$\beta$-Mn shows a large Sommerfeld coefficient of 70 mJ ${\mathrm{Mn}}^{-1}{\mathrm{K}}^{-2}$, which is one of the largest values in $3d$-electron systems. Although the origin has been attributed to the fluctuating magnetic moment, there are no material-dependent systematic studies in the $3d$-electron systems except for doped $\beta$-Mn alloys. The ordered $\beta$-Mn structure alloys ${\mathrm{Mn}}_3\mathrm{TX}$ $(\mathrm{T}; \mathrm{Co}, \mathrm{Rh}, \mathrm{Ir}, \mathrm{X}; \mathrm{Si}, \mathrm{Ge})$ also exhibit the Sommerfeld coefficient enhancement. According to the inelastic neutron scattering study of ${\mathrm{Mn}}_3\mathrm{RhSi}$, the magnetic moment of Mn is about a full $3d^5$ magnetic moment of 5 ${\mu }_{\mathrm{B}}$, whereas the long-range ordered magnetic moment $M$ is 2.6 ${\mu }_{\mathrm{B}}$ at 4 K, suggesting that a large part of the magnetic moment is fluctuating. Here, the enhanced Sommerfeld coefficients are compared with the Néel temperatures $T_{\mathrm{N}}$. The critical exponent of about 1/3 requires the further accurate approximation than the self-consistent renormalization approximation for the wide $T_{\mathrm{N}}$ range. The disappearance of the magnetic short-range order in the ${\mathrm{Mn}}_3\mathrm{TX}$ system except for ${\mathrm{Mn}}_3\mathrm{RhSi}$ and ${\mathrm{Mn}}_3\mathrm{CoSi}$ is also discussed with the critical spin-orbit coupling of about 50 meV, leading to the spin excitation gap at low energies. The required conditions for the high magnetic short-range order temperature are summarized, suggesting the close relation to the Sommerfeld coefficient enhancement conditions.

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• Density of states
• Magnetic susceptibility
• Phase diagrams
• Phase transitions
• Spin-orbit coupling