We first describe the experimental geometry and why the magnetotropic susceptibility is sensitive to the transverse magnetic susceptibility. The inset of Fig. 1a shows the experimental geometry for a magnetotropic susceptibility measurement13,14. A silicon microcantilever is driven near its fundamental bending mode, defining an axis n around which the tip of the lever oscillates by a small angle δθ. We place the cantilever in an external magnetic field B and rotate the cantilever in the plane normal to the vector n. This constrains the field angle θ to always lie in the same plane as the lever oscillation angle δθ.
The sample is placed on the tip of the cantilever with one crystallographic axis aligned along the length of the lever and another crystallographic axis aligned along n. For an orthorhombic crystal like UTe2, this places the third axis perpendicular to the surface of the lever (inset of Fig. 1a). We perform two sets of rotation experiments: one in the ac-plane and one in the bc-plane. In all experiments reported here, the c-axis is perpendicular to the surface of the cantilever, and θ is defined as the angle between the applied magnetic field and the c-axis.
When a magnetic field is applied along one of the crystallographic axes of UTe2, it produces a magnetic moment parallel to that axis. When the magnetic field is rotated away from that axis, the moment is no longer parallel to the magnetic field. This produces a magnetic torque, τ = M × B, that bends the cantilever by a small angle. The angular derivative of this torque defines the magnetotropic susceptibility: k ≡ ∂τ/∂θ. This susceptibility adds to the elastic bending stiffness of the cantilever and is measured by the shift in the cantilever resonance frequency (see SI for details of the calibration procedure). In terms of the vector n around which the cantilever oscillates in an applied field B, the magnetotropic susceptibility is
$${k}_{{{{\bf{n}}}}}({{{\bf{B}}}})=({{{\bf{n}}}}\times {{{\bf{B}}}})\cdot ({{{\bf{n}}}}\times {{{\bf{M}}}})-\frac{1}{{\mu }_{0}}({{{\bf{n}}}}\times {{{\bf{B}}}})\cdot \chi ({{{\bf{B}}}})\cdot ({{{\bf{n}}}}\times {{{\bf{B}}}}),$$
(1)
where χij(B) ≡ μ0∂Mi(B)/∂Bj is the differential magnetic susceptibility probed by the oscillating magnetic field component that is perpendicular to the applied field B14. This oscillating field component arises due to the reorientation of the sample with respect to the field due to the cantilever vibration. The first term in Eq. (1) captures how the torque changes when a fixed moment M is rotated in a field B around the axis n. The second term captures how the torque changes due to the change in the moment itself as the crystal is rotated in the field.
We can now illustrate why k is sensitive to the transverse magnetic susceptibility. The second term of Eq. (1) selects the susceptibility tensor component that is perpendicular to both n and B. For field along the c-axis, and with the sample oscillating in the ac-plane (bc-plane), this selects χaa(B∣∣c) (χbb(B∣∣c)). These are what we define as transverse magnetic susceptibility components. Note that these are not off-diagonal susceptibilities, such as χbc(B∣∣c), which are not allowed in an orthorhombic crystal structure for field along crystal axes. Instead, χaa(B∣∣c) and χbb(B∣∣c) are longitudinal (diagonal) susceptibility components that are measured perpendicular to the static, applied magnetic field. The oscillating field component perpendicular to the external field direction is generated by the oscillation of the cantilever with a period of order 20 microseconds (see δB in the inset of Fig. 1a). Further experimental details are given in the Supplementary Information (SI).
Figure 1c shows the measured magnetotropic susceptibility divided by magnetic field for two different crystal orientations on the cantilever at T = 4 K, and with field applied along two different axes: ka(B∣∣b) and kb(B∣∣a) (see SI for details of the sample orientations). Figure 1b shows the measured magnetization along each of the principal crystallographic directions for comparison (reproduced from ref. 15). The metamagnetic transition is clearly visible near 35 tesla for B∣∣b in both the magnetization and the magnetotropic susceptibility measurements.
The magnetization in Fig. 1b is the longitudinal magnetization: it is found by integrating the magnetic susceptibility measured along the applied field direction, \({M}_{i}=\int \left(\partial {M}_{i}/\partial {B}_{i}\right)\,d{B}_{i}\)15,16. We use the longitudinal magnetization and its field derivative—the longitudinal magnetic susceptibility—in conjunction with Eq. (1) to calculate the expected magnetotropic susceptibility. This procedure does not account for any nonlinear transverse component to the magnetic susceptibility tensor (these components will be important later). This calculation is shown as dashed lines in Fig. 1c for B∣∣a and B∣∣b. The overall magnitude and the qualitative features of the calculated and measured magnetotropic susceptibility are in good agreement. Small differences can be attributed to a small misalignment between the rotation vector n, the plane of the cantilever, and the sample’s crystal axes (for more details regarding alignments, see SI II). This demonstrates that, for field along the a- and b-axes, the measured magnetotropic susceptibility is largely determined by the longitudinal magnetic susceptibility. As shown below, this will not be the case for other field orientations.
Figure 2a and b show the magnetotropic susceptibility for magnetic field applied along the c-axis, for both the ka and kb configurations. We also reproduce the data and calculations from Fig. 1c for comparison. Unlike the other two field orientations, the measured magnetotropic susceptibility for B∣∣c deviates strongly from the estimate made using the longitudinal magnetization and susceptibility alone (dashed purple line). The large negative response in the magnetotropic susceptibility compared to that inferred from magnetization measurements indicates that a new susceptibility component, hidden from the longitudinal magnetization measurements, becomes active in the magnetotropic susceptibility at a field scale of around 20 tesla.
Fig. 2: Angle-dependent magnetotropic susceptibility.
The alternative text for this image may have been generated using AI.
Magnetotropic susceptibility measurements performed in pulsed magnetic fields up to 60 T. Panel a) shows kb(B∣∣c) and compares it to kb(B∣∣a), and likewise for ka(B∣∣c) and ka(B∣∣b) in panel b). The dashed lines in both panels are the calculated values of k based on the measured longitudinal magnetic susceptibility data from Fig. 1b and Eq. (1) c and d show the magnetotropic susceptibility measured at multiple angles in the ac- and bc-planes, respectively. As the magnetic field approaches the c-axis in both planes, a large decrease is observed in k that onsets at roughly 20 T. This decrease persists for a range of angles in both planes around B∣∣c, and is abruptly cut off by the metamagnetic transition into the field-polarized phase (red shaded region in d) at θ = 59∘ in the bc plane.
Panels c and d in Fig. 2 show the evolution of the magnetotropic susceptibility for a broad range of field angles in the ac- and bc-planes. The large decrease in the magnetotropic susceptibility that onsets near 20 T for field along the c-axis is observed over a broad range of angles in both planes. In the bc-plane (Fig. 2d), the decrease in the magnetotropic susceptibility is abruptly truncated by the metamagnetic transition into the field-polarized phase (red shaded region). As magnetic field is rotated away from the b-axis, the metamagnetic transition moves to higher field and the magnitude of the jump in k increases before abruptly disappearing around 56∘. Because k is a susceptibility (i.e., a second derivative of the free energy with respect to angle), the jump is expected to grow in size as the metamagnetic transition becomes more second-order on approach to a critical endpoint. The largest jump in k appears at the angle where the jump in the magnetization (a first derivative) goes to zero17. These observations confirm the critical endpoint of the field-polarized phase as first identified by ref. 17.