Introduction

Transport properties related to topological bands have received considerable attention in fundamental science and potential technological applications. The anisotropic transport coefficients of topological systems are often useful for the mapping of hidden phases with distinct symmetries1. For topological semimetals, topological bands, such as Dirac points, Weyl points and nodal lines, exhibit exciting physical properties2,3,4,5,6. In particular, these bands strongly influence the transport properties due to their ultrahigh carrier mobility5,7 and strong Berry curvature8,9,10. The longitudinal resistivity based on topological materials is mostly related to large magnetoresistance11, linear magnetoresistance12, chiral-anomaly induced negative magnetoresistance13,14, and the planar Hall effect15,16. In magnetic topological materials, unconventional magnetoresistance behavior is often observed when combining topological bands and magnetism. Anisotropic magnetoresistance (AMR) is an effective method to detect the symmetry of the Fermi surface17,18, hidden phases1,19, and topological phase transitions20. Generally, the angle-dependent magnetoresistance originates from the relativistic spin-orbit coupling (SOC) in magnetic materials, inherently determined by the variation in conductivity and density of states (DOS) at the Fermi level in a magnetic field21. The AMR symmetry is closely related to the symmetries of the crystal and magnetic sublattice. An abnormal AMR effect, including rotating AMR and symmetry breaking, has attracted research attention due to its great physics and potential applications22. The occurrence of this anomaly requires a phase transition, such as a metamagnetic transition, topological phase transition, or density of charge carriers, to modulate or break the symmetry, such as CeAlGe1,23, CeSb/Bi24,25, Sr2IrO426,27,28,29, and EuTiO3 films30. However, an abnormal AMR in nonmagnetic ordering or nonmagnetic materials is quite rare.

Half-Heusler compounds, RPtBi, are a family of magnetic topological semimetals. They have attracted much attention due to their rich and excellent topological transport properties, such as giant anomalous Hall angle31,32,33,34,35,36,37,38,39, large negative magnetoresistance induced by chiral anomalies40,41,42, planar Hall effect43, and thermoelectric effect44. Compared to the transport behavior of ferromagnetic or noncollinear antiferromagnetic topological semimetals, half-Heusler compounds usually exhibit unconventional behavior, such as a drum-like anomalous Hall effect32,33,40, magnetic-field induced sign change of AHE39, and abnormal planar AMR15. These anomalies indicate that the band of RPtBi is unstable in the presence of an external magnetic field. Previous theoretical calculations showed that the Zeeman energy41 or an exchange field31 split the band and led to crossing (Weyl points) around the Fermi level in a magnetic field. However, detailed characterization of the specific dynamic progress is difficult and has not been reported thus far because this effect is weak and appears in low magnetic fields; thus, the Fermi surface detection tool, the Shubnikov-de Haas (SdH) effect, does not occur. The AMR effect is sensitive to topological phase transitions, Fermi surface symmetry, and hidden phases and can provide good insight into the evolution of topological bands in a magnetic field and show the band modulation process of half-Heusler alloys from another perspective.

In this work, we thoroughly studied the AMR in HoPtBi with \(B\perp I\), where B rotated in certain crystallographic planes. The AMR symmetry exhibited a dramatic change when B increased. Remarkably, B caused a change from twofold to fourfold AMR at low B, and as B increased, the magnetic field induced an AMR contour rotation of 45°. Further analysis via Fourier transformation (FT) indicated that the AMR originated from the contributions of C2, C4, and C6 signals. In particular, C2 and C4 signals dominated the AMR and exhibited a phase angle transition, phenomenologically explaining the dramatic change in the AMR behavior. The 3D map analysis of the AMR further identified that the dramatic AMR change was a 3D effect. The results indicated that this phenomenon could originate from the topological band change in an external magnetic field.

Materials and methods

Sample preparation

Single crystals of HoPtBi were grown from high-purity starting materials (molar ratio Ho:Pt:Bi = 1:1:15) via a standard self-flux method39. The pure elements were placed in an alumina crucible. The whole tube was sealed in an evacuated quartz tube to prevent high-temperature oxidation. The crystal growth was as follows: first, the tube was heated from room temperature to 1150 °C and maintained at this temperature for one day, then slowly cooled to 650 °C at a rate of 2 K h−1. The excess Bi flux was removed by centrifuging the tube at 650 °C. Bulk single crystals of HoPtBi with typical dimensions of 1.5 mm × 1 mm × 1 mm were obtained. The inset in Fig. S1b in the Supplementary information shows a photograph of the HoPtBi single crystal with the (111) plane.

Physical property characterization

The magnetic properties were measured in a Quantum Design Magnetic Property Measurement System (7 T, MPMS 3). The magnetotransport measurement was carried out based on the four-probe method in a Physical Property Measurement System (9 T, PPMS) with a sample rotator and a Cryogen-Free Magnet System with a 3D rotating magnetic field. The magnetic torque measurement was carried out on the PPMS with the Torque Magnetometer. The upper limit of torque is 10−5 Nm. The crystal structure of single crystals was characterized via a Rigaku XRD system with Cu Kα radiation (λ = 1.5418 Å) at room temperature.

First principle calculations

The density of states (DOS) was calculated using the WIEN2K code based on the framework of density functional theory45. The Perdew-Burke-Ernzerhof generalized gradient approximation46 was used to calculate exchange correlation potentials. The cutoff energy was set to −6.0 Ry, defining the separation of the valence and core states. Due to heavy elements, we included spin-orbit coupling (SOC) in a second variational procedure. A large exchange parameter, Ueff = 0.6 Ry, was applied to Ho. The Fermi surface was plotted with the XCrysDen program.

Results

Anomalous AMR with B rotating in the (100) and (110) planes

From Fig. 1a, HoPtBi crystallizes in a noncentrosymmetric cubic structure with a space group F-43m. The Néel temperature is 1.2 K35, and there are no traces of metamagnetic transitions in the paramagnetic state39. The resistivity ρxx (0 T) exhibits semiconductor to metal-like behavior (Fig. S1a). The red planes in Fig. 1a and d are the (100) and (110) planes, respectively. In configuration I (inset in Fig. 1b), B rotates in the (100) plane, I along [100] direction, and B and I remain perpendicular. Figure 1b shows the angular dependence of ρxx (B), and Fig. 1c represents the AMR curves at several typical magnetic fields for configuration I. Clearly, the ρxx (B) curves show a small change at B ≤ 1 T at different angles. The AMR curve at 0.7 T shows weak anisotropy and exhibits an approximate twofold symmetry (Fig. 1c). The peak position is approximately 90°. With increasing B, the ρxx (B) curves of θ = 0°/90° and θ = 45° separate at about B = 1 T. The AMR symmetry becomes a fourfold contour at 2 T. It forms peaks in the B // [100] direction (90°) and turns to a valley in the B // [110] direction (45°). At high B, the ρxx (B) curve of θ = 45° is enhanced. At B = 4 T, the AMR symmetry is rotated by approximately 45° such that the peaks of the fourfold AMR shift from the [100] to [110] direction. These results indicate that the HoPtBi single crystal has a complex magnetic-field-induced AMR effect.

Fig. 1: Anomalous anisotropic magnetoresistance.
figure 1

a Schematic representation of the crystal structure of the HoPtBi compound and the (100) plane (red plane). b Magnetoresistance as a function of B for configuration I at 2 K and several angles. The inset is a schematic of configuration I, in which B rotates in the (100) plane and I // [100] direction. θ is the angle between the normal line of the current plane and the magnetic field direction. B is always perpendicular to the I direction. c Angular dependence of the ρxx (B) curves for configuration I at B = 0.7, 2, and 3 T. d Crystal structure of HoPtBi and the (110) plane. e Magnetoresistance as a function of B dependence for configuration II at 2 K and several angles. The inset is a schematic of configuration II, in which B rotates in the (110) plane and I // [110] direction. f AMR curves for configuration II at B = 1.5 and 3 T.

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The MR line types at different angles also show strong anisotropy. Figure S5 provides the polynomial fitting of MR at θ = 0° and 45°. The MR in B < 1 T shows similar behavior, which is consistent with the weak anisotropy of AMR. In the range of 1 T < B < 3 T, the MR cannot be scaled by the linear and quadratic terms, which is potentially related to the band change, a phenomenon similar to that observed in ZrTe5 with topological phase transition20. Notably, the change in AMR symmetry occurs in this magnetic field range. The line types in a high magnetic field also show different behaviors in that the quadratic term changes sign. Remarkably, the Berry phase at high magnetic fields also shows distinct changes at θ = 0° (B//[100]) and 45° (B//[110]) (Fig. S17). A nontrivial Berry phase is taken at B//[100], but a value (-0.4π) other than π is taken at B//[110]. Thus, the symmetry rotation at 4 T is potentially related to the anisotropy of the Berry phase and topological properties.

In configuration II (inset in Fig. 1e), I // [110], B rotates in the (110) plane, and B and I also remain perpendicular. We select several angles to assess the B-dependence of ρxx (B) (Fig. 1e). The ρxx (B) curves exhibit distinctly different B-dependent behavior for high and low fields. In a weak magnetic field, ρxx (θ = 90°) shows V-type curves, unlike those at θ = 0, 30, and 45°. Correspondingly, a twofold symmetry AMR at 1.5 T is formed with a peak along [100] (90°) in Fig. 1f. In a high magnetic field, ρxx (θ = 45°) is enhanced, and the four-peak feature is stable at B ≥ 3 T (Fig. S10g).

In the supplemental materials (SM), Fig. S10 shows the AMR results for configuration III. The abnormal AMR phenomenon also appears. These results indicate that the abnormal AMR effect is not accidental. The magnetic field rotates in the (110) plane and current I // [110] in configurations II and III. The difference is that I is along the (111) plane for configuration II, and I is along the (110) plane for configuration III. If the AMR signal depends on the crystal symmetry, then a certain angle will appear between the two samples. The rotation angle between the two samples is approximately 36°, which is consistent with the geometrical relationship between the (111) and (110) planes (see Fig. S10d, e), indicating the crystal-axis dependence of the AMR. The abnormal AMR effect with B rotating in the (100) and (110) planes indicates a possible three-dimensional (3D) effect.

Magnetic field dependence of the abnormal AMR in configuration I

We performed comprehensive AMR characterization for configuration I at a series of magnetic fields to reveal the B-dependent AMR behavior. The detailed results of AMR at 5 K are shown in Fig. 2. Here, the curves with the typical features show the AMR evolution. Figure 2a–h shows the shape of the AMR at different magnetic fields. At B = 0.3 T, the twofold symmetry curve is formed, and the slightly splitting peak approaches 90°. As B increases, a small double-peak signal appears at approximately 0° and B = 1 T (Fig. 2b), the amplitude of which increases as B increases up to 2 T (Fig. 2d). In this process, the double-peak feature gradually disappears and forms a large peak. As B continually increases, the large peak splits into a double-peak signal and gradually shrinks (Fig. 2d–f). This progress is stronger at 2 K in Fig. S3. In the range of high magnetic fields, this signal around 0° disappears completely. Interestingly, the peak at 90° begins to split into two peaks at approximately 2 T (Fig. 2d). The splitting peaks completely separate as B increases, and they become stable at approximately 45 and 135°. At high magnetic fields, the fourfold symmetry of the AMR curves with a peak at 45° only persists (Fig. 2g, h).

Fig. 2: Detailed AMR curves for configuration I with B rotating in the (100) plane, I // [100] direction, and T = 5 K.
figure 2

a–h Featured AMR curves at different magnetic fields. i AMR map collected under a series of magnetic fields at 5 K. The blue arrow is the valley of MR, and the red arrow is the peak of MR. j Peak position of the AMR curves in (a) as a function of B, in which the rotated fourfold AMR is formed at Bc1. The relative shift between two fourfold MRs is approximately 45°.

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Figure 2i maps the AMR effect at T = 5 K and 0.3 T ≤ B ≤ 7 T. The color contrast indicates the change in ρxx (B). The fourfold AMR transition is clear at approximately 3 T. Another feature in the mapping is a twofold symmetry of the AMR signal with the pale blue color, crossing the high magnetic field at B ≥ 3 T, which is distinguished from that at 0.7 T in Fig. 1c. The valley of the twofold symmetry AMR at high magnetic fields rotates 90° compared to that at low magnetic fields. To clearly show the AMR evolution, the peak positions formed in the phase diagram of AMR in Fig. 2j are used. The symmetry of the AMR exhibits a dramatic change at B ≤ 3 T and T = 5 K. For B ≤ 3 T, the peaks around 0 and 90° independently evolve as B increases. Phenomenologically, the split peaks merge into a single peak at 1 T, then split again and form a rotated fourfold AMR at B ≥ 3 T. The rotation degree between the fourfold AMR at 1.5 T and the fourfold AMR at B ≥ 3 T occur approximately 45°.

Temperature dependence of the abnormal AMR effect in configuration I

In Fig. 3, we show the temperature dependence of the AMR map (T ≤ 20 K) in configuration I. Considering the significant difference between low B and high B, the temperature dependence AMR curves at 1 and 9 T are illustrated in Figs. 3a and b, respectively. When B = 1 T, the fourfold AMR with a peak along the [100] direction is observed at T = 2 K and 5 K. When T ≥ 10 K, 1 T can induce only a quasi-twofold AMR as T increases. When B = 9 T, a stable AMR contour and the rotated fourfold AMR with a peak along the [110] direction appear at T ≤ 20 K. Remarkably, ρxx (B) in the [100] direction remains almost unchanged as T increases, while ρxx (B) in the [110] directions decreases with increasing T. This significant difference in temperature-dependent properties indicates possibly different magnetoresistance mechanisms at high fields. Figure 3c–f map the AMR effect at T = 2, 10, 15, and 20 K. The critical field required for forming the rotated fourfold AMR is defined as Bc1. The critical field for reversing the fourfold AMR increases with T. Figure S4 shows the temperature dependence of Bc1. Bc1 increases smoothly with T when T ≤ 20 K. Thus, the complex AMR behavior with B rotating in the (100) plane is confirmed.

Fig. 3: Detailed T- and B-dependent AMR curves for configuration I with B rotating in the (100) plane, I // [100] direction.
figure 3

(a) and (b) are the AMR curves at 1 and 9 T, respectively. c–f AMR maps collected under a series of magnetic fields at 2, 10, 15, and 20 K. The blue arrow is the valley of MR, and the red arrow is the peak of MR.

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Mechanism of the complex AMR behavior

The previous section clarifies the unusual AMR behavior. However, the mechanism behind the phenomenon is still elusive since the mixture of the AMR signals with different symmetries causes further quantitative analysis to be challenging. Fourier transformation is an effective analytical tool for analyzing periodic signals, causing the complex AMR behavior, symmetry, and phase angle transition of AMR curves to become clearer. Thus, we perform FT analysis of the AMR in configuration I. Figure S12b shows the period (ω) of the FT amplitude (AFT) at 5 K. ω is the reciprocal of the FT frequency. Three signals with periods ω = 180, 90, and 60° dominate the AMR curves, corresponding to C2, C4, and C6 symmetry, respectively. Here, we extract the AFT and phase angle as functions of B. Figure 4a–c shows the magnetic field dependence of the phase and AFT for C2, C4, and C6 in configuration I at 5 K. The phase and AFT of C2, C4, and C6 signals strongly depend on B. First, we select the C2 signal to show the evolution. The AFT(C2) can be divided into two sections (Fig. 4a). In the low-magnetic field range (B < 3 T), AFT(C2) initially increases and then decreases to almost zero at 3 T. In the high-magnetic field range (B > 3 T), AFT(C2) monotonically increases. Along with this progress, the phase angle of the C2 signal shows an abrupt switch from 0 to 90° at the critical field, indicating that the twofold symmetry of the AMR signal rotates 90°. The switch indicates a possible band change. However, this kind of transition is not an isolated phenomenon. The phase angles of the C4 and C6 signals also show abrupt transitions in the low B range. The contribution of the C4 signal can be divided into three sections, as shown in Fig. 4b. In low- and high-magnetic field ranges, the phase angle is 45°. In the middle range, the phase angle shifts to 90°, and the fourfold symmetry of the AMR signal rotates 45° in this range. The critical fields of the two transitions are 1 and 3 T. The AFT(C4) also swells in the middle range. Remarkably, in the same range, a Hall resistivity well forms with B // [100]39 in Fig. S6a, and this anomaly is usually ascribed to the Berry curvature-induced AHE for half-Heusler systems31,32,39,40. Remarkably, the Hall resistivity exhibits a sign change between the B // [100] and B // [110] directions39 (Fig. S6). This result indicates that the system can potentially undergo a topological band change for different directions of the magnetic fields. All transitions of the three signals are summarized as the symmetry rotation or the inversion of the peaks and valleys. By comparing the AFT values of the three signals (see Fig. S12), the C4 signal dominates, followed by C2, while the contribution of the C6 signal is negligible. The complex AMR behavior is divided into three symmetrical signals, exhibiting three typical features. First, the phase angle shows sharp transitions at low magnetic fields. Second, coupled to the phase angle transition, the AFT amplitude shows a nonpolynomial variation with B. Third, the critical magnetic fields of the three signals are independent, indicating that different symmetrical signals are nonuniform magnetic responses. In the high-magnetic field range, the phase angles of all signals remain stable, and the AFT continues to increase monotonically.

Fig. 4: Extracted FT amplitude and phase angle for configuration I with B rotating in the (001) plane.
figure 4

a–c Magnetic field dependence of the FT amplitude AFT and phase angle for C2, C4, and C6 signals at 5 K. d–f Magnetic field dependence of AFT(C2), AFT(C4), and AFT(C6) at different temperatures, respectively. The inset in Panel (e) is the temperature dependence of the critical field Bc1 for the C4 transition at high magnetic fields.

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The temperature evolution of the three signals is also extracted from the FT of the AMR. Figure 4d–f show the B-dependence of the AFT for C2, C4, and C6 signals at different T. The AFT(C2) signal peak decreases with T in the low-magnetic field range. The swell almost disappears at 10 K. Interestingly, the critical field of the transition is stable at approximately 2.5 T for T < 10 K. Above the critical field, AFT(C2) decreases slowly with T. In contrast, the C4 signal exhibits a different temperature dependence. First, the peak of the AFT(C4) swell is stable at different T. However, the magnetic field range of the swell extends, and the critical field Bc1 increases with T. Second, the AFT(C4) value above Bc1 decreases monotonically with T. The similar behavior of Bc1 in Fig. 4 and the inset in Fig. 4e indicates that the progress extracted by FT is reliable and that the C4 signal dominates the abnormal AMR behavior at low magnetic fields. For the C6 signal, the peak of the AFT(C6) swell decreases and gradually disappears at 10 K. The AFT(C6) above the swell slightly decreases with T. Compared to the C2 and C6 signals, the C4 signal is more sensitive to the external field, including T and B.

3D mapping of the complex AMR behavior for configuration I

After the evolution of the novel AMR effect in two dimensions (2D) was resolved, so how does it evolve in the three -dimensional (3D) magnetic field? We constructed detailed 3D AMR maps at T = 2 K and a series of magnetic fields to characterize the exotic AMR effect more comprehensively. Figure 5d–f are 3D AMR maps at B = 0.5, 1, and 2 T, respectively. At B = 0.5 T, the AMR contour profile around φ = 90° exhibited quasi-twofold symmetry, which was consistent with the 2D measurement in Fig. S4. From an overall perspective, there was a splitting tendency of the orange region from a twofold symmetry to a fourfold symmetry. At 1 T, the fourfold contour profile was formed at approximately φ = 90° (Fig. 5e). Typical curves with φ = 70° and 90° are shown in Fig. 5b. At 2 T, the orange region further split into 2 × 2 dots. Here, two typical AMR curves were used to show the changes in Fig. 5c. When φ = 90°, the AMR showed two small peaks at approximately θ = 0° and two large peaks at approximately θ = 90°. When φ deviated from 90°, the two small peaks disappeared immediately, and the two large peaks were enhanced. From the 3D AMR maps, we observed the reconstruction progress of the AMR from a twofold at 0.5 T to a quasi-fourfold symmetry at 1 T, followed by further splitting at 2 T. These results indicated that the exotic AMR phenomenon in HoPtBi was a 3D effect and not only in a certain crystallographic plane. Research studies on Sr2RuO4 indicated that both the size and shape of the Fermi surface could be extracted from angular magnetoresistance oscillation measurements, which was consistent with those determined based on more rigorous quantum oscillation data47,48,49. The orbital AMR and the Fermi surface exhibited a close relationship. Therefore, as a powerful tool to detect the Fermi surface symmetry of 3D and quasi-2D systems50, the AMR provided direct information on the Fermi surface at low magnetic fields. The AMR transition occurring at low field could correlate to the reconstruction of the Fermi surface.

Fig. 5: 3D mapping of AMR at 2 K.
figure 5

a Configuration of the 3D AMR measurement, I // [001] and B rotation in 3D space. The direction of B is defined by combining θ and φ. b AMR curves with φ = 90 and 70° at 1 T. The dashed lines in Panel (e) show the positions of 3D mapping. c MR curves with φ = 90 and 70° at 2 T. The dashed lines in Panel (f) show the positions of 3D mapping. d–f are detailed angular spectroscopic mapping of MR at 0.5, 1 and 2 T.

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Fermi surface topology and density of states

In this section, due to the close relationship between the orbital AMR and the Fermi surface, we show the Fermi surfaces and density of states (DOS) of HoPtBi. Figure S14 shows the band structures of HoPtBi in paramagnetic and ferromagnetic states. The bands crossing the Fermi level are indexed as 34, 35, and 36. The corresponding Fermi surfaces are shown in Fig. 6d. Figure S15 also provides the Fermi surfaces in the first Brillouin zone. The FT analysis indicates that there are three contributions, i.e., C2, C4, and C6 signals, to the AMR. According to the relationship of the symmetry of AMR signals and the Fermi surface, the AMR symmetry is consistent with the Fermi surface in a certain plane. The Fermi surfaces of bands 34 and 35 show an approximate cubic structure, which is potentially related to the C4 signal. Band 36 forms the Fermi surface with a sixfold disk and triangular columns on both sides of the disk, which potentially contributes to C2 and C6 signals. Based on the twofold magnetic anisotropy of HoPtBi in the (100) plane (Figs S8, S9), the C2 signal may also contain a contribution from anisotropic magnetic scattering.

Fig. 6: Fermi surfaces and density of states of HoPtBi.
figure 6

a Schematic of the four-band crossing of the HoPtBi compound. b Positions of the band crossing tuned by the splitting degree. c Fermi surfaces for the three bands crossing the Fermi level with the magnetic moment along the [001] direction. d Density of states for M // [001], [110], and [111]. The inset is the relative change of M // [110] and [111] from the DOS[001].

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An important feature for the magnetic half-Heusler compound is magnetic field-induced band change. The nontrivial Berry phase extracted from SdH oscillations in high magnetic fields identified topological properties of HoPtBi (Fig. S17). In our study, the tunable effect of the external magnetic field on the bands is divided into two aspects: the intensity of band splitting and anisotropy. First, Fig. 6a shows a sketch of the four-band crossing at the Γ point. A magnetic field drives band splitting via Zeeman energy41 or an exchange field31. In this process, the position and number of Weyl points are tuned by the intensity of external magnetic fields (Fig. 6b and Fig. S16). Figure S14 provides the band structures of paramagnetic and ferromagnetic states to simulate this process. Second, Fig. 6c shows the DOS of the HoPtBi compound with a magnetic moment along [001], [110], and [111]. It has a low DOS at EF. The relative change in DOS with M // [111], [110] is approximately 5–10% to M // [001] (inset in Fig. 6c). This result correlates to an anisotropic tunable effect of magnetic fields on the bands. The theoretical calculations on GdPtBi31 and HoPtBi39 indicate that the position and number of Weyl points are also anisotropic for different moment directions. In our previous work on HoPtBi, the anisotropic additional Hall resistivity39 also identified this fact (Fig. S6). In addition, the Berry phase also shows strong anisotropy at high magnetic fields. As shown in Fig. S17, the nontrivial π-Berry phase is extracted at B // [100], but a value (−0.4π) other than π is taken at B // [110]. This indicates that HoPtBi has a strong anisotropy of topological properties with B rotating in the (100) plane. The rotated fourfold AMR at high magnetic fields is potentially related to the anisotropy of the Berry phase. The band is dynamic with moment M rotating in a certain plane, and the magnetic field-induced variation also contributes to AMR. Therefore, the source of the phase angle transition can be related to the modulation of topological bands by the strength and direction of B.

Discussion

In our measurements, the AMR was measured in the configuration of B perpendicular to I. In practical measurements, a mismatch between the directions of B and I is inevitable, i.e., the directions of B and I are not strictly perpendicular, which yields a non-90° fixed angle between B and I. However, the 3D measurements of AMR in Fig. 5 show that the novel phenomenon is a 3D effect and intrinsic for HoPtBi.

The marked feature of HoPtBi is the dramatic change in the AMR at low fields. This transition shows a smooth change as B increases. The complex transitions of AMR were decomposed into three different symmetry signals with certain phase transformations using FT analysis. The anomaly at low fields is effectively shown and related to the anomalous phenomenon of the topological band change. In this section, the specific mechanism of phase angle transitions is discussed. First, as elaborated in the introduction, a metamagnetic transition accompanied by changing magnetic anisotropy is predominantly the key trigger of abnormal AMR. However, in the HoPtBi compound, the magnetic ordering temperature is TN = 1.2 K35. All measurements were performed in paramagnetic states. To exclude the metamagnetic transition as the origin of the AMR transition, we made a comparison with the AMR change and magnetic anisotropy in same plane at 2 K and 5 K. In general, the symmetry change of AMR induced by meta-magnetic transitions should company with the symmetry change of anisotropic magnetization. As shown in Figs. S8 and S9, the highly sensitive magnetic torque measurement with B rotating in the (100) plane does not show any sign of magnetic anisotropy changes at B ≤ 2 T. Therefore, the metamagnetic transition is excluded as the origin of phase angle transitions; specifically, there must be another special factor responsible for the novel phenomenon. Second, the modulation of the topological band also induces an anomalous AMR effect18,20,51. In the results section, we identified the topological band change under magnetic fields. At high magnetic fields, the stable rotated fourfold AMR is correlated with the anisotropy of topological bands and the Berry phase. At low magnetic fields, the absence of SdH prevents the detection of information on the Fermi surface and topological band change via SdH oscillations. However, the line-type change and the sign change of anomalous Hall at low magnetic fields indicate that the topological band change occurs and exhibits strong anisotropy. In topological materials, the positions of Weyl or Dirac points related to EF usually strongly influence the transport properties2,3, including chiral anomaly52, AMR51, and AHE39,53. Research on BiSbTeSe2 nanodevices51 provides another possible path to explain the phase angle transitions, indicating that shifting EF up and down to the Dirac point by the applied gate voltage can switch the peaks and valleys of the AMR. Instead of the gate voltage, the magnetic field is the main factor that regulates the topological band in HoPtBi. The split of the four bands around EF forms the band crossing and the shift of crossing around EF in an external magnetic field and the modulated anisotropy of different bands. Therefore, the inversion of AMR peaks and valleys or the phase angle transitions in HoPtBi is deduced by modulating the position of the Weyl point crossing EF. Another important feature of this mechanism is the critical field. The AMR transition in BiSbTeSe2 nanodevices requires a critical gate field to shift the EF crossing Dirac point. From the FT analysis, the C2, C4, and C6 signals also show a critical field to shift the phase angle. All results indicate that the magnetic field-modulated topological bands are the origin of this novel AMR phenomenon.

Conclusion

We found an unusual AMR effect in the topological semimetal HoPtBi, indicating the modulation of the topological band change. Since most of the AMR effects were stable and the symmetry was unchanged, the AMR with complex transitions was rare and usually underwent hidden phase transitions or symmetry breaking. The AMR of HoPtBi exhibited a dramatic change from a quasi-twofold symmetry to a fourfold symmetry and finally became a stable rotated fourfold symmetry as B increased. Further measurements identified that the abnormal AMR phenomenon in a low-magnetic field was an intrinsic 3D effect. From the FT analysis, we decomposed the complex AMR behavior into three different symmetrical signals. The AMR anomaly was ascribed to the composition of nonuniform phase angle transitions. Further analysis indicated that this phenomenon originated from the modulation of the topological band, including the formation of band crossing and the shift of crossing around EF in an external magnetic field. Our results showed the rich physics of HoPtBi. In addition to HoPtBi, other rare-earth-based half-Heusler compounds could potentially exhibit similar unusual phenomena.