# Interplay between singlet and triplet pairings in multi-band two-dimensional
oxide superconductors

L. Lepori<EMAIL_ADDRESS>Dipartimento di Fisica, Universitá della
Calabria, Arcavacata di Rende I-87036, Cosenza, Italy.  I.N.F.N., Gruppo
collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy.  D.
Giuliano  Dipartimento di Fisica, Universitá della Calabria, Arcavacata di
Rende I-87036, Cosenza, Italy.  I.N.F.N., Gruppo collegato di Cosenza,
Arcavacata di Rende I-87036, Cosenza, Italy.  A. Nava  Dipartimento di Fisica,
Universitá della Calabria, Arcavacata di Rende I-87036, Cosenza, Italy.
I.N.F.N., Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza,
Italy.  C. A. Perroni  Dipartmento di Fisica ”Ettore Pancini”, Università
degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo,
Via Cintia, I-80126 Napoli, Italy.  CNR-SPIN c/o Università degli Studi di
Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia,
I-80126 Napoli, Italy.

###### Abstract

We theoretically study the superconducting properties of multi-band two-
dimensional transition metal oxide superconductors by analyzing not only the
role played by conventional singlet pairings, but also by the triplet order
parameters, favored by the spin-orbit couplings present in these materials. In
particular, we focus on the two-dimensional electron gas at the (001)
interface between $LaAlO_{3}$ and $SrTiO_{3}$ band insulators where the low
electron densities and the sizeable spin-orbit couplings affect the
superconducting features. Our theoretical study is based on an extended
superconducting mean-field analysis of the typical multi-band tight-binding
Hamiltonian, as well as on a parallel analysis of the effective electronic
bands in the low-momentum limit, including static on-site and inter-site
intra-band attractive potentials under applied magnetic fields. The presence
of triplet pairings is able to strongly reduce the singlet order parameters
which, as a result, are no longer a monotonic function of the charge density.
The interplay between the singlet and the triplet pairings affects the
dispersion of quasi-particle excitations in the Brillouin zone and also
induces anisotropy in the superconducting behavior under the action of an in-
plane and of an out-of-plane magnetic fields. Finally, non-trivial topological
superconducting states become stable as a function of the charge density, as
well as of the magnitude and of the orientation of the magnetic field. In
addition to the chiral, time-reversal breaking, topological superconducting
phase, favored by the linear Rashba couplings and by the on-site attractive
potentials in the presence of an out-of-plane magnetic field, we find that a
time-reversal invariant topological helical superconducting phase is promoted
by not-linear spin-orbit couplings and by the inter-site attractive
interactions in the absence of magnetic field.

##  I Introduction

The transition metal oxides represent a large class of materials with
functional properties not only in the bulk but also in hetero-structures and
nano-structures. In particular, the two-dimensional electron gas (2DEG) at the
(001) interface between $LaAlO_{3}$ (LAO) and $SrTiO_{3}$ (STO) band
insulators has gained a continuously growing interest in recent years, as an
ideal playground to investigate the interplay between magnetism,
superconductivity, and spin-orbit coupling. Indeed, the 2DEG hosts a complex
phase diagram, depending on the electron density Caviglia2 ; biscaras2 , on
the temperature and  on the applied magnetic field.

It is well known that LAO/STO 2DEGs host an unconventional superconducting
regime, achievable by tuning the applied gate voltage. The origin of the
superconductivity is still not understood Gorkov2016 ; Edge2015 ; Ruhman2016 ;
gariglio2016 ; tafuri2017 , and various open questions remain unsolved about
the role of quantum  electronic correlations maniv2015strong ; Monteiro2019 ,
of multiband effects Trevisan2018 ; Wojcik2020 ; zeg2020 ; jouan2021 , and of
the spin-orbit coupling Khalsa2013 ; Diez2015 ; Zhong2013 ; Shalom2010 ;
Rout2017 . Moreover, the possibility of topological superconductivity is also
under debate Scheurer ; Mohanta ; Loder ; Fukaya ; settino2020 ;
santamaria2021 ; tafuri2017 ; perroni2019 .

Even more interestingly, the superconducting critical temperature, $T_{c}$, in
the LAO/STO 2DEG exhibits a dome-shape behavior, as a function of the applied
gate voltage Reyren ; Rout2017 ; joshua2012universal ; maniv2015strong ;
biscaras2 ; biscaras2010two ; Caviglia2 . When the carrier density increases,
$T_{c}$ first increases up to a maximum value, $T_{c}^{\mathrm{max}}$ $\simeq$
300 mK, at an optimal effettive doping, then it starts to decrease. The
resulting phase diagram is qualitatively very similar to that of high $T_{c}$
cuprates, of organic superconductors, of Fe-based superconductors, as well as
of heavy fermions taillefer ; keimer . Recently, the shape of the
superconducting dome has been qualitatively explained by assuming  a
particular real-space potential, effectively attractive in suitable windows of
momentum space, and resulting into an extended s-wave pairing zeg2020 .
Moreover, a forthcoming insightful work paramekanti2020  showed that the
formation of a similar dome (or even many of them, when multiband fermionic
models are considered) is related generically to an attractive potential with
finite range. The same work shed light on previous works, where the dome has
been suggested instead as an effect of the spin-orbit coupling Shalom2010 ;
Rout2017 ; Yin2019 ; singh2018gap . Furthermore, an asymmetric response in
shear-resistivity to an applied magnetic field (in-plane or out-of-plane) has
been observed caviglia2009 : a similar asymmetry can suggest a possible
spatial asymmetry in the pairing.

To  address the open questions above, in this paper we discuss the
superconductivity in LAO/STO 2DEG, by making a singlet-triplet mixed ansatz
for the pairing and by studying its physical consequences. This possibility
looks pretty natural, due to the inversion symmetry breaking term of the
heterostructure which gives rise to an effective Rashba-like coupling, already
known to favour mixed pairings rashba2001 ; tafuri2017 ; alidoust2021 .
Related notable effects are qualitative deviations of the standard BCS/BEC
crossover pieri2019 . The same possibility has been corroborated quite
recently, using a Monte-Carlo approach on a square lattice rosenberg2017 :
there, even a local (Hubbard) interaction proved sufficient for singlet-
triplet mixing, provided that a Rashba coupling is added. Interestingly, a
singlet-triplet mixed pairing allows (but does not imply) edge excitations,
protected by a nontrivial topology, whose presence has not been ruled out so
far by current transport experiments. Moreover, it determines an asymmetric
response to an applied magnetic field, qualitatively similar to that observed
in caviglia2009 . We finally notice that,  while a pure triplet p-wave pairing
is ruled out by previous experiments which did not detect the expected nodes
in the superconducting gap hwang2018 ; sumita2020 , instead a singlet-triplet
mixed pairing would not contradict the experimental results.

In  this paper, we employ a tight-binding model including the low-energy
electronic structure of  the LAO/STO 2DEG with the ${d_{xy},d_{xz},d_{yz}}$
orbitals of the Ti atoms popovic ; delugas ; scopigno ; salluzzo ; Khalsa2013
. Various papers have pointed out the close relation between  the onset of the
superconductivity and the filling of the degenerate $d_{xz/yz}$ sub-bands, at
an high density of states valentinis ; singh . Then, we adopt an attractive
static potential, with both local and nearest-neighbour terms, able to host
all the pairing configurations mentioned above. In addition, we include the
atomic spin-orbit and the inversion asymmetric potential associated with the
orbital Rashba interaction. Finally, we consider a magnetic field as a source
of time reversal symmetry breaking. To achieve our results, we perform a
detailed analysis of the most favorable topological superconducting phases.
This analysis is based on self-consistent computations of the order
parameters, minimizing the mean-field free energy in the Hamiltonian
parameters space, set by the electron filling, by the attraction strengths,
and by the amplitude and orientation of the magnetic field.

We point out how the interplay of singlet and triplet pairings is able to
affect the superconducting properties of LAO/STO 2DEGs. First, we show that
the singlet order parameters are strongly reduced with increasing the role of
triplet pairings, thus acquiring a non-monotonic dependence on the charge
density. Interestingly, some notable effects are found related with not-linear
corrections to the effective Rashba spin-orbit coupling. For instance, in the
absence of magnetic fields, the not-linear spin-orbit terms, combined with the
triplet pairings, favor a quite stable (time-reversal invariant) topological
helical superconducting phase. The triplet pairings are also responsible for
an anisotropic behavior of the superconducting order parameters, when the
magnetic field is applied in-plane and out-of-plane. Finally, in the presence
of out-of-plane magnetic fields, we recover (time-reversal breaking) chiral
topological superconducting phases, also when the triplet pairings are
vanishing.

The paper is organized as follows:

In Section II, we report the main electronic properties of the normal state.

In Section III, we discuss the general set-up of the mean-analysis of
superconductivity.

In Section IV, we analyze the superconducting solutions at zero magnetic
field.

In Section V,  we analyze the effects of a magnetic field.

Finally, we devote Section VI to our conclusions and outlook.

##  II Normal state

###  II.1 Model Hamiltonian

Following the derivation of Ref. [perroni2019, ], we write down the general
tight-binding Hamiltonian for the two-dimensional LAO/STO-001 system by
considering the 2DEG effectively confined on a square with lattice step
$a=3.9\,\AA\equiv 1$. The system has also a broken out-of-plane inversion
symmetry, having only the $t_{2g}$-orbitals close to the Fermi level. In
LAO/STO systems, the transition metal (TM)-oxygen bond angle is almost ideal,
and the three $t_{2g}$-bands are mainly decoupled in the momentum space ${\bf
k}$.

Figure 1: Plot of $\epsilon_{yz}{\color[rgb]{0,0,0}({\bf k})}$,
$\epsilon_{zx}{\color[rgb]{0,0,0}({\bf k})}$, and
$\epsilon_{xy}{\color[rgb]{0,0,0}({\bf k})}$ (left panel, in meV), and of the
mixed bands $\epsilon_{-,0,+}{\color[rgb]{0,0,0}({\bf k})}$ (middle and right
panels), in the absence of magnetic field, for $k_{y}=0$, and as functions of
$k_{x}$ (still around $k_{x}=0$, then for $\nu\to 0$). In the middle panel,
around $k_{x}\approx 0.35$, we have avoided crossings, due to the role of the
atomic spin-orbit coupling and the breaking inversion symmetry term.

Moreover, the $d_{xy}$ band has a truly two-dimensional character, while the
$d_{yz}$ and $d_{zx}$ bands are quasi one-dimensional. In the following, we
denote with $H_{0}({\bf k})$ the corresponding normal-state contribution to
the total system Hamiltonian in momentum space. Moreover, herewith we use the
index $\tau=\\{yz,zx,xy\\}$ to refer to the three different $t_{2g}$ orbitals
$d_{yz}$, $d_{zx}$, and $d_{xy}$, respectively, while we label the spin with
$\sigma=\\{\uparrow,\downarrow\\}$. In addition, we add a term $H_{SO}$ to the
total Hamiltonian, accounting for the atomic spin-orbit coupling of the TM
ions. Finally, we include the microscopic couplings arising from the out-of-
plane oxygen displacements around the TM, with the inversion asymmetry giving
rise to an effective hybridization of $d_{xy}$ and $d_{yz}$ or
$d_{zx}$-orbitals along the $y$ or $x$ directions, respectively. We denote
this contribution as $H_{Z}({\bf k})$.

In momentum space, we set

${\cal H}=\hat{D}^{\dagger}({\bf k})H({\bf k})\hat{D}({\bf k})\>\>,$ (1)

with $\hat{D}({\bf k})$ labelling the vector

$[c_{yz,\uparrow}({\bf k}),c_{yz,\downarrow}({\bf k}),c_{zx,\uparrow}({\bf
k}),c_{zx,\downarrow}({\bf k}),c_{xy,\uparrow}({\bf k}),c_{xy,\downarrow}({\bf
k})]_{t}\>,$ (2)

(note the different grouping of the operators corresponding to the various
orbitals with respect to perroni2019 ), and

$H({\bf k})=\sum_{\tau}H_{\tau}({\bf k})=H_{0}({\bf k})+H_{\rm SO}+H_{Z}({\bf
k})+H_{M}\>\>\>\>.$ (3)

The various terms at the right-hand side of Eq. (3) are grouped as follows.
The first term $H_{0}({\bf k})$ is the lattice band term

$H_{0}({\bf
k})=\left[\begin{array}[]{ccc}\epsilon_{yz}{\color[rgb]{0,0,0}({\bf k})}{\bf
I}_{2\mathrm{x}2}&{\bf 0}&{\bf 0}\\\ {\bf
0}&\epsilon_{zx}{\color[rgb]{0,0,0}({\bf k})}{\bf I}_{2\mathrm{x}2}&{\bf 0}\\\
{\bf 0}&{\bf 0}&\epsilon_{xy}{\color[rgb]{0,0,0}({\bf k})}{\bf
I}_{2\mathrm{x}2}\end{array}\right]\;,$ (4)

with

$\displaystyle\epsilon_{yz}{\color[rgb]{0,0,0}({\bf k})}$ $\displaystyle=$
$\displaystyle 2t_{1y}[1-\cos(k_{y})]+2t_{2x}[1-\cos(k_{x})]$
$\displaystyle\epsilon_{zx}{\color[rgb]{0,0,0}({\bf k})}$ $\displaystyle=$
$\displaystyle 2t_{1x}[1-\cos(k_{x})]+2t_{2y}[1-\cos(k_{y})]$
$\displaystyle\epsilon_{xy}{\color[rgb]{0,0,0}({\bf k})}$ $\displaystyle=$
$\displaystyle 4t_{1}-2t_{1x}\cos(k_{x})-2t_{1y}\cos(k_{y})+E_{t}\,,$ (5)

and the parameters set so that

$\displaystyle t_{1x}=t_{1y}\equiv t_{1}=300\>{\rm meV}$ $\displaystyle
t_{2x}=t_{2y}\equiv t_{2}=20\>{\rm meV}$ $\displaystyle E_{t}=-50\>{\rm
meV}\,.$ (6)

The second term in Eq.(3) is the atomic $l-s$ spin-orbit coupling term, given
by

$H_{\rm SO}=w_{{\rm SO}}\,\hat{{\bf l}}\otimes{\bf\sigma},$ (7)

with $\hat{{\bf l}}$ the vector operator describing the orbital angular
momentum and ${\bf\sigma}$ (the Pauli matrices) the spin angular momentum. In
order to make the spin-orbit term explicit, we introduce the matrices
$\hat{l}_{x}$, $\hat{l}_{y}$ and $\hat{l}_{z}$, which are the projections of
the $l=2$ angular momentum operator onto the $t_{2g}$ subspace,

$\displaystyle\hat{l}_{x}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&i\\\
0&-i&0\end{pmatrix},$ (8) $\displaystyle\hat{l}_{y}$
$\displaystyle=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\ i&0&0\end{pmatrix},$ (9)
$\displaystyle\hat{l}_{z}$ $\displaystyle=\begin{pmatrix}0&i&0\\\ -i&0&0\\\
0&0&0\end{pmatrix},$ (10)

assuming $\\{yz,zx,xy\\}$ as orbital basis. Using these operators, we write
the atomic spin-orbit coupling as

${\color[rgb]{0,0,0}H_{\rm SO}=i{\color[rgb]{0,0,0}w_{{\rm
SO}}}\left[\begin{array}[]{ccc}{\bf 0}&{\bf\sigma}_{z}&-{\bf\sigma}_{y}\\\
-{\bf\sigma}_{z}&{\bf 0}&{\bf\sigma}_{x}\\\
{\bf\sigma}_{y}&-{\bf\sigma}_{x}&{\bf 0}\end{array}\right]}\>\>\>\>,$ (11)

with ${\color[rgb]{0,0,0}w_{\rm SO}}=10\>{\rm meV}$.

The third term in Eq. (3) is the inversion symmetry breaking term $H_{Z}({\bf
k})$, given by

$H_{Z}({\bf k})=\gamma\left[\hat{l}_{y}\otimes{\bf
I}_{2\mathrm{x}2}\sin{k_{x}}-\hat{l}_{x}\otimes{\bf
I}_{2\mathrm{x}2}\sin{k_{y}}\right],$ (12)

or equivalently

$H_{Z}({\bf k})=i\gamma\left[\begin{array}[]{ccc}{\bf 0}&{\bf
0}&-\sin(k_{x}){\bf I}_{2\mathrm{x}2}\\\ {\bf 0}&{\bf 0}&-\sin(k_{y}){\bf
I}_{2\mathrm{x}2}\\\ \sin(k_{x}){\bf I}_{2\mathrm{x}2}&\sin(k_{y}){\bf
I}_{2\mathrm{x}2}&{\bf 0}\end{array}\right]\;\;\;\;,$ (13)

with $\gamma=20\>{\rm meV}$.  This important term stems from the breakdown of
a reflection symmetry along a particular axis of the LAO/STO compounds, due to
a corresponding lattice distortion. The net effect is the mixing of the
different orbitals $yz$, $zx$, and $xy$, having different parity under the
mentioned reflection symmetry. More details are in perroni2019 . Furthermore,
the effects of the coupling $\gamma$ on the stability of singlet and triplet
superconducting order parameters will be discussed in Appendix 5.

The last term in Eq. (3) describes the coupling of the electron spin and
orbital moments  with an external magnetic field B, whose direction is given
by the vector ${\bf M}=-\mu_{B}{\bf B}/\hbar$, with $\mu_{B}$ Bohr magneton:

$H_{M}=g_{s}\,{\bf I}_{3\mathrm{x}3}\otimes{\bf
M}\cdot\frac{{\bf\sigma}}{2}+{\bf M}\cdot{\bf\l}\otimes{\bf
I}_{2\mathrm{x}2}\,,$ (14)

which includes the gyromagnetic factor $g_{s}$ assumed equal to 2.  In the
absence of this term, the total Hamiltonian is time-reversal invariant:

$H({\bf k})=U_{T}^{-1}H^{*}(-{\bf k})U_{T}\,,\quad\quad\quad U_{T}={\bf
I}_{3\mathrm{x}3}\otimes\sigma_{y}$ (15)

($\sigma_{y}$ acting on the spin index $\sigma$, ${\bf I}_{3\mathrm{x}3}$ on
the $\tau$ index, and $U_{T}\,U_{T}^{*}=-{\bf I}_{3\mathrm{x}3}$), as it can
be straightforwardly checked by expressing $H({\bf k})$ in terms e. g. of
Gell-Mann matrices (see Appendix 1). Therefore, in the absence of
superconductive pairing, $H({\bf k})$ belongs to the class AII of the
classification for topological insulators and superconductors, see e.g.
ludwig2009 ; ryu2016 ; Bernevigbook .

In the following part of the Section, we will analyze the bands in the normal
state in the absence of the external magnetic field. The external field weakly
affects the electronic structure of the normal state, but its role will be
relevant in the analysis of the superconducting phases since the low energy
induced by the field competes with those due to the superconducting pairings.

###  II.2 Band structure

In the left panel of Fig. 1 we show the ”bare” $\tau$ bands, which we derived
by setting to zero both the spin-orbit coupling and the inversion symmetry
breaking term. We notice that the lowest band is  the $xy$ one, separated by
the energy $|E_{t}|$ from the $yz$ and $zx$ bands, which are degenerate at
$k_{x}=0$. In the same figure, we plot the bands, as a function of $k_{x}$, at
$k_{y}=0$. We identify the almost flat band with the $yz$ one and we also note
that the dispersion of the $zx$ band is negligible as a function of $k_{y}$
and at fixed $k_{x}$.

Introducing the spin-orbit coupling and the inversion symmetry breaking term,
the $\tau$-bands are mixed together by the rotation matrix $M(\bf{k})$ that
diagonalizes $H(\bf{k})$. Therefore, the orbital character is mixed, giving
rise to a more complex spectrum perroni2019  and to new, “mixed” bands, which
in the following we label with the indices $\eta_{\\{-,0,+\\}}$. In fact, we
see that the spin-orbit coupling and the inversion symmetry breaking mostly
affect the electronic bands at low densities. To evidence this fact, in the
central and right panels of Fig. 1, we report these bands, which at finite
values of the momentum exhibit avoided crossing. In particular, in the right
panel of Fig. 1, we focus on small values of the momenta. We see that the
$\eta_{\pm}$-bands display minima at finite momenta, which is traced back to
an emerging effective linear Rashba coupling (see Ref. [perroni2019, ] and the
next subsection). On the other hand, $\eta_{0}$ shows a single minimum at
${\bf k}=0$, indicating an enhanced importance of not-linear corrections.

In the following, we will carefully investigate the behavior of the
$\eta_{0}$-band at small values of ${\bf k}$. In fact, the minima of the
$\eta_{0}$-band mark the density values where superconductivity sets in.
Indeed, theoretical and experimental studies suggest the presence of
superconducting phases in the range for the lattice average filling $\nu$
approximately between the onset of the $\eta_{0}$ band and that of the
$\eta_{+}$ band (see e.g. gariglio2016 ).

Figure 2:  Upper panel: density of states of the $\tau$-bands, as a function
of the single-particle energy. Lower panel: carrier surface density, as a
function of the chemical potential.

Therefore, we focus on the electronic properties within this density range. In
the upper panel of Fig. 2, we plot the density of states (DOS) of the
$\eta$-bands, as a function of the single-particle energy (note that the
density has the dimensions of the inverse of an energy, since we set the
lattice step $a=1$). The minima of the bands are characterized by steps in the
density of states. In the same panel, the projection of the DOS on the orbital
basis $xy$, $yz$, and $zx$ is also reported, such to highlight their
contributions. We remark that the quasi-one-dimensional bands $yz$ and $zx$
provide the most relevant contribution to the density of states. Actually,
within this energy range, the density of states due to the $xy$ band is almost
constant, after a step at much lower energy (of the order of $E_{t}$). This
asymmetry in the density of states within this energy range will be
fundamental for the interpretation of  some superconducting properties, which
are very sensitive to the magnitude of the available electronic states.
Finally, within a slightly larger energy range, in the lower panel of Fig. 2,
we plot the carrier space density, as a function of the chemical potential.
For the parameters of the tight-binding model used in this paper, the density
at the minimum of  $yz$ and $zx$ bands is about $2\cdot 10^{13}$ $cm^{-2}$.
This value is in agreement with previous experimental results Stornaiuolo .
Moreover, we notice that, in analogy with the behavior of the density of
states, as soon as the quasi-one-dimensional $yz$ and $zx$ bands start to be
filled,  they rapidly become predominant. This occurs around zero energy,
where a crossover in the densities is visible. There, the carrier density, of
the order of $5\cdot 10^{13}$ $cm^{-2}$, results half from the $xy$ band and
half from the $yz$ and $zx$ ones.

###  II.3 Effective low-energy theories

In order to better spell out the following results, we now present an
effective theory for the $\eta_{-}$\- and $\eta_{0}$-bands, around their
common minima, at $k_{x}=k_{y}=0$. To derive the corresponding Hamiltonians we
exploit the second order degenerate perturbation theory. We present the
details of our derivation in Appendix 3. The results are respectively:

$H_{-}^{(\mathrm{eff})}({\bf k})=\epsilon_{-}({\bf k})\,{\bf
I}-(a_{1}\,k_{x}+a_{2}\,k_{x}^{3})\,\sigma_{y}+(a_{1}\,k_{y}+a_{2}\,k_{y}^{3})\,\sigma_{x}\,,$
(16)

with $a_{1}=8$ meV (in units of the lattice step $a\equiv 1$), $a_{2}=43.46$
meV, $\epsilon_{-}({\bf k})=\big{(}-54.+280.8\,(k_{x}^{2}+k_{y}^{2})\big{)}$
meV (so that $t_{-}^{(\mathrm{eff})}=280.8$ meV), and

$H_{0}^{(\mathrm{eff})}({\bf k})=\left[\begin{array}[]{c}\epsilon_{0}({\bf
k})\quad\quad i\,a_{3}\,k_{-}-i\,a_{4}\,k_{+}^{3}-a_{5}\,k_{x}k_{y}\,k_{+}\\\
{}\hfil\\\
-i\,a_{3}\,k_{+}+i\,a_{4}\,k_{-}^{3}-a_{5}\,k_{x}k_{y}\,k_{-}\quad\quad\epsilon_{0}({\bf
k})\\\ \end{array}\right]\,,$ (17)

with $k_{\pm}=k_{x}\pm ik_{y}$, $a_{3}=0.8$ meV, $a_{4}=8.627$ meV,
$a_{5}=22.8$ meV and $\epsilon_{0}({\bf
k})=\big{(}-10.8+157.2\,(k_{x}^{2}+k_{y}^{2})\big{)}$ meV (so that
$t_{0}^{(\mathrm{eff})}=157.2$ meV).

We notice that, around the minima of the $\eta$ bands, $k_{x}=k_{y}=0$, an
effective linear Rashba-like spin-orbit coupling appears in both the effective
Hamiltonians, together with a not-linear spin-orbit term. This second term is
especially relevant for the first dome on the $\eta_{0}$ band, already at
vanishing filling, as the central and right panels of Fig. 1 suggest. It is
known rashba2001  that the linear coupling, breaking space inversion symmetry
(as $H_{Z}({\bf k})$ it derives from), favours triplet components for
superconducting pairings, inducing single-triplet mixings, see e. g. annett ;
brydon2015 . Instead, not-linear spin-orbit terms have been postulated to
induce observable modifications on the spin polarization in inversion
symmetric (001) $SrTiO_{3}$ compounds nakamura2012  and for Josephson
junctions alidoust2021 . Other notable effects will be described in the
following.

In Fig. 11 of Appendix 3, we perform a direct comparison of
$H_{-}^{(\mathrm{eff})({\bf k})}$ and $H_{0}^{(\mathrm{eff})}({\bf k})$ with
the exact spectrum of $H({\bf k})$ in Eq. (1). We see that the agreement is
excellent up to $k_{x,y}\approx 0.2$. This value must be compared with the
typical momenta where the $\eta_{+}$ band starts to be populated, around
$k_{x,y}=0.4$. The corresponding filling is estimated to be approximately the
end of the superconductive dome gariglio2016 . Therefore, we expect
$H_{0}^{(\mathrm{eff})}({\bf k})$ to describe approximately a low-density half
of the first dome. Importantly, $H_{-}^{(\mathrm{eff})}({\bf k})$ and
$H_{0}^{(\mathrm{eff})}({\bf k})$, that are a central result in the present
work, still act on the $\sigma=\\{\uparrow,\downarrow\\}$ indices, that are
not mixed each others along the perturbation theory procedure.

##  III Mean-field analysis of superconductivity: general set-up

In this Section we analyze the possible presence of superconducting phases,
within the same energy and density ranges, and focusing in particular on the
interplay of singlet and triplet pairings.

###  III.1  Setting up the effective interaction Hamiltonian

In order to induce superconductivity in the 2DEG, in the following we
consider a pertinent, nearly realistic, attractive electronic interaction,
together with its effects on the bands of $H({\bf k})$. In particular, in the
absence of more specific coupling mechanism related to, e.g.,
antiferromagnetic correlations, we rely on the “natural” mechanism, based on
phonon exchange plus screened Coulomb repulsion. While the phonon attraction
is expected to be independent of the orbital index $\tau$, the Coulomb
repulsion is expected to be more important within the same orbital, due to the
larger overlaps between the electronic wavefunctions. Therefore, in the low
density regime analyzed in this paper,  we choose the realistic attractive
potential given by:

$W=-\sum_{i,j,\tau,\sigma,\sigma^{\prime}}U_{i,j}\,n_{i\tau\sigma}\,n_{j\tau\sigma^{\prime}}-\sum_{i,j,\tau,\tau^{\prime}\neq\tau,\sigma,\sigma^{\prime}}Z_{i,j}\,n_{i\tau\sigma}\,n_{j\tau^{\prime}\sigma^{\prime}}\,,$
(18)

with  the decay of $U_{i,j}$ and $Z_{i,j}$, with the separation $i-j$, ruled
by two different lengths $\xi_{U}$ and $\xi_{V}$. The same lengths depend on
the electronic density $\rho_{e}$, that is,  on the lattice average filling
$\nu$, and are expected to be of the order of a few lattice steps. Moreover,
we assume a static potential, consistently with the low charge density in the
regime of Fermi energies that we are considering. Actually, the dynamic part
of the potential from the phonon exchange can be extrapolated to the low-
energy limit, therefore the effective attractive potential at the Fermi energy
can be assumes as static mazin2011 ; cataudella2021 .

The bands $\tau$, $\tau^{\prime}$ are mixed together by the rotation matrix
$M(\bf{k})$ that diagonalizes $H(\bf{k})$;  $M(\bf{k})$ accordingly affects
the potential $W$ in Eq. (18), as well. It is difficult to  deal with the
transformed potential, since $M(\bf{k})$ depends on $\bf{k}$ and the Fourier
transform of $W$ contains 4 momenta $\bf{k}_{i}$, $i=1,\dots,4$.

Therefore, in this paper we analyze the effects, especially on the topology,
of the full potential $W$, focusing on its first term, diagonal on the orbital
index $\tau$. This choice is motivated by the energy gap between the
$\eta_{-}$ and $\eta_{0}$ bands around ${\bf k}=0$, where the superconductive
dome is located gariglio2016 , being approximately $\delta E_{\eta}=43$ meV.
This gap is an obstruction for the zero-momentum pairings between (unbalanced
species in) the bands, and forbids them from attractions such that the
superconductive gap, calculated at vanishing unbalance, is under a threshold
around $\delta E_{\eta}$ (see pethick ; marchetti2007 ; rad2007  and
references therein). Furthermore, nonzero-momentum balanced pairings are known
to require at least a subtle fine-tuning between interaction and density
pethick ; marchetti2007 ; rad2007 ; pieri2019 .

Within the tight-binding model, we focus on the on-site and the nearest
neighbor contribution of the attractive potential  between electrons in states
with the same orbital symmetry. The local interaction necessarily couples
electrons with opposite spins favoring spin-singlet symmetry. On the other
hand, we assume that the term due to the nearest neighbors takes into account
the equal spin contribution controlling the spin-triplet instability. Hence,
we simplify $W$ according to

$W=-U\sum_{i,\tau}n_{i\tau\uparrow}\,n_{i\tau\downarrow}-\frac{V}{2}\sum_{i\delta\tau\sigma}n_{i\tau\sigma}\,n_{i+\delta\tau\sigma}$
(19)

where $i$ and $j$ label two dimensional vectors associated to the lattice
sites, $U$ and $V$ are the local and nearest neighbor pairing energy,
respectively, and $n_{i\tau\sigma}=c_{i\tau\sigma}^{\dagger}c_{i\tau\sigma}$
is the local density operator for the $\sigma$ spin polarization and the
$\tau$ orbital, at a given position $i$, whose nearest neighbor sites are
indicated by $\delta$.  In Appendix 4, we discuss additional attractive terms
which have not been introduced in Eq. (19).

As specified in the subsection title, we have considered in Eq. (19) an
effective interaction, in particular an attractive local Hubbard term. We note
that in breit2010  a local repulsive Hubbard term, around 2 eV, has been
predicted from tunneling spectroscopy. In fact, this value of U is not large,
but it is comparable with the electron bandwidth. Furthermore, in LAO/STO
systems analyzed in this paper, the typical densities per spin polarization
are low compared to, e.g. , the half-filling regime. Accordingly, the net
contribution to the total energy from Hubbard interaction is limited.
Moreover, it is known that, in these density regimes, the effects from polaron
dynamics are relevant on the electronic states cancellieri2016 ;
cataudella2021  giving rise to a net lowering of the Hubbard interaction for
the 2D quantum gas. Finally, our effective model for the local interaction has
been widely used in the literature, for example Mohanta ; Loder ; Fukaya ;
perroni2019 ; settino2020  quoted in the introduction.

In the following, we will check that  $W$ in Eq. (19) qualitatively reproduces
the most interesting features of superconductivity in LAO/STO compounds
gariglio2016 ,  with the values of $U$ and $V$, yielding the superconducting
behavior discussed in this paper always, being of the order of hundreds of
meV.

###  III.2 Mean field analysis

We now analyze, within mean-field approximation, the Hamiltonian in Eq. (1),
with the interaction in Eq. (19), by mostly focusing onto the zero temperature
case.

Due to the introduction of the nearest neighbor attractive term $V$, it
becomes important to properly infer the profile of the superconducting order
parameter in real space. In the following, we do so by encompassing within our
mean-field approach both the spin and the orbital degrees of freedom. In
particular, we recover the appropriate pairing ansatz by analyzing the set of
the irreducible representations of the point-group symmetry of the square
lattice, by assuming over-all translational invariance (we provide the details
in Appendix 2). Moreover, in the absence of an externally-applied magnetic
coupling, we retain time-reversal invariance. As a result, we obtain:

$\displaystyle W$ $\displaystyle\approx$
$\displaystyle-U\sum_{i,\tau}D_{i\tau}\left[c_{i\tau\uparrow}^{\dagger}c_{i\tau,\downarrow}^{\dagger}+{\color[rgb]{0,0,0}\mathrm{H.c.}}\right]-$
(20)
$\displaystyle-\frac{V}{2}\sum_{i,\tau,\delta,\sigma}\left[F_{i\tau\sigma}(\delta)c_{i\tau\sigma}^{\dagger}c_{i+\delta\tau\sigma}^{\dagger}+{\color[rgb]{0,0,0}\mathrm{H.c.}}\right]+$
$\displaystyle+\,U\sum_{i\tau}D^{2}_{i\tau}+\frac{V}{2}\sum_{i,\tau,\delta,\sigma}|F_{i\tau,\sigma}(\delta)|^{2}.$

In Eq. (20), $D_{i\tau}=\langle c_{i\tau\downarrow}c_{i\tau\uparrow}\rangle$
is the singlet pairing amplitudes, depending on the orbital index $\tau$, with
$\langle\rangle$ denoting the ground state average, and the  over-all gauge
choice is made so that the singlet order parameters
$\Delta_{is}^{\tau}=UD_{i\tau}$ are real. The local s-wave pairing
corresponds to the most favored superconducting instability Michaeli ;
Nakamura ; Loder1 ; Mohanta1 .  Finally, $F_{i\delta\sigma}(\delta)=\langle
c_{i+\delta\tau\sigma}c_{i\tau\sigma}\rangle$ are the equal spin triplet
pairing complex amplitudes, depending on both $\tau$ and $\sigma$. Due to the
space inversion symmetry of the square lattice,
$F_{i\tau\sigma}(\delta)=F_{i\tau\sigma}(-\delta)$, and the triplet amplitudes
along the $y$-axis have only a phase different from those along the $x$-axis:
$F_{i\tau\sigma}(\delta_{y})=\theta^{\sigma}F_{i\tau\sigma}(\delta_{x})$, with
$F_{i\tau\sigma}(\delta_{x})=F_{i\tau\sigma}$ fixed real.  In Appendix 4, we
discuss additional pairings which could be considered in the system.

We point out that our ansatz for the superconducting mean field is diagonal in
bare bands $\epsilon_{i}$, therefore inter-orbital pairing between the rotated
bands $\eta_{i}$ is present since we get the corresponding gap parameters via
a self-consistent procedure, minimizing the (free) energy of the system, where
the mixing between the $\epsilon_{i}$ bands is included. The statement is
reinforced by the fact that, in the absence of inversion symmetry breaking
term $\gamma$, the structure of the gaps is changed. Indeed, as discussed in
Appendix 5, the coupling $\gamma$ is relevant to control the interplay between
singlet and triplet order parameters.

At the mean-field level, the triplet and the singlet-triplet mixed pairing are
separately determined by the nearest neighbour part of the potential in Eq.
(19), and by the spin-orbit (Rashba-like and not-linear) terms in Eqs. (16)
and (17) rashba2001 , and their onset is clearly enforced by the combined
effect of the two terms. However, in the absence of the $V$ term in Eq. (19)
(therefore with only an Hubbard attraction), mean-field decoupling does not
yield triplet components  (indeed, recover these terms requires resorting to
alternative approaches, such as Monte-Carlo simulations, see e.g.
rosenberg2017 ). Therefore, we expect that the mean-field approach
overestimates the singlet components, while the  emergence of triplet
components appears even more substantiated.

In momentum space, we describe the set of pairing configurations in Eq. (20)
within the general matrix parametrization annett

$\Delta^{\tau}({\bf k})=\left(\begin{array}[]{c}c_{\uparrow}({\bf k})\\\
c_{\downarrow}({\bf
k})\end{array}\right)^{\dagger}\,\tilde{\Delta}^{\tau}({\bf
k})\,\left(\begin{array}[]{c}c_{\uparrow}(-{\bf k})\\\ c_{\downarrow}(-{\bf
k})\end{array}\right)^{*}\,,$ (21)

with

$\tilde{\Delta}^{\tau}({\bf k})=i\,\Big{(}\Delta_{s}^{\tau}({\bf k})\,{\bf
I}_{2\mathrm{x}2}+{\bf d}^{\tau}({\bf
k})\cdot{\bf\sigma}\Big{)}\,\sigma_{y}\,.$ (22)

In particular, Eq. (22) can be rewritten in components as (see  Appendix 1 for
more details):

$\displaystyle\Big{(}\Delta_{s}^{\tau}\,,\,\Delta_{t,\uparrow\downarrow}^{\tau}\Big{[}\alpha\,(s_{x}+i\,s_{y})+\beta\,(s_{x}-i\,s_{y})\Big{]}\,,$
$\displaystyle\Delta_{t,\uparrow}^{\tau}(s_{x}+i\,s_{y})\,,\,\Delta_{t,\downarrow}^{\tau}\,(s_{x}-i\,s_{y})\Big{)}\,,$

with $\Delta_{t,\sigma}^{\tau}=-iVF_{\tau\sigma}$ and $s_{\\{x,y\\}}\equiv\sin
k_{\\{x,y\\}}$. Since no interaction term $\propto V$ between electrons with
opposite spins is present in Eq. (19), then
$\Delta_{t,\uparrow\downarrow}^{\tau}=0$ in Eq. (III.2).

In principle, all the parameters in Eq. (III.2) may have nonzero phases.
However, time-reversal invariance, with the same $U_{T}$ for the positive-
energy sector as in Eq. (15),  sets the phases to 0 (see brydon2015  and
Appendix 1). The ansatz in Eq. (III.2) perfectly reproduces the
superconducting solutions known in the literature as $p_{x}+ip_{y}$ and
$p_{x}-ip_{y}$, which are known to support topological helical
superconductivity at zero magnetic field qi2009 ; sato2017 . Moreover, from
Eq. (22), the triplet pairing vector ${\bf d}(\bf{k})$ (giving the
superconducting excitation gap, in the absence of the singlet pairing) is
given by following components:

$d_{x}({\bf k})=2VF_{r\sigma}\sin(k_{y}a),\ \ d_{y}({\bf
k})=-2VF_{r\sigma}\sin(k_{x}a),$ (24)

with $d_{z}({\bf k})=0$. Indeed, in frigeri2004  it has been shown that the
superconducting transition temperature is maximized when the spin-triplet
pairing vector $d({\bf k})$ is aligned with the polarization vector ${\bf
g}({\bf k})$ (essentially two-dimensional) parametrizing the spin-orbit
coupling.  Additional details about the properties of the pairing vector
$d({\bf k})$ will be provided in Appendix 4.

We comment finally that the possibility of helical phases samokhin , with
Cooper pairs with nonzero momentum, has not been considered in the present
work. Similar phases are expected to be suppressed in the regime of small
filling for the $\eta_{0}$ band, where, around ${\bf k}=0$, effective Rashba
terms are not linear but at least cubic in the momentum. We point out that the
most interesting results discussed in the next sections focus just on this
parameter regime.

##  IV Superconducting solutions at zero magnetic field

In this Section, we analyze the interplay between the triplet and the singlet
order parameters, in the absence of an external magnetic field.

###  IV.1 Emergence and stability of the superconducting solution

To assess the presence and  the stability of the superconducting states, we
solve the self-consistent equations by using a variational method.
Specifically, we study variationally the zero-temperature mean-field grand-
canonical free-energy $\Omega$ determined by the pairing in Eq. (III.2) plus
the single-particle Hamiltonian in Eq. (1). In particular, we numerically
minimize $\Omega$, varying $D_{i\tau}$, $F_{i\tau\sigma}$. In the minimization
procedure, we vary the superconductive amplitudes, the interaction strength
$U$ in Eq. (19) in the interval $\Delta U=[0,500]$ meV, and the interaction
strength $V$ in the interval $\Delta V=[0,1000]$ meV. We consider a square
lattice with side-length $L$, with $L=[40,240]$ sites, finding a satisfying
convergence starting from $L=120$. Furthermore, we set $\mu=E_{F}$, $E_{F}$
being the Fermi energy at a fixed $\nu$ and, typically, between the minima of
the $\eta_{0}$ and $\eta_{+}$ bands.

Figure 3: Upper panel: Phase diagram in the plane $U-V$ to distinguish the
normal state (absence of superconductivity), singlet, triplet and
singlet+triplet (S+P) superconducting phases at the chemical potential
$\mu=-9$ meV ( close to the onset of intermediate electronic bands). Lower
panel: Comparison between the grand-canonical free energies of the normal
state, of the singlet-pairing and the triplet-singlet superconductive ground-
states, at $U=350$ meV and $V=600$ meV.

Figure 4:  Left panel: Zero temperature singlet pairing amplitudes, for
$U=350$ meV and $V=0$ meV. Middle panel: The same as in the left panel, but
for $U=350$ meV and $\frac{V}{2}=300$ meV. Right panel: Zero temperature
triplet pairing amplitudes, again for $U=350$ meV, $\frac{V}{2}=300$ meV.
Notice that, in the absence of magnetic field,
$\Delta^{\tau}_{t}=\Delta^{\tau}_{t,\uparrow}=\Delta^{\tau}_{t,\downarrow}$.

By means of our variational procedure, beyond the normal state and a singlet-
pairing regime, we find a range of values for $U$ and $V$ where the singlet
and triplet pairings coexist.  Instead, as pointed out in the previous
subsection, no triplet pairing between different spins is found (we will
comment later on on this result). When the coexistence takes place, the ratio
between $U$ and $V$ can be tuned such that the triplet component of the order
parameter is not negligible  or even dominant (further details and insight on
the solutions described below will be given in Section IV.2, via the analysis
of the effective Hamiltonian in Eq. (17)).

First, we focus on the regime of values for $U$ and $V$ where only the singlet
pairing is stable.  As shown in Fig. 3, this occurs for $U=[300,400]\>{\rm
meV}$ and for $V<600$ meV. Superconductivity becomes stable once the chemical
potential $\mu$ is set above the minimum of the rotated band $\eta_{0}$,
around $-54$ meV. Conversely, no pairing is observed for lower values of
$\mu$. As reported in Fig. 3, for $U<300$ meV, superconductivity is not found
(that is true at any $\mu$).

In the left panel  of Fig. 4, corresponding to  $U=350$ meV and $V=0$, we plot
the singlet order parameters, showing that they are different from zero,
starting from the minimum of intermediate pair of bands ($\mu$ goes from about
$-10$ meV to about $20$ meV).  The obtained starting point for
superconductivity is in agreement with the appearance of a finite density of
states, shown in the previous Section, and with the experimentally measured
behavior of the superconducting pairing gariglio2016 ; singh . We remark that
we always find
${\color[rgb]{0,0,0}\Delta_{s}^{yz}}={\color[rgb]{0,0,0}\Delta_{s}^{zx}}$ in
the self-consistent superconducting solutions. Moreover, as shown in the left
panel of Fig. 4, not only ${\color[rgb]{0,0,0}\Delta_{s}^{yz}}$, but also
${\color[rgb]{0,0,0}\Delta_{s}^{xy}}$ is different from zero, starting from
the the minimum of the $\eta_{0}$ band. Due to the difference of mass and
density of states in the normal state, ${\color[rgb]{0,0,0}\Delta_{s}^{xy}}$
is smaller than ${\color[rgb]{0,0,0}\Delta_{s}^{yz}}$.  We also point out
that, increasing the carrier density, the order parameters get enhanced
continuously: apparently, there is no dome, as a function of the density. This
finding matches the general result in paramekanti2020 , relating the presence
of superconducting domes with finite-range potentials.

Then, we consider the effect of a  nonzero attractive term $\propto V$. As
reported in Fig. 3, for $V\gg U$, the triplet is the dominant pairing, while
the singlet pairing tends to vanish. However, Fig. 3 shows an intermediate
regime, from values of $V$ slightly smaller than $2U$, where the order
parameters coexist. In the middle panel of Fig. 4, we plot the singlet order
parameters for the different orbitals in the same regime, as a function of the
chemical potential. At finite $V$, the appearance of a triplet pairing
parallels a reduction of the s-wave order parameters. In particular, the
smallest one, $\Delta_{s}^{xy}$ assumes values in agreement with experimental
estimates, around $0.1$ meV. Therefore, there is a destructive interference
between singlet and triplet amplitude pairings. Moreover, this interplay is
also able to induce a dome, as a function of the density,  beginning from the
place where the density of states of the normal phase shows a step. We notice
that a similar behavior takes place at higher densities, corresponding to the
minimum of the band $\eta_{+}$, where the density of states has another step.
Therefore, our theoretical calculation predicts that, beyond the over-doped
regime of the first dome, corresponding to the band $\eta_{0}$, there is a
possibility of a second dome with similar extent, related to the higher energy
band $\eta_{+}$.  We observe finally the presence of a regime where $U\neq 0$
and $V\neq 0$ but the normal state is favoured. This effect is due to a low
density of states in the analyzed range of chemical potential, much lower (at
least two orders of magnitude) than in a standard metal. To rule out the
possibility that this would be due to finite-size effects, in our calculation,
we analyzed $L\times L$ square lattices up to $L=160$, checking numerical
convergence of the superconductive gap parameters. Also a MF analysis on the
$\eta_{-}$ and $\eta_{0}$ bands, starting from the effective theory in Eq.
(17), led to the same conclusion.

Figure 5: Upper Panel: Relative critical temperature for singlet order
parameters, at $U=350$ meV, $\frac{V}{2}=600$ meV, and $\mu$ varying, starting
from the bottom of $\eta_{0}$ as calculate in this paper. Lower Panel:
Relative critical temperature taken from Ref. Stornaiuolo

Finally, in the right panel of Fig. 4, we plot the  $\uparrow$ triplet
amplitudes for the orbital $\tau$, again at $U=350$ meV and $\frac{V}{2}=300$
meV. In analogy with the singlet pairing, we always find
$\Delta_{t,\sigma}^{yz}=\Delta_{t,\sigma}^{zx}$, therefore the orbital $yz$
and $zx$ are strongly coupled also in the triplet channel.  Moreover, since
there is no applied magnetic field,
$\Delta_{t,\uparrow}^{\tau}=\Delta_{t,\downarrow}^{\tau}$. We notice that the
triplet pairings always increase as a function of the density, then the
reduction of the singlet pairing is compensated by an enhancement of the
triplet channel, for all the values of the chemical potentials.

Figure 6:  Upper panel: Bogoliubov-de Gennes spectra at $\mu=-9$ meV, for
$U=V=0$ meV. Central panel: The same as in the upper panel, but for $U=350$
meV, $V=0$ meV. Lower panel: The same as in th upper panel, but for $U=350$
meV and $\frac{V}{2}=300$ meV.

In the lower panel of Fig. (3), we perform a comparison between the grand-
canonical free energies of the normal state, of the singlet-pairing and the
triplet-singlet superconductive ground-states, at $U=350$ meV and $V=600$ meV.
We point out that the grand-canonical potential can be lowered in a
significant way if the triplet component is included in the energy balance.

In Fig. 5 (upper panel), we plot the relative critical temperature for the
singlet order parameters, obtained by employing the mean-field approach, and
by varying the particle density and by setting $U=350$ meV and
$\frac{V}{2}=300$ meV. Moreover, the range for the single-particle energy to
starts from the bottom of the $\eta_{0}$ band. We recover a dome-shaped plot,
with the maximum critical temperature $T_{c}^{\mathrm{max}}\simeq 1$ K. At
this critical temperature, the triplet order parameters are reduced, in
comparison to their value at zero temperature, however they are not
necessarily zero.

More in detail, we locate the beginning the first dome at energy $E=\mu=-10$
meV and electronic density $n=2.14\cdot 10^{13}$ $\mathrm{cm}^{-}2$, the
maximum of $T_{c}$ at E = - 7 meV and $n=2.6\cdot 10^{13}$ $\mathrm{cm}^{-2}$,
and finally the end of the first dome at E = -2.5 meV and $n=3.3\cdot 10^{13}$
$\mathrm{cm}^{-2}$. These values can be compared e.g. with experimental data
in Stornaiuolo , summed up in the lower panel of Fig. 5. There the same dome
is measured between $n=1.8\cdot 10^{13}$ $\mathrm{cm}^{-2}$ and $n=3.1\cdot
10^{13}$ $\mathrm{cm}^{-2}$, the top of the dome being around $n=2.4\cdot
10^{13}$ $\mathrm{cm}^{-}2$. Therefore, we find a reasonable agreement between
theory and experiment. We remark that experimental data joshua2012universal ;
Stornaiuolo  have identified a critical value $n_{c}$ of the density of the
order of $1.8-1.9$ $10^{13}$ $\mathrm{cm}^{-2}$, which characterizes the
underdoped superconducting regime. Actually, in comparison with experimental
results, as reported in Fig. 5, the theoretical data show only a slight shift
of the entire phase diagram. In our approach, this effect depends only on the
energy $E_{t}$ which measures the energy distance between the band $xy$ and
$yz$ in Hamiltonian 5. If a slightly smaller value than 50 meV had been
chosen, a perfect agreement with the experimental critical temperature would
be obtained.

Since the triplet order parameter depends on the momentum ${\bf k}$, it is
interesting to plot the Bogoliubov-de Gennes spectrum obtained by using the
superconducting order parameters obtained from the minimization of the free
energy $\Omega$. In Fig. 6, we focus onto $\mu=-9$ meV, that is, a chemical
potential slightly higher than the minimum of the band $\eta_{0}$. First, in
the upper panel of Fig. 6, we plot the spectrum corresponding to the normal
state. The zeros correspond to the the center zone $\eta_{0}$ and the finite
momentum zone $\eta_{-}$ whose origin is $xy$-like. At finite momentum, there
is a small splitting of the zeroes, due to the fact that this value of the
chemical potential is close to the avoided crossing of the electronic bands.
In the middle panel of Fig. 6, we plot the spectrum corresponding to the
singlet pairing. The gaps correspond to the center zone $\eta_{0}$ and the
finite momentum zone $\eta_{-}$ whose origin is $xy$-like. Therefore, the gaps
at finite momentum are smaller than those at center zone. However, when only
the singlet is present, along the $\Gamma X$ ($k_{y}=0$ or $k_{x}=0$) and
$\Gamma M$ ($k_{x}=k_{y}$) directions, the gaps are perfectly symmetric.
Finally,  in the left panel of Fig. 6, we consider the combined effect of
singlet and triplet pairings. We notice that the spectrum drastically changes
with respect to the case where only the singlet is present. In particular, the
gaps get reduced along $\Gamma M$ direction. This behavior can be ascribed to
the  $s_{x}$ and $s_{y}$ dependence of the order parameter given in Eq.
(III.2): along the $\Gamma M$ direction, at finite wave-vector, the gap is
mainly given by singlet channel.

To conclude, we mention the interesting possibility to consider possible
superconductive solutions from the continuum effective Hamiltonians in Eqs.
(16) and (17). We analyzed their presence, by adding an attractive potential
as in Eq. (19). Working with lattice sizes up to $L=256$, we found only normal
solutions in large ranges for $U$ and $V$ (as large as in Section III). We
ascribe this outcome to relevant finite-size effects at the considered low
fillings $\nu<0.1$, required for the validities of the effective theories in
Eqs. (16) and (17).

###  IV.2 Clues of emerging topology at zero magnetic field

The pairing in Eq. (III.2), preserving time-reversal invariance, leads to a
Bogoliubov Hamiltonian in the class DIII of the classification for topological
insulators and superconductors ludwig2009 ; ryu2016 ; Bernevigbook . Indeed
$U_{T}={\bf I}_{3\mathrm{x}3}\otimes{\bf I}_{2\mathrm{x}2}\otimes\sigma_{y}$,
$U_{T}\,U_{T}^{*}={\bf I}_{12\mathrm{x}12}$ and $U_{C}={\bf
I}_{3\mathrm{x}3}\otimes\sigma_{y}\otimes\sigma_{y}$, $U_{C}\,U_{C}^{*}=-{\bf
I}_{12\mathrm{x}12}$ (the second matrices in the Kronecker products acting on
the Nambu-Gorkov indices). The chiral symmetry
$U_{S}=U_{T}\,U_{C}=U_{C}\,U_{T}$, $U_{S}^{2}={\bf I}_{12\mathrm{x}12}$, is
also realized, however not affecting the topology class (see Appendix 1). In
this class, nontrivial topological configurations are possible.  Therefore, in
the following we are going to explore this possibility.

Figure 7:  Bogoliubov spectrum for varying $k_{x}$, $U=350$ meV, $V=600$ meV,
at $\mu=-9$ meV (first dome) in the left panel, at $U=310$ meV, $V=520$ meV
$\mu=14$ meV (second dome) in the right panel. The probability P of the first
edge state corresponding to the spectrum of the left panel as a function of
the position y (in units of the lattice parameter $a$) for different values of
$k_{x}$. For all the plots, a finite size $L_{y}=950$ along the $y$-axis is
adopted.

We have analyzed the topological features of the (minimal energy) mean-field
superconducting solutions, within the tight-binding scheme exposed in the
above. In particular, we have examined different values of chemical potential.
Correspondingly, we found signatures of helical superconductivity, both in the
first and second dome. As an example, in the left and middle panels of Fig. 7,
we show the Bogoliubov spectrum at $\mu=-9$ meV (that is, in the
correspondence of the second dome, that means for the $\eta_{0}$ bands), for a
lattice that is infinite along the $x$-axis, but is finite along $y$-axis,
thus breaking translational invariance in that direction. In particular, along
the $y$-axis, we have considered a finite size of $L_{y}=950$ sites, with
hard-wall boundary conditions. Then, we have calculated the excitation
spectrum at fixed $k_{x}$, finding that two  finite-energy edge states are
present within the bulk gap, which become degenerate at $k_{x}=0$. This can be
ascribed to the fact that, at the onset of the first dome ($\mu\simeq-10$
meV), as shown in Fig. 1, the energy spectrum in the normal state does not
present a linear Rashba coupling. Therefore, in the limit of small wave-
vectors, the dispersion of the edge states is not linear, as a function of
$k_{x}$. Rather, it is nearly flat. As shown in the middle panel of Fig. 7,
for small values of $k_{x}$, the wavefunctions corresponding to the in-gap
edge states are well localized close to an edge of the system. With increasing
$k_{x}$, their wave-function tends to spread over all the bulk.

We point out that the observed behavior is mostly related to the triplet
pairing. Indeed, at zero magnetic field, the singlet pairing term alone is not
able to provide nontrivial topological phases. However, at $\mu=-9$ meV we
have checked a topological phase transition, induced by the attractive term
$V$ in Eq. (19) when the triplet order parameters are stable (see Fig. 3). In
particular, the results shown in Fig. 7 are obtained when singlet and triplet
pairings coexist above a critical value of the attractive potential $V$.  In
Section IV.3, we discuss in detail the nature of those phase transitions.

We have also analyzed some features, possibly topological, at higher values of
the carrier density. In the left panel of Fig. 7, we report the excitation
spectrum, with finite size along $y$-axis and at $\mu=14$ meV. Even if the
excitation spectrum is more complex in comparison with that shown at lower
densities, at zero magnetic field, we still find the presence of in-gap edge
states, which gradually merge in the bulk continuum, with increasing $k_{x}$.

###  IV.3 Further insights from the effective theory

Starting from the full multiband Hamiltonian in Eq. (1), it is rather
cumbersome to characterize entirely the full topology content of our system,
since  it is tough to recover the required analytical form of the band wave-
functions in momentum space ludwig2009 ; ryu2016 ; Bernevigbook . However,
this task can be achieved, at least partially, starting from the effective
Hamiltonians in Eqs. (16) and especially (17). This approaches demonstrates to
be valuable also for a better characterization  of the entire phase diagram in
the considered range of fillings, as well as of the nature of the first
superconducting dome.

Since in most of the available literature gariglio2016 , the superconducting
dome is located approximately in correspondence of the $\eta_{0}$ band, we
focus mainly on $H_{0}^{(\mathrm{eff})}({\bf k})$ in Eqs. (17),  with
superconducting pairings added. We recall that $H_{0}^{(\mathrm{eff})}({\bf
k})$ is expected to described properly at least the lower-density half of the
dome. Moreover, the resulting Bogoliubov Hamiltonian shares the same (time-
reversal and charge-conjugation) symmetry content as the multiband one in Eq.
(20), therefore allowing (but not guaranteeing) the same topological phase
structure. Indeed, a similar discussion can be performed directly for the
$\tau$ bands, obtaining the same qualitative results: for a given pair
content, the effective spin-orbit ${\bf g}({\bf k})$ (absent in the $\tau$
bands basis) has only quantitative effects on the phase diagram.

We assume again a pairing ${\bf\Delta}(\bf{k})$, in the form of Eq. (22) ( at
this stage, we do not need to specify the precise form of the corresponding
attractive potential, which instead we generically denote as $G$). The
resulting Bogoliubov Hamiltonian is of the general form
$H_{0}^{\mathrm{BG}}=\frac{1}{2}\sum_{\bf{k}}\psi^{{\dagger}}_{\bf{k}}\mathcal{H}(\bf{k})\psi^{\
}_{\bf{k}}$, with

$\mathcal{H}(\bf{k})=\left(\begin{array}[]{cc}h_{0}(\bf{k})&{\bf\Delta}(\bf{k})\\\
{\bf\Delta}^{{\dagger}}(\bf{k})&-h_{0}^{T}(-\bf{k})\end{array}\right)$ (25)

$\psi_{\bf{k}}=(c^{\ }_{{\bf k}\uparrow},c^{\
}_{\bf{k}\downarrow}c^{{\dagger}}_{-\bf{k}\uparrow},c^{{\dagger}}_{-\bf{k}\downarrow})^{\mathrm{T}}$
and

$h_{0}({{\bf k}})=\xi_{0}({\bf k})\,{\bf I}_{2\mathrm{x}2}+{\bf g}({\bf
k})\cdot\bm{\sigma}\,.$ (26)

with $\xi_{0}({\bf k})=\epsilon_{0}({\bf k})-\mu$ ($\mu$ measured from
$\epsilon_{0}({\bf k}=0)$). By time-reversal-symmetry, it turns out that
$\xi_{0}({\bf k})$ and ${\bf g}({\bf k})$ are symmetric and antisymmetric in
${\bf k}$, respectively sato2006 . The Hamiltonian in Eq. (25) has the
spectrum sato2006

$E_{\pm}({\bf k})=\sqrt{\Big{(}\xi_{0}({\bf k})\pm|{\bf g}({\bf
k})|\Big{)}^{2}+\Big{(}\Delta_{s}\pm|{\bf d}({\bf k})|\Big{)}^{2}}$ (27)

if ${\bf d}({\bf k})$ (see Eq. (22)) is aligned with ${\bf g}({\bf k})$, which
is the most likely possibility, as we noted above. In three dimensional
systems, this spectrum can give rise to topologically-protected nodal lines,
where $E_{-}({\bf k})=0$, leading to nodal (non-centrosymmetric)
superconductors. These systems require a partially different topological
classification from the ten-fold way for fully gapped ones brydon2015 ;
samokhin2015 , as well as for their edge modes samokhin2016 . However, in two
dimensions, only isolated zeros for $E_{\pm}({\bf k})=0$ can occur, not
topologically protected, and identified as transition points.

We analyze separately  three configurations for ${\bf d}({\bf k})$: i) pure
s-wave (spin singlet) pairing, ii) pure p-wave (triplet) state, iii) mixed s-p
waves pairing. In the first configuration, no topology is realized: the phase
is continuously connected with a purely BCS one, where ${\bf g}({\bf k})=0$.
Indeed, in spite of ${\bf g}({\bf k})\neq 0$, since $\Delta_{s}\neq 0$, no
zeros of $E_{\pm}({\bf k})$ can occur in between  and for a certain ${\bf k}$.
This regime clearly corresponds to that in Section IV for the multiband model
in Eq. (20), at $\mu=-9$ meV, $U<300$ meV and $V=0$.

In the normal state, due to the terms in Eqs. (11) and (13), mixing the spins,
only the sum of the spin currents is conserved, related to the global symmetry
group $U(1)_{V}$, in turn defined by the transformations: $c_{\uparrow}({\bf
k})\to e^{i\theta}\,c_{\uparrow}({\bf k})$, $\theta=[0,2\pi)$, and the same
for $c_{\downarrow}({\bf k})$. Therefore, due to the bilinear pairing
condensate $\langle c_{\uparrow}({\bf k})c_{\downarrow}(-{\bf
k})\rangle=\Delta_{s}$, the spontaneous symmetry breaking

$U(1)_{V}\to Z_{2\,V}$ (28)

occurs, where for $Z_{2\,V}\in U(1)_{V}$, $\theta=\\{0,\pi\\}$.

Similarly, in the configuration ii), when $\Delta_{s}=0$, two phases can be
realized a priori,  in this scheme discerned by the sign of $\mu$ readgreen ;
Bernevigbook . These two phases, one with non trivial and one with trivial
topologies, are continuously connected with those at ${\bf g}({\bf k})=0$.
This because the unique zeros of $E_{\pm}({\bf k})$ can occur where ${\bf
d}({\bf k})=0$, that means at ${\bf k}=0$. In this point, also ${\bf g}({\bf
k})=0$ and $\epsilon_{0}({\bf k})=0$, therefore $E_{\pm}({\bf k})=\mu$. Now,
as  $G\to 0$, then $\mu\approx k_{F}>0$ (since the filling of the $\eta_{0}$
band is positive),  while the condition $\mu=0$, separating the two phases,
occurs for $G$ large enough (similarly as in the previous Section). Again, the
breaking pattern in Eq. (28) is realized.

The data from the mean-field analysis on the lattice suggests that the
topological phase sets in for $U\geq 250$ and $V\geq 550$ meV, where we find
the triplet pairing (see Fig. 3). The same phase, a superconducting
counterpart of a quantum spin Hall phase, is the unique possible  topological
one in the DIII class  and in two dimensions ludwig2009 ; ryu2016 ;
Bernevigbook , and it is also known as helical superconducting phase.
Moreover, this is labelled by a topological index $n=1$, witnessing the number
of pairs of edge states, related by the time-reversal conjugation qi2009 ;
Bernevigbook . This index (written for the general case e.g. in sato2017 )
becomes, in the limit of decoupled spin sectors, ${\bf g}({\bf k})\to
0,\,\,\forall\,{\bf k}$, Bernevigbook :

$n=\frac{1}{2}\,(n_{\uparrow}-n_{\downarrow})\quad\mathrm{mod}\,2\,,$ (29)

where $n_{\uparrow}=1$ and $n_{\downarrow}=-1$ label the topological phases
(with broken time-reversal invariance, then in the D class of the ten-fold way
classification ludwig2009 ; ryu2016 ; Bernevigbook ) described in readgreen .
The numbers $n_{\uparrow/\downarrow}=\pm 1$ correspond to nontrivial element
of the first homotopy class nakahara ; coleman1 ; coleman2 ,
$\pi_{1}(\tilde{O)}$, on a circle $\tilde{O}$ around the high symmetry node at
${\bf k}=0$. In particular, the nontrivial homotopies are related to the phase
factors $e^{\pm i\,\phi_{\bf k}}$ parametrizing as follows the nonvanishing
p-wave gaps (connected by time reversal conjugation):

$\Delta_{t,\uparrow/\downarrow}\,(k_{x}\pm
i\,k_{y})=\Delta_{t,\uparrow/\downarrow}\,|{\bf k}|\,e^{\pm i\,\phi_{\bf
k}}\,.$ (30)

Therefore, for instance, $n=0$ for the extended s-wave case, considered in
zeg2020 . In the more general case where the spin sectors are coupled together
(as when ${\bf g}({\bf k})\neq 0$, or in the presence of a superconducting
pairing between the two spin-species, as in the case iii)), the topological
index  $n=1$ can be expressed in terms of Chern numbers of positive- and
negative-energy eigenstates sato2017 . Alternatively, the same index can be
defined as a winding number of a phase yokoyama2014 , exploiting directly the
full Bogoliubov Hamiltonian $H_{\mathrm{BG}}({\bf k})$, instead of its bands (
see notatop  for more details).

Finally, in the configuration iii), where singlet and triplet pairings
coexist, a trivial and a topological phase can be again realized a priori.
Referring to the Hamiltonian in Eq. (26), they are separated by the condition
(on $G$, $\mu$, $\Delta_{s}$, and ${\bf d}({\bf k})$) that $E_{-}({\bf k})=0$,
for a certain $|{\bf k}|$. This is expected for
$|\Delta_{s}|\sim|\Delta_{t,\uparrow}|=|\Delta_{t,\downarrow}|$. Again, the
breaking pattern in Eq. (28) is realized. Being again in the DIII class of the
ten-fold way classification, the two phases belong respectively to the same
topological classes of the phases in the case ii), then they are discerned by
the same topological index yokoyama2014 ; sato2017 .

This situation corresponds to the singlet-triplet coexisting regime, found in
Section IV. Correspondingly, the presence of edge states indicates that
between the two possible phases described above, the topological one is
realized. In the following Section, we will analyze its response to an applied
magnetic field.

Figure 8:  Upper panel: Singlet order parameter $\Delta^{xy}_{s}$, as a
function of the magnetic field along the $z$-axis for $U=350$ meV and for
different values of the attractive term $V$ and at $\mu=-9$ meV. Lower panel:
Same as in the upper panel, but with the magnetic field along the $x$-axis
(lower panel). We recall that in Eq. (19) the nearest neighbour pairing has
coupling $\frac{V}{2}$.

##  V Superconducting solutions in the presence of a magnetic field

It is important, on the experimental point of view as well, to consider how
the above scenario is modified under the application of a uniform magnetic
field, inducing the Zeeman term in Eq. (14). For this purpose, we start with
discussing how an applied magnetic field modifies the superconducting order
parameter.

###  V.1 Effects of the magnetic field on the superconducting order parameter

The first macroscopic effect is the breakdown of the time-reversal symmetry.
Along this direction, we analyze the effects of an external magnetic field on
the superconducting solutions, considering field orientations both along the
$z$-axis (out-plane) and the $x$-axis (in-plane).

First, we focus on the singlet order parameter related to the orbital $xy$,
$\Delta_{xy}^{s}$. We recall that this order parameter provides an estimate of
the minimal superconducting gap in the excitation spectrum. In the upper panel
of Fig. 8, we plot  this quantity, as a function of $M_{x}$, and for different
values of $V$. In particular, we focus on the under-doped regime of the first
dome, at $\mu=-9$ meV. For $V=0$, there is a continuous curve, as a function
of the magnetic field. However, there is a range of values for $V$ where the
order parameter shows a discontinuity. Actually, this discontinuity marks the
onset of the triplet pairing, which, on the other hand, start weakening the
singlet order parameter (see Fig. 3). As shown below,  the triplet pairing can
lower the ground-state energy, at increasing magnetic field and fixed $V$.
Therefore, there is a transition between a phase with only the singlet pairing
to a phase with combined singlet and triplet  pairings. With increasing $V$,
the triplet pairing becomes stable at lower strengths of the magnetic field
(Indeed,  we have shown above that, for $V>560$ meV, there is the formation of
the triplet pairing even at $M_{z}=0$). For these values of $V$, the reduction
of the singlet order parameter is quite rapid as a function of the strength of
$M_{z}$.

In the lower panel of Fig. 8, we plot $\Delta_{xy}^{s}$ as a function of
$M_{x}$, for different values of $V$. There is a different coupling between
the in-plane magnetic field and the triplet order parameter.  Therefore, the
role of the triplet pairing is affected not only by the density, but also by
the orientation of the magnetic field. Indeed, it can be responsible of the
experimentally measured magnetic field anisotropy, in some superconducting
properties, between the in-plane and out-of-plane field configurations
caviglia2009 . In particular, in the under-doped regime of the first dome, at
$\mu=-9$ meV, the anisotropy emerging from the comparison between the upper
and lower panels of Fig. 8 is present but not marked. Indeed, the behavior of
the order parameter, as a function of $M_{x}$, follows that as a function of
$M_{z}$, the first behavior being characterized only by slightly smaller
values of critical fields (at different $V$).

Figure 9: Singlet (left panel) and triplet (right panel) pairing amplitudes,
as a function of a constant magnetic field ${\bf M}$ along the $z$-axis (out
of plane) with $U=350$ meV and $\frac{V}{2}=300$ meV. Middle panel: same as in
the left panel, but with the magnetic field along $x$-axis (in-plane).

In order to deeply analyze the role of the anisotropy of the superconducting
solutions, we consider different fillings, comparing the behaviors of the
order parameters between the first dome, beginning at the minimum of the
$\eta_{0}$ band ($\mu\approx-10$ meV), and the second one, related to the
$\eta_{+}$ band ($\mu\approx 10$ meV). Therefore, in the left and the middle
panel of Fig. 9, we consider two values of $\mu$: $\mu=-4$ meV, corresponding
to the over-doped regime of the first dome, and $\mu=-11$ meV, corresponding
to the under-doped regime of the second dome. In the left panel we analyze the
behavior of $\Delta_{xy}^{s}$, as a function of $M_{z}$, and in the middle
panel we draw a similar plot, evidencing the dependence on $M_{x}$. Actually,
as shown in Fig. 9, at $\mu=-4$ meV, there is an anisotropy favoring the
stability along the $z$-axis. O n the other hand, as discussed above, the
triplet pairing becomes stronger with increasing density, and it
systematically weakens the singlet amplitudes. Moreover, the triplet order
parameters are more sensitive to the magnetic field along the $z$-axis (as
shown in the right panel of Fig. 9) than to the one along the $x$-axis.
Therefore, with increasing density, the anisotropy changes drastically. In
particular, at $\mu=11$ meV, the critical field along $z$ is less than 1 meV,
while that along $x$ becomes larger than unity. Therefore, the results are in
qualitative agreement with experimental data, caviglia2009  where the
anisotropy is more marked. Indeed, even if the role of the triplet pairing can
be relevant to inducing anisotropy in the superconducting order parameters,
it is not enough to explain all the relevant features found in  LAO/STO-001 as
a function of in-plane and out-of-plane magnetic fields.

Finally, in the right panel of Fig. 9, we have carefully analyzed also the
behavior of the triplet pairing, as a function of $M_{z}$,  finding a
remarkable linear dependence of the triplet order parameters, opposite for the
two spin components: one is enforced, the other one is weakened.

###  V.2 Clues of emergent topology in the presence of a magnetic field

Figure 10: Upper panel: Bogoliubov spectrum for varying $k_{x}$, $V=0$ meV,
$\mu=14$ meV and finite size $L_{y}=950$ sites along the $y$-axis. Lower
panel: probability of the edge state as a function of the position (in units
of the lattice parameter $a$) for different values of $k_{x}$ (in units of
$\pi/a$). The value of $M_{z}$, inducing a topological phase transition (see
the main text), is smaller than the critical value suppressing the self-
consistent superconducting solutions.

We now focus on the possibility of topology in the presence of a magnetic
field. In these conditions, the breakdown of the time-reversal symmetry makes
the topology class of the system to change from DIII to D. In two dimensions,
the latter class still allows nontrivial topology, still labelled by a $Z$
index ($Z_{2}$ for 1D systems; in this case the precise calculation of the
related invariant has been performed in perroni2019 ). In this way, a magnetic
field can preserve the edge states  that we discuss in Section IV.2.

Still analyzing the possible presence of edge states, we find  that
topological properties are favored by an out-of-plane magnetic field, but
disfavored by an in-plane one. Indeed, they are weakened with changing the
orientation of the magnetic field from the $z$-axis to $x$-axis, and there
here are characteristic angles (around $\frac{\pi}{4}$) where a topological
phase transition takes place. Eventually,  for a sufficiently intense in-plane
magnetic field, one only gets trivial topological phases, in agreement with
Loder .

In order to detect the chiral topological superconducting phase when $M_{x}$
is sub-relevant, we  consider again a lattice with a finite size along the
$y$-axis. In this case, the increase of $M_{z}$ is able to induce a phase
transition,  as soon as it takes a value of the order of the chemical
potential, as measured from the bottom of the sub-band:
$M_{z}>\sqrt{\tilde{\mu}^{2}+\Delta^{2}}$, where $\tilde{\mu}$ is the chemical
potential measure from the bottom of the $\eta_{+}$ sub-bands, and $\Delta$ is
the effective superconductive gap. Thus, the critical values to enter the
topological phase depend in general on $U$, $V$, and $\mu$. In the upper and
lower panels of Fig. 10, we show the results obtained at $V=0$,  with only the
singlet pairing present. We remark that, in contrast with the results of the
previous Section at zero magnetic field, now the topological phases can be
driven by the singlet order parameters only. We find that the chiral Majorana
edge states have a linear dispersion as a function of $k_{x}$, and are quite
localized at the edge for small values of $k_{x}$, similarly as in Loder . As
usual, with increasing $k_{x}$, these states merge into the bulk continuum.

Again the described picture can be understood in better qualitative details by
analyzing the effective Hamiltonians in Eqs. (16) and (17),  in analogy with
the discussion at ${\bf M}=0$ in Section III.1. Since the Zeeman term in Eq.
(14) has the form $H_{M}={\bf I}_{3\mathrm{x}3}\otimes{\bf
M}\cdot{\bf\sigma}$, it is sufficient to add this term to
$H_{0}^{(\mathrm{eff})}$, as described in the Appendix 3 (before Eq. (4)), in
spite of the fact that the $zx$ and $yz$ bands are mixed to yield the
$\eta_{0}$ band: this mixing is diagonal on the spin $\sigma$ index. In Eq.
(26), the same added term results in the shift ${\bf g}({\bf k})\to{\bf
g}({\bf k})+{\bf M}$, losing explicitly the antisymmetry required by time-
reversal invariance sato2006 .

A Zeeman term, $H^{(z)}_{M}=M_{z}\sigma_{z}$, along the $z$-axis, orthogonal
to the plane of our system, creates an effective unbalance in the chemical
potentials  with respect to $\sigma$. This imbalance is known to spoil the
s-wave pairing  $\Delta^{\tau}_{s}$ (as well as
$\Delta^{\tau}_{p,\uparrow\downarrow}$, if present), leading to the normal
phase, or at most  to a normal-superconductive mixed one (a secondary
possibility that we neglect, in the light of the stability of the p-wave
pairing, see below) pethick ; marchetti2007 ; rad2007 . This transition
occurs,  in the low-pairing limit, around the value
$M_{z}=\frac{\mu_{\uparrow}-\mu_{\downarrow}}{2}=\frac{\Delta^{\tau}_{s}}{\sqrt{2}}$,
the so called Chandrasekhar-Clogston limit (strictly valid for perturbative
Hubbard attractions) pethick . Concerning the triplet pairings
$\Delta_{t,\uparrow}$ and $\Delta_{t,\downarrow}$, they are spoiled
asymmetrically by the $M_{z}$ term, mainly via the shift of the chemical
potentials $\mu_{\uparrow,\downarrow}$. In particular, if $M_{z}>0$,
$\mu_{\downarrow}$ is decreased (at fixed  $\Delta_{t,\downarrow}$). At a
critical value of $M_{z}$, the condition $\mu_{\downarrow}=0$ is realized.
Then, the topological phase with $n_{\downarrow}=-1$ is driven to a
topologically trivial phase, still with p-wave pairing (the strongly-coupled
phase in readgreen ). At the point $\mu_{\downarrow}=0$, a zero in the
Bogoliubov gap is reached. The residual topology from the $\uparrow$
contribution is encoded in the $n_{\uparrow}=1\in Z$ phase, belonging to the D
class of the ten-fold way classification.

A qualitatively similar phase evolution, determined by the effective
unbalances induced by $M_{z}$, is valid if the phases do not host a nontrivial
topology, a possibility  allowed by symmetry considerations only, and possibly
occurring in certain coupling regions, as mentioned in Section IV.2.

##  VI Conclusions and outlook

In this paper we have discussed the onset and the physical consequences of a
singlet-triplet pairing in the two dimensional electron gas at LAO/STO-001
interfaces. This configuration looks rather natural a priori, due to the
inversion symmetry breaking term in the tight-binding Hamiltonian of the
system. We have made an extended superconducting mean-field analysis of this
multi-band tight-binding Hamiltonian, as well as of effective electronic bands
in the limit of low values of the momentum. We have included static on-site
(favoring spin-singlet pairings) and inter-site (promoting spin-triplet order
parameters) intra-band attractive potentials under applied magnetic fields. It
is interesting to notice that a singlet-triplet mixed pairing here results
robust for  extended regions of the analyzed space of parameters.

We have found various interesting features, as a reduction of the singlet
order parameter, as a function of the charge density, an asymmetric response
to in-plane and out-of-plane magnetic fields (a fact already observed
experimentally caviglia2009 ), and the possibility on nontrivial topology and
protected edge states. In particular, not-linear spin-orbit couplings and
inter-site attractive interactions make stable a time-reversal invariant
topological helical superconducting phase in the absence of a magnetic field.

In this paper, we have discussed the interplay between singlet and triplet
order pairings in the the clean limit. Effects of dilute nonmagnetic
impurities can affect the properties of non-centrosymmetric superconductors
samokhin . For example, the impurity effects on the critical temperature are
similar to those in multi-band centrosymmetric superconductors. Moreover,
Anderson’s theorem holds for singlet pairing samokhin ; anderson1959 . Indeed,
we expect that disorder effects can induce a reduction of the triplet order
parameters changing only quantitatively the interplay between singlet and
triplet pairings. This is the reason why in the paper the focus has been on
the properties related to the singlet order parameters.

A natural development of the present work is the study of the effective shape
of the dome, that means the finite-temperature dependence. We mention here
that we also performed mean-field calculations at finite temperature, finding
a first dome for the singlet order parameter, corresponding with the
$\eta_{0}$ band, and with a shape qualitatively very similar to that reported
in gariglio2016 . However, to achieve reliable quantitative details, the
inclusion of fluctuations beyond mean field is required. On the same regard,
we also mention a very recent paper jouan2021 , where two-different purely
$s$-wave phases have been predicted around the optimal doping, where the
critical temperature is maximized. The relation with the present work surely
deserves future attention. Other future generalizations of the present work
are aimed to include the effect of the second term on the multiband potential
in Eq. (18), that is inter-orbital attractive potentials. Closely related, a
relevant issue consists of the possibility of nonzero-momentum superconductive
couplings between different $\eta$ bands. This feature extremely unlikely in
translational invariant systems pethick ; marchetti2007 ; rad2007 , is known
to be more stable in lattice systems, see e. g. moore2012 ; shi2013 ;
mannarelli2018 .

Finally, we point out that effects due to unconventional superconductivity can
be confirmed not only by thermodynamic properties but also by response
properties such as the I-V characteristics of Josephson junctions tafuri2017 ;
alidoust2021 . In quasi one-dimensional systems, notable properties can be
deduced from anomalous Josephson effects tagliacozzo2015 ; tagliacozzo2018
and local spectroscopic measurements perroni_1d . These probes could be also
used to disentangle superconducting topological features from trivial ones in
the actual two dimensional case.

## Acknowledgements

The authors thank Michele Burrello, Simone Paganelli, Marco Salluzzo, and
Andrea Trombettoni for useful discussions. L. L., D. G. and C. A. P.
acknowledge financial support from Italy’s MIUR PRIN project TOP-SPIN (Grant
No. PRIN 20177SL7HC). C.A.P. acknowledges funding from the project QUAN-TOX
(QUANtum Technologies with 2D-OXides) of Quan-tERA ERA-NET Cofund in Quantum
Technologies (GrantAgreement No. 731473) implemented within the European
Union’s Horizon 2020 Programme. A. N. was financially supported by POR
Calabria FESR-FSE 2014/2020 - Linea B) Azione 10.5.12, grant no. A.5.1.

## APPENDIX 1: Hamiltonian symmetries

In this Appendix, we discuss the symmetries of the normal Hamiltonians in Eqs.
(3) and (17), as well as the corresponding Bogoliubov Hamiltonians with the
pairing in Eq. (III.2) added.

The normal Hamiltonians in Eqs. (3) (here considered at vanishing Zeeman
coupling $H_{M}$) can be expressed in terms of  Gell-Mann matrices
$\\{\lambda_{i}\\}$,  acting on the on the $\tau$ index, times the Pauli
matrices acting on the spin index $\sigma$:

$\Big{(}a({\bf k})\,{\bf I}_{3\mathrm{x}3}+b({\bf k})\,\lambda_{3}+c({\bf
k})\,\lambda_{8}\Big{)}\otimes{\bf
I}_{2\mathrm{x}2}+w_{\mathrm{SO}}\,\Big{(}-\lambda_{2}\otimes\sigma_{z}+\lambda_{5}\otimes\sigma_{y}-\lambda_{7}\otimes\sigma_{x}\Big{)}+\gamma\,\Big{(}\sin
k_{x}\,\lambda_{5}\otimes{\bf I}_{2\mathrm{x}2}+\sin
k_{y}\,\lambda_{5}\otimes{\bf I}_{2\mathrm{x}2}\Big{)}\,,$ (31)

the precise expressions for $a({\bf k})$, $b({\bf k})$, and $c({\bf k})$ being
unimportant here. Exploiting now the property for the Pauli matrices,
$\\{\sigma_{i},\sigma_{j}\\}=\delta_{ij}$, it is know immediate to prove (Eq.
(15)):

$H({\bf k})=U_{T}^{-1}H^{*}(-{\bf k})U_{T}\,,\quad\quad\quad U_{T}={\bf
I}_{3\mathrm{x}3}\otimes\sigma_{y}\,.$ (32)

In the Nambu-Gorkov basis $\Psi({\bf k})=\left(\begin{array}[]{c}{\bf c}({\bf
k})\\\ i\sigma_{y}\,{\bf c}^{\dagger}(-{\bf k})\end{array}\right)$, $c({\bf
k})=\left(\begin{array}[]{c}c_{\uparrow}({\bf k})\\\ c_{\downarrow}({\bf
k})\end{array}\right)$, $H({\bf k})$ is recast as follows:

$H({\bf k})\to H_{\mathrm{BG}}({\bf k})=\left(\begin{array}[]{cc}H({\bf
k})&{\bf 0}_{2\,\mathrm{x}\,2}\\\ {\bf 0}_{2\,\mathrm{x}\,2}&-H(-{\bf
k})\end{array}\right)$ (33)

so that $U_{T}$ becomes $U_{T}={\bf I}_{3\mathrm{x}3}\otimes{\bf
I}_{2\mathrm{x}2}\otimes\sigma_{y}$, the second identity in the Kronecker
product acting on the Nambu-Gorkov indices. The normal effective Hamiltonian
in Eq. (17) for the $\eta_{0}$ band, neglecting the out-of-diagonal
contribution with higher momentum power than one, reads (see the derivation in
the Appendix 3):

$H_{0}^{(\mathrm{eff})}({\bf
k})=\Big{(}-10.8+157.2\,(k_{x}^{2}+k_{y}^{2})\Big{)}\,{\bf
I}_{2\mathrm{x}2}-a_{3}\,k_{x}\,\sigma_{y}+a_{3}\,k_{y}\,\sigma_{x}\,.$ (34)

with $\epsilon_{0}({\bf k})=\big{(}-10.8+157.2\,(k_{x}^{2}+k_{y}^{2})\big{)}$
meV (so that $t_{0}^{(\mathrm{eff})}=157.2$ meV) and $a_{3}=0.8$ meV. It
holds:

$H_{0}^{(\mathrm{eff})}({\bf
k})=\tilde{U}_{T}^{-1}H_{0}^{(\mathrm{eff})*}(-{\bf
k})\tilde{U}_{T}\,,\quad\quad\quad\tilde{U}_{T}=\sigma_{y}\,,$ (35)

$\tilde{U}_{T}=\sigma_{y}$ acting on the spin index, as $\sigma_{y}$ in Eq.
(32). In the Bogoliubov form, obtained as above for $H({\bf k})$,
$\tilde{U}_{T}={\bf I}_{2\mathrm{x}2}\otimes\sigma_{y}$.

The next step is to include the pairing in Eqs. (22) and (III.2). We have to
consider the three spin channels:

$\langle c^{\tau}_{\downarrow}({\bf k})c^{\tau}_{\downarrow}(-{\bf
k})\rangle=e^{i\,\phi^{\tau}_{\downarrow\downarrow}}\,{\color[rgb]{0,0,0}\Delta_{t,\downarrow}^{\tau}}(k_{x}-i\,k_{y})\,,$
(36) $\langle c^{\tau}_{\uparrow}({\bf k})c^{\tau}_{\uparrow}(-{\bf
k})\rangle=e^{i\,\phi^{\tau}_{\uparrow\uparrow}}\,{\color[rgb]{0,0,0}\Delta_{t,\uparrow}^{\tau}}(k_{x}+i\,k_{y})\,,$
(37)

and

$\langle c^{\tau}_{\uparrow}({\bf k})c^{\tau}_{\downarrow}(-{\bf
k})\rangle=e^{i\,\phi^{\tau}_{\uparrow\downarrow}}\,\Big{(}{\color[rgb]{0,0,0}\Delta_{s,\uparrow}^{\tau}+\Delta_{t,\uparrow}^{\tau}}\Big{[}\alpha\,(k_{x}+i\,k_{y})+\beta\,(k_{x}-i\,k_{y})\Big{]}\Big{)}\,.$
(38)

Two global phase factors between $e^{i\,\phi^{\tau}_{\uparrow\downarrow}}$,
$e^{i\,\phi^{\tau}_{\uparrow\uparrow}}$, and
$e^{i\,\phi^{\tau}_{\downarrow\downarrow}}$ can be reabsorbed overall via a
phase redefinition of the fermionic annihilation operators. However, even in
this way the third phase cannot be reabsorbed. This phenomenon a condensed
matter counterpart to (minimal model for) the CP$\sim$T violation in particle
physics, via chiral fermion condensation weinberg2 . We choose to keep the
phase factor $e^{i\,\phi^{\tau}_{\uparrow\downarrow}}$.

We notice that time-reversal invariance, interchanging the spins, imposes
$\Delta_{t,\uparrow}^{\tau}=\Delta_{t,\downarrow}^{\tau}$, a result also found
by the minimization procedure of the mean-field free energy in the main text.
Dividing all the parameters in Eq. (38) in real an imaginary parts, it is
useful to parametrize them as follows:

$\langle c^{\tau}_{\uparrow}({\bf k})c^{\tau}_{\downarrow}(-{\bf
k})\rangle=\Big{(}\Delta_{1}^{\tau}+i\,\Delta_{2}^{\tau}\Big{)}+\Big{(}\Delta_{3}^{\tau}+i\,\Delta_{4}^{\tau}\Big{)}k_{x}+\Big{(}\Delta_{5}^{\tau}+i\,\Delta_{6}^{\tau}\Big{)}k_{y}\,.$
(39)

Therefore, for each $\tau$ band, we obtain for the $4\,\mathrm{x}\,4$ pairing
block Hamiltonian $H^{\tau}_{{\bf\Delta}}({\bf k})$ of the total one
$H_{{\bf\Delta}}({\bf k})$:

$H^{\tau}_{{\bf\Delta}}({\bf
k})=\Delta_{1}^{\tau}\,\big{(}\sigma_{x}\otimes{\bf
I}_{2\mathrm{x}2}\big{)}+\Delta_{2}^{\tau}\,\big{(}\sigma_{y}\otimes{\bf
I}_{2\mathrm{x}2}\big{)}+\Big{(}\Delta_{1}^{\tau}\,k_{x}+\Delta_{1}^{\tau}\,k_{y}\Big{)}\,\big{(}\sigma_{x}\otimes\sigma_{z}\big{)}-\Big{(}\Delta_{2}^{\tau}\,k_{x}+\Delta_{4}^{\tau}\,k_{y}\Big{)}\,\big{(}\sigma_{y}\otimes\sigma_{x}\big{)}\,.$
(40)

Imposing now time reversal invariance of the total multiband Hamiltonian
$H_{\mathrm{BG}}({\bf k})+H_{{\bf\Delta}}({\bf k})$ (see Eq. (33)),

$H_{\mathrm{BG}}({\bf k})+H_{{\bf\Delta}}({\bf
k})=V_{T}^{-1}\big{(}H^{*}_{\mathrm{BG}}(-{\bf k})+H^{*}_{{\bf\Delta}}(-{\bf
k})\big{)}V_{T}\,,$ (41)

we immediately obtain that it must hold

$V_{T}=U_{T}={\bf I}_{3\mathrm{x}3}\otimes{\bf
I}_{2\mathrm{x}2}\otimes\sigma_{y}\,,$ (42)

and importantly $\Delta_{2}=\Delta_{4}=\Delta_{6}=0$. That means that all the
pairing couplings in Eqs. (36)-(38) must be real, as claimed in the main text.
At this point is straightforward to obtain that $H_{\mathrm{BG}}({\bf
k})+H^{\tau}_{{\bf\Delta}}({\bf k})$ is also invariant under charge
conjugation:

$H_{\mathrm{BG}}({\bf k})+H_{{\bf\Delta}}({\bf
k})=-U_{C}^{-1}\big{(}H^{*}_{\mathrm{BG}}(-{\bf k})+H^{*}_{{\bf\Delta}}(-{\bf
k})\big{)}U_{C}\,,\quad\quad{\mathrm{with}}\quad\quad U_{C}={\bf
I}_{3\mathrm{x}3}\otimes\sigma_{y}\otimes\sigma_{y}\,.$ (43)

A further symmetry of the Bogoliubov Hamiltonian $\big{(}H_{\mathrm{BG}}({\bf
k})+H_{{\bf\Delta}}({\bf k})\big{)}$ is the chiral symmetry

$H_{\mathrm{BG}}({\bf k})+H_{{\bf\Delta}}({\bf
k})=U_{S}^{-1}\big{(}H^{*}_{\mathrm{BG}}(-{\bf k})+H^{*}_{{\bf\Delta}}(-{\bf
k})\big{)}U_{S}\,,\quad\quad U_{S}=U_{T}\,U_{C}=U_{C}\,U_{T}\,,\quad\quad
U_{S}^{2}={\bf I}_{12\mathrm{x}12}\,,$ (44)

composed by the product of time- and charge-conjugation symmetries. Chiral
symmetry does not change the topology class of the Bogoliubov Hamiltonian,
remaining DIII.

## APPENDIX 2: setting the pairing ansatz

On a general lattice, the mean field approach to superconductivity proceeds
identifying all the possible pairings corresponding to the irreducible
representations of the point-group symmetry of the lattice: the ansatzs for
the pairings are expressed in terms of them annett . This procedure is
analogous to the expansion in terms of spherical harmonics in the isotropic,
$SO(3)$ invariant, free space.

The spectra in Eq. (5), in the absence of the spin-orbit and inversion terms
in Eqs. (11) and (13), display a $D_{2}$ point group symmetry dress_book .
However, due to the presence of the latter two terms, in order to implement
the mean field approach, we have still to assume all the pairings allowed by
the symmetry of the isotropic square lattice. This lattice has (finite) point
group symmetry $D_{4}$, containing $D_{2}$ and composed by the group $C_{4}$
of the rotation of angles $\theta_{n}=n\,\frac{\pi}{2},\,\,n=0,\dots,3$, and
by the reflections around one (vertical or horizontal) axes (see e.g.
dress_book ).

In this case, the basis functions labelling the irreducible representations
which the eight-fold regular representation decomposes in, are
$k_{x}^{2}+k_{y}^{2}$, $k_{x}^{2}-k_{y}^{2}$, $k_{x}k_{y}$,
$\Big{(}\begin{array}[]{c}k_{x}\\\ k_{y}\end{array}\Big{)}$. The first three
(parity even) functions, together with the constant $c$, can be assumed as a
basis for the spin-singlet pairings, while the (parity odd) doublet can be
assumed as a basis for the spin-triplet pairing. Summing up, the superfluid
pairing can be generally assumed as:

$\Delta=\Delta_{s,0}+\Delta_{s,1}\,m(|{\bf k}|)+\Delta_{s,2}\,f(|{\bf
k}|)\,\frac{k_{x}^{2}-k_{y}^{2}}{|{\bf k}|^{2}}+\Delta_{s,3}\,g(|{\bf
k}|)\,\hat{k}_{x}\hat{k}_{y}+{\color[rgb]{0,0,0}\Delta_{t}}\,h(|{\bf
k}|)\,(\hat{k}_{x}+i\hat{k}_{y})\,,$ (45)

plus possible powers of these terms (required e. g. in the presence of a
strong cubic Rashba-coupling,  as in nakamura2012 ). Notice however that
pairings involving higher powers than 1 should be less relevant around
$k_{x}=k_{y}=0$. Both in the effective theories for $\epsilon_{-}$ and
$\epsilon_{0}$ bands,  it is expected that the spin-orbit interaction
(including the relevant not-linear corrections) plays a critical role in
determining the pairings that set in. The effect of a linear spin-orbit
coupling has been studied in rashba2001 , where a mixing (singlet-triplet and
s-p wave) has been identified. The same situation could be expected here.
However, the not-linear spin-orbit couplings could change the picture.

## APPENDIX 3: derivation of the effective theories in Eqs. (16) and (17)

In order to probe the presence of non-BCS pairings, suggested by recent works,
it is useful to restart considering the structure of the $\epsilon_{yz}$,
$\epsilon_{zx}$, and $\epsilon_{xy}$ bands (in the absence of Zeeman
couplings), as well as the real bands $\epsilon_{+,i}$, $\epsilon_{0,i}$, and
$\epsilon_{-,i}$ (in the following we will neglect the degeneracy index $i$,
for sake of brevity) resulting from the mixing of them, according to the
Hamiltonian in Eq. (3). The plots of them are given in Fig. 1.

It is clear that the mixing of the $\epsilon_{yz}$, $\epsilon_{zx}$, and
$\epsilon_{xy}$ is relevant only around their band-touching points, around the
momenta $k_{x}=k_{y}=0$ and $k_{x}=k_{0}\approx\pm 0.35,k_{y}=0$, as expected
from the low ratios $\frac{\gamma}{t_{1}}$ and $\frac{\Delta_{so}}{t_{1}}$ in
Eq. (3). Clearly, the importance of the same mixing relatively to the three
bands depends on which ones are touching each others: around $k_{x}=0$, the
bands $\epsilon_{yz}$, $\epsilon_{zx}$ undergo an important mixing, while
around the point $k_{x}=k_{0},k_{y}=0$ ($k_{x}=0,k_{y}\approx\pm 0.35$), the
bands $\epsilon_{zx}$, and $\epsilon_{xy}$ ($\epsilon_{yz}$, and
$\epsilon_{xy}$) do.

We focus first around the point $k_{x}=k_{y}=0$. In this region, due to the
relatively small ratios
$\lambda_{1}=\frac{\gamma}{{\color[rgb]{0,0,0}E_{t}}}=0.4$ and
$\lambda_{2}=\frac{\Delta_{so}}{{\color[rgb]{0,0,0}E_{t}}}=0.2$ effective
expressions can be obtained for $\epsilon_{+}$, $\epsilon_{0}$, and
$\epsilon_{-}$, exploiting second-order perturbation theory (in standard
notation):

$E_{n}(\lambda)=E_{n}^{(0)}+\lambda^{2}\sum_{k\neq n}\frac{|\langle
k^{(0)}|V|n^{(0)}\rangle|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}+O(\lambda^{3})\,,$
(46)

and, in the presence of two perturbations $\lambda_{1}\,V_{1}$ and
$\lambda_{2}\,V_{2}$:

$E_{n}(\lambda)=E_{n}^{(0)}+{\color[rgb]{0,0,0}E_{t}}^{2}\,\sum_{i}\lambda_{i}^{2}\sum_{k\neq
n}\frac{|\langle
k^{(0)}|V_{i}|n^{(0)}\rangle|^{2}}{E_{n}^{(0)}-E_{k}^{(0)}}+{\color[rgb]{0,0,0}E_{t}}^{2}\,\lambda_{1}\lambda_{2}\,\sum_{k\neq
n}\Bigg{(}\frac{\langle k^{(0)}|V_{1}|n^{(0)}\rangle\langle
n^{(0)}|V_{2}|k_{0}^{(0)}\rangle}{E_{n}^{(0)}-E_{k}^{(0)}}+\mathrm{H.c.}\Bigg{)}+O(\lambda^{3})\,.$
(47)

### Lower band

We start with the derivation of the effective theory for the $\eta_{-}$ band.
Exploiting Eq. (47), the couplings of $\epsilon_{xy}$ with $\epsilon_{yz}$ and
$\epsilon_{zx}$, included perturbatively, yields to

$H_{-}^{(\mathrm{eff})}=\Big{(}\epsilon_{xy}+{\color[rgb]{0,0,0}E_{t}}^{2}\,\frac{\lambda_{1}^{2}\sin^{2}k_{x}+\lambda_{2}^{2}}{\epsilon_{xy}-\epsilon_{yz}}+{\color[rgb]{0,0,0}E_{t}}^{2}\,\frac{\lambda_{1}^{2}\sin^{2}k_{y}+\lambda_{2}^{2}}{\epsilon_{xy}-\epsilon_{zx}}\Big{)}\,{\bf
I}+2\,{\color[rgb]{0,0,0}E_{t}}^{2}\,\lambda_{1}\,\lambda_{2}\,\Big{(}\frac{\sigma_{y}\sin
k_{x}}{\epsilon_{xy}-\epsilon_{yz}}-\frac{\sigma_{x}\sin
k_{y}}{\epsilon_{xy}-\epsilon_{zx}}\Big{)}+O(\lambda_{i}^{3})\,,$ (48)

and, expanding in powers of $k_{x}$ and $k_{y}$:

$H_{-}^{(\mathrm{eff})}({\bf k})=\epsilon_{-}({\bf k})\,{\bf
I}-(a_{1}\,k_{x}+a_{2}\,k_{x}^{3})\,\sigma_{y}+(a_{1}\,k_{y}+a_{2}\,k_{y}^{3})\,\sigma_{x}\,,$
(49)

with $a_{1}=8$ meV (in units of the lattice step $a\equiv 1$), $a_{2}=43.46$
meV, $\epsilon_{-}({\bf k})=\big{(}-54.+280.8\,(k_{x}^{2}+k_{y}^{2})\big{)}$
meV (so that $t_{-}^{(\mathrm{eff})}=280.8$ meV). The spectrum of the
effective Hamiltonian in Eq. (49), compared with the exact one, is shown in
Fig. 11. The agreement is excellent around $k_{x}=k_{y}=0$.

### Central band

The derivation of the effective theory for the $\eta_{0}$ band is more subtle.
Indeed, the $\epsilon_{yz}$, $\epsilon_{zx}$ are degenerate at
$k_{x}=k_{y}=0$, then before applying second-order perturbation theory, some
elaboration of the Hamiltonian (3) is required. In particular, we notice that
$H_{Z}({\bf k})$ in Eq. (13) do not mix $\epsilon_{zx}$ with $\epsilon_{yz}$.
Therefore we start diagonalizing exactly the partial Hamiltonian $H_{0}({\bf
k})+H_{SO}$ in the subspace $\epsilon_{zx}$ with $\epsilon_{yz}$, by a unitary
transformation $O=\mathrm{diag}(O_{2\mathrm{x}2},1)$). This transformation
does not mix the spins,  therefore $O$ can ne also written as
$O=\tilde{O}_{2\mathrm{x}2}\otimes{\bf I}_{2\mathrm{x}2}$. In this way, we
obtain

$H_{0}^{\prime}+H_{SO}^{\prime\,(\mathrm{rid})}=\left[\begin{array}[]{ccc}\epsilon_{a}\,{\bf
I}&{\bf 0}&{\bf 0}\\\ {\bf 0}&\epsilon_{b}\,{\bf I}&{\bf 0}\\\ {\bf 0}&{\bf
0}&\epsilon_{xy}\,{\bf I}\end{array}\right]\;\;\;\;,$ (50)

with $\epsilon_{a}<\epsilon_{b}$ around $k_{x}=k_{y}=0$.  At this level, the
spins get mixed each others. The bands $\epsilon_{a}$, $\epsilon_{b}$, and
$\epsilon_{xy}$ are coupled further by the terms from $H_{SO}$ and $H_{Z}({\bf
k})$, rotated by $O$:

$H_{0}^{\prime}+H_{SO}^{\prime}+H^{\prime}_{Z}=\left[\begin{array}[]{ccc}\epsilon_{a}\,{\bf
I}&{\bf 0}&V\\\ {\bf 0}&\epsilon_{b}\,{\bf I}&W\\\
V^{\dagger}&W^{\dagger}&\epsilon_{xy}\,{\bf I}\end{array}\right]\,,$ (51)

where

$V({\bf k})\equiv V=\left[\begin{array}[]{cc}a({\bf k})&b({\bf k})\\\
{}\hfil\\\ -b^{*}({\bf k})&a({\bf k})\\\ \end{array}\right]$ (52)

is the matrix mixing the $\epsilon_{a}$ and $\epsilon_{xy}$ bands, while $W$
is the matrix mixing $\epsilon_{b}$ and $\epsilon_{xy}$. The precise
expressions for its elements are rather involved and not immediately relevant
here. No further coupling between $\epsilon_{b}$ and $\epsilon_{a}$ occurs
from $H_{SO}$ and $H_{Z}({\bf k})$.

The effect of $V$ and $W$ is introduced perturbatively, as above, obtaining
for $\epsilon_{a}$:

$H_{0}^{(\mathrm{eff})}=\epsilon_{a}\,{\bf
I}+\frac{1}{\epsilon_{a}-\epsilon_{xy}}\,\left[\begin{array}[]{cc}|a({\bf
k})|^{2}+|b({\bf k})|^{2}&-a({\bf k})b({\bf k})+a^{*}({\bf k})b({\bf k})\\\
{}\hfil\\\ -a^{*}({\bf k})b^{*}({\bf k})+a({\bf k})b^{*}({\bf k})&|a({\bf
k})|^{2}+|b({\bf k})|^{2}\\\ \end{array}\right]\,\,$ (53)

Expanding all these expressions around $k_{x}=k_{y}=0$, we obtain
approximately (and up to momentum powers bigger than 3):

$H_{0}^{(\mathrm{eff})}({\bf k})=\left[\begin{array}[]{c}\epsilon_{0}({\bf
k})\quad\quad
a_{3}\,(ik_{x}+k_{y})-a_{4}\,(ik_{x}-k_{y})^{3}-a_{5}\,k_{x}k_{y}(k_{x}+ik_{y})\\\
{}\hfil\\\
a_{3}\,(-ik_{x}+k_{y})+a_{4}\,(ik_{x}+k_{y})^{3}-a_{5}\,k_{x}k_{y}(k_{x}-ik_{y})\quad\quad\epsilon_{0}({\bf
k})\\\ \end{array}\right]\,,$ (54)

with $a_{3}=0.8$ meV, $a_{4}=8.627$ meV, $a_{5}=22.8$ meV and
$\epsilon_{0}({\bf k})=\big{(}-10.8+157.2\,(k_{x}^{2}+k_{y}^{2})\big{)}$ meV
(so that $t_{0}^{(\mathrm{eff})}=157.2$ meV). We see that the dispersion is
very close to that before the perturbative mixing. The spectrum of the
effective Hamiltonian in Eq. (17), compared with the exact one, is shown in
Fig. 11. The agreement is excellent around $k_{x}=k_{y}=0$.

Adding a Zeeman term as in Eq. (14) is rather straightforward. Indeed, since
the Zeeman term in the same equation has the form $H_{M}={\bf
I}_{3\mathrm{x}3}\otimes{\bf M}\cdot{\bf\sigma}$, it is sufficient to add the
term $\bf{M}\cdot\bf{\sigma}$ to $H_{-}^{(\mathrm{eff})}$.

The same situation is realized for $H_{-}^{(\mathrm{eff})}$, since, as we
described before Eq. (50), the matrix $O$, mixing the $zx$ and $yz$ bands, is
diagonal in the spin index $\sigma$, $O=\tilde{O}_{2\mathrm{x}2}\otimes{\bf
I}_{2\mathrm{x}2}$. In Fig. 11, we compare the exact bands from Eq. (1) with
the bands of the effective Hamiltonians in Eqs. (16,17).

Figure 11: Comparison of the exact bands from Eq. (1) (black lines) with the
bands of the effective Hamiltonians in Eqs. (16) and (17) (dashed lines).

## APPENDIX 4: Additional pairings

In this Appendix , we discuss additional pairings which could be considered in
the system.

In Eq. (19), an inter-site attraction between particles with opposite spins
can be included, as well. Correspondingly, we considered also a triplet term
related to a pairing between opposite spins. However, the self-consistent
solution of the superconducting phase showed that this term is always zero in
the range of $U$ and $V$ that we have considered in this paper. Actually, in
the triplet pairing representation, this term would correspond to a vector
$d_{z}({\bf k})$, which comes out to vanish. This is quite typical of two-
dimensional superconductors with in-plane spin-orbit coupling frigeri2004 ;
sato2010 , such as the one we consider here, while an out-of-plane spin-orbit
coupling tends to favour a nonzero $d_{z}$. That $d_{z}({\bf k})=0$ can be
already inferred from Eq. (17). Indeed, in frigeri2004  it has been shown that
the superconducting transition temperature is maximized when the spin-triplet
pairing vector $d({\bf k})$ (see Eq. (III.2)) is aligned with the polarization
vector ${\bf g}({\bf k})$ parametrizing the spin-orbit coupling
($H_{\mathrm{SO}}({\bf k})={\bf g}({\bf k})\cdot{\bf\sigma}$). For
$H_{0}^{(\mathrm{eff})}({\bf k})$ in Eq. (17) (neglecting the out-of-diagonal
terms with power in the momenta higher than one, subleading around ${\bf
k}=0$), we have

$H_{0}^{(\mathrm{eff})}({\bf k})\approx\epsilon_{0}({\bf k})\,{\bf
I}_{2\mathrm{x}2}+g_{1}({\bf k})\,\sigma_{x}+g_{2}({\bf k})\,\sigma_{y}\,,$
(55)

with $g_{1}({\bf k})=0.8\,k_{x}$ and $g_{2}({\bf k})=-0.8\,k_{y}$, and
$g_{3}({\bf k})=0$. Therefore it is expected that $d_{z}({\bf k})=0$.

Finally, it is worth commenting on the fact that, strictly speaking, we make
the ansatz in Eq. (20) for the unrotated $\tau$-bands. However, the same
ansatz can be adopted at least for the low-density regime of the $\eta_{-}$
and especially $\eta_{0}$ bands, the latter doublet being the regime where
superconductivity is postulated gariglio2016 . Indeed, around ${\bf k}=0$, one
gets (see Appendix 3 for details),

$\eta_{0,\sigma}({\bf k})=\alpha_{(yz),\sigma}({\bf k})\,c_{(yz),\sigma}({\bf
k})+\alpha_{(zx),\sigma}({\bf k})\,c_{(zx),\sigma}\,({\bf k})\,,$ (56)

with $\big{(}\tau=(yz,zx)$):

$|\alpha_{\tau,\sigma}({\bf k}\to
0)|=\frac{1}{\sqrt{2}}+O(k_{x}^{2})+O(k_{y}^{2})\,.$ (57)

Therefore, since the same mapping is also diagonal in $\sigma$, it preserves,
up to phases, the structure of the s-p pairing (at most linear in the
momenta), at least around ${\bf k}=0$. This behaviour is even strengthened for
the $\eta_{-}$ band, that results from the mixing around ${\bf k}=0$ of the
$xy$ bands with the others: in fact, this mixing is suppressed by the energy
gap $E_{t}$ in Eq. (5).

It is also worth stressing that we do not add a spin-singlet intersite
contribution. Indeed, our analysis focuses on the regime of low filling, where
the occupied electronic states are close to ${\bf k}=0$. In the small momentum
limit, the spin-singlet intersite term would give rise to extended s-wave
order parameter (ruled by a sum of cosines of the momentum) which would result
in just a subleading additive contribution to the standard s-wave parameter
induced by on-site attractive terms. Therefore, we neglected it, adding
instead the leading nonzero contribution with equal spins (that is also more
relevant once one turns on a magnetic field).

## APPENDIX 5: Effects of inversion-symmetry breaking term

In Section IV of the main text, we have found a large regime for $U$ and $V$,
where triplet and singlet pairings coexist in the first dome. Then we inferred
that this mixing, allowed by $V$, stems from the parallel contributions of the
spin-orbit term in Eq. (11) and of the inversion breaking one in Eq. (13).
Indeed, at the effective level, they result together in the Rashba-like
coupling of Eq. (17), known to induce a mixing of pairings with different
parity rashba2001 . More in detail, in the same regime of energies, the
contribution of Eq. (11) is dominant on that of Eq. (13) (vanishing at ${\bf
k}=0$), a fact can be inferred also in the direct construction of Eq. (17).

It is important to investigate directly this effect of the spin-orbit term in
Eq. (11) on the mixing of the pairings. For this purpose, we repeat the mean
field procedure performed above, switching off the same term. In the resulting
Fig. 12, again at fixed $U=350$ meV, $\frac{V}{2}=290$ meV, and $\mu=-9$ meV,
it appears clear that Eq. (11) collaborates to enforce the triplet pairing,
correspondingly lowering the singlet component. However, this effect does not
look critical, changing the previously found behaviours only quantitatively.
Therefore, the major role to the singlet-triplet mixing seems, in the analyzed
regimes, to result from the $V$ term of the potential in Eq. (19), even for
chemical potentials close to the Lifshitz point where the first dome starts.
In turn, the $V$ term can be ascribed to the relatively low charge densities
at the location of the dome. For these reasons, similar results are obtained
at higher chemical potentials going from the first to the second dome. We
ascribe the present result to the ability, described in Section III, of the
mean-field approach to grasp the interplay between singlet and triplet
components.

Figure 12:  Singlet (Left panel) and triplet (right panel) pairing amplitudes
(in meV) for the $\tau=zx$ bands, as a function of the chemical potential (in
meV). The attractive couplings in Eq. (19) are set as $U=350$ meV and
$\frac{V}{2}=290$ meV.

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