Probing multipolar order in the candidate altermagnet MnF$_2$ through the elastocaloric effect under strain

TL;DR

This work demonstrates a thermodynamic probe of multipolar altermagnetic order in MnF$_2$ by leveraging the elastocaloric effect under strain. By combining elastocaloric measurements, Landau free-energy modeling, and first-principles calculations, the authors identify a finite-temperature altermagnetic critical point evidenced by a cusp in the crossover temperature $T^*$ as a function of the conjugate field $h=oldsymbol{oldsymbol{ abla}} h = oldsymbol{oldsymbol{ u}}$. The study links the observed AM coupling constant $oldsymbol{ abla}$ to the piezomagnetic response, showing how small carrier doping and SOC can enhance the effect, and provides a roadmap for exploring AM quantum criticality in d-wave altermagnets, including metallic candidates. Overall, the work establishes elastocalorics as a sensitive thermodynamic probe of multipolar AM order and its fluctuations, with implications for strain-tuned quantum critical phenomena and AM-driven functionalities.

Abstract

Altermagnets break a combination of time-reversal and rotational symmetries without generating a net magnetization. As such, the order parameter of $d$-wave altermagnets has the same symmetry as magnetic multipoles, and couples to the product of a magnetic field and uniaxial strain. We combine elastocaloric experiments, free-energy modeling, and first-principles calculations on MnF$_2$ to establish a thermodynamic probe of the predicted finite-temperature altermagnetic critical point. These results pave the way to explore altermagnetic quantum criticality in $d$-wave materials and beyond.

Figures (13)

• Figure 1: (a) Emergent ferro-octupolar order Bho24Buiarelli2025 of the AM candidate MnF$_2$. The pink arrows depict the antiferromagnetic ordering of dipoles, whereas the ferroically ordered octupoles are depicted in purple and cyan. (b) A top view of the crystal structure. The F ions (gray circles) create an octahedral environment around the Mn atoms (purple circles); as a result, the two Mn sublattice sites are related by a non-symmorphic symmetry involving a $90^\circ$ rotation and a half-translation. A strain $\varepsilon_{xy}$ will break this rotational symmetry. The conjugate field to the AM order, $\hat{h}$, is composed of $\varepsilon_{xy}$ and a magnetic field along the $c$-axis, $\mu_0 H_z$. (c) Experimentally determined crossover lines, $T^*-T_c$, of MnF$_2$ as a function of conjugate field $\hat{h}=\mu_B\mu_0 H_z \varepsilon_{xy}$. The experimental data, extracted from the data in Fig. \ref{['fig:2']} and in the End Matter, follow the expectation for an AM critical point at $\mu_B\mu_0 H_z \varepsilon_{xy}\,=\,0$. The data are well described by a fit to Eq. \ref{['eq:crossover-expanded']} with the mean-field exponent $1/(\beta \delta)\,=\,2/3$. For clarity, error bars are shown only for representative data points.
• Figure 2: (a) The application of a uniaxial stress along the [110] direction, $\sigma_{110}$, leads to an induced strain that can be described by the superposition of a symmetric strain and an antisymmetric strain. The latter is denoted by $\varepsilon_{xy}$ and $\nu$ is the Poisson's ratio. (b,c) ECE data, $\eta_{xy}$, on MnF$_2$ as a function of the relative temperature, $T-T_c$. $T_c$ is the transition temperature including non-AM shifts in strain and field (see text). While for $\mu_0 H_z\,=\,$ 0 T (b) the ECE features occur at the same temperature (see dotted lines, representing the point of steepest slope), they move to higher temperature with higher compression for 6.44 T (c), as expected for a system with AM order.
• Figure 3: Comparison of (a) experimental $\eta_{xy}-\eta_n$ (with $\eta_n$ the high-temperature background, see End Matter) of MnF$_2$ as a function of $T$ at a fixed strain of $\varepsilon_{xy}\,=\,-0.11\,\%$ and different fields up to 6.44 T with (b) mean-field simulations for $\eta_{xy}$, based on the free energy Eq. \ref{['eq:free-energy-expanded']}. The contributions to $\eta_{xy}$ due to $a_1$ and $\lambda$ are marked by vertical bars. The AM contribution associated with $\lambda$ leads to a negative change in $\eta_{xy}$ under finite fields, as the entropy $S$ grows toward a maximum at zero strain (see inset).
• Figure 4: (a) Magnitude of $\lambda$ per unit cell as a function of additional carrier concentration from DFT calculations. Introduction of $\sim 0.001$ holes per unit cell ($\sim 10^{19}\, {\rm cm}^{-3}$) in the DFT calculations provides agreement with the experimental observation (dotted pink line) for reasonable values of the Hund's coupling $J$. (b) Net magnetic moment $M_z$ per unit cell of MnF$_2$ as a function of $\epsilon_{xy}$ shear strain for different magnitudes of SOC calculated from DFT. Even though the spin group analysis allows the $M_z$ PZM response to be finite, our calculations indicate that the corresponding component, $\lambda$, is suppressed at zero temperature because of the large band gap when SOC is not taken into account. (c) Simulations of $\eta_{xy}$ vs. $T-T_c$ for a higher value of $\lambda$ (dotted red line in panel (a)), which may be realized in other materials with metallic character or larger SOC (see (b) and the End Matter). The predicted $\eta_{xy}$ values should be readily observable in experiments over a wider temperature range, enabling the direct measurement of the AM susceptibility.
• Figure 5: Elastocaloric effect, $\eta_{xy}=\Delta T/\Delta \varepsilon_{xy}$, of MnF$_2$ vs. temperature, $T$, at different constant strains and zero magnetic field. The inset shows an enlarged view of the data around the phase transition. The pink arrow indicates that the phase transition shifts to higher temperatures with increasing compression (negative strains). The change in the magnitude of the elastocaloric effect at high temperatures likely results from small strain-induced changes of the phononic contributions and is denoted by $\eta_n$ in the main text