# Observation of terahertz second harmonic generation from Dirac surface
states in the topological insulator Bi2Se3

Jonathan Stensberg  Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A  Xingyue Han  Department
of Physics and Astronomy, University of Pennsylvania, Philadelphia,
Pennsylvania 19104, U.S.A  Zhuoliang Ni  Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A  Xiong Yao
Department of Physics and Astronomy, Rutgers, The State University of New
Jersey, Piscataway, New Jersey 08854, U.S.A.  Xiaoyu Yuan  Department of
Physics and Astronomy, Rutgers, The State University of New Jersey,
Piscataway, New Jersey 08854, U.S.A.  Debarghya Mallick  Department of Physics
and Astronomy, Rutgers, The State University of New Jersey, Piscataway, New
Jersey 08854, U.S.A.  Akshat Gandhi  Department of Physics and Astronomy,
Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854,
U.S.A.  Seongshik Oh  Department of Physics and Astronomy, Rutgers, The State
University of New Jersey, Piscataway, New Jersey 08854, U.S.A.  Liang Wu
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Pennsylvania, Philadelphia, Pennsylvania 19104, U.S.A

###### Abstract

We report the observation of second harmonic generation with high conversion
efficiency $\sim 0.005\%$ in the terahertz regime from thin films of the
topological insulator Bi2Se3 that exhibit the linear photogalvanic effect,
measured via time-domain terahertz spectroscopy and terahertz emission,
respectively. As neither phenomena is observable from topologically trivial
In-doped Bi2Se3, and since no enhancement is observed when subject to band
bending, the efficient thickness-independent nonliear responses are
attributable to the Dirac fermions of topological surface states of Bi2Se3.
This observation of intrinsic terahertz second harmonic generation in an
equilibrium system unlocks the full suite of both even and odd harmonic orders
in the terahertz regime and opens new pathways to probing quantum geometry via
intraband nonlinear processes.

Introduction

Harmonic generation (HG) has been an invaluable nonlinear optical technique
since its first demonstration [1] and continues to power recent advances
ranging from the imagining of microscopic magnetic domains [2, 3] to the
development of tabletop sources of extreme ultraviolet and x-ray light for
attosecond science [4, 5]. Nevertheless, employing HG to study phenomena below
$\sim$ 100 meV has been severely impeded by the historical terahertz (THz) gap
[6], traditionally $\sim$ 0.1-30 THz (1 THz $\approx$ 4.1 meV), where
technical challenges have impeded the development of intense light sources.
Recent progress in intense THz generation [7, 8], however, has enabled the
first applications of HG to the THz regime. Since its first demonstration [9],
THz third harmonic generation (THG) has rapidly become a standard tool for
characterizing the Higgs mode [9, 10, 11, 12, 13] and other nonlinear optical
processes [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27] in a
variety of superconductors [28, 29, 30, 31, 32, 33, 34, 35]. Yet more
recently, odd-order THz-HG has been reported in doped Si [36, 37, 38] and
materials hosting Dirac fermions, namely graphene [39, 40, 41, 42], Cd3As2
[43, 44, 45], and the bismuth chalcogenide family of topological insulators
[46, 47, 42]. The latest studies have explored effectively tuning the
nonlinear process of THz-HG via gating [40] and metasurfacing [41, 42].

Despite these advances, THz-HG remains highly constrained, limited to odd-
order low harmonics. Most strikingly, intrinsic even-order THz harmonics,
which are only generated in systems that break inversion symmetry, have never
been demonstrated in an equilibrium material, having been observed only in
superconductors with a net propagating supercurrent [48, 49] or in carefully
engineered devices [50]. Moreover, second harmonic generation (SHG) has
different origins than odd harmonics, including conventional asymmetric
scattering [51], skew scattering of chiral Bloch fermions [52, 53], and
quantum geometry [54, 55, 56]. In the THz regime, scattering contributions are
diminished due to the frequency exceeding the scattering rate [57, 58];
however, the full derivation of intraband SHG from quantum geometry still
needs to be developed beyond terms the shared with nonlinear rectification
current [52]. The lack of THz-SHG in the studies of the bismuth chalcogenides
[46, 47, 42] is of particular note, as SHG–and even-order HG in
general–originating from the surface has been a well-established feature of
the nonlinear optical response outside of the THz regime [59, 60, 61, 62, 63,
64, 65]. Furthermore, as the bismuth chalcogenides are prototypical
topological insulators [66, 67, 68, 69] with a centrosymmetric bulk and
inversion symmetry-breaking surfaces, the second-order intraband response of
THz-SHG offers a nonlinear pathway to measuring properties of the topological
surface state and the quantum geometry while intrinsically avoiding properties
of the bulk band, without resorting to doping [47].

Here, we report the observation of THz-SHG from Bi2Se3 samples exhibiting the
linear photogalvanic effect (LPGE). With LPGE determined by THz emission [70]
and THz-SHG measured via intense time-domain THz spectroscopy (TDTS) [71],
thin films of Bi2Se3 that display LPGE are found to produce THz-SHG that is
highly efficient and independent of the sample thickness. As these phenomena
are not observed in topologically trivial In-doped Bi2Se3, and since there is
no enhancement observed when samples are subject to band bending, the
efficient response is attributable to the Dirac fermions of the topological
surface states. We further show that as both LPGE and SHG result from second-
order nonlinear processes, both effects originate from unequal populations of
twinned domains of the three-fold symmetric surface of Bi2Se3, which motivates
the future development of techniques to preferentially control the orientation
of crystal growth on millimeter scales, particularly for materials that break
various symmetries. This observation of intrinsic SHG in the THz regime in an
equilibrium system enables the future THz-HG investigation of nonlinear
intraband quantum processes via the full suite of harmonic orders, both even
and odd.

Results and Discussion

Thin film samples of Bi2Se3 are grown via molecular beam epitaxy on c-axis
Al2O3 substrates (10 mm x 10 mm x 0.5 mm), following standard growth
procedures, utilizing the two-step growth process [72, 73]. As each van der
Waals unit of Bi2Se3 forms a quintuple layer (QL), samples of 16 QL, 32 QL, 64
QL, and 100 QL are grown to form a thickness series (1 QL $\approx$ 1 nm).
Band bending can occur at the surface of Bi2Se3 [74, 75, 76], which introduces
an internal electric field $E_{int}$ near the surface and results in an
effective $\chi^{(2)}$ for second-order processes of
$\chi_{eff}^{(2)}=\chi_{ijk}^{(2)}+\chi_{ijkl}^{(3)}E_{int}$ [59, 61, 77].
Therefore, the samples are capped in situ with 50 nm of Se, which greatly
mitigates band bending, isolates the intrinsic second-order response, and
protects against damage [78]. Finally, as sufficient In-doping suppresses the
topological surface states of Bi2Se3 while being subject to the same band
bending effects [67, 46], a Se-capped 100 QL sample of (Bi0.9In0.1)2Se3 is
grown to directly discriminate between topological and trivial origins.

Figure 1: a. Schematic of the THz emission spectrometer. The NIR and THz beam
paths are depicted in magenta and green, respectively, and the THz beam path
is contained in a dry air-purged box. Both sample (S) and polarizer (P) are
mounted in rotating stages to enable characterization of the azimuthal angle
dependence of the THz emission. Labeled optical elements include beam splitter
(BS), pelical (Pel), ZnTe crystal (ZnTe), quarter wave plate (QWP), Wollaston
prism (WP), photodiodes (PD), and delay stage (DS). b. Normalized electric
field profile of the emitted THz pulse from Se-capped 100 QL Bi2Se3 and
(Bi0.9In0.1)2Se3 obtained by electro-optic sampling mapped in the time domain.
c. The peak normalized electric field as the Bi2Se3 azimuthal angle $\phi$ is
rotated. Figure 2: a. Schematic of the intense TDTS system. The NIR and THz
beam paths are depicted in magenta and green, respectively, and the THz beam
path is contained in a dry air-purged box. Labeled optical elements include
sample (S), LiNbO3 crystal (LiNbO3), THz filters (F1 and F2), diffraction
grating (DG), beam splitter (BS), pelical (Pel), ZnTe crystal (ZnTe), quarter
wave plate (QWP), Wollaston prism (WP), photodiodes (PD), and delay stage
(DS). b. Harmonic generation spectra for Se-capped Bi2Se3 and (Bi0.9In0.1)2Se3
samples under a 0.5 THz fundamental pump with respect to a reference
substrate. The change between spectra taken with 1.0 THz-specific and 1.5 THz-
specific filters are indicated by breaks in the spectra, and the Bi2Se3 band
structure is sketched with the Fermi level (inset). c. Bi2Se3 peak spectral
weight at the 2nd and 3rd harmonic as a function of the peak 0.5 THz pump
field $E_{pump}$, with fits to $E_{pump}^{2}$ and $E_{pump}^{3}$ respectively.
d. Peak spectral weight at the 2nd and 3rd harmonics as function of sample
thickness.

The samples of Bi2Se3 are evaluated for their room temperature LPGE response
by measuring the THz emission [70] of the samples under normal incidence, near
infrared (NIR) pumping. When a single domain of Bi2Se3 is pumped above the
bulk band gap of $\sim$ 0.3 eV, photocurrent current generation moves free
carriers across the domain [79, 77, 80]. These carriers are generated from the
so-called shift vector from quantum geometry, and also scatter asymmetrically
off of the wedge-shaped potential of the trigonal domains [51, 81], yielding a
low-frequency LPGE current with directionality defined by the domain
orientation and which couples out to free space as a THz pulse. This emitted
THz pulse is generated and detected by a THz emission spectrometer depicted
schematically in Fig 1.a and described in previous works [82, 83]. In brief,
the sample is pumped over a spot size of order 1 mm by linearly polarized,
broadband 1530 nm (0.81 eV), 50 fs pulses with a repetition rate of 1 kHz. A
quasi-single cycle THz pulse is emitted from the sample in transmission
geometry; collected, collimated, and focused onto a ZnTe crystal by a pair of
off-axis parabolic mirrors in $4f$ geometry; and measured via electro-optic
sampling [84]. By varying the optical path length of the NIR probe pulse via
the delay stage, the electric field profile of the emitted THz pulse $E_{THz}$
is mapped out in the time domain.

THz emission data is depicted in Fig 1.b,c for typical 100 QL Bi2Se3 and
(Bi0.9In0.1)2Se3 samples. As shown in 1.b, the (Bi0.9In0.1)2Se3 emits no
signal whereas a pronounced quasi-single cycle THz pulse is emitted from the
Bi2Se3, the polarity of which changes sign throughout the duration of the
pulse when the sample is rotated azimuthally by 180 degrees. By tracing out
the peak value of $E_{THz}$ as the sample is rotated, as shown in Fig 1.c, the
azimuthal angle dependence clearly follows
$E_{THz}^{max}=E_{0}\sin{(3\phi+\phi_{0})}$, where $E_{0}$ is the peak
electric field strength, $\phi$ is the azimuthal angle, and $\phi_{0}$ is an
arbitrary angle difference between the crystalline axes and the lab frame for
a given sample.

This $\sin{(3\phi+\phi_{0}})$ dependence of the emitted $E_{THz}$ is precisely
the azimuthal angle dependence expected for THz emission from a single domain
of Bi2Se3 due to LPGE under normal incidence [59, 60, 61, 62]. LPGE is only
permitted in systems that break inversion symmetry [51]. As bulk Bi2Se3 is
centrosymmetric, only the three-fold symmetric surface of Bi2Se3 breaks
inversion and two-fold rotation symmetries and contributes to the LPGE,
yielding a $\sin{(3\phi+\phi_{0})}$ dependence of the LPGE current for a
single domain under normal incidence (See supplemental information (SI) for
derivation). Since the spot size of the NIR pump (order 1 mm) vastly exceeds
the domain size of Bi2Se3 (order 1 $\mu$m; see Fig 3.c,d), the THz emission
method measures the net LPGE produced by a large ensemble of Bi2Se3 domains.
Since twinned domains in the sample produce oppositely-signed LPGE responses,
as demonstrated in Fig 1.b, the azimuthal dependence of the single domain LPGE
can be generalized to the case of many domains as
$(f_{+}-f_{-})\sin{(3\phi+\phi_{0})}$, where $f_{+}+f_{-}=1$ indicate the
fraction of oppositely-oriented domains, respectively (See SI for derivation).
The observation of a clear three-fold LPGE signal from the Bi2Se3 therefore
indicates the presence of a dominant domain orientation over millimeter length
scales, such that $f_{+}\neq f_{-}$.

Figure 3: a,b. Comparison of THz-SHG and LPGE, respectively, for two bare 100
QL Bi2Se3 samples. The band bending effect on the uncapped samples is sketched
in (a, inset), and the azimuthal angle in (b) is offset for clarity. c,d.
Atomic force microscopy images of representative areas of bare 100 QL Bi2Se3
for Sample 1 and Sample 2, respectively, where oppositely-oriented domains on
the surface are highlighted with blue and red boxes.

As SHG is limited by the same symmetry considerations as LPGE and expected to
be generated from the surface of Bi2Se3 [59, 60, 61, 62], the THz-HG of the
samples is measured via intense TDTS [71] at room temperature as shown
schematically in Fig 2.a. Intense broadband, quasi-single cycle THz pulses are
generated from LiNbO3 via the tilted pulse front method [85, 86, 87] by
pumping with linearly polarized, broadband 800 nm, 35 fs pulses with a
repetition rate of 1 kHz. The generated intense THz pulses are collected,
directed through the sample at a waist of order 1 mm, and focused onto a ZnTe
crystal by a quartet of OAPs in $8f$ geometry. Prior to the sample, optical
filters (F1) convert the broadband pulse into a narrow-band few cycle pulse
centered at 0.5 THz (spectral width $\sim$ 20$\%$). After transmitting through
the sample, the resulting THz pulse is passed through optical filters (F2) to
suppress the spectral weight of the 0.5 THz fundamental pulse and pass the
frequency range around the harmonic to be observed: 1.0 THz for SHG or 1.5 THz
for THG. The remaining THz that impinges upon the ZnTe crystal is measured by
standard electro-optic sampling [84], allowing the electric field profile to
be mapped out in the time domain by varying the delay stage of the probe
pulse. Finally, taking the Fourier transform of the THz pulse in the time
domain yields the spectral weight of the pulse as a function of frequency.

The HG spectra for the Bi2Se3 samples shown in Fig 2.b exhibit clear THz-SHG
at 1.0 THz and THz-THG at 1.5 THz when pumping with the 0.5 THz fundamental;
however, no HG above the background level is observed for (Bi0.9In0.1)2Se3.
Such THz-HG has been attributed [39, 40, 41, 42, 43, 44, 45, 46, 47, 63, 64,
65] to the ultrafast dynamics of Dirac fermions as they are accelerated by the
THz light field through the surface Dirac cone. This Dirac fermion origin of
the efficient THz-HG in the bismuth chalcogenides is confirmed here by direct
comparison with the topologically trivial (Bi0.9In0.1)2Se3 sample, where the
THz-HG is dramatically reduced in the absence of Dirac fermions. Furthermore,
three key features demonstrate strong agreement with the THz-THG observed in
previous studies [46, 47, 42] of bismuth chalcogenides: First, the THG
conversion efficiency is $\sim 0.04\%$ (accounting for the THG-specific
filters), which closely matches the conversion efficiency in previous reports.
Second, the yield of the THz-THG scales perturbatively as $E_{pump}^{3}$, as
shown in Fig 2.c which likewise agrees with previous results and contrasts
sharply with the saturation observed in graphene [39, 40, 41, 42] and Cd3As2
[43, 44]. Third, the THz-THG yield is nearly thickness-independent, as shown
in Fig 2.d, which is consistent with the dominant contribution to the THz-THG
originating from the topological surface state. Together, these results
reaffirm the conclusions of the previous studies, indicate that the intrinsic
nonlinear properties of the Bi2Se3 samples measured here are consistent with
those of the previous studies, and demonstrate the dominance of the surface
Dirac fermions in the nonlinear THz response even when bulk contributions are
not symmetry-forbidden.

Returning to Fig 2.b, a clear THz-SHG peak is observed at 1.0 THz for the
Bi2Se3 samples, in addition to the THz-THG peak at 1.5 THz. As shown in Fig
2.c, the 1.0 THz peak scales according to the $E_{pump}^{2}$ expectation for a
perturbative second-order response (See SI for comparison to linear fit).
Furthermore, the 1.0 THz peak is thickness independent, as shown in Fig 2.d,
matching the expectation for Bi2Se3 where only the surface breaks inversion
symmetry and two-fold rotation symmetry as required for a second-order
process. Since bulk contributions are symmetry-forbidden [59, 60, 61, 62], the
band bending contribution is suppressed by the Se capping layer [78] (sketched
in Fig 2.b inset) irrespective of the sample topology, Dirac fermions are
known to dominate the nonlinear THz response when present [46, 47, 42, 39, 40,
41, 42, 43, 44, 45, 63, 64, 65], and no THz-SHG is observed from the
topologically trivial Se-capped (Bi0.9In0.1)2Se3, this clear THz-SHG response
from Bi2Se3 can be attributed to the topological surface state. This
observation of THz-SHG reaching a high conversion efficiency of $\sim 0.005\%$
(accounting for the SHG-specific filters) is consistent with HG studies
outside of the THz regime [59, 60, 61, 62, 63, 64, 65], but contrasts sharply
with the previous THz studies [46, 47, 42] of bismuth chalcogenides, which
failed to report THz-SHG.

Finally, we verify these conclusions by measuring capless Bi2Se3 samples. Fig
3.a,b respectively compare the THz-SHG and THz emission results for two 100 QL
bare Bi2Se3 samples taken from the same batch to ensure similar growth
quality, with both samples exposed to atmosphere for a similar duration of
several hours to ensure stable and comparable $\chi_{eff}^{(2)}$ due to band
bending [59, 61, 63, 77, 80] (sketched in Fig 3.a inset). The capless samples
exhibit no enhancement of either LPGE or THz-SHG compared to the Se-capped
samples, confirming that the band bending contribution is small and that the
large second-order nonlinear response in Bi2Se3 is indeed due to the Dirac
fermions. The lack of a cap permits the orientation of surface domains to be
determined by atomic force microscopy (AFM). As shown in Fig 3.c,d,
respectively, AFM clearly reveals twinned domains on the surface of both
samples. As variations in domain size and quality average out over many
domains, the ratio of many domains yields a rough but sufficiently accurate
proxy for comparing the relative populations of oppositely-oriented domains. A
careful counting of these domains shows that the ratio of oppositely-oriented
domains is $\sim 1.5:1$ in Sample 1 and $\sim 1.8:1$ in Sample 2,
demonstrating that both samples have unequal (not $1:1$) populations of
twinned domains. These domain ratios correlate with the magnitude of the
second-order responses for the samples, agreeing with the theoretical
expectation and confirming the dependence of both LPGE and THz-SHG upon the
presence of unequal populations of twinned domains. This dependence of the
second-order processes upon unequal populations of twinned domains presents
one potential reason why the previous studies [46, 47, 42] of bismuth
chalcogenides failed to report THz-SHG, and it highlights the importance of
improving control over crystal growth for materials that break various
symmetries.

To summarize, we have observed THz-SHG from the topological surface state of
Bi2Se3 featuring a highly efficient conversion rate of $\sim 0.005\%$. These
results extend beyond previous studies [46, 47, 42] of similar topological
insulator bismuth chalcogenides and demonstrate intrinsic SHG in the THz
regime for an equilibrium system. This advance motivates the extensive
theoretical elaboration of nonlinear intraband quantum processes, as has been
accomplished for the interband [56], and enables further development of THz-HG
techniques to discriminate between nonlinear quantum effects. While various
scattering [51, 52, 53] and quantum geometry [88, 89, 90, 91, 92]
contributions to intraband SHG involving Dirac fermions have been proposed, a
full treatment of intraband SHG is lacking [52] and remains for future
theoretical work. Altogether, these results vastly expand the range of future
studies by unlocking even-order HG in the THz regime, open new experimental
and theoretical pathways to the low-energy study of topological surface states
and nonlinear intraband quantum processes, and motivate further efforts to
develop efficient THz optical elements and material growth techniques that
yield untwinned domains.

Acknowledgement

We thank F. de Juan, A. Grushin, T. Morimoti, and J. Lu for helpful
discussions. This project was mainly sponsored by the Army Research Office
under the YIP award under Grant No. W911NF-19-1-0342, and is supported under
the grant No. W911NF-20-2-0166. J.S. was also partially supported by the
Gordon and Betty Moore Foundation’s EPiQS Initiative under the grant GBMF9212
to L.W.. X.H. is supported by the NSF EPM program under grant no. DMR-2213891.
J.S. and Z.N. acknowledges support from the Dissertation Completion Fellowship
at the University of Pennsylvania. The work at Rutgers by X. Yao, X. Yuan, D.
M., A. G. and S. O. was also supported by NSF DMR2004125, and the center for
Quantum Materials Synthesis (cQMS), funded by the Gordon and Betty Moore
Foundation’s EPiQS initiative through grant GBMF10104.

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#  Quantum critical fluctuations generate intensely magnetic field–resilient
superconductivity in UTe2

Z. Wu,1† T. I. Weinberger,1† A. J. Hickey,1 D. V. Chichinadze,2
D. Shaffer,3 A. Cabala,4 H. Chen,1 M. Long,1 T. J. Brumm,2 W. Xie,5
Y. Lin,5, Y. Skourski,6 Z. Zengwei,5 D. E. Graf,2 V. Sechovský,4
G. G. Lonzarich,1 M. Vališka,4 F. M. Grosche,1 A. G. Eaton1∗.

1Cavendish Laboratory, University of Cambridge,
JJ Thomson Avenue, Cambridge, CB3 0HE, UK
2National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA
3Department of Physics, University of Wisconsin-Madison,
Madison, Wisconsin 53706, USA
4Charles University, Faculty of Mathematics and Physics, Department of
Condensed Matter Physics, Ke Karlovu 5, Prague 2, 121 16, Czech Republic
5Wuhan National High Magnetic Field Center, Wuhan 430074, China
6Hochfeld-Magnetlabor Dresden (HLD-EMFL),
Helmholtz-Zentrum Dresden-Rossendorf, Dresden, 01328, Germany

∗Email<EMAIL_ADDRESS>
†These authors contributed equally to this work

> Quantum critical phase boundaries (QCPBs) – where a continuous phase
> transition occurs at zero temperature – have been found to nucleate novel
> electronic states in a number of strongly correlated materials [1, 2, 3].
> Emergent electronic phases, such as unconventional superconductivity,
> frequently occur in close proximity to a QCPB [4, 5, 6, 7, 8, 9, 10].
> However, the antagonism between magnetic field and superconductivity
> generally suppresses such superconductive phases to modest magnetic field
> ranges, except in notable cases such as the high-$T_{\text{c}}$ cuprates
> [10, 11]. Here we show that the heavy fermion Kondo-lattice system UTe2
> possesses a QCPB at remarkably high magnetic fields $\sim$ 50 T,
> underpinning an extremely high field superconducting state that persists to
> fields $>$ 70 T despite its relatively modest transition temperature of
> $T_{\text{c}}\approx$ 2 K. Whereas previous studies [12, 13, 14, 15, 16, 17,
> 18, 19] found this superconducting state to exist exclusively inside a
> field-polarised paramagnetic host-phase accessed following a first-order
> metamagnetic transition, for magnetic field tilt angles in the rotation
> plane connecting the reciprocal (010) and (101) vectors, we find an extended
> region of this superconductivity outside the field-polarised state. In this
> special rotation plane we also observe a pronounced suppression of the
> metamagnetic transition towards zero temperature, revealing that the
> metamagnetic transition surface ends at a QCPB. The superconducting
> $T_{\text{c}}$ exhibits a strong angular dependence and is enhanced close to
> the QCPB, where the onset of superconductivity is stretched over a
> surprisingly large magnetic field range reaching as low as 30 T. We model
> our data by a phenomenological Ginzburg-Landau theory, and show how an
> extended quantum critical line – rather than a more conventional QCPB at a
> singular point – anchors the remarkable high magnetic field phase landscape
> of UTe2.

## Introduction

At absolute zero temperature, classical phases of matter are completely frozen
in place. By contrast, in emergent electronic phases quantum zero-point motion
allows for fluctuations to persist down to temperature $T=$ 0 K. For
sufficiently strong fluctuations a quantum phase transition may occur,
typically from an ordered to a disordered state [1, 2, 3]. Depending on the
dimensionality of the underlying phase space, such a transition may occur at a
quantum critical point (QCP) or, in higher dimensions, across a continuous
line or surface composed of QCPs.

Near a QCP, the enhancement of magnetic or density fluctuations can provide
the pairing glue for unconventional superconducting phases to condense [4].
For example, $d$-wave superconductivity is often found in proximity to
antiferromagnetic (AFM) QCPs [20, 11, 10], whereas ferromagnetic (FM) QCPs can
promote the formation of exotic $p$-wave superconductive phases [5, 21, 22].
In a magnetic field the spin-degree of freedom of $p$-wave superconductors can
yield complex multi-component superconducting phase diagrams [23, 24]. In some
rare cases, such as the FM superconductor URhGe, the application of magnetic
fields can even induce superconductivity close to a high magnetic field QCP
[25, 8, 26].

The heavy fermion paramagnet UTe2 crystallises in an orthorhombic body-centred
lattice, with no evidence of long-range magnetic order above $T=$ 40 mK (ref.
[27]). In zero applied magnetic field, $\mu_{0}H$, a superconducting
groundstate forms below a critical temperature $T_{\text{c}}=$ 2.1 K (refs.
[28, 29, 12, 30]). Despite the absence of FM order, this superconductive state
exhibits multiple characteristics of odd-parity $p$-wave pairing [28, 12, 31]
analogous to its FM uranium-based cousins UGe2, URhGe and UCoGe [5, 21, 22].
The superconducting phase diagram of UTe2 under applied magnetic field and
pressure is astonishingly complex, with up to six distinct superconductive
phases having been reported [32, 12, 13, 33, 14, 34]. The most remarkable of
these phases is only found at extremely high magnetic field strengths,
starting at $\mu_{0}H\approx$ 40 T and persisting to $\mu_{0}H>$ 73 T, with a
strong sensitivity to the orientation of magnetic field tilt angle [14, 18,
13, 19].

This high-$H$ superconductivity has been found to be entirely confined inside
a field-polarised paramagnetic (PPM) host-phase, nestling against the PPM
phase boundary within a narrow angular range for field tilted in the
crystallographic $b-c$ plane [14, 15, 13, 16, 17, 18]. Recent measurements
have also confirmed its existence for a magnetic field component tilted away
from the $b-c$ plane towards the $a$-axis [19] – but again, this exotic
high-$H$ superconductivity was exclusively found inside the PPM state. The
metamagnetic transition into the PPM state has been found to be strongly
first-order at low temperatures for the orientations measured so far.

Here, we probe the high-$H$ phase diagram of UTe2 with contacted and
contactless electrical conductivity measurements on a new generation of
pristine quality samples [30, 35, 36, 37]. We find an extended range of the
high-$H$ superconductive state in the multi-dimensional $H_{a,b,c}$ phase
space, where $H_{i}$ denotes the magnetic field oriented parallel to the
crystallographic $i$ direction. Strikingly, for rotations in the $b-abc-ac$
plane (explicitly defined as tilting the magnetic field to point along each of
the reciprocal (010), (111) and (101) vectors) we uncover an extended range of
superconductivity at $\mu_{0}H\approx$ 40 T that extends below the PPM. Non-
Ohmic current–voltage characteristics identifying the onset region of SC3
extend as low as $\mu_{0}H=$ 30 T. This angular range at which the onset is
pushed to minimal field coincides with the orientation at which we observe the
first-order metamagnetic (MM) transition to the PPM being suppressed towards
zero temperature, producing a MM quantum critical end point. In combination,
our high-$H$ measurements (i) identify a quantum critical boundary (the QCPB)
to the surface of first-order metamagnetic transitions in $\vec{H}-$space at
the remarkably high magnetic field scale of $\sim$ 50 T, (ii) anchor the
field–resilient superconductivity in UTe2 close to this boundary, and (iii)
reveal that in a narrow field range close to the QCPB the onset of
field–resilient superconductivity can reach to fields that lie far below the
MM transition field.

## Results

Figure 1 summarises our high-$H$ measurements of the superconducting phase
landscape of UTe2. We label the superconducting groundstate accessed at
$\mu_{0}H=$ 0 T as SC1 (in blue), while the pink region identifies a distinct
superconducting state [38] attained for $\mu_{0}H_{b}\lesssim$ 35 T (SC2) that
shares some region of coexistence with SC1 [39]. The focus of the present
study is the purple SC3 phase, which spans an extended region in the
$H_{a,b,c}$ phase space both inside the PPM state (coloured orange) and below
it.

The phase landscape of Fig. 1 is constructed in part from the data in Figure
3. Up to the maximum applied field of the magnet system utilised for this
suite of measurements, of 41.5 T, for incremental magnetic field tilt angles
rotating from $b$ towards $ca$ at $T=$ 0.4 K we suddenly access the SC3 state
at $\theta_{b-ac}=$ 17.5$\degree$. In Fig. 3b we define the purple SC3 region
by the observation of zero resistivity $\rho$ (in the low current limit).
Interestingly, we observe a broadening of the onset of the SC3 phase for
successively higher rotation angles $\theta_{b-ac}$, identified as the turning
over of $\rho(H)$ from a positive to a negative gradient. This onset extends
down to a minimum magnetic field of $\approx$ 30 T at $\theta_{b-ac}\approx$
34$\degree$, before progressing back up to higher fields for further
inclination in the $b-ac$ plane. This wide magnetic field domain of negatively
sloped magnetoresistance exhibits a highly non-linear relationship between
excitation current and probed voltage (Fig. 3e,f), indicating the likely
presence of exotic vortex dynamics. We therefore identify a broad range of the
UTe2 high field phase diagram, in this special rotation plane, to be occupied
by the onset of the SC3 phase.

Figure 1: The high magnetic field phase landscape of UTe2. a, Three
superconducting phases are identified, labelled SC1 (blue), SC2 (pink) and SC3
(purple). Orange and red points demarcate a metamagnetic transition to the
field–polarised paramagnetic state only accessible in high magnetic field
strengths of $\mu_{0}H_{b}>$ 34 T. Data points identifying superconducting
phase boundaries are from measurements at $T=$ 0.3-0.7 K while those for the
polarised paramagnet (PPM) are taken at $T=$ 4-6 K. Each data point
corresponds either to a magnetic field sweep of a dc magnet, or to a pulsed
field measurement. Lines and surfaces are guides to the eye. The quantum
critical phase boundary (QCPB) stems from where the first-order transition to
the PPM is suppressed to 0 K, identified by the green star and marked by the
green line that bends upwards out of the plane of the page. Figure 2: (cont.)
b, Bird’s-eye view of the phase diagram down along the $H||b$ direction,
illustrating the extent of the superconducting phases and the location of the
critical boundary to the metamagnetic transition surface. The approximate
location of the critical boundary has been inferred from our high-field
measurements along the rotation trajectory, and is indicated by the green line
(and solid star) and from high field transport data in ref. [19] (open stars).
Figure 3: Spillover of SC3 outside the polarised paramagnetic phase. a,
Electrical resistivity $\rho$ versus $\mu_{0}H$ at successive magnetic field
tilt angles in the $b-abc-ca$ rotation plane at $T=$ 0.4 K. At
$\theta_{b-ca}=$ 17.5$\degree$ zero resistance is observed over the field
interval of 0 T $\leq\mu_{0}H\leq$ 41 T as each of SC1, SC2 and SC3 are
successively accessed. b, The phase landscape of UTe2 plotted radially in
$\theta_{b-ca}$. An extended region of the onset of SC3 is observed outside
the PPM state, identified by the turning over from positively to negatively
sloped $\rho(H)$ in panel (a). The red circles defining the PPM boundary were
measured above $T_{c}^{SC3}$. c, A comparison of the temperature sensitivity
of SC3 at $\theta_{b-ca}=$ 18$\degree$ (left) and $\theta_{b-ca}=$ 35$\degree$
(right). On the edge of the SC3 dome at $\theta_{b-ca}=$ 18$\degree$ the
$T_{\text{c}}$ is only $\approx$ 0.6 K, whereas at $\theta_{b-ca}=$
35$\degree$ SC3 is much more robust to elevated temperatures. d, The evolution
of the SC1, SC2, SC3 and PPM phases for $\theta_{b-ca}=$ 0$\degree$ (left) and
$\theta_{b-ca}=$ 35$\degree$ (right). At $\theta_{b-ca}=$ 35$\degree$ the
melting of PPM is not observed at 5 K (marked by a cross – see Supplementary
Information for a detailed discussion of the raw data that construct this
panel). This shows that SC3 extends outside the PPM state in this special
rotational plane. e, Effective resistivity at $\theta_{b-ca}=$ 35$\degree$ for
different excitation currents as indicated. f, Log-log plot of voltage $V$
versus current $I$ from 0.4-1.8 K at $\theta_{b-ca}=$ 35$\degree$. Ohmic $I-V$
characteristics are observed in the normal paramagnetic state at $\mu_{0}H=$
25 T, whereas at $\mu_{0}H=$ 41 T the $I-V$ behaviour is highly non-Ohmic. We
therefore identify the turning over of the $\rho(H)$ traces in (a) as marking
the onset of SC3, characterised by this markedly non-linear relationship
between $I$ and $V$.

For $H\parallel b$, the first-order MM transition to the PPM is observed at
$\mu_{0}H_{m}\approx$ 34.5 T in the zero temperature limit [40, 16, 41, 36].
On warming, this transition passes a critical end point at $\approx$ 10 K and
evolves into a broad crossover, characterised by e.g. a maximum in $\rho(H)$
(refs. [12, 13, 40, 16, 42]). The transition gradually melts, moving down to
lower values of $H_{m}$ with increasing temperature up to $\approx$ 30 K
whereat it is suppressed to 0 T. We plot this schematically in Fig. 3d for
$H\parallel b$ (i.e. $\theta_{b-ca}=0\degree$), along with the corresponding
phase landscape for $\theta_{b-ca}=35\degree$. Strikingly, upon warming above
$T_{c}^{SC3}$ at this angle, we do not observe a signature of the PPM state in
our accessible field range of $\mu_{0}H\leq$ 41.5 T until $T>10$ K (see
Supplementary Information for raw data and extended discussion). Furthermore,
we performed a series of rotations in the $b-ca$ plane at 6 K – to suppress
all superconducting states and therefore solely probe the PPM phase – to map
out the boundary of the MM transition in this plane, which we plot as red
circles in Fig. 3b (see Supplementary Information). The MM transition clearly
bisects a substantive region of SC3. Therefore, our measurements have
uncovered that an extended range of SC3 – and its broad onset region – extends
considerably outside the PPM state in the $b-ca$ plane. This is in stark
contrast to prior measurements of SC3 in the $b-c$ plane that observed SC3
exclusively within the PPM host-state [14, 15, 13, 16, 17, 18].

This motivates a closer examination of the nature of the field-induced MM
phase transition in UTe2, to which we now turn. In the FM superconductor
URhGe, magnetic field–induced superconductivity is found at a QCPB
characterised by the suppression of a first-order MM transition to zero
temperature [8]. In Figure 4 we map the evolution of the UTe2 MM transition
through contactless resistivity measurements by the PDO technique (see
Methods) in pulsed magnetic fields. All data are at $T=4.2$ K, with three
sequences of rotations performed tilting $H$ away from $b$ towards each of
$a$, $c$, and $ac$. We performed these measurements on the same sample, to
minimise any variations in sample properties.

Figure 4: Suppression of the first order metamagnetic phase transition towards
zero temperature. Contactless conductivity measurements measured in pulsed
magnetic fields by the PDO technique, rotating from $H\parallel b$ towards a,
$H\parallel a$, b, $H\parallel c$ and c, $H\parallel ca$. All data were
collected on the same sample at $T=$ 4.2 K. When the MM transition field
$\mu_{0}H_{m}\approx$ 50 T, in the $b-a$ and $b-c$ planes the transition is
still relatively sharp and first order-like. By contrast, in the $b-ca$ plane
it has softened markedly and looks considerably more second order-like at
$\theta_{b-ca}$ = 42$\degree$. d, Schematic defining the $b-abc-ca$ rotation
plane as the arc traced by rotating through the (010), (111) and (101)
reciprocal-space vectors. e, Intensity of the MM transition for the three
rotation planes in (a-c) (see Supplementary Information). f, Temperature
evolution for $H$ oriented $\parallel b$ and at $\theta_{b-c}$ = 47$\degree$
and $\theta_{b-ca}$ = 42$\degree$. The 1.5 K curve at $\theta_{b-ca}$ =
42$\degree$ looks similar in profile to the 12 K trace at $\theta_{b-ca}$ =
0$\degree$; furthermore, while $\theta_{b-ca}$ = 42$\degree$ and
$\theta_{b-c}$ = 47$\degree$ have very similar values of $H_{m}$, it is clear
that $\theta_{b-ca}$ = 42$\degree$ is softer and thus lies closer to the QCPB.

At $\theta_{b-a}=$ $\theta_{b-c}=\theta_{b-ca}=$ 0$\degree$ (i.e. $H\parallel
b$) the MM transition in the contactless resistivity is sharp and first order-
like in character, consistent with prior studies [14, 36, 43, 44]. Tilting the
magnetic field in the $b-a$ and $b-c$ planes, the magnitude of the jump in the
PDO frequency shift $\Delta f_{\text{PDO}}$ decreases at higher rotation
angles. However, it is still identified as a sharp jump even at
$\mu_{0}H_{m}\approx$ 50 T in both of these planes. In marked contrast, in the
$b-ca$ plane the MM transition undergoes a pronounced softening at inclination
angles of $\theta_{b-ca}\approx$ 40$\degree$, with the jump being strongly
suppressed and becoming much more second order-like (Fig. 4c). We plot the
temperature variation of three $H$-orientations in Fig. 4f. It is clear that
in the $b-ca$ rotation plane the first-order MM transition is suppressed
towards zero temperature much more rapidly than in either the $b-a$ or $b-c$
planes. Similar to the case of URhGe, this is characteristic of crossing a
QCPB [8]. We note that recent high-$H$ measurements of UTe2 reported in ref.
[19] are consistent with our finding, and indeed provide corroborating
evidence for the existence of an extended quantum critical line in the
multidimensional phase landscape. This is seen by the suppression of the MM
transition upon rotating towards $a$ at fixed $\theta_{b-c}=$ 23$\degree$ and
$\theta_{b-c}=$ 30$\degree$ (see Supplementary Information for a detailed
comparison of our measurements with those of ref. [19]).

Strikingly, this angular range over which the MM transition is markedly
suppressed coincides with the magnetic field orientation over which the SC3
onset occurs at minimal $H$ (Fig. 3a,b). Therefore, our measurements provide
strong evidence for the presence of an extended range of quantum critical
fluctuations, at the remarkably high magnetic field scale of $\sim$ 50 T,
underpinning the formation of the exotic SC3 superconductive state of UTe2.

## Discussion

As prior high-$H$ measurements of UTe2 in the $b-c$ plane only observed SC3
inside the PPM, the field-induced compensation of magnetic exchange was
proposed as a possible microscopic origin [18]. However, our finding of the
zero resistance SC3 state extending outside the PPM – accompanied by an
extended field region of the onset of SC3 even further away from the MM phase
transition line – argues strongly against this scenario being the sole
mechanism. The extremely high-$H$ scale of SC3 led to another interesting
theoretical proposal based on Hofstadter physics – similar to prior
considerations of Landau quantization driving $H$-reentrant superconductivity
[45] – but the recent observation of SC3 surviving in exceptionally disordered
crystals [46] argues strongly against coherent Landau quantization.

To identify the likely source of the magnetic fluctuations driving the
formation of SC3, we model the MM phase transition to the PPM state using a
Ginzburg-Landau theory:

$\displaystyle\mathcal{F}[\mathbf{M}](\mathbf{H})$
$\displaystyle=\frac{1}{2}\chi^{-1}_{i}M_{i}^{2}+\frac{1}{4}\beta_{ij}M_{i}^{2}M_{j}^{2}+\frac{1}{6}\gamma
M_{y}^{6}-\mathbf{M\cdot H}$ (1)

where $i,j=x,y,z$, $\mathbf{M}$ is the magnetic order parameter
(magnetization), and $\chi^{-1}_{i},\beta_{ij},$ and $\gamma$ are Ginzburg-
Landau parameters [47, 48, 43] . The parameters are chosen such that at zero
applied field the free energy has two minima: a global paramagnetic minimum at
$\absolutevalue{\textbf{M}}$ = 0, and a minimum with higher energy at
$\mathbf{M}=\mathbf{M}_{*}$ pointing along the $b$ direction, corresponding to
the PPM phase. Applying an external magnetic field lowers the minimum at
$\mathbf{M}_{*}$ until it becomes the new global minimum at the MM phase
transition field $H_{m}$. The condition for the local minimum to exist is
$\beta_{yy}<0$; a critical end point (CEP) thus arises when $\beta_{yy}$
changes sign at some temperature $T_{0}$. The CEP generally appears at
$T_{CEP}\lesssim T_{0}$, and in UTe2 is located at $\approx$ 10 K for
$H\parallel b$ (refs. [40, 42]).

Figure 5: QCPB phenomenological theory. Cartoon sketch of the QCPB line (red)
in the space of magnetic field components $H_{j}$ in arbitrary units,
determined by the intersection of the MM phase transition boundary (orange)
and the locus of points at which $T_{0}=0$ (blue).

To leading order close to $T_{0}$ the temperature dependence of the GL
coefficients can be modeled phenomenologically as [47]

$\displaystyle\chi^{-1}_{i}(T)$ $\displaystyle\propto 1+bT^{2}-bcT^{4}$
$\displaystyle\beta_{ij}(T)$ $\displaystyle\propto 1-cT^{2}$ (2)

In principle, the $b$ and $c$ coefficients carry $i,j$ indices, but the only
ones relevant for $T_{CEP}$ are $i=j=y$. Generally, $b$ and $c$ are also
functions of the direction of the applied magnetic field, such that
$T_{0}(\mathbf{H})=1/\sqrt{c(\mathbf{H})}=0$ at some $\mathbf{H}_{QCP}$ (at
which $c$ diverges), leading to a QCPB with $T_{CEP}\propto T_{0}\rightarrow
0$ K. Importantly, following this treatment we expect a line of critical
points since $c^{-1}(\mathbf{H}_{*})=0$ defines a 2D surface that intersects
the MM transition surface along an extended quantum critical line. Thereby,
this model accurately captures the geometrical qualities of the high-$H$
quantum critical landscape of UTe2 probed by our experiments. Furthermore, as
for the case of FM uranium-based superconductors such as URhGe, high-field re-
entrant superconductivity in such a setting is expected to occur on both sides
of the QCPB [8, 26, 48]. Our observation of SC3 on both sides of the MM
transition of UTe2 therefore provides strong evidence, in addition to the
pronounced suppression of the MM transition towards zero temperature, in
favour of a quantum critical origin for SC3.

Our study of the high-$H$ phase space of UTe2 at ambient pressure poses the
natural corollary of: how does this quantum critical phase landscape evolve
under pressure? The pressure dependences of the specific heat and electrical
resistivity of UTe2 have led to the proposal that an AFM QCP is located at a
critical pressure of $p_{c}\approx$ 1.45 GPa, beyond which the superconducting
groundstate(s) are abruptly truncated [49, 32, 12] and incommensurate AFM
order is stabilised [50]. $\mu_{0}H_{m}$ decreases under pressure, and an
intriguing $H$-induced superconducting state – which appears to bear numerous
similarities with the ambient pressure SC3 state and persists to high fields –
has been observed at pressures above $p_{c}$ for $H\parallel c$ [33]. Given
the remarkable complexity of the ambient pressure phase landscape uncovered by
our measurements, a promising new route to understanding the multitude of
distinct superconducting states [32, 33, 31] in UTe2 lies in tracking the
evolution of the QCPB we report here, and its associated Widom lines, in
relation to the numerous superconducting phase boundaries spanning the
multidimensional pressure–temperature–$H_{a,b,c}$ phase landscape.

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relation in the bulk, giving exponential solutions, and then selecting those
solutions satisfying the boundary conditions. Let us sketch this explicitly
for $s=1$, to keep the example concise:

Solving $\det\big{(}\widetilde{W}(\lambda)-{\leavevmode\rm
1\mkern-5.4muI}\big{)}=0$ (with $\widetilde{W}$ as in Prop. 4.1) for the decay
coefficients $\lambda$ results in exactly two solutions, one for each value
$\chi$ of the chirality (both are non-degenerate), with
$\theta_{\pm}=\frac{1}{2}(\theta_{1}\pm\theta_{2})$ [46]:

$\lambda_{\chi}=\frac{2\cos\theta_{-}}{\cos\theta_{-}+\chi\sin\theta_{+}}-1.$
(61)

Since $\lambda_{+}\lambda_{-}=1$, we know that there is always at most one
candidate for an exponentially decaying solution to the right. Prop. 4.1 then
yields the ansatz

$\phi(x)=\lambda_{\chi}^{x}\phi_{0},\quad x\geq 0,$ (62)

where $\phi_{0}=(1,\chi)$ is determined (up to normalization) by
$\gamma\phi=\chi\phi$. The splitting coin now decides, which of these
potential solutions can be made to fit at the boundaries, e.g., by shifting
this solution to the right and inserting a finite number of transition
components to match the boundary conditions. In this example, the walk affects
only its direct neighbours, hence there is at most one transition element.

Since we know that every eigenvector of a decoupled walk can be chosen to be
localized on one side of the cut, we can choose $\phi(x)=0$ for all $x<0$, and
check if $\phi$ fulfils the eigenvalue equation for each of the four
differently decoupled walks. We note that in this simple example, inserting
transition elements could never make a modified $\phi^{\prime}$ solve an
eigenvalue equation, that $\phi$ did not solve before.

This, together with the eigenvectors for the eigenvalue $s=-1$, analyzed on
both sides of the cut, leads to Fig. 2. Consistent with the general theory of
our classification [11], each eigenvector found this way contributes its
chirality $\chi$ to the corresponding indices
$\mathop{\mathrm{s}\mkern-1.0mu\mathrm{i}}\nolimits_{\pm},\mathop{\hbox{\raisebox{4.49997pt}{$\scriptstyle\rightharpoonup$}}\mathop{\mathrm{s}\mkern-1.0mu\textup{\i}}\nolimits},\mathop{\hbox{\raisebox{4.49997pt}{$\scriptstyle\leftharpoonup$}}\mathop{\mathrm{s}\mkern-1.0mu\textup{\i}}\nolimits}$,
depending on whether they stem from $s=\pm 1$ and whether they are
exponentially decaying to the left or right (while vanishing on the other
side). This is an explicit demonstration of bulk-boundary-correspondence. A
web-application allows to check these results dynamically [59].

##  5 Summary and Outlook

We gave a homotopy classification of the systems with non-trivial index group
considered in [11] under the additional assumption of translation invariance,
i.e. essentially gapped one-dimensional quantum walks with discrete symmetries
which commute with lattice translations. In doing so we generalized the
correspondence between strictly local walks and analytic band structures and
proved that the continuity of the bands implies essentially locality. This
classification is complete within the class of translation invariant systems,
i.e. two translation invariant walks obeying the same symmetries can be
deformed into each other if and only if their invariants agree. For the
symmetry types
$\textnormal{{S}}=\textnormal{{D}},\textnormal{{BD}}\textnormal{I}$ we found
an additional invariant, which depends on the cell structure and trivializes
after regrouping neighbouring cells or adding trivial systems. For each
symmetry type with non-trivial index group, concrete index formulas were
derived. In particular, the classification was obtained without referring to
topological K-theory.

Possible directions for future research include the generalization of the
results obtained here to higher dimensional lattices. For Hamiltonian systems,
such a program leads to the now famous periodic table for the index groups
[31]. This corresponds to the classification of vector bundles over higher-
dimensional Brillouin zones. Proving a bulk-boundary correspondence in higher
dimensions, from analogy with the Hamiltonian case one expects to observe
(directed) transport along edges if the edge is chosen along one of the
principal directions of the lattice.

## Acknowledgements

We would all like to thank Alberto Grünbaum for stimulating discussions.

C. Cedzich acknowledges partial support by the Excellence Initiative of the
German Federal and State Governments (ZUK 81) and the DFG (project B01 of CRC
183).

T. Geib, C. Stahl and R. F. Werner acknowledge support from the ERC grant
DQSIM, the DFG SFB 1227 DQmat, and the European project SIQS.

The work of L. Velázquez is partially supported by the research projects
MTM2014-53963-P and MTM2017-89941-P from the Ministry of Science and
Innovation of Spain and the European Regional Development Fund (ERDF), and by
Project E-64 of Diputación General de Aragón (Spain).

A. H. Werner thanks the Humboldt Foundation for its support with a Feodor
Lynen Fellowship and the VILLUM FONDEN via the QMATH Centre of Excellence
(Grant No. 10059).

The publication of this article was funded by the Open Access fund of the
Leibniz Universität Hannover.

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