mediated exchange.