N\'{e}el and stripe ordering from spin-orbital entanglement in $\alpha$-Sr$_2$CrO$_4$

The rich phenomenology engendered by the coupling between the spin and orbital degrees of freedom has become appreciated as a key feature of many strongly-correlated electron systems. The resulting emergent physics is particularly prominent in a number of materials, from Fe-based unconventional superconductors to transition metal oxides, including manganites and vanadates. Here, we investigate the electronic ground states of $\alpha$-Sr$_2$CrO$_4$, a compound that is a rare embodiment of the spin-1 Kugel-Khomskii model on the square lattice -- a paradigmatic platform to capture the physics of coupled magnetic and orbital electronic orders. We have used resonant X-ray diffraction at the Cr-$K$ edge to reveal N\'{e}el magnetic order at the in-plane wavevector $\mathbf{Q}_N = (1/2, 1/2)$ below $T_N = 112$ K, as well as an additional electronic order at the 'stripe' wavevector $\mathbf{Q}_s = (1/2, 0)$ below T$_s$ $\sim 50$ K. These findings are examined within the framework of the Kugel-Khomskii model by a combination of mean-field and Monte-Carlo approaches, which supports the stability of the spin N\'{e}el phase with subsequent lower-temperature stripe orbital ordering, revealing a candidate mechanism for the experimentally observed peak at $\mathbf{Q}_s$. On the basis of these findings, we propose that $\alpha$-Sr$_2$CrO$_4$ serves as a new platform in which to investigate multi-orbital physics and its role in the low-temperature phases of Mott insulators.