# Normal state quantum geometry and superconducting domes in (111) oxide
interfaces

Florian Simon<EMAIL_ADDRESS>Laboratoire de
Physique des Solides, Université Paris-Saclay, CNRS UMR 8502, F-91405 Orsay
Cedex, France  Mark O. Goerbig  Laboratoire de Physique des Solides,
Université Paris-Saclay, CNRS UMR 8502, F-91405 Orsay Cedex, France  Marc
Gabay  Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS UMR
8502, F-91405 Orsay Cedex, France

###### Abstract

We theoretically investigate the influence of the normal state quantum
geometry on the superconducting phase in (111) oriented oxide interfaces and
discuss some of the implications for the $\text{LaAlO}_{3}/\text{SrTiO}_{3}$
(LAO/STO) heterostructure. From a tight-binding modeling of the interface, we
derive a two-band low-energy model, allowing us to analytically compute the
quantum geometry and giving us access to the superfluid weight, as well as to
showcase the role of two particular relevant energy scales. One is given by
the trigonal crystal field which stems from the local trigonal symmetry at the
interface, and the other one is due to orbital mixing at the interface. Our
calculations indicate that the variation of the superfluid weight with the
chemical potential $\mu$ is controlled by the quantum geometry in the
low-$\mu$ limit where it presents a dome. At higher values of $\mu$ the
conventional contribution dominates. In order to make quantitative comparisons
between our results and experimental findings, we rely on an experimentally
observed global reduction of the superfluid weight that we apply to both the
conventional and geometric contributions. Furthermore, an experimentally
measured non-monotonic variation of $\mu$ with the gate voltage $V_{g}$ is
taken into account and yields a two-dome scenario for the superconducting
critical temperature as a function of $V_{g}$. The observed dome in the
low-$V_{g}$ regime is explained by the non-monotonic evolution of a dominant
conventional part of the superfluid density. In contrast, the expected second
dome at larger values of $V_{g}$ limit would be due to a dominant quantum-
geometric contribution.

##  I Introduction

Superconductivity has, since 1911, become a flagship of condensed-matter
physics. The main paradigm is given by the Bardeen-Cooper-Schrieffer (BCS)
theory [1] which, in its standard form, describes the mechanism whereby
quasiparticles in a single, partially filled band, pair and condense in a
single collective dissipationless state. This single-band approximation has
its limits. Indeed, since the 1950s [2, 3], it has been realized that in a
multiband situation, even in the adiabatic limit, each band carries a
signature of the other bands in the form of two geometric contributions,
namely in what has later been identified as the Berry curvature and the
quantum metric [4]. These quantities form what we call band/quantum geometry.
In the context of superconductivity, this means that even if the Cooper
pairing takes place within a single band, it is a priori affected by the other
electronic bands of the normal state, particularly through the normal state
quantum geometry. While BCS theory does not take these geometric effects into
account, recent studies have theoretically pointed out the relevance of the
quantum metric for the superfluid weight of flat-band models [5, 6, 7, 8, 9],
as well as of the Berry curvature of Dirac-like systems [10], such as 2D
transition metal dichalcogenides.

Our study emphasizes the impact of the normal state quantum geometry on
superconductivity for (111)-oriented oxide interfaces, and more specifically
for the LaAlO3-SrTiO3 (LAO/STO) heterostructure [11]. Let us point out that
the results which we present here may be relevant for other materials,
including other (111) oxide interfaces. The LAO/STO heterostructure hosts an
electron gas (2DEG) on the STO side, confined to a few layers in the vicinity
of the interface [12]. For the (111) interface, carriers in the 2DEG move on a
honeycomb structure with three orbitals per site and, from that point of view,
this may be seen as a three-orbital version of graphene [13].

Starting from a tight-binding (TB) modeling of the interface, we develop a
low-energy model around the $\Gamma$ point in two steps. First, we consider
the spinless case for which the system has six bands. The dominance in energy
of intra-orbital nearest-neighbor hoppings allows us to decouple the six bands
into a bonding triplet and an anti-bonding triplet of bands. Focusing on the
bonding triplet, we derive a low-energy three band model to quadratic order in
$\mathbf{k}$. The latter presents two upper bands, degenerate at the $\Gamma$
point, that represent the $|e_{\pm g}\rangle$ states. The other band is lower
in energy, because of the trigonal crystal field, and represents the
$|a_{1g}\rangle$ state. The three bands have isotropic and quadratic
dispersions but the lowest band is significantly heavier than the other two.
The latter presents a strong peak of the quantum metric, driven by inter-
orbital nearest neighbor hoppings, which we refer to as orbital mixing.
Second, we consider the spinful case in which a spin-orbit coupling term is
taken into account. We then have a twelve-band TB model, from which we also
derive a two-band low-energy model. The latter reveals an isotropic Dirac cone
at the $\Gamma$ point, which stems from the orbital mixing.

From the low-energy model, we can then draw a qualitative scenario for the
chemical potential ($\mu$) dependence of the conventional and geometric
contributions to the superfluid weight [6, 9]. The conventional contribution
is essentially linear in $\mu$, because of the isotropy of the dispersion, and
the geometric contribution presents a dome as a function of $\mu$, since the
Dirac cone produces a strong peak in the quantum metric.

Finally, we compare the results of the qualitative scenario to the
experimental data, in the case of the LAO/STO interface. Taking into account
thermal and disorder effects, we compute the superfluid weight both for the
low-energy model, and the spinful TB model. In both cases, we find the correct
order of magnitude for the superfluid weight (and the associated BKT
temperature). The most significant difference between the two models is the
value of $\mu$ below which the geometric contribution becomes dominant. In the
low-energy model, the $\mu$ functional form of this quantity displays such a
steep variation that it seems hardly possible to experimentally observe the
regime when it dominates. In the spinful TB model, bands from the anti-bonding
group, though $6-7$eVs away, contribute significantly to the quantum metric of
the lowest Kramers’ partners, producing a smoother variation of the geometric
term and increasing the value of chemical potential below which the
geometrical contribution dominates.

While the evolution of the superfluid weight is naturally discussed in terms
of the chemical potential within a theoretical approach, experiments use the
gate voltage $V_{g}$ as a tuning parameter rather than $\mu$. Oftentimes the
variation of $\mu$ with $V_{g}$ is monotonic both quantities are related by a
monotonic function but Hall transport experiments [14, 15] have indicated that
this is not the case for the LAO/STO (111) interfaces, where a non-monotonic
dependence of the Hall carrier density on $V_{g}$ has been observed. Assuming
that the Hall carrier density is a monotonic function of the chemical
potential, this suggests that the chemical potential is non monotonic in
$V_{g}$. Taking this as an experimental fact, we then infer a qualitative
dependence of the superfluid weight (and thus BKT temperature) on the gate
voltage. The experimentally observed dome is explained by the conventional
contribution and the non monotonic dependence of $\mu$ on $V_{g}$. As $V_{g}$
is further increased, we expect a saturation effect followed by the appearance
of a second superconducting dome, due purely to the geometric contribution to
the superfluid weight.

The paper is organized as follows. In section II, we present the tight-binding
model, both in the spinless and spinful cases. In section III, we derive low-
energy models and discuss their structure, both in the spinless and the
spinful cases. We then build on the spinful low-energy model to discuss the
dependence of the superfluid weight on the chemical potential. Finally, in
section IV we make contact with experiments for the spinful case. We
numerically compute the superfluid weight versus the chemical potential, for
the low-energy model and then for the full TB model, and we discuss the extent
to which the results that are obtained are consistent with the scenario
presented in section III.3.

##  II Tight-binding model

We first introduce the relevant TB modeling of the $(111)$ interface and
discuss its various terms. The values of the relevant energy scales, presented
in detail in Sec. III, are mainly taken from Refs. [16, 17, 18, 12, 15]. The
structure of the (111) oriented STO substrate is shown in Fig. 1.

Figure 1: STO side, just below the (111) LAO/STO interface, adapted from [16].
Left: cubic lattice cell the corners of which are occupied by Ti4+ ions. The
gray areas indicate planes normal to the (111) direction. Right: projection
onto (111) planes. Two layers of Ti4+ ions (blue and red) form a honeycomb
lattice, where the two triangular sublattices are displaced by the vector
$\mathbf{a}_{0}$. Lastly, $a_{\text{STO}}=3.905\text{\AA}$ and
$a_{0}=\sqrt{2/3}a_{\text{STO}}$.

The two-dimensional electron gas (2DEG) is located on the STO side of the
LAO/STO interface [12]. From a structural point of view, the three-dimensional
(3D) SrTiO3 crystal has a ABO3 cubic perovskite structure (left panel, Fig.
1). In the (111) orientation, see Fig. 1, two consecutive (111) planes contain
Ti ions for one and SrO3 ionic groups for the other. Focusing on the Ti (111)
planes, the atomic arrangement consists of layers of two-dimensional (2D)
triangular lattices displaced by the vector $\mathbf{a}_{0}$ (see Fig. 1).
Consequently, the Ti atoms form ABC-stacked two-dimensional 2D triangular
lattices in the (111) planes (see Fig. 1, left panel). From an electronic
point of view, the charge carriers hop precisely between neighboring Ti atoms
through direct orbital overlap or via the O sites.

While the basic building block for the description of the 2DEG would in
principle contain three (ABC) layers of Ti atoms (red, blue and green in Fig.
1), the location of the Fermi energy for the (111) direction, as seen in Ref.
[12], allows us to reduce the model to only two layers shown in Fig. 1 (right
panel), as we have checked explicitly numerically in Appendix A. Indeed, for a
unit trilayer stack, the TB Hamiltonian describing the kinetics of the 2DEG
parallel to the (111) interface produces nine bands (not counting spin)
organized in three groups (bonding, non-bonding, anti-bonding). Numerical
inspection, using representative values of hopping amplitudes, shows that the
energy difference between consecutive groups is on the order of several eV
such that the three non-bonding bands which come from the third (green) layer
are several eVs away from the Fermi energy, as discussed in Appendix A.
Additionally, the dispersions of the occupied bonding triplet bands show very
little difference with those of a bilayer model, where only the first (blue)
and second (red) layers are considered (see Fig. 10 of Appendix A).

We can therefore leave out the third layer and consider the system shown on
the right in Fig. 1, i.e. two triangular layers (red and blue) displaced by
the vector $\mathbf{a}_{0}$ that form a honeycomb lattice characterized by the
layer/sublattice index $\big{\\{}1,2\big{\\}}$. On each site, we have the
three conducting $t_{2g}$ Ti orbitals,
$\big{\\{}d_{yz},d_{xz},d_{xy}\big{\\}}$, which, in the following, we denote
$\big{\\{}\text{X},\text{Y},\text{Z}\big{\\}}$, respectively.

Accounting for spin, we then have a honeycomb lattice with two spins and three
orbitals per site, yielding a twelve-band system. Next, we discuss each type
of hopping entering our TB model.

###  II.1 Kinetic term

The kinetic part of the model takes into account hoppings between the
different lattice sites and orbitals. This term describes carrier motions
conserving the orbital character, with amplitudes $t$ and $t_{d}$ for nearest
and third nearest neighbors belonging to two different layers, respectively.
The general form of the kinetic term is thus diagonal in terms of the orbitals
but off-diagonal in terms of the layers. Therefore, in the
$\\{1,2\\}\otimes\big{\\{}\text{X},\text{Y},\text{Z}\big{\\}}$ basis the
kinetic term reads

$\begin{pmatrix}0&H_{\text{cin}}\\\
H_{\text{cin}}^{*}&0\end{pmatrix}=\tau_{x}\otimes\text{Re}(H_{\text{cin}})-\tau_{y}\otimes\text{Im}(H_{\text{cin}}),$
(1)

with $H_{\text{cin}}=t\operatorname{diag}(e,f,g)$ in the orbital subspace. The
Pauli matrices $\tau_{x}$ and $\tau_{y}$ in Eq. (1) act on the layer subspace.
Explicit expressions for the functions $e,f$ and $g$ may be found in appendix
B.

###  II.2 Orbital mixing terms

While the kinetic term does not couple different orbitals, such couplings are
generated at the interface by orbital mixing. In appendix C, we show by
symmetry considerations that a natural choice is

$\tau_{x}\otimes H_{\text{om}}=\tau_{x}\otimes
c_{0}\begin{pmatrix}0&i\delta&-i\alpha\\\ -i\delta&0&i\beta\\\
i\alpha&-i\beta&0\end{pmatrix},$ (2)

where $\alpha=\sin(\sqrt{3}/2k_{x}+3/2k_{y})$,
$\beta=\sin(\sqrt{3}/2k_{x}-3/2k_{y})$, $\delta=-\sin(\sqrt{3}k_{x})$ and
$c_{0}$ the strength of the orbital mixing. Here, we measure the wave vectors
in units of the inverse $a_{0}^{-1}$ of the distance between nearest-neighbor
sites in the (111) plane (see Fig. 1), and $\tau_{x}$ is again a Pauli matrix
acting on the layer degree of freedom. Note that with inversion symmetry,
these terms are prohibited. But in reality, interfaces between
$\text{LaAlO}_{3}$ and $\text{SrTiO}_{3}$ always have corrugation [19, 20],
such that inversion symmetry is broken and orbitals that would have been
orthogonal are not, resulting in non-zero overlap and allowed interorbital
hoppings. It will give rise to an orbital Rashba effect.

###  II.3 Trigonal crystal field

Note that the (111) interface has a different point symmetry than the orbitals
whose symmetry is governed by the (cubic) bulk symmetry of LAO and STO.
Therefore the $t_{2g}$ orbitals are not orthogonal to each other in the
hexagonal lattice, resulting in a trigonal crystal field, where the couplings
have the same value because of the hexagonal symmetry. It lifts the degeneracy
between the $e_{\pm g}$ orbitals and the $a_{1g}$ orbital within the
conducting $t_{2g}$ orbitals of Ti. This trigonal crystal field, of strength
$d$, thus couples the different orbitals in the same layers so that it may be
written as

$H_{d}=-d\tau_{0}\otimes\begin{pmatrix}0&1&1\\\ 1&0&1\\\ 1&1&0\end{pmatrix},$
(3)

where $\tau_{0}$ is the identity matrix indicating that the trigonal crystal
field is diagonal in the layer index.

###  II.4 Confinement energy

Finally, we need to take into account a confinement term that reflects the
different onsite potentials for the two sublattices, which reside in different
layers. It is equivalent to the Semenoff mass in graphene, breaking the
$\mathcal{C}_{6}$ symmetry down to $\mathcal{C}_{3}$. We have $-V\Lambda_{0}$
for layer 1 and $V\Lambda_{0}$ for layer 2, so that this term may be written
as $\tau_{z}\otimes(-V\Lambda_{0})$, in terms of the $3\times 3$ identity
matrix noted $\Lambda_{0}$. While this term may be important for other
properties of the LAO/STO interface, we will see that it does not affect those
studied in this paper, and we will later omit it when reducing the TB model to
a low-energy model.

###  II.5 Spinless six-band model

With these four terms, the six-band TB model is written, in
$\\{1,2\\}\otimes\big{\\{}\text{X},\text{Y},\text{Z}\big{\\}}$, as

$H_{6}=\begin{pmatrix}-VI_{3}+H_{d}&H_{\text{cin}}+H_{\text{om}}\\\
H_{\text{cin}}^{*}+H_{\text{om}}&VI_{3}+H_{d}\end{pmatrix}.$ (4)

A more convenient basis is the trigonal basis in which the trigonal crystal
field term is diagonal. The latter is detailed in appendix E. Hereafter, we
discuss the band structure described by $H_{6}$ in the trigonal basis. We now
add spin to our problem.

###  II.6 Spin-orbit coupling

The spin-orbit coupling (SOC) term is identical for the two layers, and thus
naturally diagonal in the layer subspace. In the
$\\{\uparrow,\downarrow\\}\otimes\\{\text{X},\text{Y},\text{Z}\\}$ basis, it
reads [21]

$H_{\text{SOC}}=-\lambda\sigma_{x}\otimes\Lambda_{7}+\lambda\sigma_{y}\otimes\Lambda_{5}-\lambda\sigma_{z}\otimes\Lambda_{2},$
(5)

where the $\Lambda$-matrices denote the Gell-Mann matrices, explicited in
appendix D. The energy scale $\lambda$ is given by $\lambda\simeq 8$meV [15].

###  II.7 Spinful twelve-band model

The spinless Hamiltonian $H_{6}$ is identical for each spin, the corresponding
term in the
$\\{\uparrow,\downarrow\\}\otimes\\{1,2\\}\otimes\big{\\{}\text{X},\text{Y},\text{Z}\big{\\}}$
basis, will be $\sigma_{0}\otimes H_{6}$. We thus finally obtain our twelve-
band TB model, as

$\displaystyle H_{12}$ $\displaystyle=\sigma_{0}\otimes
H_{6}-\lambda\sigma_{x}\otimes\tau_{0}\otimes\Lambda_{7}+\lambda\sigma_{y}\otimes\tau_{0}\otimes\Lambda_{5}$
$\displaystyle-\lambda\sigma_{z}\otimes\tau_{0}\otimes\Lambda_{2},$ (6)

where the last three terms correspond to $\tau_{0}\otimes H_{\text{SOC}}$.

##  III Low-energy model

We now aim to derive a low-energy expression from the TB model. We first
discuss the problem without spin, and derive a low-energy three-band model,
valid for each spin orientation. We then add the and find that the low-energy
limit leads to a two-band model. To do so, we only apply momentum independent
unitary transformations, that leave the quantum geometry invariant.

###  III.1 Spinless problem

Numerical diagonalization shows that the low-filling regime occurs near the
$\Gamma$ point. Moreover, in the vicinity of the latter, there are two groups
of three bands separated by several eV. This is because the gap between the
two groups at the $\Gamma$ point is $2(2t+t_{d})\sim 6.5$eV, and the kinetic
energy is clearly the largest energy scale. Therefore, for low fillings, it
appears possible to reduce the above six-band expression to two effective
three-band models, one for each group. To make a similar structure appear
explicitly in $H_{6}$, we apply the following unitary transformation

$U=U_{l}\otimes\Lambda_{0}=\frac{1}{\sqrt{2}}\begin{pmatrix}-1&1\\\
1&1\end{pmatrix}\otimes\Lambda_{0},$ (7)

which maximally entangles the two layers, in a symmetric $|s_{l}\rangle$ and
an anti-symmetric combination $|a_{l}\rangle$. The six-band Hamiltonian is
then transformed to

$U^{\dagger}H_{6}U=\begin{pmatrix}H_{d}-H_{\text{om}}-\text{Re}(H_{\text{cin}})&-VI_{3}+i\text{Im}(H_{\text{cin}})\\\
-VI_{3}-i\text{Im}(H_{\text{cin}})&H_{d}+H_{\text{om}}+\text{Re}(H_{\text{cin}})\end{pmatrix}.$
(8)

Numerical inspection confirms that the diagonal blocks pertain to the two
groups. Thus, we may focus on the lower diagonal block, which corresponds to
the symmetric entanglement of the two layers, and take it as a low-energy
three-band model that reads

$H_{3}=H_{d}+H_{\text{om}}+\text{Re}(H_{\text{cin}}).$ (9)

A discussion of the validity of this approximation, done in appendix F, shows
that with a precision of a few meV, this three-band approximation is valid
over an area centered at $\Gamma$ and covering approximately ten percent of
the Brillouin zone (BZ). To be consistent with this approximation, we need to
expand $H_{3}$ to quadratic order in $k$.

####  III.1.1 Quadratic three-band model

In appendix G, we show that to quadratic order, we have

$H_{3}=-(2t+t_{d})\Big{(}1-\frac{1}{4}k^{2}\Big{)}\Lambda_{0}+\begin{pmatrix}d-t_{\text{eff}}(k_{x}^{2}-k_{y}^{2})&-2t_{\text{eff}}k_{x}k_{y}&ick_{x}\\\
-2t_{\text{eff}}k_{x}k_{y}&d+t_{\text{eff}}(k_{x}^{2}-k_{y}^{2})&ick_{y}\\\
-ick_{x}&-ick_{y}&-2d\end{pmatrix},$ (10)

with $t_{\text{eff}}=(t-t_{d})/8$ and $c=3c_{0}/\sqrt{2}$. Note that $H_{3}$
is expressed in the trigonal basis (see appendix E). The trigonal crystal
field lifts the threefold degeneracy at the $\Gamma$ point (between $a_{1g}$
and $e_{\pm g}$ states). The linear and quadratic terms arise from the orbital
mixing and kinetic terms, respectively. Inspection of Eq. (10), reveals that
the $a_{1g}$ state is only coupled to the $e_{\pm g}$ states through the
orbital mixing contribution, which generates its quantum geometry.

$H_{3}$ can then be exactly diagonalized, and we find the following
eigenvalues for the last term:

$\epsilon_{1}=d+t_{\text{eff}}k^{2},\quad\epsilon_{2}=d+\bigg{(}\frac{c^{2}}{3d}-t_{\text{eff}}\bigg{)}k^{2},$
(11)

and

$\epsilon_{3}=-2d-\frac{c^{2}}{3d}k^{2},$ (12)

to quadratic order in the wave-vector components. The values taken hereafter
are those corresponding to Refs. [16, 17, 18, 12, 15], i.e. $t=1.6$ eV,
$t_{d}=70$ meV, $V=100$ meV, $d=3$ meV. Additionally, we estimate $c_{0}=40$
meV. We thus find an isotropic electron-like band structure.

We point out that these results may apply to other (111) oxide interfaces. In
Section IV, we discuss the relevance of our results to the experimental
context, illustrated with the case of the LAO/STO (111) interface. The lowest
energy band ($\epsilon_{3}$) is substantially flatter than the other two.
Indeed, its band mass can be computed to be

$m_{B}=\frac{\hbar^{2}}{2a_{0}^{2}}\bigg{(}\frac{2t+t_{d}}{4}-\frac{c^{2}}{3d}\bigg{)}^{-1}\simeq
21m_{0},$ (13)

with $m_{0}\simeq 9.1\times 10^{-31}$ kg the rest mass of an electron. This
band presents a peak in its quantum metric at the value given by
$g_{3,\mu\mu}(\Gamma)=c_{0}^{2}/9d\simeq 90a_{0}^{2}$.

Note that going beyond the low-energy model, which is obtained using a
quadratic order expansion in k, we find that the interorbital contributions to
cubic order, give rise to an orbital Rashba effect which moves the minimum
away from the $\Gamma$ point and therefore the actual band mass differs from
Eq. (13). We then plot this band structure and contrast it with the one we get
from the TB form of the kinetic and orbital mixing terms in Fig. 2. The band
structure of the full TB model in the full Brillouin Zone (BZ) is shown in
Appendix H.

Figure 2: Energy dispersions near $\Gamma$, in the $k_{x}=k_{y}$ direction.
Each color corresponds to one of the three bands, blue is the lowest band, red
is the second lowest and green the third lowest band. Dashed lines correspond
to the dispersions coming from the low-energy model, Eq. (10). Solid lines
come from the full Tight-binding Hamiltonian, Eq. (4).

We indeed get the aforementioned precision of a few meVs. Note that the offset
of 2 meV between the TB and low-energy bands as seen in Fig. 2, is due to the
fact that $V$ does not enter the low-energy expression of the Hamiltonian.
Such a global shift does not have a physical relevance on the quantum geometry
and superfluid weight as it can be compensated by a redefinition of the
chemical potential with respect to the $\Gamma$ point value of the lowest
band. We then get a lower band that is substantially flatter than the other
ones and that is close in energy to a level crossing at the $\Gamma$ point.

###  III.2 Spinful problem

We now derive a low-energy model starting from the spinful Hamiltonian
$H_{12}$.

####  III.2.1 Derivation of the low-energy model

We start by applying, once again, the unitary matrix $U$ given in Eq. (7) ,
and we use the matrix $P$ to express the orbital part in the trigonal basis.
The SOC term is left unchanged by $U$, while the $\sigma_{0}\otimes H_{6}$
term gets us two copies of Eq. (8). We then make the same approximation, and
by switching the layer and spin degrees of freedom, we get a Hamiltonian which
is the counterpart of the bonding and anti-bonding groups of the spinless
case, only now with one copy for each spin as well as the accompanying SOC. We
then restrict $H_{12}$ to the bonding subspace, i.e. the symmetric combination
of the two layers $|s_{l}\rangle$, and then get a six-band Hamiltonian
$\tilde{H}_{6}$, written as

$\tilde{H}_{6}=\sigma_{0}\otimes H_{3}+H_{\text{SOC}},$ (14)

where $H_{\text{SOC}}$ is expressed in the trigonal basis, and $H_{3}$ is
given by Eq.(10). Further manipulation detailed in appendix I, using the
experimentally relevant simplifying approximation $\lambda=3d$, yields the
two-band Hamiltonian
$H_{2}=h_{0}(\mathbf{k})\sigma_{0}+\mathbf{h}(\mathbf{k})\cdot\mathbf{\sigma}$,
with

$h_{0}(\mathbf{k})=-2t-t_{d}+\big{(}1-3\sqrt{3}\big{)}d+\frac{2t+t_{d}}{4}k^{2},$
(15)

and

$\mathbf{h}(\mathbf{k})=\frac{c_{0}}{2}\Big{(}-2k_{x},k_{x}+\sqrt{3}k_{y},k_{x}-\sqrt{3}k_{y}\Big{)}.$
(16)

Note that the spin-orbit coupling mixes all three bonding spinless bands,
producing three Kramers pairs of energy branches separated by an energy gap at
the $\Gamma$ point. The low-energy model focuses on the lowest Kramers pairs.

####  III.2.2 Dispersions

The dispersions are given by

$\epsilon_{\pm}=-2t-t_{d}+\big{(}1-3\sqrt{3}\big{)}d\pm\sqrt{\frac{3}{2}}c_{0}k+\frac{2t+t_{d}}{4}k^{2}.$
(17)

As in the spinless case, we compare the dispersions of the latter with those
of the full TB model $H_{12}$ in Fig. 3.

Figure 3: Energy dispersions of the low-energy two-band model $H_{2}$ overlaid
with the two lowest band of the full TB model $H_{12}$, near $\Gamma$ and in
the $k_{x}=k_{y}$ direction. The energy is defined with respect to the
crossing point of the solid lines.

####  III.2.3 Quantum/band geometry

##### Quantum metric

The quantum metric is the same for the two bands, being equal to [6]

$g_{\mu\nu}=\frac{k^{2}\delta_{\mu\nu}-k_{\mu}k_{\nu}}{4k^{4}}.$ (18)

##### Berry curvature

As for the Berry curvature, the upper and lower parts of the Dirac cone yield
opposite $\delta(k)$ contributions at the contact point. Note that in the full
model, there are two sources of Berry curvature. One is, as we just mentioned,
the spin-orbit coupling, which results in the mentioned Dirac cone. The other
is the mixing of the lowest band with the others which drives a Berry
curvature the order of unity in $a_{0}^{2}$, as mentioned in Sec. IV.3.

###  III.3 Superfluid weight in the low-energy model

During the past decade, a significant number of papers have discussed the role
of the normal-state quantum metric on the superconducting state [6, 9, 8]. It
was found that the tensor relating the supercurrent to the electrodynamic
perturbation of a superconductor produces two types of terms. One type is the
well-known BCS contribution (see Ref. [22], for example), and the other is a
geometric contribution which stems from interband couplings when the normal-
state electronic structure involves several bands. In the isolated band limit,
and in two-band models, this geometric contribution directly involves the
normal-state quantum metric [6].

Initially, the theory was developed for flat bands where the conventional
contribution vanishes and the geometric contribution then dominates. While we
do not have flat bands, the Dirac cone structure of our low-energy model gives
a strong quantum metric near the $\Gamma$ point. It thus seems relevant to
investigate whether the normal-state quantum metric produces a sizeable effect
on the superconducting state through this geometric superfluid weight. In the
following two sections, we discuss the two contributions in the context of our
low-energy model. We begin with a qualitative discussion aimed at explaining
generic scenarios for the Berezinskii-Kosterlitz-Thouless (BKT) temperature
versus the gate voltage $V_{g}$. For the superconducting state, we assume a
conventional $s$-wave pairing, which can accommodate the disordered nature of
oxide interfaces. As for the value of the $s$-wave gap, it has been measured
to be $\Delta=40\mu$eV at optimal doping for the $(001)$ LAO/STO interface
[23], and a similar value for the (111) interface was reported in Ref. [24].

####  III.3.1 BKT Temperature

In addition to the superfluid weight (which has the dimension of an energy in
2D), we consider the associated Berezinskii-Kosterlitz-Thouless (BKT)
temperature, using the (isotropic) Nelson-Kosterlitz criterion [6],

$T_{\text{BKT}}=\frac{\pi}{8k_{B}}D(T_{\text{BKT}}),$ (19)

where $D(T)$ is the superfluid weight at temperature $T$. The BKT temperature
$T_{\text{BKT}}$ is the temperature above which vortex-antivortex pairs start
to unbind and thus destroy superconductivity. It is smaller than the critical
temperature calculated in the framework of a mean-field approach. For
$T_{\text{BKT}}$ not too close to $T_{c}$, we may approximate
$D(T_{\text{BKT}})$ by $D(T=0)$. This defines a “mean-field” BKT temperature
which is larger than the actual one and may thus give an upper bound estimate,

$T_{\text{BKT}}=\frac{\pi}{8k_{B}}D(T=0).$ (20)

####  III.3.2 Conventional contribution

The conventional contribution to the superfluid weight, at $T=0$, is given by
[6, 22]

$D_{\mu\nu,\text{conv}}=\int_{\mathscr{S}_{\text{occ}}(\mu)}\mathcal{D}_{2}\mathbf{k}\frac{\Delta^{2}}{E^{3}}(\partial_{\mu}\epsilon)(\partial_{\nu}\epsilon),$
(21)

where $\mathscr{S}_{\text{occ}}(\mu)$ denotes the set of occupied states in
the BZ at the chemical potential $\mu$. $\mathcal{D}_{2}\mathbf{k}$ is the
integration measure for the hexagonal BZ. For isotropic linearly dispersing
bands, Eq. (21) gives $D_{\mu\nu,\text{conv}}=\delta_{\mu\nu}D_{\text{conv}}$
with $D_{\text{conv}}\propto\sqrt{\Delta^{2}+\mu^{2}}$ [6], where $\Delta$ is
the s-wave superconducting gap. In appendix J, we show that a similar result
holds for a general isotropic quadratic band. Consequently, the conventional
superfluid weight is essentially proportional to the chemical potential $\mu$
for $\mu\gtrsim\Delta$.

####  III.3.3 Geometric contribution

##### General expression

The geometric contribution at $T=0$ for a two-band Hamiltonian of the form
$h_{0}\sigma_{0}+\mathbf{h}\cdot\bm{\sigma}$, is given by [6]

$D^{\pm}_{\text{geom},\mu\nu}=\pm\int_{\mathscr{S}_{\text{occ}}(\mu)}\mathcal{D}_{2}\mathbf{k}\frac{4\Delta^{2}}{E_{\pm}}\frac{h}{\mu-
h_{0}}g_{\mu\nu},$ (22)

and more generally the geometric contribution is linked to the quantum metric
in the so-called isolated limit [6], noting that Eq. (22) does not rely on it.
Furthermore, note the additional factor of two with respect to the expression
given in Ref. [6]. This is because the definition of the metric in Ref. [6] is
twice the usual one [4, 25].

##### Qualitative $\mu$ dependence

Because of the Dirac cone at the $\Gamma$ point, the quantum metric of the two
lowest bands in our model will exhibit a strong peak. Even in the spinless
case, the lowest band is accompanied by a strong peak at the $\Gamma$ point.
We therefore consider the case where the bands exhibit a peaked quantum metric
around where the zero of the chemical potential is defined, hereafter called
the zero-filling point.

The $1/E(\mathbf{k})$ factor in Eq. (22) enhances the contribution at the
Fermi contour, making it dominant. Focusing on this contribution, we can
propose a scenario explaining the emergence of a dome in the geometric
superfluid weight when the metric has a peak at the zero-filling point. We
sketch this scenario in Fig. 4.

Figure 4: Emergence of a dome in the geometric superfluid weight from the
Fermi contour contribution (in red) in Eq.(22) and its corresponding location
in the dome (red dot). (a) Low-$\mu$ regim. (b) Intermediate-$\mu$ regime. (c)
High-$\mu$ regime

.

At low $\mu$, the band starts to be filled around $\Gamma$. The Fermi contour
is thus at the top of the peak, but it is also narrow, such that
$D_{\text{geom}}$ is low. However, as the filling increases, the Fermi contour
gets wider while still being high and thus $D_{\text{geom}}$ becomes larger.
This is the low-$\mu$ regime, shown in Fig. 4a.

The chemical potential $\mu$ then reaches a value where the trade-off between
the height and the extent of the Fermi contour is optimal, and
$D_{\text{geom}}$ reaches its maximal value. This is the intermediate-$\mu$
regime in Fig. 4b.

Beyond the latter, the Fermi contour still gets wider but not enough to
compensate the smaller values of $g_{\mu\nu}$, resulting in a decrease of
$D_{\text{geom}}$. This is the high-$\mu$ regime in Fig. 4c.

####  III.3.4 Qualitative $\mu$-dependence of the BKT temperature

Following the above discussion, we sketch the qualitative evolution of the BKT
temperature given by Eq. (20) as shown in Fig. 5.

Figure 5: Qualitative dependence of the geometric contribution, with a dome
behavior, and the conventional contribution, with a linear behavior, to the
BKT temperature of Eq.(20).

In order to showcase the role of the geometric part, Fig. 5 is a plot of the
conventional and geometric contributions in a situation when both quantities
are of the same order. We show below that this is the case for the low-energy
and full TB model.

##  IV  Relating the superfluid weight to the experimental data

###  IV.1 Thermal and disorder effects

Experimental studies of the superconducting transition in (001) and (111)
oriented LAO/STO interfaces indicate that the BKT scenario is indeed relevant
[24, 26, 27, 28]. The superfluid weight is proportional to the superfluid
carrier density $n_{s}$. Temperature effects could be included through the
temperature dependence of the SC gap and the Fermi factor [29], however
disorder effects would also need to be properly included, a task beyond the
scope of the present work. Microwave measurements of the London penetration
depth by Lesne et al. [28] show that the value of $n_{s}$ at $T_{BKT}$ is one
order of magnitude smaller than it is at "zero temperature". These experiments
contain both disorder and thermal effects. Therefore, one way to take these
factors into account would be applying such a renormalisation to the
superfluid density and thus to the superfluid weight. While we do not provide
a full-fledged theory of the origin of this reduction of the superfluid
density, we take it as an experimental fact, and hereafter we qualitatively
renormalize both the conventional and geometric contributions to the
superfluid weight by a factor $1/10$.

###  IV.2 Superfluid weight from the low-energy model

We now numerically compute both contributions of the superfluid weight for the
low-energy model, using Eq. (22). The result is plotted in Fig. 6, and is
indeed consistent with the analysis made in Sec.III.3. Note that the chemical
potential is defined with respect to the Dirac point of the low-energy model.

Figure 6: Conventional (orange) and geometric (blue) contributions to the
superfluid weight from the low-energy model of Eqs.(15,16) using Eq.(22). The
chemical potential $\mu$ is defined with respect to the Dirac point of the
low-energy model. $T_{\text{BKT}}$ is defined in Eq.(20).

The conventional contribution is essentially linear in the chemical potential
$\mu$. Note that it is non-zero below the Dirac point because the minimum of
the lower band is located away from the $\Gamma$ point, as seen in Fig. 3.
Notice the small dip seen near $\mu=0$ in Fig. 6, which might be due to a
reduction of the density of states, because of the Dirac point.

The evolution of the geometric contribution is consistent with the scenario
depicted in Sec.III.3 and Fig. 4. Because of the Dirac point, the quantum
metric exhibits a strong (in fact divergent) peak at $\Gamma$, ie $\mu=0$ in
Fig. 6. For $\mu<0$, the Fermi contour is at the bottom of the metric’s peak,
so the geometric contribution is low. But then as $\mu$ gets closer to zero,
the Fermi contour moves to a higher position on the metric’s peak. The metric
diverges at the degeneracy point so that the geometric contribution diverges
at $\mu=0$. For $\mu>0$, the Fermi contour gradually moves to the bottom of
the peak, and the geometric contribution goes back to zero. All this happens
on a scale of 1meV around the Dirac point, as seen in Fig. 6. We note that the
variation of the geometric contribution is so steep around $\mu=0$ that one
would need to get to very low values of $\mu$ in order to reach the regime
where the geometric contribution dominates.

###  IV.3 Superfluid weight from the tight-binding model

Figure 7: Conventional (orange) and geometric (blue) contributions to the
superfluid weight from the spinful TB model. The chemical potential $\mu$ is
defined with respect to the Dirac point of the spinful TB model.

As a test of the accuracy of the low-energy model, we can also compute the
superfluid weight from the spinful TB model, as shown in Fig. 7. The chemical
potential $\mu$ is defined with respect to the Dirac point of the spinful TB
model. We see that our qualitative arguments of section III.3 hold here as
well. The geometric contribution shows a divergent behaviour in the $\mu\ll
1$meV regime, and precipitously drops to zero when we increase $\mu$. The
conventional contribution is linear in the chemical potential. One difference
between Fig. 7 and the superfluid weight of the low-energy model is that the
two contributions become equal for a larger value of $\mu$. This may be due to
the fact that bands in both the anti-bonding group, despite being $7$eV higher
in energy, and the other bonding bands have a non-negligible overlap with the
two bands of interest. Note that this is also the case for the Berry
curvature, where this overlap between the lowest and the highest bands
generates a Berry curvature of order $a_{0}^{2}$.

###  IV.4 Gate voltage dependence: Two-dome scenario

We have discussed the qualitative dependence of the BKT temperature [Eq.(20)]
on the chemical potential. However, the experimentally observed
superconducting dome [30, 31, 32, 11, 14] is measured when one tunes the gate
voltage $V_{g}$ or the conductivity of the interface. There are strong
indications [14, 15] suggesting that the (Hall) carrier density (or the
chemical potential) has a non-monotonic dependence on $V_{g}$ or on the
conductivity. Therefore, there is no simple correspondence between the
superconducting domes that result from changing $\mu$ and the phase diagram
that one obtains upon changing $V_{g}$. Based on the dependence of the Hall
number on $V_{g}$, and that of $\mu$ on the carrier density, we propose that
the variation of $\mu$ with $V_{g}$ is as depicted in Fig. 8a.

Figure 8: (a) Proposed dependance of the chemical potential on the gate
voltage. (b) Correspondence between the qualitative $\mu$ and $V_{g}$
dependencies of the geometric and conventional contributions.

The scenario depicted in Fig. 8 may be understood as follows. The initial
value of the chemical potential (at $V_{g}=0$) is at a point where the
conventional contribution dominates, indicated by $(1)$. At first, increasing
the gate voltage also increases the chemical potential so that the BKT
temperature also increases. This is the underdoped regime from $(1)$ to $(2)$.
It is followed by the optimal doping region near $(2)$, starting around the
top of the dome. Further increase of the gate voltage leads to a decrease of
the chemical potential, due to the non-monotonic relation between $V_{g}$ and
$\mu$, and therefore to lower values of the BKT temperature, in the overdoped
regime, from $(2)$ to $(3)$.

The resulting qualitative gate-voltage dependence of the BKT temperature
(Eq.(20)) is then sketched in Fig. 9.

Figure 9: Schematic view of the proposed qualitative evolution of the BKT
temperature as a function of the gate voltage.

We can draw further conclusions from this scenario. The experimentally
observed dome happens in a regime when the conventional contribution
dominates. But if we follow the experimental evidence indicating that $\mu$
decreases as $V_{g}$ increases, we could reach the low-$\mu$ regime at higher
$V_{g}$ and reveal the dome due to the geometric contribution. In other words,
the measured domes would be caused by the non-monotonic variation of the
chemical potential with respect to the gate voltage such that there should be
a secondary superconducting dome, coming from the geometric contribution, for
higher values of the gate voltage. The evolution of the critical temperature
(or superfluid density, BKT temperature) would be similar to that sketched in
Fig. 9, as long as only the lowest energy band contributes to the superfluid
condensate.

According to our picture, the two superconducting domes that one expects upon
increasing the gate voltage have thus different origins. The first one
corresponds to the regime when the conventional superfluid weight dominates
and the second one to the regime when the geometric weight dominates.

##  V Conclusion

Our study underscores the impact of the normal state quantum geometry on the
superconducting state of the (111) $\text{LaAlO}_{3}/\text{SrTiO}_{3}$
interface. Building upon a tight-binding modeling of the interface, we
developed a two-band low-energy model around the $\Gamma$ point. This low-
energy model reveals an isotropic Dirac cone at the $\Gamma$ point, driven by
the orbital mixing. From the low-energy model, we then drew a qualitative
scenario for the chemical potential ($\mu$) dependence of the conventional and
geometric contributions to the superfluid weight. The conventional
contribution is suggested to be linear in $\mu$, because of the dispersion’s
isotropy. We also argued that the strong peak in the quantum metric coming
from the Dirac cone results in a dome-shaped behavior of the geometric
contribution as a function of $\mu$.

Finally, we probed the relevance of our scenario to experiments in the case of
the LAO/STO interface. We first effectively took into account thermal and
disorder effects by renormalizing the superfluid weight. Then, we numerically
computed the superfluid weight from the two-band low-energy model and the
spinful TB model. In both cases, we find the right order of magnitude for the
superfluid weight (and the associated BKT temperature). The most significant
difference between the two models is the value of $\mu$ below which the
geometric contribution becomes dominant. In the low-energy model, the
geometric contribution exhibits a very narrow peak around $\mu=0$ and
therefore the regime where it dominates would seem hard to observe, in
practice. In the spinful TB model, bands from the anti-bonding group, despite
being $6-7$ eVs higher up, and other bonding bands, contribute significantly
to the quantum metric of the lowest Kramers’ partners, thereby making the
geometric contribution less steep and increasing the value of chemical
potential below which the geometric contribution dominates. We then argue
that, despite its simplicity, our low-energy model captures the evolution of
the superfluid weight reasonably well.

Experiments tune the 2DEG with a gate voltage $V_{g}$, so that we aimed to
qualitatively describe the behaviour of the superfluid weight as a function of
said $V_{g}$. Hall transport experiments [14, 15] find a non-monotonic
dependence of the Hall carrier density on $V_{g}$. Assuming that the Hall
carrier density is monotonic in the chemical potential, this means that the
chemical potential is non-monotonic in $V_{g}$. Taking this as an experimental
fact, we then inferred a qualitative dependence of the superfluid weight (and
thus BKT temperature) on the gate voltage. The experimentally observed dome is
suggested to come from the conventional contribution and the non-monotonic
dependence of $\mu$ on $V_{g}$. Extrapolating further, we also suggest the
saturation of the dome mentioned earlier and the appearance of a second
superconducting dome, due purely to the geometric contribution to the
superfluid weight.

Given the ubiquituousness of quantum geometry, this hidden influence on the
superconducting state might be apparent in other classes of materials.
Finally, this positive effect of the normal-state quantum metric on
superconductivity needs to be contrasted with a previous theoretical
discussion [10] suggesting a negative impact of the normal-state Berry
curvature on superconductivity. This could suggest a normal state curvature-
metric competition towards superconductivity.

## Acknowledgements

We wish to acknowledge Frédéric Piéchon for his insightful input on our work
and careful reading of our manuscript. We thank Andrea Caviglia and Roberta
Citro for valuable discussions.

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##  Appendix A Bilayer model

Here, we explain in more detail the choice shown in Fig. 1 to only include two
layers of Ti atoms in the TB model. As said in the main text, one should a
priori consider three layers, so a nine band TB model (without spin). In Fig.
10 we show a comparison of the band dispersions obtained from the bilayer TB
model (on the left) and a similarly obtained nine band trilayer TB model.

Figure 10: Comparison between the bilayer (left) and trilayer (right) TB
models.

First, we see that the bonding and anti-bonding triplets are not visibly
modified by the addition of the third layer. Furthermore, the three additional
bands are approximately at zero energy, and therefore several eVs away from
the band we are concerned about, in the bonding triplet. Second, also the
eigenstates of the bonding triplet should not be modified by the additional
bands because, again, of their energy difference. Finally, from Eq.(23), we
also see that following the same argument, the quantum geometric tensor [4]
should not be modified by the additional bands.

$Q^{n}_{\mu\nu}=\sum_{m\neq
n}\frac{\bra{u_{n}}\partial_{\mu}H\ket{u_{m}}\bra{u_{m}}\partial_{\nu}H\ket{u_{n}}}{(E_{n}-E_{m})^{2}}.$
(23)

In conclusion, we can reasonably discard the third layer and consider a
bilayer model of Ti atoms, as shown in Fig. 1.

##  Appendix B Expression of the kinetic terms

Hoppings are between two neighboring layers, with amplitude $t$ for
$\pi$-hoppings and $t_{d}$ for $\delta$-hoppings between blue and red sites
(Fig. 1). The origin of the basis lattice vectors is chosen at the center of
an hexagon. $e$, $f$ and $g$ have the following expressions [33, 16],

$\begin{cases}e=-\bigg{\\{}\exp(ik_{y})+\exp[i\Big{(}\frac{\sqrt{3}}{2}k_{x}-\frac{1}{2}k_{y}\Big{)}\bigg{]}\bigg{\\}}-r\exp[-i\Big{(}\frac{\sqrt{3}}{2}k_{x}+\frac{1}{2}k_{y}\Big{)}\bigg{]}\\\
f=-\bigg{\\{}\exp(ik_{y})+\exp[-i\Big{(}\frac{\sqrt{3}}{2}k_{x}+\frac{1}{2}k_{y}\Big{)}\bigg{]}\bigg{\\}}-r\exp[i\Big{(}\frac{\sqrt{3}}{2}k_{x}-\frac{1}{2}k_{y}\Big{)}\bigg{]}\\\
g=-\exp\Big(-\frac{i}{2}k_{y}\Big{missing})\times
2\cos\Big(\frac{\sqrt{3}}{2}k_{x}\Big{missing})-r\exp(ik_{y})\end{cases}\hskip
5.69054pt\text{with}\hskip 5.69054ptr=\frac{t_{d}}{t}.$ (24)

As example, we schematize the case of the $d_{xy}$ orbitals corresponding to
the hopping term $g$ in Fig. 11, and we derive $g$.

Figure 11: Diagram of the inter-layer and intra-orbital hoppings for the
$d_{xy}$ orbitals. On the left, the 3D cubic lattice and the $d_{xy}$ orbitals
of the Ti atoms. Hopping paths and corresponding amplitudes are shown as green
color lines. On the right, the associated projection in the $(111)$
orientation.

First, consider the hopping pictured on the right of Fig. 11, going from the
top red atom to the neighbouring left blue atom, with an amplitude of $t$. As
seen in Fig. 11, the associated overlap in the cubic lattice is a
$\pi$-overlap through the intermediate O site. The amplitude of this hopping
is $t=1.6$eV. The associated phase is
$\text{exp}(-i\mathbf{k}\cdot\mathbf{a}_{2})$ with
$\mathbf{a}_{1}=(-\sqrt{3}/2,3/2)$, by equivalence with the red atom in the
same motif. The similar hopping to the right also has amplitude $t$ and is
associated to the phase $\text{exp}(-i\mathbf{k}\cdot\mathbf{a}_{1})$ with
$\mathbf{a}_{1}=(\sqrt{3}/2,3/2)$. The other hopping from the top red atom to
the lower blue atom is, as seen on the left of Fig. 11, the result of the
direct overlap of the $d$ orbitals of the two atoms. It is therefore a
$\delta$-hopping, associated with an energy amplitude $t_{d}=70$meV. As for
its phase, it does not come with a phase shift, as the two atoms are part of
the same motif. This way, and shifting every term by $\text{exp}(ik_{y})$ so
that the origin is at the central green atom, the interlayer $d_{xy}-d_{xy}$
hopping term is given by

$\displaystyle g$
$\displaystyle=-te^{-i/2(-\sqrt{3}k_{x}+k_{y})}-te^{-i/2(\sqrt{3}k_{x}+k_{y})}-t_{d}e^{ik_{y}}$
$\displaystyle=-2te^{-ik_{y}/2}\cos\bigg(\frac{\sqrt{3}}{2}k_{x}\bigg{missing})-t_{d}e^{ik_{y}}.$
(25)

##  Appendix C Derivation of the orbital mixing term

In the orbital basis, $H_{\text{om}}$ can be written as $A\otimes B$ with $A$
and $B$ acting in the layer and orbital subspaces respectively. The orbital
mixing term consists of the interlayer and interorbital couplings so the
diagonal elements of $A$ and $B$ must vanish. The hexagonal lattice structure
seen in Fig. 1 has the $\mathcal{C}_{3v}$ symmetry. In order to respect the
latter, we assume that all the couplings have the same magnitude in energy and
that those between layer 1 to layer 2 are the same as those from layer 2 to
layer 1. Therefore we can write

$A\otimes B=c_{0}\begin{pmatrix}0&a\\\
a&0\end{pmatrix}\otimes\begin{pmatrix}0&b_{1}&b_{2}\\\ b_{3}&0&b_{4}\\\
b_{5}&b_{6}&0\end{pmatrix},$ (26)

where the coefficients are complex, and of modulus 1 in order to have the same
magnitude $c_{0}$ in energy. They are further constrained by the fact that the
term must be Hermitian. Using $(A\otimes B)^{\dagger}=A^{\dagger}\otimes
B^{\dagger}$, this means that we must have $a^{*}=a$, $b_{3}=b_{1}^{*}$,
$b_{5}=b_{2}^{*}$ and $b_{6}=b_{3}^{*}$. We choose $a=1$, such that

$A\otimes B=c_{0}\tau_{x}\otimes\begin{pmatrix}0&b_{1}&b_{2}\\\
b_{1}^{*}&0&b_{3}\\\ b_{2}^{*}&b_{3}^{*}&0\end{pmatrix}.$ (27)

We then introduce $(\phi_{i},\psi_{i})$ such that
$b_{i}(\mathbf{k})=\operatorname{cos}(\phi_{i}(\mathbf{k}))+i\operatorname{sin}(\psi_{i}(\mathbf{k}))$.
We look for $\phi_{i}$ and $\psi_{i}$ that are linear combinations of $k_{x}$
and $k_{y}$, which is natural for TB models. These hoppings are also
antisymmetric under an inversion operation $\mathbf{r}\longmapsto-\mathbf{r}$
[19], which adds the constraint $b_{i}(-\mathbf{k})=-b_{i}(\mathbf{k})$ since
in $\operatorname{exp}(i\mathbf{k}\cdot\mathbf{r})$ doing
$\mathbf{r}\longmapsto-\mathbf{r}$ is equivalent to
$\mathbf{k}\longmapsto-\mathbf{k}$. Hence,
$\operatorname{cos}(\phi_{i}(\mathbf{k}))=0$. Writing the allowed hoppings
between the red and blue sites explicitly (Fig. 1) with the above requirements
then gives
$b_{i}(\mathbf{k})=i\operatorname{sin}(\psi_{i}(\mathbf{k}))=i\operatorname{sin}(\alpha_{i}k_{x}+\beta_{i}k_{y})$,
with $\psi_{i}(\mathbf{k})=\alpha_{i}k_{x}+\beta_{i}k_{y}$. Next, the orbital
mixing term needs to obey the $\mathcal{C}_{3v}$ symmetry, i.e. a $2\pi/3$
rotation with an axis perpendicular to the $(111)$ plane and a mirror symmetry
parallel to the $(\overline{11}2)$ orientation. The $2\pi/3$ rotation
transforms $\mathbf{r}=(x,y,z)$ into $\mathbf{r}^{\prime}=(z,x,y)$ in the
original cubic unit cell. Therefore the orbitals are transformed as
$(d_{yz},d_{xz},d_{xy})\longmapsto(d_{xy},d_{yz},d_{xz})$. In order to obey
this $\mathcal{C}_{3}$ symmetry, we must therefore have
$b_{1}(\mathbf{k}^{\prime})=b_{3}(\mathbf{k})$,
$b_{2}(\mathbf{k}^{\prime})=b_{1}^{*}(\mathbf{k})$ and
$b_{3}(\mathbf{k}^{\prime})=b_{2}^{*}(\mathbf{k})$ with
$\mathbf{k}^{\prime}=(-1/2k_{x}-\sqrt{3}/2k_{y},\sqrt{3}/2k_{x}-1/2k_{y})$.
The mirror operation maps $\mathbf{r}=(x,y,z)$ to
$\mathbf{r}^{\prime}=(y,x,z)$ and $\mathbf{k}=(k_{x},k_{y})$ to
$\mathbf{k}^{\prime}=(-k_{x},k_{y})$, so that the orbitals transform as
$(d_{yz},d_{xz},d_{xy})\longmapsto(d_{xz},d_{yz},d_{xy})$. In order to obey
this symmetry, we must have
$b_{1}(\mathbf{k}^{\prime})=b_{1}^{*}(\mathbf{k})$,
$b_{2}(\mathbf{k^{\prime}})=b_{3}(\mathbf{k})$ and
$b_{3}(\mathbf{k}^{\prime})=b_{2}(\mathbf{k})$. These constraints on the
$b_{i}$s put constraints on the coefficients $(\alpha_{i},\beta_{i})$ by
taking the low-$k$ limit and identifying the $k_{x}$ and $k_{y}$ components
(this is allowed since the constraints must be valid for all $\mathbf{k}$).
The resulting system of equations puts five independent constraints such that
$\beta_{1}=0$, $(\alpha_{2},\beta_{2})=(1/2\alpha_{1},\sqrt{3}/2\alpha_{1})$
and $(\alpha_{3},\beta_{3})=(-1/2\alpha_{1},\sqrt{3}/2\alpha_{1})$. We then
recover Eq. (2) with $\alpha_{1}=-\sqrt{3}$.

##  Appendix D Gell-Mann matrices

$\displaystyle\Lambda_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},\hskip 2.84526pt\Lambda_{2}=\begin{pmatrix}0&-i&0\\\
i&0&0\\\ 0&0&0\end{pmatrix}$ $\displaystyle\Lambda_{3}=\begin{pmatrix}1&0&0\\\
0&-1&0\\\ 0&0&0\end{pmatrix},\hskip
2.84526pt\Lambda_{4}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\ 1&0&0\end{pmatrix}$
$\displaystyle\Lambda_{5}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},\hskip 2.84526pt\Lambda_{6}=\begin{pmatrix}0&0&0\\\
0&0&1\\\ 0&1&0\end{pmatrix}$ $\displaystyle\Lambda_{7}=\begin{pmatrix}0&0&0\\\
0&0&-i\\\ 0&i&0\end{pmatrix},\hskip
2.84526pt\Lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&-2\end{pmatrix}.$ (28)

##  Appendix E Trigonal basis

Let $U$ be the following unitary transformation in the
$\\{1,2\\}\otimes\big{\\{}\text{X},\text{Y},\text{Z}\big{\\}}$ basis

$U=\tau_{0}\otimes P,\quad
P=\begin{pmatrix}-\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\\\
\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{3}}\\\
0&\frac{2}{\sqrt{6}}&\frac{1}{\sqrt{3}}\end{pmatrix}.$ (29)

In this basis, the trigonal crystal field becomes diagonal,

$P^{\dagger}dH_{d}P=\begin{pmatrix}d&0&0\\\ 0&d&0\\\ 0&0&-2d\end{pmatrix}.$
(30)

The first two-fold degenerate eigenvalues represent the $e_{+g}$ and $e_{-g}$
orbitals while the third one represents the $a_{1g}$ orbital. They are
separated by an energy gap of $3d\simeq 10$ meV. The kinetic term is
transformed as

$H_{\text{cin}}=t\begin{pmatrix}\frac{e+f}{2}&\frac{e-f}{2\sqrt{3}}&-\frac{e-f}{\sqrt{6}}\\\
\frac{e-f}{2\sqrt{3}}&\frac{e+f+4g}{6}&-\frac{e+f-2g}{3\sqrt{2}}\\\
-\frac{e-f}{\sqrt{6}}&-\frac{e+f-2g}{3\sqrt{2}}&\frac{e+f+g}{3}\end{pmatrix},$
(31)

while the orbital mixing term becomes

$H_{\text{om}}=c_{0}\begin{pmatrix}0&iD&-iA\\\ -iD&0&iB\\\
iA&-iB&0\end{pmatrix},$ (32)

with

$\begin{cases}A=-\frac{1}{\sqrt{6}}\big{(}\alpha+\beta-2\delta\big{)}\\\
B=\frac{1}{\sqrt{2}}\big{(}\alpha-\beta\big{)}\\\
D=\frac{1}{\sqrt{3}}\big{(}\alpha+\beta+\delta\big{)}.\end{cases}$ (33)

##  Appendix F Validity of the quadratic three-band approximation

We now discuss the validity of the three-band approximation. Doing so amounts
to neglecting the off-diagonal blocks in Eq. (8) which contain the confinement
energy and the imaginary part of the kinetic term. From Ref. [34], the effect
of such off-diagonal terms is in
$\mathcal{O}\Big{(}\frac{||E||^{2}}{\text{gap}}\Big{)}$, where $E$ is the off-
diagonal perturbation. The numerically observed gap (with our choice of
parameters) is around $6-7$ eV. The biggest contribution of the two terms is
at zeroth and first order in $k$, i.e. in terms of scalar quantities, we have
$E\sim V\pm itk$ to linear order. Therefore the intrinsic error of the three-
band approximation is roughly given by

$\frac{V^{2}+t^{2}k^{2}}{6.5\text{eV}}\simeq(1.5+400k^{2})\text{meV}.$ (34)

This means that if we want a precision on the order of 1 meV, we find that the
approximation holds until $k\sim 0.1$, so about a tenth of the BZ. This is
indeed what we find when we compare the band structure with and without the
off-diagonal blocks. More precisely, the confinement energy globally shifts
every band by 1 to 2 meV while the imaginary part of the kinetic term breaks
the isotropy of the band structure obtained within the low-energy model and
gives rise to the $\mathcal{C}_{3}$ symmetric structure of ellipses seen in
experimental studies (see Ref. [16] for example). So the validity of the low-
energy model is restrained to the first tenth of the BZ around the $\Gamma$
point. Knowing this, what is the natural order of expansion we can do to the
low-energy model ? The relevant terms will be the ones above or around our
precision of a few meVs. For the orbital mixing term, the first two
corrections are linear and cubic in $k$. We then have $c_{0}k\sim 4$ meV and
$c_{0}k^{3}\sim 0.04$ meV for $k\sim 0.1$, so we only take the linear term. As
for the kinetic term, the first two corrections are of order $tk^{2}$ and
$tk^{4}$. This gives $tk^{2}\sim 10$ meV and $tk^{4}\sim 0.1$ meV, we
therefore only keep the quadratic term. In conclusion, we can thus expand our
three-band model to quadratic order while being coherent with the three-band
approximation.

##  Appendix G Quadratic expansion of $H_{3}$

Here, we derive the quadratic expansion of $H_{3}$ in Eq. (10). We remind the
reader that the matrices are written in the trigonal basis. For the orbital
mixing term, we have

$H_{\text{om}}=\begin{pmatrix}0&0&ick_{x}\\\ 0&0&ick_{y}\\\
-ick_{x}&-ick_{y}&0\end{pmatrix}+\mathcal{O}(k^{3}),$ (35)

with $c=(3/\sqrt{2})c_{0}$. For the kinetic term, we have

$\text{Re}(H_{\text{cin}})=-t(2+r)\bigg{(}1-\frac{1}{4}k^{2}\bigg{)}\Lambda_{0}+t\begin{pmatrix}-\frac{1}{8}(1-r)(k_{x}^{2}-k_{y}^{2})&-\frac{1}{4}(1-r)k_{x}k_{y}&\frac{1}{2\sqrt{2}}(1-r)k_{x}k_{y}\\\
-\frac{1}{4}(1-r)k_{x}k_{y}&\frac{1}{8}(1-r)(k_{x}^{2}-k_{y}^{2})&\frac{1}{4\sqrt{2}}(1-r)(k_{x}^{2}-k_{y}^{2})\\\
\frac{1}{2\sqrt{2}}(1-r)k_{x}k_{y}&\frac{1}{4\sqrt{2}}(1-r)(k_{x}^{2}-k_{y}^{2})&0\end{pmatrix}+\mathcal{O}(k^{4}),$
(36)

where we separated the traceful and traceless parts using

$\begin{pmatrix}a&0&0\\\ 0&b&0\\\
0&0&c\end{pmatrix}=\frac{a+b+c}{3}\Lambda_{0}+\frac{a-b}{2}\Lambda_{3}+\frac{a+b-2c}{2\sqrt{3}}\Lambda_{8},$
(37)

in terms of the Gell-Mann matrices. Now, if we define
$t_{\text{eff}}=(t-t_{d})/8$, we indeed find Eq. (10), neglecting the
quadratic terms whenever there already exists a linear term.

##  Appendix H Band structure of Eq. (4)

Figure 12: Lower triplet of the TB model in Eq. (4). (a) Lowest band in the
full BZ. (b) Second lowest band in the full BZ. (c) Third lowest band in the
full BZ.

An important issue for our discussion here, from the electronic point of view,
is the absence of other electron or hole pockets in the first BZ that would
provide other and possibly different metallic and superconducting properties
of the system. We therefore plot the dispersion of the lowest three bands in
the entire BZ (see Fig. 12), where we clearly see that the band minima are
situated at the $\Gamma$ point and that there are no other pockets in the BZ
for our range of chemical potential. Note that Fig.12 was obtained using the
spinless TB Hamiltonian, but the result also holds for the spinful TB
Hamiltonian, there is no electron pocket other than the one we consider at the
$\Gamma$ point.

##  Appendix I Derivation of the spinful two-band low-energy model

We begin things at Eq.(14),

$\tilde{H}_{6}=\sigma_{0}\otimes H_{3}+H_{\text{SOC}},$ (38)

where we also apply $U_{s}=(\sigma_{0}-i\sigma_{y})/\sqrt{2}$ on the spin
part, so that $H_{\text{SOC}}$ reads

$H_{\text{SOC}}=i\lambda\sigma_{x}\otimes\begin{pmatrix}0&-\frac{1}{\sqrt{3}}&\frac{2}{\sqrt{6}}\\\
\frac{1}{\sqrt{3}}&0&0\\\
-\frac{2}{\sqrt{6}}&0&0\end{pmatrix}+i\lambda\sigma_{y}\otimes\begin{pmatrix}0&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}\\\
-\frac{1}{\sqrt{3}}&0&\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&-\frac{1}{\sqrt{2}}&0\end{pmatrix}-i\lambda\sigma_{z}\otimes\begin{pmatrix}0&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{6}}\\\
-\frac{1}{\sqrt{3}}&0&-\frac{1}{\sqrt{2}}\\\
-\frac{1}{\sqrt{6}}&\frac{1}{\sqrt{2}}&0\end{pmatrix}.$ (39)

We then use that fact that close to the $\Gamma$ point, the exact eigenbasis
should not be far from the one exactly at the $\Gamma$ point. Consequently, we
exactly diagonalize $\tilde{H}_{6}(\Gamma)$, for the particular case
$\lambda=3d$. This choice is in line with the experimentally determined values
of the parameters and happens to greatly simplify the resulting expressions.
We thus determine the eigenstates of the two lowest energies, which we denote
as $|u_{1}\rangle$ and $|u_{2}\rangle$. We then define a two-band Hamiltonian
$H_{2}$ in the latter’s basis, ie

$H_{2}=\sum_{i,j=1}^{2}\langle
u_{i}|\tilde{H}_{6}|u_{j}\rangle|u_{i}\rangle\langle u_{j}|$ (40)

This way we get to Eqs.(15,16).

##  Appendix J Calculation of $D_{\text{conv}}$ for a general quadratic and
isotropic band dispersion.

Let us consider a general isotropic and quadratic band
$\epsilon=\epsilon_{0}+\alpha k^{2}$. Then, we can readily show that
$D^{\text{conv}}_{xx}=D^{\text{conv}}_{yy}=D_{\text{conv}}$ and
$D^{\text{conv}}_{xy}=0$. We also see that
$\mathscr{S}_{\text{occ}}(\mu)=B\big{(}0;\sqrt{\mu/\alpha}\big{)}$. We then
have

$\displaystyle D_{\text{conv}}$
$\displaystyle=\frac{1}{2}\int_{\mathscr{S}_{\text{occ}}(\mu)}\frac{\Delta^{2}}{\big{[}\Delta^{2}+(\alpha
k^{2}-\mu)^{2}\big{]}^{3/2}}4\alpha^{2}k^{2}\mathcal{D}_{2}\mathbf{k}$
$\displaystyle\propto\frac{2\alpha^{2}}{(2\pi)^{2}}\int_{0}^{2\pi}\int_{0}^{\sqrt{\mu/\alpha}}\frac{\Delta^{2}}{\big{[}\Delta^{2}+(\alpha
k^{2}-\mu)^{2}\big{]}^{3/2}}k^{3}dkd\theta$
$\displaystyle=\frac{2\alpha^{2}}{2\pi}\int_{0}^{\sqrt{\mu/\alpha}}\frac{\Delta^{2}}{\big{[}\Delta^{2}+(\alpha
k^{2}-\mu)^{2}\big{]}^{3/2}}k^{3}dk$
$\displaystyle=\frac{\alpha^{2}}{\pi}\frac{1}{2\alpha^{2}}\int_{0}^{\mu}\frac{\Delta^{2}}{\big{[}\Delta^{2}+(\epsilon-\mu)^{2}\big{]}^{3/2}}\epsilon
d\epsilon$
$\displaystyle=\frac{1}{2\pi}\Big{(}\sqrt{\Delta^{2}+\mu^{2}}-\Delta\Big{)}.$
(41)