Landau theory and exchange instabilities in Mn$_5$Si$_3$: A case against altermagnetism
Thin-film Mn$_5$Si$_3$ is one of the most studied altermagnetic candidates thanks to its metallicity, demonstrated anomalous transport properties, and assumed $d$-wave exchange splitting pattern enabling spin-polarized transport and various spintronic applications. Its postulated altermagnetic structure has zero propagation vector, in contrast to the collinear antiferromagnetic bulk phase (AFM2) which orders at the $M$ star. In this work, the two phases are analyzed using Landau theories, first-principles calculations of the paramagnetic instabilities, and Monte Carlo simulations. AFM2 appears in a Landau theory as a symmetry-protected inversion-even, permutation-odd mode at a single arm of the $M$ star. At $Γ$, the same intracell ordering pattern belongs to the collinear branch of an $E_{2g}$ order parameter. In both cases, higher-order terms are required for the phase selection. First-principles calculations for the paramagnetic, disordered-local-moment state correctly identify the leading exchange instability at the $M$ star, and the resulting classical Heisenberg model orders at a reasonable temperature into the orthogonal $3M$ phase favored by single-site entropy. The $Γ$-point $E_{2g}$ mode, whose Landau theory contains the altermagnetic sector, is substantially weaker and further suppressed by epitaxial strain representative of Mn$_5$Si$_3$ films exhibiting anomalous transport. The same strain reduces the leading magnetic exchange scale. These results provide a natural explanation for the bulk $M$-point instability but strongly disfavor the postulated relocation of the propagation vector from $M$ to $Γ$ in a moderately strained bulklike Mn$_5$Si$_3$ film, suggesting that the corresponding altermagnetic phase is unlikely to be stabilized without additional physics.