## Abstract

Lifshitz transition (LT) refers to an abrupt change in the electronic structure and Fermi surface and is associated to a variety of emergent quantum phenomena. Amongst the LTs observed in known materials, the field-induced LT has been rare and its origin remains elusive. To understand the origin of field-induced LT, it is important to extend the material basis beyond the usual setting of heavy fermion metals. Here, we report on a field-induced LT in PrAlSi, a magnetic Weyl semimetal candidate with localized 4*f* electrons, through a study of magnetotransport up to 55 T. The quantum oscillation analysis reveals that across a threshold field *B*^{*} ≈ 14.5 T the oscillation frequency (*F*_{1} = 43 T) is replaced by two new frequencies (*F*_{2} = 62 T and *F*_{3} = 103 T). Strikingly, the LT occurs well below the quantum limit, with obvious temperature-dependent oscillation frequency and field-dependent cyclotron mass. Our work not only enriches the rare examples of field-induced LTs but also paves the way for further investigation of the interplay among topology, magnetism, and electronic correlation.

## Introduction

The Lifshitz transition (LT) has received renewed attention in condensed matter physics. An LT^{1} is an electronic topological transition of the Fermi surface (FS) driven by the variation of the band structure and/or the Fermi energy. Since such a transition does not necessarily require simultaneous symmetry breaking, and meanwhile, it can occur at *T* = 0, and be tuned by parameters other than temperature (such as pressure, strain, doping, magnetic field, etc.)^{1,2}, it, therefore, can be deemed as a topological quantum phase transition. In the vicinity of Lifshitz transitions, many peculiar emergent phenomena may appear, such as van-Hove singularity, non-Fermi-liquid behavior, unconventional superconductivity, and so on (e.g. refs. ^{3,4}).

Compared with a number of cases tuned by doping or pressure that have been widely seen in topological systems^{5,6}, cuprate superconductors^{7,8}, iron pnictides superconductors^{9,10}, and other strongly correlated materials^{3,11}, the examples of LT driven by a magnetic field are rare. This is because the energy scale of a laboratory magnetic field, in the order of 1–10 meV, is much smaller than the characteristic energy scale of most metals (~10^{2}−10^{3} meV). Only in a few cases, mostly limited in heavy-fermion (HF) metals^{12,13,14,15,16,17,18,19,20}, the hybridization between conduction electrons and localized *f* electrons leads to narrow renormalized bands with a small Fermi energy and thus the Zeeman term can be sufficiently strong to shift the spin-split FS^{15}. Recently, field-induced LTs were also observed in some low Fermi energy non-magnetic semimetals, such as bismuth^{21}, TaP^{22}, and TaAs^{23}, wherein magnetic field beyond the quantum limit can empty a Dirac or Weyl pocket with small Fermi energy. However, in these cases, no additional Fermi pocket emerges and the carriers of the empty pocket were transferred to other pockets (previously existing).

Here we present a new example of field-induced LT beyond heavy-fermion systems in the magnetic Weyl semimetal candidate PrAlSi, by a systematic study of quantum oscillation (QO) effect with the magnetic field extending up to 55 T. We observe a single frequency (*F*_{1} = 43 T) below a critical field of *B*^{*} = 14.5 T, in agreement with what was previously reported^{24}. Above *B*^{*}, we see clearly the emergence of two new frequencies (*F*_{2} = 62 T and *F*_{3} = 103 T) and the disappearance of the original *F*_{1}. We exclude the possibility of magnetic breakdown and identify *B*^{*} as a critical point where the field-induced LT occurs. By comparing the reported Fermi surface of NdAlSi and the theoretical calculation of PrAlSi, we conclude that the LT occurs in the hole-like Weyl pockets along the direction of Γ−X of the Brillouin zone (BZ, hereafter). Our work not only enriches the rare examples of field-induced LTs but also paves the way for further investigation on the interplay among topology, magnetism, and electronic correlation.

The *R*Al*X* family (*R* = rare earth and *X* = Si or Ge) compounds have recently been proposed to host ideal candidates of magnetic Weyl semimetals^{25,26}, and provide a platform to investigate the interaction between magnetism and Weyl physics^{27,28,29,30,31}. They crystallize in a tetragonal structure with the noncentrosymmetric space group symmetry of I4_{1}/md (No. 109). One advantage of this family is that Weyl nodes generated by inversion breaking are robust and can be shifted by the Zeeman coupling in the *k* space^{25}. Several intriguing physical properties have been observed in this family. The list includes the coexistence of type-I and type-II Weyl fermion in LaAlSi^{32} and LaAlGe^{33}, topological Hall effect in CeAlGe^{34}, anisotropic anomalous Hall effect in CeAlSi^{35}, and Weyl-driven collective magnetism in NdAlSi^{36}. The compound PrAlSi studied here is a ferromagnetic semimetal with Curie temperature *T*_{C} ~ 18 K. A recent work based on static field (9 T) transport measurements by Lyu et al. revealed a large anomalous Hall conductivity ~2000 Ω^{−1} cm^{−1} and an unusual temperature dependence of QO with a single frequency^{24}.

## Results

### Experimental results and analysis

High-quality single crystals of PrAlSi were synthesized using the flux method. The inset of Fig. 1a shows the crystal structure of PrAlSi with the magnetic moments of Pr easily orientated along the *c*-axis after the application of a small field^{24}. In our transport measurements, the magnetic field was along the *c*-axis and the electrical current was along the *b*-axis (more details are presented in the “Methods” section and Supplemental Material). Figure 1a shows magnetoresistance measured in the static field at 2 K (red curve) and pulsed field at 1.8 K (black curve). Normalized magnetoresistance (*ρ*(*B*)−*ρ*(0))/*ρ*(0) reaches 116 at 55 T and remains non-saturating. The purple dashed line corresponds to *B*^{1.7}. Figure 1b presents the oscillatory part of the longitudinal resistivity Δ*ρ*(*B*), obtained by subtracting a smooth background from the measured *ρ*(*B*) (see Supplementary Fig. 2 for more raw data). At low temperatures, oscillations are visible above the field as low as 3 T, indicating the good quality of the sample. The rough Dingle mobility of 1/*B*_{c} = 0.33 T^{−1} is close to the average mobility of 0.26 m^{2}/V s yielded from the amplitude of the quadratic low-field magnetoresistance (see the inset of Supplementary Fig. 2a). The single-frequency QO for static field measurements retains until temperature down to 80 mK. A more complex pattern emerges when the larger magnetic field is applied, as seen in Fig. 1c.

Figure 2a and b show the results of the fast Fourier transformation (FFT) of the oscillatory part of the magnetoresistance Δ*ρ* as a function of 1/*B*. The SdH frequencies extracted from low-field (*B* < 14.5 T) and high-field (*B* > 14.5 T) data display a dramatic difference. Note that, since this compound is ferromagnetic, we took into the demagnetization factor to correct the applied field in all the analyses of SdH effect (see Section 9 of the Supplemental Material). We identified *B*^{*} = 14.5 T as a critical field, after checking several fields close to *B*^{*}. As shown in Supplementary Fig. 3, we have intercepted a field with different values, 10, 14, 14.5, 15, and 20 T for the segmented FFT analyses. We can see the crossover from the lower QO frequency to the two higher QO frequencies as the segment changes between the 5.7 T and the selected field. Two higher QO peaks emerge when we segment the field at 20 T and we note that the low QO peaks persist because of the inclusion of the low-field QOs below 14.5 T. From the comparison of the segmented FFT analyses, 14.5 T was determined as the field of the LT. There is only one fundamental frequency in the FFT spectrum until down to 80 mK below *B*^{*} (see Fig. 2a). It should be noted that *F*_{1} gradually decreases with temperature, changing from 43 T at 80 mK to 32 T at 35 K, as shown in the inset. Similar temperature dependence in *F*_{1} was also reported in earlier work on PrAlSi^{24}, whereas the values of *F*_{1} are relatively smaller than ours. We attribute this discrepancy to the difference in stoichiometry^{19}. Figure 2b shows that above *B*^{*} there are two new QO frequencies (*F*_{2} = 62 T, *F*_{3} = 103 T) and their high-order harmonics. Such a change in the Fermi surface is also manifested in the effective cyclotron mass *m*^{*}. The value of *m*^{*} for each frequency can be deduced from the fitting of FFT amplitude according to the temperature damping factor, and this yields the small *m*^{*} = 0.08*m*_{e} for *F*_{1} (with field range 5.7–14.5 T), and 0.23*m*_{e} and 0.28*m*_{e} for *F*_{2} and *F*_{3}, respectively, where *m*_{e} is the mass of a free electron. Interestingly, a careful look into the temperature dependence of the amplitude of the oscillatory peak leads to the fact that the effective mass is enhanced 2-fold between 2.8 T and *B*^{*}, as shown in Fig. 2c. This feature is reminiscent of HF systems displaying an LT and will be discussed more later on. The effective masses of *F*_{2} and *F*_{3} as the function of a field higher than *B*^{*} are absent here, mainly because it is rather difficult to extract the exact amplitude of entangled peaks in the oscillatory part from two frequencies and their harmonic terms. More explanation about the effective mass mentioned above is exhibited in Supplementary Fig. 5. The Fermi energy of the band corresponding to *F*_{1} is then estimated *ε*_{F} ~ 125 meV.

To further demonstrate the field-induced Fermi surface change near *B*^{*} = 14.5 T, we performed Lifshitz–Kosevich (LK) fitting on Δ*ρ*(*B*) measured under the pulsed field. We notice that all the analyzed FFT data are from the pulsed fields. More details can be found in Section 4 of the Supplementary Material. As is shown in Fig. 3a, the Δ*ρ* for the field below *B*^{*} can be well reproduced by the LK fitting with a single *F*_{1} (cf. the red dot line). However, such a fitting collapse when the field exceeds *B*^{*}. This problem can be fixed in an alternate fitting by employing both *F*_{2} and *F*_{3}, seeing the blue dot line in Fig. 3b. Noteworthy that this 2-frequency LK fitting fails in the low-field window, implying that *F*_{2} and *F*_{3} appear only in the high-field range.

Thus, the variation of QO frequencies with the disappearance of 43 T and the emergence of 62 and 103 T clearly point to the change in Fermi surface topology, viz an LT. Firstly, we can rule out magnetic breakdown as the origin, because there is no extra QO frequency (19 T and 60 T) in the low-field range even for temperature as low as 80 mK. Moreover, the peak with the frequency of 43 T does not persist under a high magnetic field, either. Secondly, we can also exclude a metamagnetic transition as the driver of this process. Figure 4a shows the field dependence of magnetization at 2 K with the magnetic field applied along the *c*-axis. One clearly finds that the magnetization saturates to ~3.1*μ*_{B}/Pr at a small field 0.48 T, and no additional transition can be resolved nearby 14.5 T except for some traces of de Haas–van Alphen oscillations (inset of Fig. 4a). This is different from the case of NdAlSi, where a magnetic transition to the final Weyl-mediated helical magnetism leads to a change in QO frequency^{36}.

In order to further clarify this field-induced LT, it is helpful to estimate the density of carriers of each Fermi pocket. According to the Lifshitz–Onsager relation, *F* = (ℏ/2*π**e*)*A*_{F}, where ℏ is Planck’s constant and \({A}_{{\rm {F}}}=\pi {k}_{{\rm {F}}}^{2}\) is an extremal cross-sectional area of the Fermi surface perpendicular to the field with Fermi wave vector *k*_{F}. The bands become non-degenerate due to spin–orbit coupling^{36}. Assuming these Fermi pockets are spheres, we find that the LT wipes out \({n}_{{{\rm {F}}}_{1}}\)= 3.2 × 10^{18} cm^{−3} and produces \({n}_{{{\rm {F}}}_{2}}\)= 5.5 × 10^{18} cm^{−3} and \({n}_{{{\rm {F}}}_{3}}\)= 1.2 × 10^{19} cm^{−3} per four pockets. Note that the total number of pockets would be a multiple of 4, due to the symmetric requirement (see below).

The total carrier density of the hole and electron can be also extracted by fitting the Hall resistivity with the two-band model,

Here, *n* and *μ* represent carrier density and mobility, and the subscripts h and e denote hole and electron, respectively. We obtain *n*_{h} = 4.3 × 10^{19} cm^{−3}, *n*_{e} = 5.3 × 10^{19} cm^{−3}, *μ*_{h} = 0.18 m^{2}/V s and *μ*_{e} = 0.22 m^{2}/V s. These values fit both Hall resistivity and magnetoresistivity reasonably well up to *B*^{*} (see Fig. 4c and Supplementary Fig. 7). The deduced mobilities are also close to the value obtained from quantum oscillations and magnetoresistance. The average zero-field mobility 〈*μ*〉 extracted from the residual resistivity *ρ*_{0} = 15 μΩ cm is about 0.5 m^{2}/V s, slightly larger than the finite-field mobility. Such a discrepancy has been observed in other semimetals^{37,38} and attributed to the field-induced mobility reduction. The carrier densities of hole and electron are within 10% of the compensation 2[*n*_{e}*n*_{h}/(*n*_{e} + *n*_{h})], compared to ~4% in bismuth^{39} and WTe_{2}^{40}. This near compensation would explain the observed unsaturated magnetoresistance. The slight excess in the hole may result from uncontrollable doping which could be also the reason for the sample dependence of *F*_{1} as discussed above.

This fit, which properly works up to *B*^{*} (shown by a black arrow in Fig. 4c), fails above *B*^{*}. The change occurring at *B*^{*} is evident in Fig. 4b, which shows the first derivative of the longitudinal and Hall resistivities. The violet dotted lines demonstrate the apparent change of slope in MR and Hall resistivities at 14.5 T. We let ∣*n*_{h}−*n*_{e}∣ to stay constant across *B*^{*}, in order to respect the Luttinger theorem^{41}. We infer that *F*_{2} and *F*_{3} should correspond to carriers of opposite signs with a density difference of \(| {n}_{{{F}}_{3}}-{n}_{{{F}}_{2}}| =6.5\times 1{0}^{18}\) cm^{−3}. Assuming that there are 8 pockets for *F*_{1} (\(2{n}_{{F}_{1}}\)= 6.4 × 10^{18} cm^{−3}), would be compatible with the Luttinger theorem. Both types of carriers increase by about 0.6 × 10^{19} cm^{−3} (\({n}_{{F}_{3}}-2{n}_{{F}_{1}}\) and \({n}_{{F}_{2}}\), respectively). By fitting the Hall resistivity curve with *n*_{h} = 4.9 × 10^{19} cm^{−3} and *n*_{e} = 5.9 × 10^{19} cm^{−3}, we obtain *μ*_{h} = 0.05(1) m^{2}/V s and *μ*_{e} = 0.16(1) m^{2}/V s. This is shown in Fig. 4c with a blue line. The mobility (\(\mu =\frac{e\tau }{{m}^{* }}\)) of holes drop (~72%) more than that of electron (~27%), implying the sign of *F*_{1} with a small mass should be hole-like, since its mass increases by 2.5 times by assuming the same scattering time *τ*. We conclude that *F*_{2} has an electron-like sign and *F*_{3} is a hole-like one. Thus, each hole pocket (*F*_{1}) evolves into a larger hole pocket (*F*_{3}) and an additional electron pocket (*F*_{2}). This indicates the existence of a van-Hove singularity (saddle point) in this system.

To get more information about the Fermi surface of PrAlSi, we performed the measurements of angular-dependent MR with a pulsed magnetic field at *T* = 1.8 K. Figure 5a shows the oscillatory component extracted by subtracting the smooth background from the MR measured at different *θ*, which is defined as the angle between the *c*-axis and the magnetic field, and the current was along *b*-axis as shown in the inset (see the Supplementary Fig. 10 for the raw MR data). The SdH oscillations evolve systematically and can be observed in all angles as the magnetic field is rotated from *θ* = 0° to *θ* = 87°. Figure 5b and c present the segmented FFT spectra for different angles *θ*. The inset of Fig. 5b shows the critical fields at various angles, which indicates that the LT also envolves with *θ*. We determined the *B*^{*} as the same method mentioned above and the detail is shown in Supplementary Fig. 4. The oscillation frequency *F*_{1} shown in Fig. 5b is weakly dependent on the angle, indeed indicating a nearly spherical Fermi surface of this band and the evolution of angle-dependent oscillation frequencies *F*_{2}, *F*_{3} and their harmonic terms shown in Fig. 5c. According to the experimental results, Fig. 5d presents the angle dependence of the quantum oscillation *F*_{1} which is shown as black symbols as well as *F*_{α}, *F*_{β}, *F*_{γ} and *F*_{η} are the calculated frequencies obtained from the SKEAF program^{42}. The Fermi level was shifted with 3 meV in the calculation because of the uncertain doping. The *F*_{α} is quite close to the observed *F*_{1}. Whereas the higher frequencies obtained from the calculation are not been observed by experiment in our case, may be due to the low mobility of these bands.

Next, we tried to locate these pockets in the BZ, and our similar calculation results (see Supplementary Fig. 11). The electronic structure also resembles that of NdAlSi^{36}. Since the center of the BZ is not occupied, symmetry imposes four-fold degeneracy of each pocket. According to the calculation, the schematic of the positions of pockets *F*_{1}, which are represented by orange spheres along the Γ−X direction of the BZ in Fig. 5e. The pockets along Γ−S are large and cannot be easily modified, hence for clarity they are not shown here and the Fermi pockets are plotted in Supplementary Fig. 9. Instead, the bands along Γ−X are shallow and host Weyl points and they should be susceptible to the magnetic field. The new *F*_{2} and *F*_{3} pockets appear after the disappearance of *F*_{1}. So it is a reasonable assumption that they locate on the Γ−X line as well where the pockets *F*_{1} lie on, as shown in Fig. 5f. Indeed, some Weyl signatures were observed for the pockets *F*_{1}. All the values extracted from the peak and valley positions of SdH oscillation fall on a line with an intercept of ~−0.01 in the Landau fan diagram, indicating a non-trivial *π* phase (see Supplementary Fig. 6b). Its cyclotron mass is a very small and changes with a magnetic field. Note that a normal parabolic band (\(E=\frac{\hslash {A}_{{\rm {F}}}}{2\pi m}\)) would not change its cyclotron mass (\({m}_{{\rm {CR}}}=\frac{{\hslash }^{2}}{2\pi }\frac{\partial {A}_{{\rm {F}}}(E,{k}_{\parallel })}{\partial E}\))^{43} as the Fermi energy changes. The increasing of the effective mass may imply the increase of spin fluctuation of the pockets with LT transition, similar to the early report^{14} instead of the response from a normal parabolic band. In order to better understand the Weyl nature of the Fermi pockets, it is indispensable for confirming the Weyl points along the direction of Γ−X line of the BZ. As shown in Fig. 5e, we obtained two pairs of Weyl points along the Γ−X line by band structure calculation of selected paths of the BZ and the details are listed in the table of Supplementary Fig. 11. The results agree with the prior work on Weyl points and Fermi surface of PrAlSi^{44}.

## Discussion

Take together, three prominent features of the LT in PrAlSi can be found: (i) the QO frequencies are strongly temperature dependent, (ii) the quasiparticle effective mass increases gradually when approaching *B*^{*}, and (iii) the critical field for such LT is far below the quantum limit. In conventional metals, only a small change in QO frequency of the order of (*k*_{B}*T*/*E*_{F})^{2} is expected upon warming^{45}. Temperature-dependent QO frequency was observed in some HF metals, which is typically ascribed to the sensitivity of the *f*–*c* (*c*-conduction electrons) hybridization and the corresponding Kondo resonance state with a variation of temperature^{12,13,46}. In this framework, the change of *m*^{*} was also attributed to an itinerant-localized transition of 4*f* electrons^{14}. However, in PrAlSi, DFT calculations manifested that the Pr-4*f* bands locate well below the Fermi level (see Supplementary Fig. 8), notable *f*–*c* hybridization is unlikely. Furthermore, PrAlSi is a low-carrier density semimetal, the Kondo screening is also expected to be weak according to Nozières exhaustion idea^{47,48}. Therefore, it is unlikely that the observed LT in PrAlSi originates from a competition between the Zeeman term and a characteristic Kondo coherence energy as in HF systems. In the cases of bismuth^{21}, TaP^{22}, and TaAs^{23}, the quantum limits have been reached to empty a Dirac or Weyl pocket and caused a field-induced LT. In our case, however, a rough estimate yields the quantum limit field in the range of 40–100 T, much larger than the critical field *B*^{*} ~ 14.5 T.

Another possibility for the observed LT in PrAlSi might be related to the crystal-electric-field (CEF) effect. This arises because the nine-fold-degenerate *j* = 4 multiplet of Pr^{3+} in a *D*_{2d} (\(\bar{4}\)2m) point symmetry CEF splits into two non-Kramers doublets and five singlets^{49}. A recent analysis based on specific heat measurements revealed that the ground state is probably a doublet, while the magnetic entropy gain reaches \(R\ln 3\) at about 20 K, \(R\ln 4\) at about 30 K, and saturates to \(R\ln 9\) at a temperature as low as ~95 K^{24}. This suggests that at least one excited state sitting not far above the ground doublet, potentially in the order of 10 K. It is reasonable to speculate that a magnetic field of ~ 10 T might be sufficient to modify the CEF energy levels and the orbital characters, which possibly changes the Fermi surface topology. In addition, this field-induced evolution of CEF levels is also qualitatively consistent with the temperature-dependent QO frequency and field-dependent *m*^{*} as observed experimentally. Actually, this scenario was also proposed recently for the field-induced Fermi surface reconstruction in CeRhIn_{5}^{50}. To further address this possibility, more experiments like inelastic neutron scattering are needed to figure out the diagram of the CEF splitting.

In summary, we grew high-quality single crystals and observed pronounced SdH oscillations in PrAlSi with a magnetic field up to 55 T. A LT transition occurs around 14.5 T. The change in carrier densities and Fermi pockets revealed by QO and the Hall effect are consistent with each other. By comparison with theoretical calculations, we propose that LT occurs along the Γ–X orientation and involves the Weyl pockets. One hole pocket becomes an electron pocket and a hole pocket, which indicates the existence of a van-Hove singularity. PrAlSi, therefore, represents a unique case of field-induced LT beyond the HF systems.

## Methods

### Sample growth and characterizations

Single crystals of PrAlSi used in our studies were synthesized using the flux method. The starting materials are high-purity chunks of praseodymium, silicon, and aluminum, mixed into an alumina crucible. Then, the alumina crucible and quartz wool were placed in a quartz tube, which was sealed under a high vacuum, and heated to 1100 °C at 3 °C/min. And holding for 12 h and then the tube was cooled down to 750 °C in 100 h and dwelt for 2 days. The excess Al flux was removed by centrifuging. Powder and single crystal x-ray diffraction (XRD) have been used to obtain the XRD pattern and confirm the structure and the orientation. The atomic proportion was determined by energy-dispersive x-ray spectroscopy (EDS).

### Measurements

The low-field magnetic transport measurements were performed on an Integra AC (Oxford Instruments) with a 16 T superconducting magnet and a Leiden dilution refrigerator with a 14 T superconducting magnet. Temperature- and field-dependent resistivity measurements were made in the standard four-probe method with a pair of the current source (Keithley 6221) and DC-Nanovoltmeter (Keithley 2182A). The high-field magnetic transport measurements were carried out under a pulsed magnetic field at Wuhan National High Magnetic Field Center (WHMFC). Golden wires were attached using silver paste on the rectangular sample and every contact resistance was maintained to be <2 Ω in the measurements.

## Acknowledgements

We thank Kamran Behina for insightful discussions. This work is supported by the National Key Research and Development Program of China (Grant No. 2022YFA1403503), the National Science Foundation of China (Grant Nos. 12004123, 51861135104, and 11574097), the Fundamental Research Funds for the Central Universities (Grant no. 2019kfyXMBZ071), the open research fund of Songshan Lake Materials Laboratory (2022SLABFN27) and National Key R&D Program of China (2022YFA1602602).

## Data availability

The data supporting the present work are available from the corresponding authors upon request.

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Z.Z. and L.Z. conceived and oversaw this work. L.W., H.Z., and L.Z. performed the experiments. Z.Z., G.X., L.W., and S.C. performed band structure calculations. L.W., L.Z., Y.L., and Z.Z. wrote the manuscript with inputs from co-authors.

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Wu, L., Chi, S., Zuo, H. *et al.* Field-induced Lifshitz transition in the magnetic Weyl semimetal candidate PrAlSi.
*npj Quantum Mater.* **8**, 4 (2023). https://doi.org/10.1038/s41535-023-00537-y

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DOI: https://doi.org/10.1038/s41535-023-00537-y