Landau theory and exchange instabilities in Mn$_5$Si$_3$: A case against altermagnetism

Abstract: Thin-film Mn$<em>5$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 $E</em>{2g}$ 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.

• The paper combines Landau theory, DLM exchange calculations, and Monte Carlo simulations to show that stoichiometric Mn₅Si₃ favors the M-point AFM2 instability over the proposed Γ-point altermagnetic mode.
• DLM calculations identify the M-point AFM2 pattern as the leading instability across LDA-based models, while 1% tensile strain further suppresses Γ-point ordering and reduces the leading M-point eigenvalue by about 20%.
• Monte Carlo simulations yield an ordering temperature near 73 K and show that magnetoelastic coupling can favor AFM2, indicating that thin-film transport above 200 K likely requires altered stoichiometry, reconstruction, or other physics beyond weakly strained bulk Mn₅Si₃.

Motivation and scope

Thin films of hexagonal Mn$_5$Si$_3$ have become one of the most prominent altermagnetic candidates, owing to their metallicity and the anomalous Hall and Nernst responses observed well above 200 K, which have been attributed to a hypothetical collinear $d$-wave altermagnetic phase with a zone-center ($\Gamma$-point) propagation vector (2608.13483). This phase is postulated to arise from the bulk collinear AFM2 phase by relocating the ordering vector from the zone-boundary $M$ point to $\Gamma$ while retaining the same $(1,-1,0)$ intracell Mn2 pattern. Belashchenko tests this assignment using three complementary methods: Landau theory for both candidate ordering patterns, first-principles disordered-local-moment (DLM) calculations of the paramagnetic exchange instability, and classical Monte Carlo simulations. The central conclusion is that the $\Gamma$-point instability is strongly disfavored relative to the $M$ point, and that epitaxial strain representative of the films suppresses it further, so the postulated altermagnetic phase is unlikely to be stabilized without additional physics beyond bulklike stoichiometric Mn$_5$Si$_3$0.

Landau theory of the $_3$1-star AFM2 phase

The paramagnetic space group $_3$2 contains Mn2 atoms on a $_3$3 orbit forming six sublattices arranged in nearly octahedral clusters, labeled as inversion-related pairs $_3$4, $_3$5, $_3$6. Restricting to the inversion-even sector, the analysis of the quadratic Landau functional $_3$7 shows that at a single arm of the $_3$8 star the little cogroup (orthorhombic, $_3$9) contains no threefold rotations, so the three effective sublattices are no longer equivalent. The only symmetry-protected eigenvector is permutation-odd — for example $d$0 at $d$1 — which is precisely the experimentally observed AFM2 pattern, including the feature that one third of the Mn2 sites remains disordered by symmetry rather than by accident. The vanishing of this mode's exchange field on the Mn1 atoms follows from its odd parity under the retained in-plane twofold axis, and holds for any superposition of the three arms.

Phase selection within the $d$2 star is controlled by quartic invariants: $d$3 distinguishes single-arm from equal-amplitude ordering, and $d$4 selects orthogonal versus collinear multi-$d$5 configurations. Notably, the single-site entropic quartic terms of the classical Heisenberg model give $d$6 and $d$7, placing the mean-field Heisenberg description in the orthogonal $d$8 phase rather than the observed single-arm AFM2. Stabilizing AFM2 therefore requires additional physics; the paper identifies magnetoelastic coupling of $d$9 (transforming as $\Gamma$0) to in-plane shear strain as a natural mechanism contributing a negative $\Gamma$1, estimated in a later section.

Landau theory of the zone-center altermagnetic mode

At $\Gamma$2, the inversion-even ordering amplitudes decompose into the uniform $\Gamma$3 magnetization and a two-dimensional $\Gamma$4 irrep with the constraint $\Gamma$5. The quartic free energy $\Gamma$6 is degenerate between collinear ($\Gamma$7) and 120-degree "clock" ($\Gamma$8) branches; only the collinear branch is relevant here. Within the collinear sector, the free energy is $\Gamma$9-degenerate up to fourth order, and a sixth-order term $M$0 selects between the altermagnetic $M$1 pattern ($M$2) and the ferrimagnetic $M$3 pattern. The altermagnetic phase is thus allowed but requires both $M$4 and $M$5 — two sign conditions on higher-order terms. The paper notes that this ordering on a tripartite lattice under a two-dimensional irrep differs from the standard Landau description of collinear altermagnetism, in which the Néel vector transforms as a one-dimensional irrep; here the magnetic transition itself reduces tripartite to bipartite permutation symmetry.

DLM exchange instabilities

Using the linear-response technique in the DLM paramagnetic state (TB-LMTO, Questaal), the exchange kernel $M$6 was computed with LDA (Mn2 local moment 2.54 $M$7), scaled LDA (2.14 $M$8), and GGA (2.85 $M$9). The LDA moment is favored on the basis of neutron-diffraction moments (1.48 $\Gamma$0 ordered at 70 K, implying fluctuating moments of 1.9–2.3 $\Gamma$1). The dominant exchange is the frustrated antiferromagnetic $\Gamma$2 mRy within nearest-neighbor $\Gamma$3 triangles, followed by ferromagnetic $\Gamma$4 mRy and antiferromagnetic $\Gamma$5 mRy. The strong $\Gamma$6 between neighboring octahedral chains directly suppresses zone-center ordering in the parity-even sector.

The key numerical findings are:

Potential	Local moment ($\Gamma$7)	Leading instability	$\Gamma$8 mode at $\Gamma$9
LDA	2.54	$(1,-1,0)$0, AFM2 pattern	Well below $(1,-1,0)$1
Scaled LDA	2.14	$(1,-1,0)$2, AFM2 pattern	Well below $(1,-1,0)$3
GGA	2.85	$(1,-1,0)$4 (nearly ferromagnetic in cell)	Only slightly unstable, far below zone boundary

For both LDA variants, the leading eigenmode is the symmetry-protected inversion-even, permutation-odd mode at $(1,-1,0)$5 — the DLM calculation independently reproduces the experimentally observed bulk instability. The $(1,-1,0)$6 mode at $(1,-1,0)$7, which contains the altermagnetic sector, is substantially weaker and, importantly, $(1,-1,0)$8 has a global minimum at $(1,-1,0)$9 in the entire $\Gamma$0 plane. Under a 1% in-plane tensile strain with 0.3% $\Gamma$1-axis contraction, representative of Mn$\Gamma$2Si$\Gamma$3 films on Si(111), the $\Gamma$4 mode becomes even less competitive, and the leading $\Gamma$5-point eigenvalue is reduced by roughly 20%. The paper is explicit that this result contradicts any explanation of the films' high-temperature anomalous transport as a strain-stabilized bulklike altermagnetic phase: no mechanism in the strained $\Gamma$6 can sustain a Hall-active state above 200 K.

Monte Carlo results and ordering temperature

Mean-field theory gives $\Gamma$7–225 K from $\Gamma$8–2.14 mRy, but strong frustration of $\Gamma$9 implies a large mean-field overestimate. Classical Monte Carlo simulations (UppASD, up to 18432 spins) with the LDA exchange parameters yield $M$0 K, in reasonable agreement with the experimental $M$1 K, and the scaled-LDA potential (27% larger $M$2) would improve the agreement further. The simulations select the equal-amplitude orthogonal $M$3 phase (arm chirality $M$4 in the ordered state), consistent with the purely entropic quartic terms of the Heisenberg model. The paper argues this does not undermine the quadratic instability analysis: both $M$5 and AFM2 condense from the same $M$6-star mode and are distinguished only by quartic terms, and the transition temperature should be of the same order in either case. A limitation acknowledged here is that the Heisenberg pair-exchange model cannot capture the interactions that select the single-arm AFM2 phase in the real material.

Magnetoelastic coupling and exchange striction

The magnetoelastic coefficient $M$7 was extracted from the strain-induced splitting of the $M$8-arm eigenvalues, giving $M$9 mRy and $_5$0 eV per primitive cell. Combined with the calculated $_5$1 GPa and the measured spontaneous orthorhombic exchange striction of 0.481% at 70 K, this yields a self-consistent estimate of the ordered amplitude $_5$2 at $_5$3, roughly consistent with the measured 1.48 $_5$4. The resulting magnetoelastic correction $_5$5 meV per primitive cell is about a quarter of the entropic $_5$6 near $_5$7, providing a substantial push toward the single-arm AFM2 phase. The paper notes that epitaxial clamping in a film could suppress this exchange striction, potentially shifting the system toward the orthogonal $_5$8 phase — which, however, carries no orthorhombic striction and no higher exchange scale, so it cannot explain the high-temperature anomalous transport either.

Limitations and open questions

Several assumptions bound the conclusions. The DLM analysis is restricted to the quadratic (pair-exchange) instability of a stoichiometric, bulklike paramagnet; non-Heisenberg electronic interactions and correlations beyond mean field are invoked only qualitatively to explain the single-arm selection of AFM2, and the contribution of frozen $_5$9 phonons to $_3$00 is not quantified. The magnetoelastic estimate neglects internal structural relaxations and uses a ferromagnetic-state elastic tensor. Most significantly, the analysis cannot rule out that the film's Hall-active magnetic state involves a substantial departure from homogeneous bulklike Mn$_3$01Si$_3$02 — altered site occupation, off-stoichiometry, surface reconstruction, or intercalation on the empty $_3$03 sites (as in carbon-doped Mn$_3$04Si$_3$05, which becomes ferromagnetic). The open question the paper leaves is therefore what microscopic phase, distinct from weakly strained stoichiometric Mn$_3$06Si$_3$07, produces anomalous transport persisting above 200 K in the films, given that the film anomalies near 80 K indicate a transition at approximately the bulk magnetic energy scale.

Conclusion

The paper demonstrates that the exotic bulk AFM2 phase of Mn$_3$08Si$_3$09 is a symmetry-protected inversion-even, permutation-odd mode at the $_3$10 star, correctly identified as the leading instability by DLM calculations and accompanied by a quantitatively reasonable Monte Carlo ordering temperature of about 73 K. The zone-center $_3$11 mode that would support the proposed altermagnetic phase is substantially weaker in the paramagnetic exchange kernel and is further suppressed, not stabilized, by epitaxial strain representative of the films. The high-temperature Hall-active state in thin Mn$_3$12Si$_3$13 films therefore cannot be understood as a weakly strained perturbation of bulk stoichiometric Mn$_3$14Si$_3$15.