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

Technical Summary: Probing Multipolar Order in the Candidate Altermagnet MnF₂ through the Elastocaloric Effect under Strain

Problem and Motivation
Altermagnets (AMs) represent a recently classified phase of matter that breaks a combination of time-reversal and rotational symmetries while maintaining zero net magnetization. Unlike conventional dipolar ferromagnets or antiferromagnets, the order parameter of $d$-wave altermagnets corresponds to higher-rank multipole orders (specifically, emergent ferroic octupolar order in rutile structures like MnF₂). A key theoretical prediction for these systems is that the application of a "conjugate field" ($\hat{h}$)—a perturbation transforming like the order parameter—should convert a sharp phase transition at zero field into a crossover at finite field. However, identifying and applying this conjugate field is experimentally challenging. In AMs with $d$-wave symmetry, piezomagnetism theoretically allows the conjugate field to be realized as a product of a magnetic field ($\mu_0 H_z$) and a symmetry-breaking shear strain ($\epsilon_{xy}$). While the elastocaloric effect (ECE) has been proposed as a sensitive thermodynamic probe for such symmetry breaking, it has not yet been experimentally utilized to characterize the multipolar order of altermagnets.

Methodology
The authors investigate MnF₂, a rutile insulator predicted to host $d_{xy}$-wave altermagnetism, using a multi-faceted approach:

1. Experimental Elastocalorics: The team applies uniaxial stress along the crystallographic [110] direction to induce the necessary shear strain ($\epsilon_{xy}$) and measures the adiabatic temperature change ($\eta_{xy}$) as a function of temperature and magnetic field ($\mu_0 H_z$). This allows for the extraction of the entropy derivative with respect to strain.
2. Thermodynamic Modeling: They construct a free-energy model incorporating the AM order parameter ($\Phi$) coupled to the conjugate field $\hat{h} = \mu_B \epsilon_{xy} \mu_0 H_z$. This model predicts specific scaling relations for the crossover temperature ($T^*$) and the behavior of the ECE near the critical point.
3. First-Principles Calculations: Density Functional Theory (DFT) calculations are performed to determine the microscopic origin of the coupling constant ($\lambda$) between the order parameter and the conjugate field. These calculations explore the role of spin-orbit coupling (SOC) and carrier concentration (off-stoichiometry) in generating the piezomagnetic response.

Key Results

• Identification of Crossover Lines: The experiments successfully identify the predicted crossover lines in MnF₂. The data shows that the characteristic temperature of the ECE feature ($T^*$) shifts with the applied field and strain. When plotted as $T^* - T_c$ versus the conjugate field $\hat{h}$, the data collapses onto a single curve following the scaling relation $T^* - T_c \propto \hat{h}^{1/(\beta\delta)}$. The fit yields a mean-field exponent of $1/(\beta\delta) = 2/3$, consistent with the theoretical expectations for an altermagnetic critical point.
• Entropy Accumulation Signatures: The magnitude of the ECE ($\eta_{xy}$) exhibits distinct behaviors. In zero field, a step-like change occurs at $T_c$. Under finite fields, the ECE shows a stronger negative response as the system approaches $T^*$ from below. This is attributed to the accumulation of entropy near the critical point ($\epsilon_{xy}\mu_0 H_z = 0$), which is suppressed by the application of the conjugate field.
• Microscopic Origin of Coupling: DFT calculations reveal that the piezomagnetic coefficient ($\lambda$) is strongly suppressed in pristine MnF₂ at zero temperature due to the material's insulating nature and large band gap, which prevents the induction of magnetic moments without exciting electrons across the gap. However, the calculations demonstrate that introducing a small concentration of charge carriers (approximately 0.001 holes per unit cell, corresponding to $\sim 10^{19}$ cm$^{-3}$) significantly enhances $\lambda$ by orders of magnitude, bringing the theoretical values into agreement with experimental observations. This suggests that slight off-stoichiometry or defects in the samples play a crucial role in the observed thermodynamic response.

Significance and Claims
The paper establishes the elastocaloric effect as a viable thermodynamic probe for detecting and characterizing the multipolar order in altermagnets. By experimentally verifying the scaling behavior of the crossover temperature and the entropy accumulation near the critical point, the authors provide direct evidence for the predicted symmetry breaking in MnF₂.

The authors claim that this methodology opens a pathway for exploring altermagnetic quantum criticality in both insulators and metals. They note that while the coupling in MnF₂ is relatively weak (requiring off-stoichiometry to be observable), the effect should be substantially larger in metallic altermagnets (e.g., KV₂Se₂O) or systems with stronger spin-orbit coupling. Consequently, the ECE approach offers a powerful tool to measure the altermagnetic susceptibility and investigate fluctuations near quantum critical points, potentially aiding in the understanding of phenomena such as altermagnetism-induced superconductivity. The work also highlights the sensitivity of altermagnetic responses to minute deviations from stoichiometry, rationalizing discrepancies in prior piezomagnetic measurements.