# Resonant torsion magnetometry in anisotropic quantum materials

## TL;DR

This paper introduces a novel, sensitive method using resonant torsion magnetometry to measure the magnetotropic coefficient in anisotropic quantum materials, providing new insights into their thermodynamic properties and phase transitions.

## Contribution

The paper presents a simple experimental technique to directly measure the magnetotropic coefficient using cantilever frequency shifts, enhancing the study of anisotropic quantum materials.

## Key findings

- Quantitative measurement of the magnetotropic coefficient $k$ from frequency shifts.
- Application of the method to NbP and RuCl$_3$ demonstrating broad applicability.
- Identification of discontinuities in $k$ at second-order phase transitions.

## Abstract

Unusual behavior of quantum materials commonly arises from their effective low-dimensional physics, which reflects the underlying anisotropy in the spin and charge degrees of freedom. Torque magnetometry is a highly sensitive technique to directly quantify the anisotropy in quantum materials, such as the layered high-T$_c$ superconductors, anisotropic quantum spin-liquids, and the surface states of topological insulators. Here we introduce the magnetotropic coefficient $k=\partial^2 F/\partial \theta^2$, the second derivative of the free energy F with respect to the angle $\theta$ between the sample and the applied magnetic field, and report a simple and effective method to experimentally detect it. A sub-$\mu$g crystallite is placed at the tip of a commercially available atomic force microscopy cantilever, and we show that $k$ can be quantitatively inferred from a shift in the resonant frequency under magnetic field. While related to the magnetic torque $\tau=\partial F/\partial \theta$, $k$ takes the role of torque susceptibility, and thus provides distinct insights into anisotropic materials akin to the difference between magnetization and magnetic susceptibility. The thermodynamic coefficient $k$ is discontinuous at second-order phase transitions and subject to Ehrenfest relations with the specific heat and magnetic susceptibility. We apply this simple yet quantitative method on the exemplary cases of the Weyl-semimetal NbP and the spin-liquid candidate RuCl$_3$, yet it is broadly applicable in quantum materials research.