# Fermi Surface and Lifshitz Transitions of a Ferromagnetic Superconductor
under External Magnetic Fields

R. Leenen  High Field Magnet Laboratory (HFML-EMFL), Radboud University,
Toernooiveld 7, 6525 ED Nijmegen, The Netherlands.  Radboud University,
Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen,
The Netherlands.  D. Aoki  Institute for Materials Research, Tohoku
University, Ibaraki 311-1313, Japan  G. Knebel  Univ. Grenoble Alpes, CEA,
Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France  A. Pourret  Univ.
Grenoble Alpes, CEA, Grenoble INP, IRIG, PHELIQS, F-38000 Grenoble, France  A.
McCollam<EMAIL_ADDRESS>High Field Magnet Laboratory (HFML-EMFL),
Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands.
Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135,
6525 AJ Nijmegen, The Netherlands.

###### Abstract

Lifshitz transitions are being increasingly recognised as significant in a
wide variety of strongly correlated and topological materials, and
understanding the origin and influence of Lifshitz transitions is leading to
deeper understanding of key aspects of magnetic, transport or quantum critical
behaviour. In the ferromagnetic superconductor UCoGe, a magnetic field applied
along the $c$-axis has been shown to induce a series of anomalies in both
transport and thermopower that may be caused by Lifshitz transitions. The need
to understand the subtleties of the relationship between magnetism,
superconductivity and a heavy electron Fermi surface in the ferromagnetic
superconductors makes it important to explore if and why a series of magnetic-
field-induced Lifshitz transitions occurs in UCoGe. Here we report magnetic
susceptibility measurements of UCoGe, performed at temperatures down to 45 mK
and magnetic fields ($\mu_{0}H||c)$ up to 30 T. We observe a series of
clearly-defined features in the susceptibility, and multiple sets of strongly
field-dependent de Haas-van Alphen oscillations, from which we extract
detailed field-dependence of the quasiparticle properties. We complement our
experimental results with density functional theory bandstructure
calculations, and include a simple model of the influence of magnetic field on
the calculated Fermi surface. By comparing experimental and calculated
results, we determine the likely shape of the Fermi surface and identify
candidate Lifshitz transitions that could correspond to two of the features in
susceptibility. We connect these results to the development of magnetization
in the system.

##  I Introduction

Uranium-based intermetallics have partially filled 5f electron orbitals that
give rise to competition between Kondo hybridization, leading to heavy
fermions, and RKKY interaction, leading to magnetic ordering. A small subset
of the uranium heavy fermion systems are the ferromagnetic superconductors.
$\mathrm{UGe_{2}}$ was the first material discovered to support coexistence of
ferromagnetism and superconductivity, under applied pressure between $p$ = 1.1
GPa and 1.6 GPa[1]. Later, coexistence of ferromagnetism and superconductivity
was found in URhGe[2] and UCoGe[3] at ambient pressure. The recent discovery
of remarkable high-field re-entrant superconductivity in $\mathrm{UTe_{2}}$ in
the presence of ferromagnetic flutuations has added further interest to this
research field [4, 5, 6, 7].

In this article, we focus on UCoGe, which is a weak, uniaxial ferromagnet with
a moment of $\sim 0.06\mu_{B}$ along the easy $c$-axis, and a Curie
temperature of $T_{C}=2.7$ K [5]. Superconductivity coexists with
ferromagnetism below a transition temperature of $T_{sc}=0.5$ K[3, 8].
Previous work showed that superconductivity is strongly influenced by
ferromagnetic fluctuations associated with the 5$f$ electrons [9, 10],
leading, for example, to highly anisotropic superconducting properties [11].
Because the 5f and conduction electrons are hybridized into the Fermi sea,
knowledge of the Fermi surface is highly desirable as a source of information
about the magnetic and superconducting quasiparticles in this and other
ferromagnetic or nearly ferromagnetic superconductors [12, 13, 14].

Quantum oscillation studies are a powerful way to probe the Fermi surface.
Previous magnetoresistance and thermopower measurements of UCoGe showed
several quantum oscillation frequencies, and identified a series of magnetic-
field-induced features, proposed to be Lifshitz transitions [15, 16, 17], that
is, topological changes of the Fermi surface. Lifshitz transitions [18] are
known to be important in heavy fermion systems, where they can be associated
with a variety of phenomena, such as van Hove singularities [12, 19, 20, 21,
13], or localization of $f$ electrons [22, 23, 24, 25, 26, 27], and often
drive definitive features of magnetic, superconducting and quantum critical
behavior [25, 28, 29, 30, 31, 32, 33, 34]. As well as a general knowledge of
the Fermi surface of UCoGe, it is therefore also crucial to understand the
nature of any observed Lifshitz transitions and the magnetic field evolution
of the Fermi surface.

Here we report on measurements of a.c. magnetic susceptibility ($\chi$) and
the de Haas-van Alphen (dHvA) effect in UCoGe, in magnetic fields parallel to
the crystal $c$-axis. We see six clear features in susceptibility that
correspond to the previously proposed Lifshitz transitions [15, 17]. We also
observe multiple, well-resolved dHvA oscillation frequencies.

As a complement to our experimental results, we performed density functional
theory (DFT) calculations to determine the bandstructure and expected Fermi
surface of UCoGe. We included a simple model of the spin-splitting effect of a
magnetic field in our calculations. By comparing the measured and predicted
dHvA frequencies, we are able to determine the likely shape of the Fermi
surface. By examining the magnetic field dependence of the Fermi surface, we
identify possible Lifshitz transitions consistent with the experimentally
observed features in magnetic susceptibility. We suggest that only two of the
observed features are due to Lifshitz transitions.

##  II Methods

UCoGe crystallizes in the orthorhombic crystal structure of the TiNiSi type
(space group Pnma). The high-quality single crystals used for this work were
grown in a tetra-arc furnace using the Czochralski method [35]. We measured
two samples from different growth batches. Sample 1 (S1) has a residual
resistance ratio (RRR = $R(300K)/R(\xrightarrow{}0K)$) of 110, and dimensions
of $0.3$ mm $\times\leavevmode\nobreak\ 0.5$ mm $\times\leavevmode\nobreak\
2.0$ mm in the a $\times$ b $\times$ c directions, respectively. Sample 2 (S2)
has RRR $\sim 40$ and dimensions of $1.0$ mm $\times\leavevmode\nobreak\ 1.0$
mm $\times\leavevmode\nobreak\ 1.4$ mm along the a $\times$ b $\times$ c
directions.

We performed magnetic susceptibility experiments up to 30 T, at temperatures
down to 45 mK in a dilution refrigerator at the High Field Magnet Laboratory
(HFML) in Nijmegen. We used the field modulation method, with modulation
fields between $1.95$ and $3.51$ mT at frequencies between 100 and 150 Hz,
with the modulation field and the d.c. field parallel to the $c$-axis of the
sample. For the DFT calculations we used the APW+lo program WIEN2k[36]. We
performed calculations using the GGA-PBE potential[37] and included spin-orbit
coupling. All uranium 5f electrons were modelled as itinerant, that is they
contributed to the Fermi volume. We used 30000 $k$-points to ensure good
resolution at the Fermi energy, and from the calculated Fermi surface we could
extract the expected dHvA frequencies[38] (see Supplementary Information). The
lattice parameters were taken from a room temperature x-ray refinement study
of a single crystal [39].

##  III Results and discussion

###  III.1 Experimental results

Figure 1:  (a) The real part of the susceptibility ($\chi^{\prime}$) of sample
S1 as a function of magnetic field ($\mu_{0}H$) for different temperatures;
curves are shifted vertically for clarity. $\chi^{\prime}$ shows distinct
features in the background shape, which are indicated by the arrows. For the
lowest temperatures, one can clearly see quantum oscillations superimposed on
the spin susceptibility. (b) The imaginary part of the susceptibility
($\chi^{\prime\prime}$) for the 45 mK curve: $\chi^{\prime\prime}$ is small
and there are no clear features. The susceptibility is given in arbitrary
units, but the units and the scale are the same in panels (a) and (b).

Fig. 1(a) shows the real part of the magnetic susceptibility $\chi^{\prime}$
of sample S1 at various temperatures, as a function of the applied magnetic
field. Data for sample S2 are similar, and are shown in the Supplementary
Information. The curves are shifted vertically relative to each other for
clarity. The superconducting transition occurs at $\sim 1$ T 111It is
difficult to resolve the upper critical field value very accurately from our
susceptibility data because the imaginary susceptibility does not show a clear
peak at $H_{c2}$. It seems that the peak is smeared out, possibly because the
modulation field affects the pinning of vortices in the mixed state [74].
However, we note that a sharp feature at $H_{c2}$ is also not observed in
magnetic torque measurements [75], but is not shown to allow us to zoom in on
the other features in the susceptibility. Fig. 1(b) shows the imaginary part
of the susceptibility $\chi^{\prime\prime}$ at 45 mK. The imaginary signal is
small, with no clear features. $\chi^{\prime}$ shows distinct features at
field values indicated in the figure by arrows labelled $1-6$. The precise
locations are $\mu_{0}H_{1}=5.4$ T, $\mu_{0}H_{2}=9$ T, $\mu_{0}H_{3}=11.5$ T,
$\mu_{0}H_{4}=16$ T, $\mu_{0}H_{5}=22.5$ T, and $\mu_{0}H_{6}=28$ T, which are
close to the values previously reported in thermopower and resistivity [17,
15], and proposed to be field-induced Lifshitz transitions. The
demagnetization factors for our samples are negligible 222The demagnetization
factor $D_{z}$ was determined for both S1 and S2, using the expression
described by Aharoni[76]. For S1 we obtain $D_{z}=0.084$ and for S2 we obtain
$D_{z}=0.26$. For S1 this contribution is negligible, which means we can
interchange $B$ and $H$ in the analysis. For S2, the difference between $H$
and $B$ is larger. However, the analysis of data from S2 neglecting $D_{z}$
(Supplementary Information) shows results that are very similar to the results
for S1 presented here, so we again conclude that the demagnetization factor is
not of high importance.. We find that the features 1-6 are significantly
broadened with increasing temperature, but most of them remain clearly visible
to temperatures well above 1 K.

Superimposed on the spin susceptibility, we also observed well-defined dHvA
oscillations in the lowest temperature $\chi^{\prime}$ curves. According to
the Onsager relation, $F=\hbar A_{\text{ext}}/2\pi e$ [42], the frequency $F$
of the oscillations corresponds to the extremal area $A_{\text{ext}}$ of the
Fermi surface in the plane perpendicular to the applied magnetic field. By
extracting and tracking the dHvA frequencies we can therefore obtain
information about the Fermi surface evolution in an external magnetic field.

To follow the field dependence of the frequencies we carried out Fourier
transform (FT) analysis over a moving magnetic field window of fixed size in
$1/\mu_{0}H$. dHvA oscillations are periodic in inverse magnetic field [42,
43], so this approach fixes the number of oscillations of each frequency in a
given window, ensuring that the amount of information and resolution in the
Fourier transform remains comparable over the whole field range. The field
window moves with small steps, such that two consecutive windows will
partially overlap.

Fig. 2 shows the extracted dHvA frequencies from the 45 mK curve as a function
of the effective magnetic field,
$H_{\text{eff}}=(\frac{1}{2}(\frac{1}{H_{L}}+\frac{1}{H_{H}}))^{-1}$, which is
the average inverse field for the window over which we performed the Fourier
transform. $H_{L}$ is the low field boundary and $H_{H}$ is the high field
boundary of the analysed window.

We track each dHvA frequency in the region where it is visible. The series of
closely spaced field-induced transitions makes it difficult to track the
frequencies continuously with field, so our procedure was to analyse the data
in four different effective field regions $A,B,C,D$, where $A$ covers
$4.9-8.9$ T, $B$ is $9.3-14.3$ T, $C$ is $16.2-20.5$ T and $D$ is $17.3-26.0$
T, indicated in Fig. 2 with different colors. The size of the inverse field
window we used for the FTs was constant within each of these regions. The
locations of the field-induced transitions 1-5 are also indicated in Fig. 2 by
arrows. The top left inset shows an example of the FT performed for each
different field region, $A,B,C,D$.

Figure 2:  dHvA frequencies extracted from the $\chi^{\prime}$ vs $\mu_{0}H$
curve at T = 45 mK, as function of the effective field
$\mu_{0}H_{\text{eff}}$. The data are analyzed over the four regions $A$ to
$D$ indicated by the different colors and the letters at the top of the
figure. The locations of the field-induced transitions 1-5 are indicated by
arrows. The inset shows an example of the FT performed for each field region,
$A,B,C,D$, with the frequency peaks we extracted highlighted in different
colors.

Following the data shown in Fig. 2: in the lowest field region, $A$, we
detected only one frequency ($F_{\beta}\sim$ 240 T) 333Our results are in
agreement with the observed frequencies reported previously from thermopower
and magnetoresistance data [17, 15]. We use the same labels, $\alpha$,
$\beta$, $\gamma$ and $\omega$, for the observed frequencies.444The FTs shown
in the inset of Fig. 2 also seem to show a small peak at $F\sim 100$ T in
regions $A$ and $B$. However, this peak was not reproducible with repeated
analysis over different field windows, so we do not consider it to be a real
dHvA frequency., which grows with increasing magnetic field; in region $B$, we
see two frequencies at $F_{\gamma}=240$ T and $F_{\beta}=310$ T; in region
$C$, the low frequencies disappear and a high frequency of $F_{\omega}=580$ T
appears; finally, an even higher frequency of $F_{\alpha}=1$ kT is detected in
region $D$ and part of region $C$. $F_{\omega}$ disappears when crossing the
transition at $\mu_{0}H_{5}=22.5$ T. The detailed magnetic field dependence we
have extracted allows us to examine the field-evolution of the different Fermi
surface pockets. Combination of these experimental results with calculations
of the UCoGe bandstructure further allows us to consider various scenarios for
Lifshitz transitions. In the following sections, we describe our DFT
calculations and then move on to a dicussion of the magnetic field dependence
of the Fermi surface.

###  III.2 DFT Calculations

Figure 3:  (a) The calculated paramagnetic Fermi surface of UCoGe. Some
pockets are extended beyond the first Brillioun zone to show their shape.
There are three two-fold degenerate bands that cross the Fermi level. A hole-
like band (number 243-244) is indicated in blue/yellow, and two electron-like
bands are indicated in purple/green (bands 245-246) and red/light blue (bands
247-248), respectively. (b) shows the effect of a rigid band splitting by a
Zeeman term with $B_{\text{nom}}=60$ T, as described in the text, (b)(i) shows
the minority-spin (min) Fermi surface and (b)(ii) the majority-spin (maj)
Fermi surface. In panels (c)(i) and (c)(ii) we show the calculated dHvA
frequency evolution, $F$, with external magnetic field along the c-direction
for the minority- and majority-spin carriers, respectively.

We performed DFT calculations with and without spin-polarization. The
calculated bandstructures and Fermi surface details are given with further
discussion in the Supplementary Information. We found that Fermi surface
orbits derived from the calculations performed without spin polarization are
in significantly better agreement with the experimental data, so we focus on
these results, which we refer to as ‘paramagnetic’ calculations.

The resulting paramagnetic Fermi surface is shown in Fig. 3(a). It is very
similar to the paramagnetic Fermi surface previously reported by Samsel-
Czekała et al.[39]. Three two-fold degenerate bands cross the Fermi level. The
hole band that lies lowest in energy (band 243-244) is indicated in
blue/yellow, the two electron bands are shown in purple/green (band 245-246)
and red/light blue (band 247-248). At some places in the figure, the pockets
are extended beyond the first Brillioun zone to clearly show their shape.

The ferromagnetism of UCoGe is understood to be of weak itinerant nature [3,
46, 47]. The good agreement between our experimental dHvA frequencies and the
paramagnetic DFT calculations is therefore a significant result, as it
reinforces this view, implying a Fermi surface that is simply spin-split at
zero magnetic field, without marked other effects due to strong polarization.
On this basis, we have modelled the field dependence of the UCoGe Fermi
surface by splitting the paramagnetic Fermi surface at zero field, and then
increasing the splitting with a Zeeman-like term
($\pm\frac{1}{2}g_{0}\mu_{B}B_{\text{nom}}$). $B_{\text{nom}}$ is a nominal
magnetic field that captures both the ferromagnetic exchange field and the
applied external magnetic field. Our susceptibility data suggest weak
metamagnetic transitions associated with the observed features 2, 3, 4 and 6,
on a monotonically increasing, approximately linear, background (see
Supplementary Information, Fig. SI.2), a result supported by previous
magnetization data over a similar field range, which shows significant
curvature developing only above $\sim 20$ T [48]. Although our model of a
linearly spin-split Fermi surface is highly simplified, we therefore believe
it provides a reasonable starting point for this first investigation of the
magnetic field dependence of the UCoGe bandstructure and Fermi surface.
Similar models applied to other heavy fermion systems have yielded valuable
information [13, 32, 33].

We note that our model gives only a qualitative picture of the effect of an
external magnetic field on the Fermi surface, as we cannot relate
$B_{\text{nom}}$ directly to the experimentally applied field. The main
difficulty is that we do not have information about the $g$-factor, either the
zero-field value or any possible magnetic field dependence that may arise from
complex spin-orbit coupling in this material. We have used the free electron
value, $g_{0}=2$, for the model, but the real value of $g$ may be considerably
different, and the ratio $g/g_{0}$ is absorbed in $B_{\text{nom}}$.

In Fig. 3(b), we show the effect of an applied magnetic field
($B_{\text{nom}}=60\leavevmode\nobreak\ T$) on the Fermi surface in Fig. 3(a):
the polarized bands move either up or down in energy, leading to minority-spin
(i) and majority-spin (ii) Fermi surfaces. Comparing the Fermi surfaces in
Fig. 3(a) and (b), the most striking differences are that the small
ellipsoidal pockets from bands 245-246 have disappeared in the minority Fermi
surface, and the middle, cushion-shaped pocket of bands 243-244 has
disappeared in the majority Fermi surface. Another notable difference is in
the topology of the large pocket of band 243-244 around the T point, which
starts to touch the Brillouin zone boundary in the minority Fermi surface.
This leads to the hole or tube, visible at the T point in Fig. 3(a), being
broken open in Fig. 3(b)(i). These changes of the Fermi surface between Fig.s
3(a) and (b) clearly represent magnetic field driven Lifshitz transitions.

Fig. 3(c) shows the calculated dHvA frequencies for $\mathbf{H}\|c$, and how
they evolve with the field $B_{\text{nom}}$. The Fermi surface and
quasiparticle orbits were determined at the values of $B_{\text{nom}}$
indicated by the points, and the lines are guides to the eye. Fig. 3(c)(i) and
(ii) correspond to the field dependence of the minority and majority Fermi
surface orbits, respectively.

The disappearances of the 0.071 kT frequency of band 246 in Fig. 3(c)(i) and
the 0.034 kT frequency of band 243 in Fig. 3(c)(ii) correspond, respectively,
to the Lifshitz transitions associated with the disappearances of the
ellipsoidal (purple) and cushion-shaped (blue) Fermi surface pockets described
above. The disappearances of the 0.59 T frequency (Fig. 3(c)(i)) and $\sim$
1.29 kT frequency (Fig. 3(c)(ii)) of band 243-244 are both related to the
change of topology of the (blue/yellow) Fermi surface near the T-point.

###  III.3 Comparison of experiment with DFT calculations

We now return to the experimentally determined field dependence of the dHvA
frequencies shown in Fig. 2, and use the DFT calculations to help us
understand the experimental data. We first focus on the appearance and
disappearance of certain frequencies as the magnetic field is increased.

In general, the appearance or disappearance of a dHvA frequency can be
ascribed to a Lishitz transition. However, the presence and amplitude of dHvA
oscillations depends strongly on the applied magnetic field [43](see
Supplementary Information), and it is also possible for a frequency to
‘appear’ with increased magnetic field because its amplitude has become large
enough to detect above the noise level. The appearance of a dHvA frequency,
although suggestive, is therefore not enough by itself to identify a Lifshitz
transition. An unambiguous Lifshitz transition occurs when a dHvA frequency
disappears upon increasing the magnetic field.

In the $B_{\text{nom}}$ range shown in Fig. 3(c), our DFT calculations give
four candidate Lifshitz transitions associated with the disappearance or
change of shape of Fermi surface pockets. In the minority-spin Fermi surface
(Fig. 3(c)(i)), a frequency from band 244 ($F\sim 0.59$ kT) disappears for
$20<B_{\text{nom}}<25$ T (change in topology of blue/yellow pocket at the
T-point). This frequency matches the experimentally observed $F_{\omega}$
(Fig. 2), such that the disappearance of $F_{\omega}$ could correspond to this
predicted Lifshitz transition on band 244. If this is the case, it means that
$B_{\text{nom}}$ is close to our experimentally applied magnetic field. The
zero-field exchange splitting is very weak (see Supplementary Information), so
the factor accounting for the difference between $B_{\text{nom}}$ and the
experimentally applied field comes primarily from the ratio of the real
$g$-factor of the system and $g_{0}$. If we match the range $20\
\text{T}<B_{\text{nom}}<25$ T to the applied magnetic field of
$\mu_{0}H_{5}=22.5$ T, at which we observe the disappearance of $F_{\omega}$,
we can estimate the $g$-factor to be $1.78<g<2.22$ for $B\parallel c$.

The disappearance of pockets from band 246 ($F=0.071$ kT, small, ellipsoidal,
purple Fermi surface), and band 243 ($F=0.034$ kT, blue, cushion-shaped
pocket) also occur in a similar $B_{\text{nom}}$ range, and could correspond
to the disappearances of $F_{\beta}$ and $F_{\gamma}$. These small Fermi
surface pockets are not those which match best to the dHvA frequencies of
$F_{\beta}$ and $F_{\gamma}$ (see Supplementary Information), but it is
possible that the disappearance of one Fermi surface pocket has an effect on
the properties of remaining pockets, particularly those on the same band. In
our calculation, band 243 disappears between $7.5$ T$<B_{\text{nom}}<15$ T,
while band 246 disappears between $30$ T$<B_{\text{nom}}<45$ T. In the
experiment, $F_{\beta}$ and $F_{\gamma}$ disappear simultaneously, at $\sim
14$ T, so this is a significant discrepancy in the field location of the
Lifshitz transition for one of the calculated Fermi surfaces. We still
consider the transition on band 246 to be a candidate, however, as we do not
expect our simple model to perfectly capture the behavior of this complicated
Fermi surface. For example, our model assumes that all Fermi surface pockets
have similar, linear magnetic field dependence, which, as we discuss below, is
unlikely to be the case in reality.

The fourth Lifshitz transition predicted for $B_{\text{nom}}<100$ T, as shown
in Fig. 3(c), is on the majority-spin Fermi surface, where a frequency ($F\sim
1.29$ kT) from band 243 disappears for $45<B_{\text{nom}}<60$ T. If our
$B_{\text{nom}}$ is indeed similar to the experimentally applied magnetic
field, with a $g$-factor in the range estimated above, this Lifshitz
transition on band 243 would be outside of our measured magnetic field range.

Our calculations do not indicate the sudden appearance of any Fermi surface
pockets over most of the $B_{\text{nom}}$ range shown in Fig. 3(c), so we
cannot account for the experimental appearance of $F_{\gamma}$ above 9 T or
$F_{\omega}$ and $F_{\alpha}$ above $\sim 16$ T. As mentioned above, the
experimental appearance of the $\gamma$, $\omega$ and $\alpha$ frequencies may
simply be due to the improved resolution given by increasing magnetic field.
In our calculation at $B_{\text{nom}}=100$ T, a low frequency appears in band
245, as the large ellipsoidal pocket around the S point starts to extend to
the R corner of the first Brillioun zone and creates a small orbit centred on
the R point. This frequency is much lower than any of those observed
experimentally.

If we extend our calculations to much higher $B_{\text{nom}}>100$ T, the shape
of the Fermi surface changes significantly, such that several Lifshitz
transitions leading to new dHvA frequencies occur on different pockets of the
Fermi surface (see Supplementary Information, Fig. 8). These new frequencies
are in the correct range to correspond to $F_{\gamma}$, $F_{\omega}$ and
$F_{\alpha}$. If we consider the experimental disappearance of $F_{\omega}$,
and match $B_{\text{nom}}$ with $\mu_{0}H_{5}$, as we did above, this higher
$B_{\text{nom}}$ range would imply a minimum $g$-factor of $g=8.89$ for
magnetic field parallel to the $c$-axis.

We are not aware of any reports of the $g$-factor in UCoGe. For comparison
among other members of the family of U-based superconductors, Shick et al.
calculate a value of $g=0.72-0.78$ for $\mathrm{UTe_{2}}$[49], and in
$\mathrm{UGe_{2}}$, Abram et al. [50] found their model of the tricritical
phase diagram to be in best agreement with experimental data for a value of
$g\sim 2$ associated with the $f$ states. The $g$-factor in
$\mathrm{URu_{2}Si_{2}}$ was determined from the observation of spin zeros and
shows a large anisotropy from $g\|c=2.65\pm 0.05$ to $g\|a=0.0\pm
0.1$[51][52]. Future work to determine detailed $g$-factor information for
UCoGe is extremely desirable, as it would allow us to be sure of the
quantitative relation between $B_{\text{nom}}$ and the experimentally applied
magnetic field. This would, in turn, allow a quantitative appraisal of the
agreement between the theoretical and experimental results.

As well as the disappearance and appearance of dHvA frequencies, Fig. 2 also
shows whether the frequencies increase or decrease as a function of magnetic
field. Experimentally, however, we measure the back-projection of the true
frequency $F_{\text{true}}$, which is related to the measured or observed
frequency $F_{\text{obs}}$ according to the following relation[53]:

$F_{\text{obs}}=F_{\text{true}}-B\frac{\partial F_{\text{true}}}{\partial B}$
(1)

Because of this relation, we cannot directly relate an increase (decrease) of
a frequency as the magnetic field increases to the growth (shrinking) of a
Fermi surface pocket.

It is difficult to extract the true frequency from measured dHvA data, but it
is worth considering the field dependence of the frequencies in more detail to
gain some insight into the general behavior. Fig. 4 shows the different field
regions $A-D$ in separate panels. The experimentally measured frequencies,
$F_{\text{obs}}$, are indicated as scatter points. A possible true frequency
in each case is indicated by a solid line, with the corresponding
backprojected frequency shown as a red dashed line to illustrate the agreement
with $F_{\text{obs}}$. It is particularly striking that the strong curvature
of $F_{\omega}$ in region $C$ can be described with a true frequency that is
smoothly, and rather weakly, increasing. We note that the true frequencies
shown in Fig. 4 are not definitive, and our observed frequencies can also be
reproduced by alternative field dependences of $F_{\text{true}}$ (see
Supplementary Information). However, in matching our measured dHvA data to the
paramagnetic Fermi surface calculations, we have tentatively assigned the
measured $F_{\omega}$ frequency (at $\sim 580$ T) to the quasiparticle orbit
on the minority hole Fermi surface (band 244), which has a predicted frequency
of 590 T undergoes a Lifshitz transition for $20\ \text{T}<B_{\text{nom}}<25\
\text{T}$. In this case, we expect the Fermi surface to grow, and hence the
true frequency to increase with increasing magnetic field, as indicated in
Fig. 4C.

Our observation of field-dependent dHvA frequencies indicates that the
magnetic field evolution of the Fermi surface does not simply depend on
conventional, linear Zeeman-like spin-splitting. This is clear, for example,
from expression (1), where a linear field dependence of $F_{\text{true}}$,
which has a general shape $(F_{\text{true}}=fB+F_{0})$, would result in
$F_{\text{obs}}=F_{0}$, which is the zero-field frequency and is a constant.

Non-linear magnetic field dependence of the Fermi surface can originate in a
ferromagnetic material from Stoner excitations[54]. Additional contributions
to the magnetization, that might give more abrupt Fermi surface changes, can
arise from metamagnetic transitions, which cause sudden increases in the
quasiparticle density of states[55, 33]. Another scenario important in some
heavy fermion systems is a decrease in the strength of Kondo coupling with
increasing magnetic field, causing the $f$-electrons to localize and
eventually no longer contribute to the Fermi surface [56, 57, 27]. All of
these scenarios could, in principle, be relevant to UCoGe, and could manifest
differently on different pockets of the Fermi surface. On localization of
$f$-electrons, however, the effective mass $m^{*}$ would be expected to
decrease with increasing magnetic field [26] which is not the case, as can be
seen in Fig. SI.3 of the Supplementary Information. Another possible cause of
anomalous magnetization in UCoGe is the field-dependent cobalt moment, which
is neglegible at zero field, and starts to grow non-uniformly when magnetic
field is applied [58, 59, 60, 61].

Figure 4: The measured magnetic field evolution of the dHvA frequencies
($F_{\text{obs}}$) shown in Fig. 2 is shown again for the four different
regions $A$ to $D$ as scatter points. A possible true frequency,
$F_{\text{true}}$, is depicted in each panel as a solid line, and its
backprojection is shown as a dashed red line to demonstrate agreement with the
measured data.

We now turn to the additional information we can extract from the quantum
oscillations. We measured the oscillations at different temperatures, and can
determine the effective mass $m^{*}$ of the quasiparticles from the
temperature dependence of the oscillation amplitudes [42]. We obtained masses
ranging from $9.7\pm 1.4$ $m_{e}$ (where $m_{e}$ is the bare electron mass)
for $F_{\beta}$ in field region $A$, to $13.2\pm 1.0$ $m_{e}$ for $F_{\alpha}$
in region $D$, without any strong magnetic field dependence. The full set of
results for $m^{*}$ is shown in the Supplementary Information.

We also tracked the amplitude of the dHvA oscillations as a function of
magnetic field, and used this to extract the Dingle temperatures, which give a
measure of the scattering on the Fermi surface [42]. The Dingle temperatures
range from $0.30\pm 0.01$ K for $F_{\beta}$ in field region $A$ to $0.85\pm
0.01$ K for $F_{\alpha}$ in region $D$. These values indicate similar
scattering on all Fermi surface pockets and relatively long quasiparticle mean
free paths of a few tens of nanometers, as we would expect from samples with
high RRR’s. The full results and discussion of the Dingle factors are also
given in the Supplementary Information.

In the foregoing discussion, we considered the field dependence of the
experimentally observed dHvA frequencies and quasiparticle properties in
relation to the predicted field dependence of the Fermi surface extracted from
our calculations. We now want to explicitly relate the field dependence of the
dHvA frequencies to the features in $\chi^{\prime}$ (numbered $1-6$ in Fig.
1).

Features $\mu_{0}H_{1}$ and $\mu_{0}H_{6}$ in $\chi^{\prime}$ are at the edges
of our resolution and field range. We cannot resolve quantum oscillations in
our data below $\mu_{0}H_{1}\sim 5.4$ T, and the available magnetic field was
not high enough for us to to reliably extract dHvA frequencies above
$\mu_{0}H_{6}\sim 28$ T, so we are unable to comment on the field dependence
of the Fermi surface across these two features.

Fig. 2 shows that $F_{\gamma}$ first appears in our data at $\mu_{0}H_{2}\sim
9$ T. However, evidence for a Lifshitz transition at $\mu_{0}H_{2}$ is weak.
Previous measurements of quantum oscillations in the transverse
magnetoresistance identify $F_{\gamma}$ both above and below $\mu_{0}H_{2}$
[17], and this, combined with the smooth field dependence of $F_{\beta}$ that
we show in Fig. 2, suggests a normal field-evolution of the Fermi surface
through this region. The origin of the large peak in $\chi^{\prime}$ at
$\mu_{0}H_{2}$ therefore remains an open question. Earlier work suggested that
a feature near 9 T$||c$ in several properties of UCoGe, including magnetic
susceptibility, was due to a ferro- to ferrimagnetic transition, driven by a
field-induced moment on the Co site, aligned antiparallel to the U moment [58,
62, 60, 61]. These suggestions were apparently ruled out in reference [59],
where the authors provided evidence that the U and Co moments are parallel,
and proposed that a Lifshitz transition is responsible for the observed
amomalous behaviours near 9 T. Our results, showing weak metamagnetism and
little evidence for a Lifshitz transition at $\mu_{0}H_{2}\sim 9$ T, have re-
opened at least some aspects of this discussion, and prompt further work to
understand the development and influence of magnetization associated with the
Co $3d$ orbitals. Inclusion of the field-dependent Co moment in bandstructure
calculations could be informative in this respect.

The peak in $\chi^{\prime}$ at $\mu_{0}H_{3}\sim 11.5$ T is well-defined up to
temperatures of 1.3 K, and there is no evidence from our dHvA data of a Fermi
surface change at this field. Apart from its significantly smaller amplitude,
the feature at $\mu_{0}H_{3}$ is therefore similar to that at $\mu_{0}H_{2}$,
and further investigation of the contributions to magnetization in UCoGe is
needed to understand its origin.

Our experimental data suggest that the features in $\chi^{\prime}$ at both
$\mu_{0}H_{4}$ and $\mu_{0}H_{5}$ are associated with Lifshitz transitions.
This is supported by our model of the field dependence of the Fermi surface.
We note however, that the shapes of the features at $\mu_{0}H_{4}$ and
$\mu_{0}H_{5}$ are very different: a large peak at $\mu_{0}H_{4}$,
corresponding to a clear step in the magnetization (Supplementary Information,
Fig. SI.2), compared to a broad minimum in $\chi^{\prime}$ that translates to
a weak inflection point in the magnetization at $\mu_{0}H_{5}$. In our
interpretation, both Lifshitz transitions involve Fermi surfaces of a single
spin orientation, but the Lifshitz transition at $\mu_{0}H_{5}$ is a change of
topology of the largest Fermi surface pocket, whereas the transition or
transitions at $\mu_{0}H_{4}$ are the disappearances of small Fermi surface
pockets. It is interesting to consider if the larger feature at $\mu_{0}H_{4}$
reflects a direct effect on the itinerant moment when a spin-polarized Fermi
surface pocket disappears. It would be useful to study in more detail how
different types of Lifshitz transition influence the magnetic susceptibility
and wider properties of the system, as this can shed light on the relative
contributions of specific bands to the overall magnetization.

##  IV Conclusion

In summary, we measured the magnetic susceptibility of high quality single
crystals of UCoGe at low temperatures and in high magnetic fields. For
magnetic field parallel to the crystal $c$-axis, we observe a series of
clearly-defined features in $\chi^{\prime}$, as well as de Haas-van Alphen
oscillations with multiple, strongly field-dependent frequencies, which
suggest that field-induced Lifshitz transitions occur.

We combined our experimental results with DFT calculations of the UCoGe
bandstructure, specifically to study the shape and field-evolution of the
Fermi surface. We find that calculations of a paramagnetic Fermi surface
better match our experimental dHvA results than a fully spin-polarized Fermi
surface, and argue that this is in keeping with the weak itinerant
ferromagnetic moment in this material. On this basis, we used a simple model
of a spin-split paramagnet to estimate the magnetic field dependence of the
Fermi surface.

Through comparison of the measured and calculated magnetic field dependence of
the dHvA frequencies, we identified candidate Lifshitz transitions that could
account for two of the experimentally observed features in $\chi^{\prime}$:
disappearance of the $\beta$ and $\gamma$ frequencies in the region of
$\mu_{0}H_{4}$, and disappearance of the $\omega$ frequency at $\mu_{0}H_{5}$.
We show how these Lifshitz transitions correspond to changes in topology or
disappearance of specific Fermi surface pockets. To confirm that these
specific Lifshitz transitions are responsible for the experimental
observations would require information about the $g$-factor of the system,
which is needed to make a definite relation between the nominal magnetic field
in our model and the actual applied field in the experiments. The candidate
Lifshitz transition we discuss for $F_{\omega}$ would fix $1.78<g<2.22$.
However, further Lifshitz transitions, which could also be in keeping with the
experimental data, occur at much higher $B_{\text{nom}}$, and would imply a
$g$-factor above 8.89. This high value of $g$ seems unrealistic, and
calculations or additional experiments aimed at determining $g$ would
therefore be highly valuable.

Through further detailed analysis of the dHvA oscillations, we determined the
field dependence of the quasiparticle effective masses, and also extracted
Dingle temperatures to gauge the scattering on different pockets of the Fermi
surface. The effective masses are moderate for a heavy fermion material and
are similar on all Fermi surfaces, of order $10\leavevmode\nobreak\ m_{e}$, in
keeping with previous results [15, 17]. The Dingle temperatures and estimated
quasiparticle mean free paths, in the range 20-50 nm, reflect the relatively
low scattering in our high quality single crystal samples. Importantly, our
results for the effective masses and the Dingle factors show that there is
only a weak magnetic field dependence of the quasiparticle properties, as well
as a weak variation of properties between Fermi surface pockets, even as the
system is tuned through significant Lifshitz transitions.

Understanding the Fermi surface and intinerant quasiparticles in the U-based
ferromagnetic or nearly ferromagnetic superconductors is a key challenge in
the field of strongly correlated electron systems, as they play a crucial
connecting role between superconductivity and magnetism. Our results on UCoGe
give the first detailed information about the magnetic field dependence of its
Fermi surface properties. The insight we have provided into the nature of the
Lifshitz transitions in UCoGe helps us to understand the relation of the Fermi
surface to the magnetic response of the system, and identifies important
questions for further study.

Efforts to develop an improved model of the spin-splitting and magnetic field
dependence of the UCoGe Fermi surface are strongly motivated by the present
results. Incorporating the real, non-linear field dependence of the
magnetization, as a reflection of the field-dependent exchange splitting,
would be one way to do this [13, 63, 64]. It would require more precise
magnetization data over a wide field and temperature range than are currently
available, but could remove much of the approximation in identifying the
field-location of the Lifshitz transitions in the model. Another approach is
to determine the field-dependent bandstructure with advanced theoretical
techniques, such as the renormalized band method [65, 66, 21]. Combining an
improved model with an investigation into the contribution of cobalt to the
bandstructure and magnetization as the magnetic field is tuned would be
particularly informative.

The results we present here are also relevant in a wider context, as there is
an increasingly extensive tableau of strongly correlated and topological
materials, spanning from graphene to iron-based superconductors and archetypal
heavy fermion systems, [67, 68, 69, 70, 71, 72, 21, 33, 22, 20, 13, 73] in
which Lifshitz transitions are found to have significant influence on behavior
and properties. This creates a pressing need to explore and understand
Lifshitz transitions in diverse settings.

###### Acknowledgements.

We would like to thank S. R. Julian and G. Zwicknagl for valuable discussions.
This work was supported by HFML-RU/NWO-I, a member of the European Magnetic
Field Laboratory (EMFL) and the French National Agency for Research ANR within
the project FRESCO No. ANR-20-CE30-0020 and FETTOM No. ANR-19- CE30-0037.

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