# Lifshitz transition enabling superconducting dome around the quantum
critical point in TiSe2

R. D. H. Hinlopen  HH Wills Physics Laboratory, University of Bristol, Tyndall
Avenue, Bristol, BS8 1TL, UK  Owen Moulding  HH Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  Institut Néel
CNRS/UGA UPR2940, 25 Avenue des Martyrs, Grenoble, 38042, France  Will Broad
HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol,
BS8 1TL, UK  Jonathan Buhot  HH Wills Physics Laboratory, University of
Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  High Field Magnet Laboratory
(HFML-EMFL), Radboud University, Toernooiveld 7, Nijmegen, 6525 ED, The
Netherlands  Femke Bangma  High Field Magnet Laboratory (HFML-EMFL), Radboud
University, Toernooiveld 7, Nijmegen, 6525 ED, The Netherlands  Alix McCollam
High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7,
Nijmegen, 6525 ED, The Netherlands  Jake Ayres  HH Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK  High Field Magnet
Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, Nijmegen, 6525 ED,
The Netherlands  Charles Sayers  Department of Physics, University of Bath,
Bath, BA2 7AY, UK  Enrico Da Como  Department of Physics, University of Bath,
Bath, BA2 7AY, UK  Felix Flicker  School of Physics and Astronomy, Queen’s
Buildings North Building, 5 The Parade, Newport Road, Cardiff, CF24 3AA, UK
Jasper van Wezel  Institute for Theoretical Physics, University of Amsterdam,
Science Park 904, Amsterdam, 1098 XH , The Netherlands  Sven Friedemann  HH
Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8
1TL, UK<EMAIL_ADDRESS>
###### Abstract

Superconductivity often emerges as a dome around a quantum critical point
(QCP) where long-range order is suppressed to zero temperature. So far, this
has been mostly studied in magnetically ordered materials. By contrast, the
interplay between charge order and superconductivity at a QCP is not fully
understood. Here, we present resistance measurements proving that a dome of
superconductivity surrounds the charge-density-wave (CDW) QCP in pristine
samples of $1T$-TiSe2 tuned with hydrostatic pressure. Furthermore, we use
quantum oscillation measurements to show that the superconductivity sets in at
a Lifshitz transition in the electronic band structure. We use density
functional theory to identify the Fermi pockets enabling superconductivity:
large electron and hole pockets connected by the CDW wave vector $\vec{Q}$
which emerge upon partial suppression of the zero-pressure CDW gap. Hence, we
conclude that superconductivity is of interband type enabled by the presence
of hole and electron bands connected by the CDW $\vec{Q}$ vector. Earlier
calculations show that interband interactions are repulsive, which suggests
that unconventional $s_{\pm}$ superconductivity is realised in TiSe2 – similar
to the iron pnictides. These results highlight the importance of Lifshitz
transitions in realising unconventional superconductivity and help understand
its interaction with CDW order in numerous materials.

##  I Main

Density waves (DWs) underlie the magnetic and charge order in many materials,
with well-known examples including CePd2Si2, BaFe2As2, and TiSe2 [1, 2, 3].
DWs may be formed by a periodic spatial variation of charge (charge-density
wave, CDW) or spin (spin-density wave, SDW) at a characteristic wave vector
$\vec{Q}$ [4]. The new periodicity breaks the translational symmetry of the
underlying atomic lattice and causes a reconstruction of the electronic Fermi
surface. Continuous DW phase transitions are accompanied by diverging
fluctuations at $\vec{Q}$. In the vicinity of a QCP (where the transition is
suppressed to zero temperature), the critical fluctuations associated with the
suppressed DW order can give rise to non-Fermi liquid behaviour, and are
strong contenders for mediating superconductivity in many unconventional
superconductors including iron pnictides, heavy fermion materials, and
possibly cuprate high-temperature superconductors [5, 6, 7]. For instance, in
iron-pnictides SDW fluctuations promote unconventional $s_{\pm}$
superconductivity by coupling electron and hole pockets connected by $\vec{Q}$
[5].

A dome of superconductivity around a QCP is taken as one of the hallmarks of
unconventional superconductivity. Prominent SDW systems demonstrate such a
dome, including the heavy fermion compound CePd2Si2 [1], iron-pnictide
BaFe2As2 [2], and the organic superconductor (TMTSF)2PF6 [8]. This is in clear
contrast to simple competition between DW order and superconductivity as
reported for instance in NbSe2, where the dome of superconductivity is absent
[9]. Complex interaction between CDW order and superconductivity has been
observed in underdoped [10] and overdoped [11] cuprates and nickelates [12] as
well as in frustrated and topologically non-trivial AV3Sb5 (A=K, Rb, Cs) [13].
We report that TiSe2 stands apart from all these by being the cleanest example
in which a superconducting dome surrounds the CDW QCP.

Figure 1: Superconductivity around the CDW QCP in TiSe2. (a) Resistance
measurements using a hydrostatic pressure transmitting medium (PTM)
demonstrate superconductivity over an extended pressure range in TiSe2. (b)
Superconducting transition temperatures $T_{\text{c}}$ defined as the
temperature at which the resistance is reduced to $90\text{\,}\mathrm{\char
37\relax}$ of the normal state resistance. The dashed vertical line indicates
the pressure at which we observe a Lifshitz transition. $T_{\text{CDW}}$ is
also shown as obtained from high-pressure X-ray diffraction measurements (blue
points) using a similarly high-quality pressure medium [14, 15]. Solid lines
and shading are guides to the eye.

In this article, we report on the interplay between CDW order,
superconductivity, and Lifshitz transitions in the transition metal
dichalcogenide $1T$-TiSe2. A Lifshitz transition marks a topological change in
the Fermi surface [16]. TiSe2 hosts a prototypical CDW associated with a
doubling of the unit cell in all directions
($\vec{Q}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$) and a transition
temperature of $T_{\text{CDW}}=$202\text{\,}\mathrm{K}$$. Evidence exists for
an excitonic contribution to the CDW mechanism [17, 18, 19, 20] and the CDW
may be chiral [3]. Previous studies found the CDW is suppressed and
superconductivity can be induced by Cu intercalation [21], high pressure [22,
14], or gating [23]. Combined with its structural simplicity and lack of
magnetic order, TiSe2 thus provides an ideal setting to investigate the
emergence of superconductivity around a CDW QCP. Under hydrostatic conditions,
the CDW transition is continuously suppressed to zero temperature at a
pressure of $P_{\text{CDW}}=$5\text{\,}\mathrm{GPa}$$, as observed by X-ray
diffraction and magnetotransport measurements [14, 15]. Hence, this provides
evidence for a CDW QCP at $P_{\text{CDW}}$. Yet, the link between
superconductivity and CDW order in TiSe2 remains elusive, as the exact
location of superconductivity relative to the CDW QCP remains debated [21, 22,
15, 14, 24, 25, 26, 27].

We study the intrinsic behaviour of pristine samples by using hydrostatic
pressure to fully suppress the CDW phase of TiSe2. We measure the low-
temperature resistance to map the superconducting dome, observe quantum
oscillations across the full CDW phase and beyond, and combine these results
with density functional theory (DFT) and a tight-binding analysis to map the
evolution of the electronic structure identifying two Lifshitz transitions. We
find that the onset of superconductivity coincides with the emergence of a
major hole and electron pocket at one of the Lifshitz transitions. Combined
with earlier theoretical work [28], this suggests that TiSe2 hosts
unconventional superconductivity with interband pairing at the CDW $Q$-vector,
and may be the first case of superconductivity with $s_{\pm}$ character in a
CDW compound.

##  II Superconductivity at the CDW QCP in TiSe2

Our resistance measurements on pristine samples tuned with hydrostatic
pressure show that the superconducting transition temperature $T_{\text{c}}$
forms a dome around the CDW QCP (see Figure 1). While TiSe2 remains a normal
metal at ambient and low pressures down to at least 60 mK, we find sharp
superconducting transitions at pressures $P>$
$2.0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. Previously, superconductivity
was observed between $2.0\text{\,}\mathrm{GPa}$ and $3.5\text{\,}\mathrm{GPa}$
with a maximum transition temperature
$T_{\text{c}}^{\text{max}}=$1.8\text{\,}\mathrm{K}$$, in a study using a solid
pressure medium [22]. By using the liquid pressure transmitting medium (PTM)
1:1 pentane-isopentane, we improve on the hydrostatic conditions. In
hydrostatic conditions, we find that superconductivity still sets in at
$2\text{\,}\mathrm{GPa}$, but now extends to at least
$5.6\text{\,}\mathrm{GPa}$, with an enhanced maximum transition temperature
$T_{\text{c}}^{\text{max}}=$2.9\text{\,}\mathrm{K}$$ close to the CDW QCP. The
presence of superconductivity beyond $P_{\text{CDW}}$ rules out the earlier
suggestion [14] that superconductivity is confined to the domain walls of the
CDW and requires new understanding.

The comparison with previously published results underlines the sensitivity of
both superconductivity and CDW order to pressure conditions. Our study shows a
$50\text{\,}\mathrm{\char 37\relax}$ higher $T_{\text{c}}^{\text{max}}$ in
hydrostatic conditions than previous studies using solid pressure media [22,
25, 26]. This sensitivity of the superconductivity to non-hydrostatic
conditions hints at an unconventional mechanism with sign change of the
superconducting gap. A similar sensitivity to pressure conditions has been
summarised for the CDW order: under the best hydrostatic conditions, the
critical pressure of the CDW order is enhanced by more than
$50\text{\,}\mathrm{\char 37\relax}$ and reaches
$P_{\text{CDW}}=$5\text{\,}\mathrm{GPa}$$ [14, 15] .

The observed dome of superconductivity around the CDW QCP in TiSe2 raises the
question to what extent the vicinity of the CDW QCP influences its normal
superconducting properties? Superconductivity is conventionally mediated by
phonons, whilst in many unconventional superconductors magnetic or nematic
fluctuations are suggested to contribute to or even dominate the binding of
Cooper pairs [6]. In heavy-fermion and iron-pnictide superconductors, these
magnetic and nematic fluctuations are believed to arise from a QCP [5]. In
TiSe2, the fluctuations around the CDW QCP involve both the lattice and the
electrons, and possibly excitons [17, 18, 19, 20]. The presence of the QCP and
its associated fluctuations is known to influence the electrons, as manifested
for TiSe2 in the normal-state resistivity $\rho=\rho_{0}+AT^{n}$, which has a
dip in the exponent $n$ in the vicinity of the QCP [15]. The value $n=3$ away
from the QCP is consistent with interband scattering driven by phonons [29].
The reduction to $n\approx 2$ near the QCP can be interpreted as evidence for
an increase of the effective electron-phonon coupling $\lambda$ [30]. Notice
that the influence of disorder, which can also cause a dip in $n$, can be
ruled out in this case as we maintain constant chemical purity with pressure
and observe quantum oscillations at all pressures studied. Hence, the dip in
$n$ suggests that the conduction electrons predominantly couple to phonon
modes that are softened to zero energy at the CDW QCP [31]. Extending this
reasoning, the maximum value $T_{\text{c}}^{\text{max}}$ of the
superconducting transition temperature at the QCP may be similarly enhanced by
an increased peak in $\lambda$. This is consistent with a reduced dip of $n$
in previous studies, with non-hydrostatic conditions, suggesting a smaller
$\lambda$ underlying the reduced $T_{\text{c}}^{\text{max}}$ [22, 25, 26].

The effective quasiparticle mass $m^{\star}$ is expected to be enhanced over
the band mass $m_{\text{band}}$ due to electron-phonon coupling as
$m^{\star}=(1+\lambda)m_{\text{band}}$. We observe an enhancement of
$m^{\star}$ for $F_{\alpha}$ rising from $\approx$20\text{\,}\mathrm{\char
37\relax}$$ to $\approx$200\text{\,}\mathrm{\char 37\relax}$$ on approaching
$P_{\text{CDW}}$. We also observe an enhancement of $m^{\star}$ by
$\approx$80\text{\,}\mathrm{\char 37\relax}$$ for $F_{\delta}$ and
$F_{\epsilon}$ (cf. black and purple symbols compared to same color lines).
Whilst an increase of the mass enhancement is thus only seen for $F_{\alpha}$,
the expected change in $\lambda$ may be small enough to be non-discernible for
$F_{\delta}$ and $F_{\epsilon}$ (compare for instance [32]).

This mechanism of enhancing $T_{\text{c}}$ is similar to that observed in
elemental uranium and LuPt2In, in which an increase of $T_{\text{c}}$
correlates with the softening of phonons as observed by inelastic X-ray
scattering and specific heat measurements, respectively [32, 33]. Yet, TiSe2
remains the first case of a dome of superconductivity around a CDW QCP where
we can study the onset of superconductivity. This allows us to answer the
question: What initiates the superconductivity in TiSe2 at
$2\text{\,}\mathrm{GPa}$? We employ quantum oscillation measurements and
electronic structure calculations to address this question.

##  III High-pressure quantum oscillation measurements

The presence of quantum oscillations in our TiSe2 samples presented in Figure
2 demonstrate that we retain the high purity of our samples under hydrostatic
pressure tuning [31]. Analysing the quantum oscillation measurements affords
us the highest resolution and most reliable method to obtain information about
the Fermi surface and electronic structure. Previous quantum oscillation
measurements have been decisive for studies of electronic structure and Fermi-
surface reconstruction as well as the quasiparticle renormalisation at quantum
phase transitions in a range of materials [34, 35, 36].

Figure 2: Quantum oscillations in TiSe2 at high pressures. (a) Resistance
after subtraction of a polynomial background plotted versus inverse magnetic
field (top axis) across a wide range of pressure. (b) Fourier transformation
of the data gives the oscillation amplitude as a function of frequency. Dashed
lines trace quantum oscillation frequencies across pressure. (c) The
temperature dependence of the amplitudes for selected frequencies and
pressures. Solid lines represent Lifshitz-Kosevich fits used to extract
effective masses.

At ambient pressure and low temperature, TiSe2 is fully described by a single
electron pocket, which manifests as a single quantum oscillation frequency
$F_{\alpha}=$0.26\text{\,}\mathrm{kT}$$ [31]. This pocket is the result of
Fermi surface reconstruction inside the CDW phase and was previously observed
using ARPES [37]. In particular, the agreement of quantum oscillations with
heat capacity data establishes the fact that $F_{\alpha}$ corresponds to the
only Fermi surface pocket present at ambient pressure [38]. We observe no
other orbits up to $2\text{\,}\mathrm{GPa}$. Within this pressure range,
$F_{\alpha}$ decreases slightly and the mass evolves smoothly (see Figure 2
and Figure 3).

Figure 3: Comparison between modelled and measured quantum oscillations. a)
Quantum oscillation frequencies from experiment (symbols) and band-structure
calculations (lines). Labels refer to frequencies identified in Figure 2, with
subscripts corresponding to orbits in Figure 4. Faint lines correspond to
small frequencies which are not observed experimentally. Neck orbits of the Se
band at $k_{z}=\pi/2$ are omitted as they are not observed experimentally. b)
Corresponding effective masses as a function of pressure. Grey background
indicates range of superconductivity observed. Open triangle may be the result
of magnetic breakdown, as discussed in Supplementary Information D.

The sudden emergence of new frequencies at higher pressures provides evidence
for two Fermi surface reconstructions taking place at $2\text{\,}\mathrm{GPa}$
and $5\text{\,}\mathrm{GPa}$ (see Figure 3). At $2\text{\,}\mathrm{GPa}$ we
identify two new frequencies $F_{\beta}\approx$1\text{\,}\mathrm{kT}$$ and
$F_{\gamma}\approx$2\text{\,}\mathrm{kT}$$ (Figure 2). These are one order of
magnitude larger than $F_{\alpha}$ and prove the emergence of at least one
large new Fermi surface (notice that heat capacity measurements rule out the
presence of such a large pocket at ambient pressure). The emergence of such a
large Fermi surface is surprising, as it happens well before the critical
pressure at which CDW order disappears.

A second Fermi surface reconstruction is observed at
$P_{\text{CDW}}=$5\text{\,}\mathrm{GPa}$$. Here, $F_{\alpha}$ ceases to exist
and a new frequency $F_{\delta}\approx$0.55\text{\,}\mathrm{kT}$$ emerges.
$F_{\alpha}$ and $F_{\delta}$ are clearly distinguished by the difference in
mass, establishing these as orbits on different Fermi surface pockets. By
contrast, $F_{\gamma}$ evolves smoothly through $P_{\text{CDW}}$ and continues
to increase up to the highest pressure of our study
($6.2\text{\,}\mathrm{GPa}$). The electronic structure calculations described
below model the complex evolution of the Fermi surfaces detected by quantum
oscillations, including the two Fermi surface reconstructions.

Figure 4: Pressure evolution of the TiSe2 Fermi surface. a-d) Simulated band
structure inside (a-c) and outside (d) the CDW phase. Blue indicates tight
binding fits to the Ti-3$d$ electron band back-folded from the L point, green
and orange the Se-4$p$ hole bands. e-h) Corresponding Fermi surface contours
in the tight-binding model. The color maps in the left half of each Brillouin
zone indicate the energies of the Se tight-binding band with red lines marking
the corresponding Fermi surface. The right half of each Brillouin zone with
black lines correspond to the Ti bands. Orbits observed in experiment are
labeled with Greek letters. h) Normal state Fermi surface outside the CDW
regime, with one of the three CDW wavevectors
$\vec{Q}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ indicated by the
black arrow. We label points on the high-pressure Brillouin zone with
$\Gamma,M,L,K,H$ where $L$ is mapped onto $M$ and $H$ onto $K$ in our two-
dimensional model. Points in the reconstructed Brillouin zone inside the CDW
phase are labelled with $\Gamma,M^{\prime},K^{\prime}$.

##  IV Electronic structure calculations

We start our analysis of the observed quantum-oscillation frequencies by
employing DFT calculations in WIEN2k [39] including spin-orbit coupling and
based on the well-established crystal structure of TiSe2 without the CDW
(space group $P\bar{3}m1$). The pressure dependence of lattice parameters is
calculated with DFT and matches the experimental results (see Methods and
Supplementary Information A). The DFT band structure is then used to
unambiguously assign the quantum oscillation frequencies above
$P_{\text{CDW}}$ (Figure 3), and to calibrate band shift corrections of the
electronic structure in the absence of CDW order as detailed in Supplementary
Information A.

Outside the CDW phase ($P\geq P_{\text{CDW}}$), DFT calculations provide an
excellent match to the experimental frequencies. The three experimental
frequencies $F_{\gamma}$, $F_{\delta}$, and $F_{\epsilon}$ are reproduced in
both magnitude and pressure dependence. Based on this, we can clearly identify
$F_{\gamma}$ and $F_{\delta}$ to correspond to the outer and inner hole
pockets at $\Gamma$, while $F_{\epsilon}$ corresponds to the electron pocket
at $L$ (cf. Figure 4). The calibration of band shifts in the DFT calculations
(see Methods and Supplementary Information A) to these observed frequencies
provides us with a reliable basis to model the effect of the CDW using a
tight-binding approach.

###  IV.1 Pressure dependence

Inside the CDW phase ($P\leq P_{\text{CDW}}$), we model the Fermi surface by
employing a two-dimensional tight-binding model with band parameters extracted
from our calibrated DFT calculations. We include a simple $k$-independent
orbitally non-selective CDW gap $\Delta$ between the Se and Ti bands (see
Methods and Supplementary Information A). The gap size $\Delta(P)$ is taken
proportional to $T_{\text{CDW}}(P)$ at all pressures, with an estimate of
$\Delta(0)=$40\text{\,}\mathrm{meV}$$ at ambient pressure based on TiSe2 being
a moderately strong coupling CDW with a transition temperature of
$T_{\text{CDW}}=$202\text{\,}\mathrm{K}$$ (see Methods). Our model
quantitatively accounts for the observed quantum oscillation frequencies over
the full pressure range studied, as shown in Figure 3.

The tight-binding model naturally reproduces the Fermi surface reconstruction
at $P_{\text{CDW}}=$5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ identified by
$F_{\delta}$ outside the CDW and $F_{\alpha}$inside the CDW phase, as shown in
Figure 3. The tight binding model also predicts the disappearance of
$F_{\epsilon}$ and emergence of $F_{\beta}$ when entering the CDW phase from
high pressures. In experiment, we observe $F_{\epsilon}$ to continue below
$P_{\text{CDW}}$ and $F_{\beta}$ is only detected below
$4\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. The fact that $F_{\beta}$ is not
observed in experiment right up to $P_{\text{CDW}}$ may be due to the low
intensity of this frequency as a result of either geometrical factors or
reduced quasiparticle weight. The continued observation of $F_{\epsilon}$ into
the CDW phase as well as the additional frequency observed at
$3.7\text{\,}\mathrm{GPa}$ are possibly results of magnetic breakdown as
discussed in Supplementary Information D. The new orbits $F_{\alpha}$ and
$F_{\beta}$ inside the CDW phase are identified as the inner and outer closed
orbits around $\Gamma$ arising from the hybridisation between the three
elliptical electron bands originally at $L$, and the smaller hole band
originally at $\Gamma$. In our model, the $\alpha$ pocket persists down to
ambient pressure, where it encompasses the entire Fermi surface in excellent
agreement with experiments [31]. While the calculated $F_{\beta}$ is seen in
Figure 3 to match the experimental pressure trend, the
$20\text{\,}\mathrm{\char 37\relax}$ difference in value between model and
experimental data is a result of the limited expansion of the employed tight
binding model (see Supplementary Information C). Similarly, we assign the
difference between observed and calculated frequencies for $F_{\alpha}$ to the
limitations of our model. Indeed, quantitative prediction of small frequencies
is usually most susceptible to details of the model.

The tight-binding model reproduces the continuity of the highest frequency,
$F_{\gamma}$, deep into the CDW phase and matches the evolution of its
effective mass. The large hole pocket associated with $F_{\gamma}$ does not
hybridise immediately upon entering the CDW. Only at
$P\leq$2\text{\,}\mathrm{GPa}$$ do the electron pockets hybridise with the
hole band $\gamma$, because only below this pressure the CDW gap overcomes the
energy difference and separation in $k$-space between $\gamma$ and $\epsilon$.
The hybridisation at $P\approx$2\text{\,}\mathrm{GPa}$$ drives the
disappearance of $F_{\beta}$ and $F_{\gamma}$, and marks a Lifshitz
transition. Thus, our tight-binding model confirms and identifies the
experimental Lifshitz transition at
$2\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$.

The onset of superconductivity at $2\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$
is observed to coincide exactly with the emergence of large electron and hole
pockets around $\Gamma$. These pockets originate from $L$ and $\Gamma$
respectively, and are connected by the CDW wave vector $\vec{Q}$ in the
unreconstructed Brillouin zone. The abrupt emergence of superconductivity at
this Lifshitz transition therefore suggests a close relationship between the
superconducting and CDW order, as well as a likely interband character for its
Cooper pairs. This tangible possibility for the emergence of interband
superconductivity in TiSe2 under pressure raises the question whether the same
mechanism may also be at play in Cu-intercalated TiSe2.

Figure 5: Doping evolution of the CuxTiSe2 Fermi surface. (a) Calculated
evolution of the carrier concentration (right axis) for the electron (grey)
and hole (black) Fermi surface pockets, as a function of copper intercalation
$x$. Blue and red shaded areas mark the experimental transition temperatures
(left axis) of the CDW and superconductivity, respectively , as reported in
Ref. [21]. (b) Electronic structure and Fermi surface topology illustrating
emergence of hole pocket when crossing the Lifshitz transition at
$x\approx$3\text{\,}\mathrm{\char 37\relax}$$. Blue indicates the Ti-3$d$
electron band back-folded from the $L$ point whilst green and orange mark the
Se-4$p$ hole bands. (c) Total spectral weight at the Fermi level showing the
reconstructed Fermi pockets at the same doping as panel (b). The full pockets
are visible as a result of finite broadening of the bands for the illustration
of spectral weight.

###  IV.2 Doping dependence

In order to model the Fermi-surface evolution of Cu-intercalated TiSe2, we
investigate the doping dependence in our tight-binding model and compare with
experimental results by Morosan et al. [21]. In our tight-binding model, the
ambient-pressure semimetallic electronic structure is established from the
match to quantum oscillation measurements (see Supplementary Information E for
further discussion of the semimetallic versus semiconducting ground state).
The Cu intercalation is modeled by raising the chemical potential to reflect
an electron doping of 0.45 electrons per Cu, as determined from supercell DFT
[40] and consistent with Hall effect and ARPES measurements [41, 42]. The
doping leads to the growth of the electron pockets originating from the $L$
point (compare Figure 5(b) with Figure 4(a)). Combined with the suppression of
the CDW gap $\Delta(x)$ (scaled to the experimental $T_{\text{CDW}}(x)$ [21,
24, 43] as before), this leads to the occurrence of two similar Lifshitz
transition as observed under pressure.

The first Lifshitz transition is predicted for Cu intercalation at
$x\approx$3\text{\,}\mathrm{\char 37\relax}$$, in close proximity to the
experimentally observed onset of superconductivity at
$x\approx$4\text{\,}\mathrm{\char 37\relax}$$ [21]. In Figure 5, we plot the
carrier concentrations for individual electron and hole bands (grey and black
lines). The electron concentration grows with Cu intercalation as expected. At
low $x<$3\text{\,}\mathrm{\char 37\relax}$$ only a single electron pocket is
present because the CDW gap is large and the separation of the underlying
electron pockets from $L$ and hole pockets from $\Gamma$ is small. Above
$3\text{\,}\mathrm{\char 37\relax}$ the CDW gap becomes too small to overcome
the growing separation and a hole pocket emerges marking a Lifshitz transition
(Figure 5). The proximity of the Lifshitz transition with the onset of
superconductivity suggests that superconductivity in CuxTiSe2 is dependent on
the presence of both electron and hole bands and thus also of interband
character.

##  V Discussion

We have established that superconductivity exists in a dome around the
pressure-induced CDW QCP in TiSe2 with
$T_{\text{c}}^{\text{max}}=$2.9\text{\,}\mathrm{K}$$. This observation unifies
the pressure and doping phase diagram [21] and suggests superconductivity is
mediated by the CDW fluctuations which soften at the CDW QCP under both tuning
parameters. Soft fluctuations near a QCP can manifest as an enhancement and
divergence of the effective masses of electrons. In TiSe2, we observe a
general enhancement of about a factor 2 over the DFT masses testifying the
presence of interactions across the phase diagram. For one frequency,
$F_{\alpha}$, we observe the mass enhancement increasing towards the CDW QCP
up to a factor 3 over the tight-binding model (Figure 3), providing evidence
for a moderate enhancement of interactions at the CDW QCP. Combined with the
evidence for a peak in electron-phonon coupling deduced from the minimum in
the exponent of the temperature dependence of the resistivity [15], this
indicates the CDW fluctuations play a role in the peak of the
superconductivity. These fluctuations may involve phonons, electrons, excitons
or a combination. This highlights that the peak of the superconductivity is
related to the CDW QCP similar to other quantum critical superconductors.

In addition, for both high-pressure and Cu-intercalated TiSe2, we established
the coincidence of a Lifshitz transition with the onset of superconductivity.
In both cases we conclude that the presence of electron and hole pockets
originating from $L$ and $\Gamma$ respectively are a prerequisite, suggesting
that the superconductivity is of interband nature, with pairing enabled by the
presence of a CDW with wave vector $\vec{Q}$. A previous renormalisation group
study found that interband pairing mechanisms are repulsive in TiSe2 [28].
This implies that the interband superconductivity in TiSe2 is likely of
unconventional $s_{\pm}$ type in which the superconducting order parameter
changes sign between the electron and hole pockets. The sensitivity of
superconductivity to strain discussed above is consistent with the sign
reversal of $s_{\pm}$ superconductivity. The emergence of superconductivity in
TiSe2 thus closely parallels the established mechanism for interband $s_{\pm}$
superconductivity in the iron pnictides, but is the first system where
$s_{\pm}$ pairing is promoted by a CDW.

These results are relevant well beyond the study of TiSe2. For instance, the
Lifshitz transition in Ba(Fe1-xCox)2As2 results in electron and hole pockets
connected by the $Q$ vector of the SDW fluctuations at the onset of
$s_{\pm}$-superconductivity [44], closely resembling the situation discussed
here. Likewise, superconductivity in magic-angle trilayer graphene is known to
be bounded by Lifshitz transitions [45]. In quasi-1D Bechgaard salts,
superconductivity is similarly driven by increasingly mismatched nesting under
pressure [46]. In the heavy fermion compound URhGe the magnetic field-induced
superconductivity has been proposed to result from a Lifshitz transition [47].
Across this broad range of physical systems with varying charge and spin
interactions considered for Cooper pair formation, Lifshitz transitions
demarcating the onset of superconductivity are surprisingly universal. This
includes systems with or without divergent effective masses, with a
prototypical Fermi liquid dispersion or in the presence of Dirac cones, and in
quasi-1D, 2D or 3D. The specific case of TiSe2 marks the first material in
this class for which charge order, as opposed to spin order, underlies the
emergence of a superconducting dome in the phase diagram. Our results suggest
that Lifshitz transitions are an overlooked prerequisite for the emergence of
unconventional superconductivity at the edge of DW phases in other materials,
especially concerning regions of co-existence.

## Methods

###  V.1 High-pressure quantum oscillation measurements

High-pressure measurements used moissanite anvil cells with a culet size of
$800\text{\,}\mu\mathrm{m}$. Metallic gaskets were prepared by indenting
$450\text{\,}\mu\mathrm{m}$ thick BeCu to approximately
$60\text{\,}\mu\mathrm{m}$ followed by drilling a $450\text{\,}\mu\mathrm{m}$
hole. These were insulated with a mixture of alumina and 1266 stycast, which
was cured at high pressure between the anvils such that the total thickness of
the gasket was less than $100\text{\,}\mu\mathrm{m}$. A
$400\text{\,}\mu\mathrm{m}$ hole was then drilled in the insulation to act as
the sample chamber. Six bi-layer electrodes were deposited on the anvil by
firstly argon sputtering $20\text{\,}\mu\mathrm{m}$ of nichrome followed by
thermally evaporating $150\text{\,}\mu\mathrm{m}$ of gold; excess nichrome was
removed with TFN etchant. Finally, gold electrodes were deposited directly
onto the sample which were electrically connected to the electrodes on the
culet with H21D silver epoxy, which was left to cure at room temperature to
avoid degrading the sample.

The pressure medium used was a 1:1 mixture of pentane and isopentane which is
hydrostatic until $7.4\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ [48]. The
pressure was determined at room temperature by ruby fluorescence from multiple
ruby flakes within the sample chamber. The uncertainty in the pressure was
taken as the standard deviation between all the rubies across the sample
chamber, both before and after the measurement.

At the HFML, Nijmegen, a helium-3 cryostat was used to achieve a lowest
temperature of $0.35\text{\,}\mathrm{K}$. In cell 2 where the measurements
were performed, the maximum magnetic field applied was $35\text{\,}\mathrm{T}$
and the field was swept at different rates below
$60\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{s}$; the slowest rate used
was $10\text{\,}\mathrm{m}\mathrm{T}\mathrm{/}\mathrm{s}$ in order to resolve
the highest frequency oscillations. A four-probe method with SR830 lock-in
amplifiers was used to measure the resistance, and a Keithley 6221 applied a
maximum current of $0.5\text{\,}\mathrm{m}\mathrm{A}$ to avoid heating the
sample. Either a model 1900 or SR544 transformer in conjunction with a SR560
pre-amplifier were used to amplify the measured voltage. Effective masses were
determined from Lifshitz-Kosevich fits to the resistance between 26 and 35 T
for all pressures and frequencies.

###  V.2 DFT calculations

We use WIEN2k with spin-orbit coupling in the Perdew-Burke-Ernzerhof basis
set. First, we calculate the pressure dependence of the lattice parameters.
This is done by fitting the equation of state to the evolution of the total
energy as a function of unit cell volume. The total energy is obtained by
relaxing the internal degrees of freedom for fixed unit cell volume. In
particular, we minimize the total energy to find the $z$ and $c/a$ parameters
for each unit cell volume. Here, $zc$ is the distance of the Se atoms out of
the plane. Subsequently, we fit the Birch-Murnaghan equation of state to the
evolution of energy versus unit cell volume to convert the volume to pressure.
Example fits and a comparison between lattice parameter evolution and X-ray
diffraction under pressure are shown in Supplementary Information A. For the
doping evolution of the ambient pressure electronic structure, see
Supplementary Information B.

In the absence of a CDW above 5 GPa we unambiguously assign the quantum
oscillations with the DFT results. We use band shifts up to 150 meV to get the
best match between DFT and experiment, see Supplementary Information A for
details. We maintain a slight electron doping throughout and although the band
shifts reduce the semimetallic overlap compared to natural DFT, we obtain
clear semimetallic character at all pressures. We note that quasi-2D neck
orbits of the hole bands at the $A$ point predicted by DFT are not detected in
quantum oscillations, see Supplementary Information B.

###  V.3 Implementation of the CDW

To implement the CDW, we fit a tight binding model to the DFT bandstructure as
detailed in Supplementary Information C. We restrict ourselves to the
$k_{z}=0$ plane at $\Gamma$ and the $k_{z}=\pi/2$ plane at $L$ within $\pm$
$150\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ of the Fermi level. Subsequently,
we implement a CDW reconstruction using the following Hamiltonian:

$\hat{H}\left(\vec{k}\right)=\begin{pmatrix}E^{4p}_{\vec{k}}&0&\Delta&\Delta&\Delta\\\
0&E^{4p^{\prime}}_{\vec{k}}&\Delta&\Delta&\Delta\\\
\Delta&\Delta&E^{3d}_{\vec{k}-{\vec{Q}}}&0&0\\\
\Delta&\Delta&0&E^{3d}_{\vec{k}-R\left({\vec{Q}}\right)}&0\\\
\Delta&\Delta&0&0&E^{3d}_{\vec{k}-R^{2}\left({\vec{Q}}\right)})\end{pmatrix}.$
(1)

Here, $\vec{Q}$ indicates the vector from $\Gamma$ to one of the $L$ points.
$R$ denotes rotation under 120 degrees with respect to the $c$-axis to cover
the three inequivalent $L$ points. The CDW gap $\Delta$ is approximated as
independent of $\vec{k}$ or orbital. This choice was made because no
orthogonality exists between out-of-plane Se 2$p_{z}$ orbitals and in-plane Ti
3$d$ orbitals, unlike the $p_{x,y}$ orbitals relevant for e.g. monolayers of
the sister compound TiTe2 [49], see Supplementary Information C and E. We add
a small band-independent linear-in-$P$ energy shift of $\pm$12 meV/GPa to all
bands keeping the bands unchanged at $P_{\text{CDW}}$ such that the band
overlap reduces at low pressure, which was applied to fit the ambient pressure
quantum oscillation frequency.

We now outline our choice for gap size at ambient pressure. BCS theory
predicts $\Delta=Ak_{B}T_{\text{CDW}}$ with $A=1.764$ in the weak coupling
limit and to our knowledge typically no higher than 3 in the strong-coupling
regime. ARPES results commonly estimate a 100 meV separation between
conduction and valence bands in the CDW phase, starting from a semiconducting
gap of 75-150 meV [37, 42, 50]. We thus use
$\Delta(0)=$40\text{\,}\mathrm{meV}$$ as a reasonable estimate of the CDW
order parameter (observed separation in the CDW minus the pre-existing
semiconducting gap), which we find to reproduce the Lifshitz transition at 2
GPa, and to be in accordance with TiSe2 being in the strong-coupling regime (A
$\approx$ 2.3). Under pressure we scale
$\Delta(P)=\Delta(0)*T_{\text{CDW}}(P)/T_{\text{CDW}}(0)$ [15]. Using these
steps, we obtain a model with minimal degrees of freedom that can account for
the CDW.

In summary, after calibrating the DFT to quantum oscillation measurements
above $P_{\text{CDW}}$, we consider only two parameters in our tight binding
model: the magnitude of the CDW gap $\Delta(0)$ and the linear pressure shift
of the bands. These two parameters are determined by two fixed points: the
size of the Fermi surface at ambient pressure and the pressure at which the
Lifshitz transition is observed. All other quantitative agreement follows
naturally, including $F_{\gamma}(p)$, the presence of $F_{\beta}$ and its
pressure dependence, and the persistence of $F_{\alpha}$. These are
independent confirmations for our model. The CDW bandstructure shown in Figure
4 may be topological as the bands are inverted.

## Additional information

Data are available at the University of Bristol data repository.

## References

## References

  * Mathur  _et al._ [1998] N. D. Mathur, F. M. Grosche, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lonzarich, Magnetically mediated superconductivity in heavy fermion compounds, Nature 394, 39 (1998).
  * Shibauchi  _et al._ [2014] T. Shibauchi, A. Carrington, and Y. Matsuda, A Quantum critical point lying beneath the superconducting dome in iron-pnictides, Annu. Rev. Condens. Matter Phys. 5, 15 (2014), 1304.6387 .
  * Ishioka  _et al._ [2010] J. Ishioka, Y. H. Liu, K. Shimatake, T. Kurosawa, K. Ichimura, Y. Toda, M. Oda, and S. Tanda, Chiral charge-density waves, Phys. Rev. Lett. 105, 176401 (2010).
  * Grüner [1994] G. Grüner,  _Density Waves in Solids_ (Addison-Wesley Pubishing Company, 1994).
  * Si  _et al._ [2016] Q. Si, R. Yu, and E. Abrahams, High-temperature superconductivity in iron pnictides and chalcogenides, Nature Reviews Materials 1, 16017 (2016).
  * Monthoux  _et al._ [2007] P. Monthoux, D. Pines, and G. G. Lonzarich, Superconductivity without phonons, Nature 450, 1177 (2007).
  * Taillefer [2010] L. Taillefer, Scattering and pairing in cuprate superconductors, Annu. Rev. Condens. Matter Phys. 1, 51 (2010).
  * Vuletić  _et al._ [2002] T. Vuletić, P. Auban-Senzier, C. Pasquier, S. Tomić, D. Jérome, M. Héritier, and K. Bechgaard, Coexistence of superconductivity and spin density wave orderings in the organic superconductor (TMTSF)2PF6, Eur. Phys. J. B 25, 319 (2002).
  * Moulding  _et al._ [2020] O. Moulding, I. Osmond, F. Flicker, T. Muramatsu, and S. Friedemann, Absence of superconducting dome at the charge-density-wave quantum phase transition in 2$H$-NbSe2, Phys. Rev. Research 2, 043392 (2020), 2006.03422 .
  * Putzke  _et al._ [2018] C. Putzke, J. Ayres, J. Buhot, S. Licciardello, N. E. Hussey, S. Friedemann, and A. Carrington, Charge order and superconductivity in underdoped YBa2Cu3O7-δ under Pressure, Phys. Rev. Lett. 120, 117002 (2018).
  * Tam  _et al._ [2022a] C. C. Tam, M. Zhu, J. Ayres, K. Kummer, F. Yakhou-Harris, J. R. Cooper, A. Carrington, and S. M. Hayden, Charge density waves and Fermi surface reconstruction in the clean overdoped cuprate superconductor Tl2Ba2CuO6+d, Nature Communications 13, 570 (2022a).
  * Tam  _et al._ [2022b] C. C. Tam, J. Choi, X. Ding, S. Agrestini, A. Nag, M. Wu, B. Huang, H. Luo, P. Gao, M. García-Fernández, L. Qiao, and K.-J. Zhou, Charge density waves in infinite-layer NdNiO2 nickelates, Nature Materials 21, 1116 (2022b).
  * Ortiz  _et al._ [2020] B. R. Ortiz, S. M. L. Teicher, Y. Hu, J. L. Zuo, P. M. Sarte, E. C. Schueller, A. M. M. Abeykoon, M. J. Krogstad, S. Rosenkranz, R. Osborn, R. Seshadri, L. Balents, J. He, and S. D. Wilson, CsV3Sb5: A $\mathbb{Z}_{2}$ Topological Kagome Metal with a Superconducting Ground State, Phys. Rev. Lett. 125, 247002 (2020).
  * Joe  _et al._ [2014] Y. I. Joe, X. M. Chen, P. Ghaemi, K. D. Finkelstein, G. A. de la Pena, Y. Gan, J. C. T. Lee, S. Yuan, J. Geck, G. J. MacDougall, T. C. Chiang, S. L. Cooper, E. Fradkin, and P. Abbamonte, Emergence of charge density wave domain walls above the superconducting dome in 1$T$-TiSe2, Nat. Phys. 10, 421 (2014).
  * Moulding  _et al._ [2022] O. Moulding, T. Muramatsu, C. Sayers, E. Da Como, and S. Friedemann, Suppression of charge-density-wave order in TiSe2 studied with high-pressure magnetoresistance, Electron. Struct. 4, 035001 (2022).
  * Lifshitz [1960] I. M. Lifshitz, Anomalies of electron characteristics of a metal in the high pressure region, Sov. Phys. JETP 11, 1130 (1960).
  * Jérome  _et al._ [1967] D. Jérome, T. M. Rice, and W. Kohn, Excitonic insulator, Phys. Rev. 158, 462 (1967).
  * Rossnagel [2011] K. Rossnagel, On the origin of charge-density waves in select layered transition-metal dichalcogenides, J. Phys. Condens. Matter 23, 213001 (2011).
  * Hellmann  _et al._ [2012] S. Hellmann, T. Rohwer, M. Kalläne, K. Hanff, C. Sohrt, A. Stange, A. Carr, M. M. Murnane, H. C. Kapteyn, L. Kipp, M. Bauer, and K. Rossnagel, Time-domain classification of charge-density-wave insulators., Nat. Commun. 3, 1069 (2012).
  * Kogar  _et al._ [2017a] A. Kogar, M. S. Rak, S. Vig, A. A. Husain, F. Flicker, Y. I. Joe, L. Venema, G. J. MacDougall, T. C. Chiang, E. Fradkin, J. van Wezel, and P. Abbamonte, Signatures of exciton condensation in a transition metal dichalcogenide, Science 358, 1314 (2017a).
  * Morosan  _et al._ [2006] E. Morosan, H. W. Zandbergen, B. S. Dennis, J. W. G. Bos, Y. Onose, T. Klimczuk, A. P. Ramirez, N. P. Ong, and J. V. Cava, Superconductivity in CuxTiSe2, Nat. Phys. 2, 544 (2006).
  * Kusmartseva  _et al._ [2009] A. F. Kusmartseva, B. Sipos, H. Berger, L. Forró, and E. Tutiš, Pressure induced superconductivity in pristine 1$T$-TiSe2, Phys. Rev. Lett. 103, 236401 (2009).
  * Li  _et al._ [2015] L. J. Li, E. C. T. O’Farrell, K. P. Loh, G. Eda, B. Özyilmaz, and A. H. Castro Neto, Controlling many-body states by the electric-field effect in a two-dimensional material, Nature 529, 185 (2015).
  * Kogar  _et al._ [2017b] A. Kogar, G. A. de la Pena, S. Lee, Y. Fang, S. X.-L. Sun, D. B. Lioi, G. Karapetrov, K. D. Finkelstein, J. P. C. Ruff, P. Abbamonte, and S. Rosenkranz, Observation of charge density wave incommensuration near the superconducting dome in CuxTiSe2, Phys. Rev. Lett. 118, 027002 (2017b).
  * Saqib  _et al._ [2021] H. Saqib, S. Rahman, Y. Zhao, C. Cazorla, D. Errandonea, R. Susilo, Y. Zhuang, Y. Huang, B. Chen, and N. Dai, Evolution of structural and electronic properties of TiSe2 under high pressure, J. Phys. Chem. Lett. 12, 9859 (2021).
  * Xia  _et al._ [2022] W. Xia, J. Wu, Z. Li, J. Yuan, C. An, X. Wang, N. Yu, Z. Zou, G. Liu, C. Zhou, J. Feng, L. Zhang, Z. Dong, B. Chen, Z. Yang, Z. Yu, H. Chen, and Y. Guo, Pressure-induced superconductivity reentrant in transition metal dichalcogenide TiSe2, arXiv:2202.06244 (2022).
  * Lee  _et al._ [2021] S. Lee, T. B. Park, J. Kim, S.-G. Jung, W. K. Seong, N. Hur, Y. Luo, D. Y. Kim, and T. Park, Tuning the charge density wave quantum critical point and the appearance of superconductivity in $\mathrm{Ti}{\mathrm{se}}_{2}$, Phys. Rev. Res. 3, 033097 (2021).
  * van Wezel  _et al._ [2011] J. van Wezel, P. Nahai-Williamson, and S. S. Saxena, Exciton-phonon interactions and superconductivity bordering charge order in TiSe2, Phys. Rev. B 83, 024502 (2011).
  * Wilson [1938] A. H. Wilson, The electrical conductivity of the transition metals, Proc. R. Soc. Lond. A 167, 580 (1938).
  * Gurvitch [1986] M. Gurvitch, Universal disorder-induced transition in the resistivity behavior of strongly coupled metals, Phys. Rev. Lett. 56, 647 (1986).
  * Knowles  _et al._ [2020] P. Knowles, B. Yang, T. Muramatsu, O. Moulding, J. Buhot, C. J. Sayers, E. Da Como, and S. Friedemann, Fermi Surface Reconstruction and Electron Dynamics at the Charge-Density-Wave Transition in TiSe2, Phys. Rev. Lett. 124, 167602 (2020).
  * Raymond  _et al._ [2011] S. Raymond, J. Bouchet, G. H. Lander, M. Le Tacon, G. Garbarino, M. Hoesch, J.-P. Rueff, M. Krisch, J. C. Lashley, R. K. Schulze, and R. C. Albers, Understanding the complex phase diagram of uranium: The role of electron-phonon coupling, Phys. Rev. Lett. 107, 136401 (2011).
  * Gruner  _et al._ [2017] T. Gruner, D. Jang, Z. Huesges, R. Cardoso-Gil, G. H. Fecher, M. M. Koza, O. Stockert, A. P. Mackenzie, M. Brando, and C. Geibel, Charge density wave quantum critical point with strong enhancement of superconductivity, Nature Physics 13, 967 (2017).
  * Shishido  _et al._ [2005] H. Shishido, R. Settai, H. Harima, and Y. Onuki, A drastic change of the fermi surface at a critical pressure in CeRhIn5: dHvA study under pressure, J. Phys. Soc. Jpn. 74, 1103 (2005).
  * Ramshaw  _et al._ [2015] B. J. Ramshaw, S. E. Sebastian, R. D. McDonald, J. Day, B. S. Tan, Z. Zhu, J. B. Betts, R. Liang, D. A. Bonn, W. N. Hardy, and N. Harrison, Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor, Science 348, 317 (2015), 1409.3990 .
  * Semeniuk  _et al._ [2022] K. Semeniuk, H. Chang, J. Baglo, S. Friedemann, A. Grockowiak, W. A. Coniglio, M. B. Gamza, P. Reiss, P. Alireza, I. Leermakers, A. McCollam, S. Tozer, and F. M. Grosche, Truncated mass divergence in a mott metal, arXiv (2022), arXiv:2202.04024 [cond-mat.str-el] .
  * Watson  _et al._ [2019] M. D. Watson, O. J. Clark, F. Mazzola, I. Marković, V. Sunko, T. K. Kim, K. Rossnagel, and P. D. C. King, Orbital- and ${k}_{z}$-selective hybridization of Se $4p$ and Ti $3d$ States in the charge density wave phase of TiSe2, Phys. Rev. Lett. 122, 076404 (2019).
  * Craven  _et al._ [1978] R. A. Craven, F. J. Di Salvo, and F. S. L. Hsu, Mechanisms for the 200 K transition in TiSe2: A measurement of the specific heat, Solid State Commun. 25, 39 (1978).
  * Blaha  _et al._ [2019] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz,  _WIEN2k_ , 19th ed. (2019).
  * Jishi and Alyahyaei [2008] R. A. Jishi and H. M. Alyahyaei, Electronic structure of superconducting copper intercalated transition metal dichalcogenides: First-principles calculations, Phys. Rev. B 78, 144516 (2008).
  * Wu  _et al._ [2007] G. Wu, H. Yang, L. Zhao, X. Luo, T. Wu, G. Wang, and X. Chen, Transport properties of single-crystalline CuxTiSe2 (0.015$\leq$x$\leq$0.110), Phys. Rev. B 76, 024513 (2007).
  * Zhao  _et al._ [2007] J. F. Zhao, H. W. Ou, G. Wu, B. P. Xie, Y. Zhang, D. W. Shen, J. Wei, L. X. Yang, J. K. Dong, M. Arita, H. Namatame, M. Taniguchi, X. H. Chen, and D. L. Feng, Evolution of the electronic structure of 1$T$-CuxTiSe2, Phys. Rev. Lett. 99, 146401 (2007).
  * Kitou  _et al._ [2019] S. Kitou, A. Nakano, S. Kobayashi, K. Sugawara, N. Katayama, N. Maejima, A. Machida, T. Watanuki, K. Ichimura, S. Tanda, T. Nakamura, and H. Sawa, Effect of Cu intercalation and pressure on excitonic interaction in $1T-\mathrm{TiS}{\mathrm{e}}_{2}$, Phys. Rev. B 99, 104109 (2019).
  * Liu  _et al._ [2010] C. Liu, T. Kondo, R. M. Fernandes, A. D. Palczewski, E. D. Mun, N. Ni, A. N. Thaler, A. Bostwick, E. Rotenberg, J. Schmalian, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, Evidence for a lifshitz transition in electron-doped iron arsenic superconductors at the onset of superconductivity, Nature Physics 6, 419 (2010).
  * Park  _et al._ [2021] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Tunable strongly coupled superconductivity in magic-angle twisted trilayer graphene, Nature 590, 249 (2021).
  * Bourbonnais and Sedeki [2011] C. Bourbonnais and A. Sedeki, Superconductivity and antiferromagnetism as interfering orders in organic conductors, C. R. Physique 12, 532 (2011).
  * Yelland  _et al._ [2011] E. A. Yelland, J. M. Barraclough, W. Wang, K. V. Kamenev, and A. D. Huxley, High-field superconductivity at an electronic topological transition in urhge, Nat. Phys 7, 890 (2011).
  * Klotz  _et al._ [2009] S. Klotz, J. C. Chervin, P. Munsch, and G. L. Marchand, Hydrostatic limits of 11 pressure transmitting media, Journal of Physics D: Applied Physics 42, 075413 (2009).
  * Antonelli  _et al._ [2022] T. Antonelli, W. Rahim, M. D. Watson, A. Rajan, O. J. Clark, A. Danilenko, K. Underwood, I. Markovic, E. Abarca-Morales, S. R. Kavanagh, P. Fevre, F. Bertran, K. Rossnagel, D. O. Scanlon, and P. D. C. King, Orbital-selective band hybridisation at the charge density wave transition in monolayer TiTe2, npj Quantum Materials 7, 98 (2022).
  * Monney  _et al._ [2012] C. Monney, G. Monney, P. Aebi, and H. Beck, Electron–hole instability in 1$T$-TiSe2, New J. Phys. 14, 075026 (2012).