# High magnetic field superconductivity in a two-band superconductor

Tancredi Salamone  Center for Quantum Spintronics, Department of Physics,
NTNU, Norwegian University of Science and Technology, NO-7491 Trondheim,
Norway  Henning G. Hugdal  Center for Quantum Spintronics, Department of
Physics, NTNU, Norwegian University of Science and Technology, NO-7491
Trondheim, Norway  Sol H. Jacobsen  Center for Quantum Spintronics, Department
of Physics, NTNU, Norwegian University of Science and Technology, NO-7491
Trondheim, Norway  Morten Amundsen  Nordita, KTH Royal Institute of Technology
and Stockholm University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

###### Abstract

When applying an external magnetic field to a superconductor, orbital and
Pauli paramagnetic pairbreaking effects govern the limit of the upper critical
magnetic field that can be supported before superconductivity breaks down.
Experimental studies have shown that many multiband superconductors exhibit
values of the upper critical magnetic field that violate the theoretically
predicted limit, giving rise to many studies treating the underlying
mechanisms that allow this. In this work we consider spin-splitting induced by
an external magnetic field in a superconductor with two relevant bands close
to the Fermi level, and show that the presence of interband superconducting
pairing produces high-field reentrant superconductivity violating the Pauli-
Chandrasekhar-Clogston limit for the value of the upper critical magnetic
field.

##  I Introduction

Over the past two decades, multiband superconductors have been attracting
great interest because of increasing experimental evidence of interesting
effects not achievable in single band systems. For instance, in the first-ever
discovered multiband superconductor, MgB2, Leggett modes have been observed
[1] and more recently optically-controlled [2]. Moreover, spontaneous time
reversal symmetry breaking has been reported in Ba$1\\!-\\!x$K$x$Fe2As2,
Sr2RuO4, UPt3 and many other multiband systems (for an exhaustive review see
[3]).

Since the extension of the Bardeen–Cooper–Schrieffer (BCS) theory to multiband
systems by Suhl, Matthias and Walker [4] and Moskalenko [5], many studies have
focused on a theoretical understanding of the effects of a multiband
description of superconductors [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
In general, if two bands are close to each other or hybridized, it is possible
to form interband Cooper pairs (see _e.g._ [18]), where the electrons
comprising the pairs come from two distinct bands. Research on interband
pairing has been rather limited, but studies have found that it affects
Josephson tunneling [7], it is an important factor in obtaining an anomalous
Hall effect [19], it can produce gapless states [13], and it influences the
BCS-BEC crossover [20]. We note that, in the context of this work, the term
interband will always refer to Cooper pairs formed by electrons in distinct
bands, and it must not be confused with the same term often found in the
literature, also called pair-hopping, referring to the hopping of intraband
pairs, _i.e._ formed by electrons in the same band, between different bands.

Among multiband superconductors, MgB2 and Fe-Based Superconductors (FeBS) are
particularly interesting because of their high critical temperatures and upper
critical magnetic fields. For instance, the critical temperature is
$39$\mathrm{K}$$ for MgB2 [21], $55$\mathrm{K}$$ for SmO$1\\!-\\!x$F$x$FeAs
[22] and $65$\mathrm{K}$$ in FeSe films on SrTiO3 substrate [23], whereas the
zero temperature estimated values of the upper critical field are
$25$\mathrm{T}$$ in single crystal MgB2 [24], $70$\mathrm{T}$$ in C-doped MgB2
thin films [25] and up to $300$\mathrm{T}$$ in FeBS [26, 27, 28, 29, 30, 31].
It is worth noting, however, that these values are often extrapolated from
low-magnetic field data obtained close to the critical temperature. Therefore,
the extrapolated low-temperature dependence of the upper critical field and
its $T=0$ value may be a bad estimate. However, the application of pulsed
fields allows to reach higher magnetic field values and obtain more reliable
estimations [30].

The simultaneous presence of superconductivity and magnetism is of great
interest for the field of superconducting spintronics [32] where the proximity
effect is exploited to achieve dissipationless information transport. However,
these two phenomena are often mutually exclusive since two effects contribute
to destroying superconductivity when an external magnetic field is applied:
the orbital and Pauli paramagnetic pairbreaking effects [33, 34]. The orbital
effect describes the breaking of Cooper pairs when the kinetic energy of
electrons, resulting from the momentum acquired in a magnetic field, exceeds
the superconducting gap. On the other hand, paramagnetic pairbreaking occurs
when Cooper pairing becomes energetically unfavourable as the Zeeman energy of
the electrons overcomes the superconducting gap. This happens when the
exchange energy reaches a value given by the Pauli, or equivalently the
Chandrasekhar-Clogston, limit $h_{c}=\Delta_{0}/\sqrt{2}=1.86T_{c}$, where
$\Delta_{0}$ is the value of the superconducting gap at zero temperature and
zero applied field, and $T_{c}$ is the superconducting critical temperature.

Figure 1: Illustration of some of the superconducting pairing processes
possible in a two-band system for (a) intraband and (b) interband Cooper
pairs. The red and blue circles represent the Fermi surfaces of the two bands,
the green and yellow lines identify the electrons forming the Cooper pairs and
the dotted lines indicates the scattering processes.

Various mechanisms producing the limit violation, or enhancement of upper
critical fields have been proposed, _e.g._ scattering by non-magnetic
impurities [35, 36], spin-triplet superconductivity [37], Fulde-Ferrell-
Larkin-Ovchinnikov (FFLO) pairing [38, 39], strong superconducting coupling,
spin-orbit coupling [40, 41], application of a voltage bias [42, 43], the
proximity of two bands to each other [44, 45, 46], and pair-hopping in three
band superconductors [47, 48]. Furthermore, many experimental works have
reported evidence for upper critical magnetic field violating the above limits
in multiple superconductors, _e.g._ NbSe2 [40, 49], iron pnictides [50, 51],
lanthanide infinite-layer nickelate [52], moiré graphene [53], organic
superconductors [54, 55], Eu-Sn molybdenum chalcogenide [56], URhGe [57].
Refs[53, 54, 55, 56, 57] are particularly interesting for the purpose of this
work because their results present two disconnected superconducting domains,
for small and large external magnetic field, due to non-spin-singlet Cooper
pairs in Ref.[53] and to the Jaccarino-Peter effect in Refs.[54, 55, 56].

In this work, we present a simple mechanism which allows to overcome the
conventional limit. We consider a two band s-wave superconductor in the
presence of an external magnetic field and demonstrate that the inclusion of
interband pairing allows to overcome the limiting value of the critical field
found in conventional superconductors. Within a BCS framework we show that for
a certain range of parameters, our system exhibits superconductivity for
significantly high values of the external magnetic field. Furthermore, we show
that the system can exhibit two separate superconducting domains, for small
and large external magnetic field, and we provide an explanation of the
mechanism producing these results.

##  II Theory

We consider the following mean-field Hamiltonian for a two bands spin-split
superconductor with both intra- and interband spin-singlet superconducting
coupling:

$\begin{split}\mathcal{H}=&\sum_{\bm{k}\sigma}(\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}\\!-\\!E_{c}\\!-\\!\sigma
h)c_{\bm{k}\sigma}^{\scaleto{(1)}{6pt}\dagger}\\!c_{\bm{k}\sigma}^{\scaleto{(1)}{6pt}}\\!+\\!\sum_{\bm{k}\sigma}(\xi_{\bm{k}}^{\scaleto{(2)}{6pt}}\\!+\\!E_{c}\\!-\\!\sigma
h)c_{\bm{k}\sigma}^{\scaleto{(2)}{6pt}\dagger}\\!c_{\bm{k}\sigma}^{\scaleto{(2)}{6pt}}\\\
&-\sum_{\bm{k}}\sum_{\alpha,\alpha^{\prime}=1,2}\left(\Delta_{\alpha\alpha^{\prime}}(\bm{k})c_{\bm{k}\uparrow}^{\scaleto{(\alpha)}{6pt}\dagger}c_{-\bm{k}\downarrow}^{\scaleto{(\alpha^{\prime})}{6pt}\dagger}+\mathrm{h.c.}\right)\\\
\end{split}$ (1)

where the operator $c_{\bm{k}\sigma}^{\scaleto{(\alpha)}{6pt}\dagger}$
($c_{\bm{k}\sigma}^{\scaleto{(\alpha)}{6pt}}$) creates (destroys) an electron
in band $\alpha$ with dispersion
$\xi_{\bm{k}}^{\scaleto{(\alpha)}{6pt}}=\varepsilon^{\alpha}_{\bm{k}}-\mu$ and
spin $\sigma$, $E_{c}$ is half the band separation and $h$ is the externally
applied in-plane magnetic field. The superconducting order parameters
$\Delta_{\alpha\alpha^{\prime}}(\bm{k})$ are defined by:

$\Delta_{\alpha\alpha^{\prime}}(\bm{k})=\frac{T}{V}\sum_{\omega_{n}}\sum_{\beta,\beta^{\prime}=1,2}\sum_{\bm{k}^{\prime}}g_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\bm{k},\bm{k}^{\prime})F^{\beta\beta^{\prime}}(\bm{k}^{\prime},\omega_{n}),$
(2)

where $\omega_{n}=(2n+1)\pi/\beta$ is the fermionic Matsubara frequency and
$F^{\beta\beta^{\prime}}$ is the anomalous component of the Green’s function.

The superconducting coupling matrix
$g_{\alpha\alpha^{\prime},\beta\beta^{\prime}}(\bm{k},\bm{k}^{\prime})$
defines the different coupling processes, the terms
$g_{\alpha\alpha,\alpha^{\prime}\alpha^{\prime}}$ describe processes involving
intraband Cooper pairs: formed by electrons in the same band, hopping between
the same band ($\alpha=\alpha^{\prime}$) or different bands
($\alpha\neq\alpha^{\prime}$). This last term is often referred to in the
literature as interband scattering, or pair hopping, and must not be confused
with the use we make of the term interband. Processes involving interband
pairs instead are described by those elements
$g_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ with
$\alpha\neq\alpha^{\prime}$ and/or $\beta\neq\beta^{\prime}$. The
superconducting pairing processes relevant for the purpose of this work are
illustrated in Fig. 1 for a two-band system.

The terms $\Delta_{11}$ and $\Delta_{22}$ represent the intraband order
parameters, while $\Delta_{12}=\Delta_{21}$ is the interband order parameter.

In the basis defined by
$\hat{\psi}_{\bm{k}}^{\dagger}=\left(c_{\bm{k}\uparrow}^{\scaleto{(1)}{6pt}\dagger},c_{-\bm{k}\downarrow}^{\scaleto{(1)}{6pt}},c_{\bm{k}\uparrow}^{\scaleto{(2)}{6pt}\dagger},c_{-\bm{k}\downarrow}^{\scaleto{(2)}{6pt}}\right)$
the inverse Green’s function for the Hamiltonian of LABEL:eq:hamiltonian is:

$\bm{\mathcal{G}}_{0}^{-1}=\begin{pmatrix}(i\omega_{n}+h)\hat{\tau}_{0}-\tilde{\xi}_{1}\hat{\tau}_{3}+\Delta_{11}\hat{\tau}_{1}&\Delta_{12}\hat{\tau}_{1}\\\
\Delta_{12}\hat{\tau}_{1}&(i\omega_{n}+h)\hat{\tau}_{0}-\tilde{\xi}_{2}\hat{\tau}_{3}+\Delta_{22}\hat{\tau}_{1}\end{pmatrix}.$
(3)

where $\tilde{\xi}_{1}=\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}-E_{c}$,
$\tilde{\xi}_{2}=\xi_{\bm{k}}^{\scaleto{(2)}{6pt}}+E_{c}$ and we restricted to
s-wave pairing (real order parameter). The two branches of the BCS
quasiparticle excitation spectrum, $E_{+}$ and $E_{-}$, are obtained by
defining
$\mathrm{det}\bm{\mathcal{G}}_{0}^{-1}\equiv\left(\tilde{\omega}_{n}^{2}+E_{+}^{2}\right)\left(\tilde{\omega}_{n}^{2}+E_{-}^{2}\right)$,
where $\tilde{\omega}_{n}=\omega_{n}-ih$. Inverting Eq. 3 we obtain the
Green’s function of the system which allows us to analyze when
superconductivity occurs. Details of the form of the Green’s function are
given in Appendix A.

Figure 2: Band structure of the system in the presence of an exchange field,
for (a) small and (b) large values of the field and generic values of $E_{c}$,
$\gamma$, $\omega_{c}$, $\mu$. The rectangles indicate the regions where
Cooper pair formation can occur.

##  III Results

We consider a two bands system with spin-split hole-like parabolic bands:
$\xi_{\bm{k}}^{\scaleto{(\alpha)}{6pt}}=-k^{2}/(2m_{\alpha})-\mu$ with
$m_{\alpha}$ the effective mass of the band $\alpha$ and $\mu$ the chemical
potential. Defining the ratio between the two band masses as
$\gamma=m_{1}/m_{2}$ we can write
$\xi_{\bm{k}}^{\scaleto{(2)}{6pt}}=\gamma\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}+(\gamma-1)\mu$.
The band structure of the system for generic values of $E_{c}$, $\gamma$,
$\mu$ and two different limits of $h$ is represented in Fig. 2, where we also
show the energy cutoff of the effective attractive interaction $\omega_{c}$.
By examining the case of small spin-splitting represented in Fig. 2(a), we
note that intraband pairing is favored, since bands with the same band index
and opposite spin are close to each other. On the other hand, the presence of
a large spin-splitting field can bring two bands with different band and spin
indices closer to each other, favoring an interband pairing mechanism, as
shown in Fig. 2(b).

To simplify the problem, we neglect those scattering processes connecting
interband to intraband Cooper pairs and vice versa, _i.e._ we set
$g_{\alpha\alpha,\alpha\beta}=g_{\alpha\beta,\alpha\alpha}=0$ with
$\alpha\neq\beta$. This simplification is justified by the fact that, for
sufficiently large $E_{c}$, either interband or intraband pairing processes
will dominate over the other, depending on $h$ and $\gamma$. Simultaneously,
the processes of intraband pairs hopping in interband pairs, or vice versa,
are either forbidden due to energy or momentum conservation, or strongly
suppressed compared to the non-mixing processes, since they would meet energy
and momentum conservation criteria for significantly smaller ranges.

With these simplifications, from Eq. 2, it is clear that the gap equation for
the intraband and interband order parameters are decoupled and we can solve
them separately. We set the chemical potential $\mu$ as the energy scale and
choose the energy cutoff for the effective attractive interaction and the band
separation to be $\omega_{c}=0.2\left|\mu\right|$ and
$E_{c}=0.05\left|\mu\right|$, respectively. Furthermore, we consider
dimensionless superconducting coupling constants:
$\lambda^{\mathrm{intra}}_{\alpha\beta}=N_{\alpha}(0)g_{\alpha\alpha,\beta\beta}$
and
$\lambda^{\mathrm{inter}}=N_{1}(0)g_{\alpha\beta,\alpha\beta},\,(\alpha\neq\beta)$,
where $N_{\alpha}(0)$ is the density of states at the Fermi energy for the
band $\alpha$. Their values are chosen to be
$\lambda^{\mathrm{intra}}_{11}=0.3$,
$\lambda^{\mathrm{inter}}=\lambda^{\mathrm{intra}}_{22}=2\lambda^{\mathrm{intra}}_{12}=0.2$.
We assume that $\lambda^{\mathrm{inter}}$ can be taken to be larger than some
$\lambda^{\mathrm{intra}}_{\alpha\beta}$ because of the external magnetic
field, which may bring two different bands with opposite spin very close to
each other in energy, see _e.g._ Fig. 2(b).

Having chosen the values of our parameters we can establish whether the system
exhibits superconductivity. We determine the critical values of temperature
$T_{c}$ and exchange field $h_{c}$ of our system for different values of the
effective mass ratio $\gamma$ by linearizing Eq. 2 with respect to
$\Delta_{\alpha\alpha^{\prime}}$, separately for interband
($\alpha\neq\alpha^{\prime}$) and intraband ($\alpha=\alpha^{\prime}$)
superconducting pairing. The linearized gap equations are derived in
Appendices B and C. In the interband case we have the following equation

$\begin{split}\frac{1}{\lambda^{\mathrm{inter}}}=\int_{-\omega_{c}}^{\omega_{c}}&\frac{d\xi}{2(\tilde{\xi}_{1}+\tilde{\xi}_{2})}\sum_{\alpha=1,2}\biggl{\\{}\tanh\left[\frac{\beta_{c}}{2}\left(\tilde{\xi}_{\alpha}+h_{c}\right)\right]\\\
&+\tanh\left[\frac{\beta_{c}}{2}\left(\tilde{\xi}_{\alpha}-h_{c}\right)\right]\biggr{\\}},\end{split}$
(4)

where $\beta_{c}=1/T_{c}$, $\tilde{\xi}_{1}=\xi-E_{c}$, and
$\tilde{\xi}_{2}=\gamma\xi+(\gamma-1)\mu+E_{c}$. The interband critical
parameters are the values satisfying Eq. 4. We note that by setting $E_{c}=0$
and $\gamma=1$, we get $\tilde{\xi}_{1}=\tilde{\xi}_{2}=\xi$, and Eq. 4
reduces to that of a single band superconductor in external magnetic field,
reported in Appendix B. In the intraband case we have the following system of
equations

$\begin{pmatrix}\delta_{1}\\\
\delta_{2}\end{pmatrix}\begin{pmatrix}\lambda_{11}I_{1}(T_{c},h_{c})-1&\lambda_{12}I_{2}(T_{c},h_{c})\\\
\lambda_{21}I_{1}(T_{c},h_{c})&\lambda_{22}I_{2}(T_{c},h_{c})-1\end{pmatrix}=0,$
(5)

where the terms $\delta_{\alpha}$ are defined through
$\Delta_{\alpha\alpha}=\epsilon\delta_{\alpha}$, $\epsilon(T=T_{c})=0$, and

$I_{\alpha}(T,h)\\!=\\!\\!\int_{-\omega_{c}}^{\omega_{c}}\\!\frac{d\xi}{4\tilde{\xi}_{\alpha}}\\!\left\\{\\!\tanh\\!\left[\\!\frac{\beta}{2}\\!\left(\\!\tilde{\xi}_{\alpha}\\!+\\!h\right)\\!\right]\\!+\\!\tanh\\!\left[\\!\frac{\beta}{2}\\!\left(\\!\tilde{\xi}_{\alpha}\\!-\\!h\\!\right)\\!\right]\\!\right\\}\\!,$
(6)

with $\alpha=1,2$, and $\beta=1/T$. The intraband critical parameters are
found by setting to zero the determinant of the matrix in Eq. 5. More details
on the procedure used to obtain the critical parameters are reported in
Appendix E.

The results are shown in Fig. 3, displaying reentrant superconductivity. The
figure shows the inter- and intraband superconducting domains, delimited by
the $h_{c}(T)$ curves, for different values of $\gamma$, the lines correspond
to the critical values of temperature and exchange field, while the colored
area identifies the superconducting region, with dark gray color identifying
the interband regions and light gray the intraband. From Fig. 3, we
immediately note a substantial difference between the superconductivity
induced by each type of coupling. The intraband region develops around $h=0$
for all the values of $\gamma$, with the highest critical temperature obtained
in absence of any external field. On the other hand, the interband regions
show an interesting behavior: except for the case $\gamma=1.1$, they are not
present in the zero-field case but around a certain finite value of $h$, thus
their appearance is conditioned to the presence of an external field.
Therefore, analyzing Fig. 3, we note that for certain values of $\gamma$ it is
possible to find two distinct/disconnected superconducting regions: the
intraband one for low exchange field values and the interband one developing
around substantially higher values of the exchange field. We note that the
value $\gamma=1.1$ as a certain significance, since it approximately
corresponds to the point where the difference in band curvature compensate for
the band separation $2E_{c}$, so that the Fermi momenta of the two bands
become the same. Hence, any $h\neq 0$ has a detrimental effect for
superconductivity.

Figure 3: Critical field $h_{c}$ as a function of temperature $T$ for
different values of the band mass ratio $\gamma$, for
$E_{c}=0.05\left|\mu\right|$, $\lambda^{\mathrm{intra}}_{11}=0.3$,
$\lambda^{\mathrm{inter}}=\lambda^{\mathrm{intra}}_{22}=2\lambda^{\mathrm{intra}}_{12}=0.2$,
and $\omega_{c}=0.2\left|\mu\right|$. The dark (light) gray areas represent
the region of interband (intraband) superconducting coupling. Except for the
case $\gamma=1.1$, we note a reentrant superconductivity, with the two
disconnected intra- and interband superconducting domains.

These results can be explained by considering the band structure provided in
Fig. 2 and the conditions for superconducting pairing to occur. In the BCS
theory, Cooper pairs are formed by electrons with opposite momenta close to
the Fermi momentum, and energies in a “shell” of width $\omega_{c}$ around the
Fermi energy. Therefore, only electrons meeting these two criteria can form
Cooper pairs. In a multiband system with spin-split bands, each band will have
its own intervals of energies and momenta where this can be realized. When the
intervals of two different bands overlap, Cooper pair formation among them is
feasible. These overlap regions are represented by the rectangles in Fig. 2.

In the absence of spin-splitting, and in the case of intraband pairing Cooper
pair formation is of course always possible. When an external field is
applied, instead, the spin-up and spin-down bands separate from each other,
with their distance increasing with $h$, thus reducing the size of the overlap
region where the formation of Cooper pairs is possible. Consequently, for
purely intraband pairing, the critical temperature of the system has a maximum
at zero field.

This picture changes when considering the case of interband pairing. While in
the intraband case the exchange field pulls the two pairs of spin-split bands
apart from each other, in the interband case it can have the opposite effect.
Superconductivity involving interband pairs can result from different cases.
With spin-split bands, we can have formation of Cooper pairs with (i), one
electron in band $1\downarrow$ and the other in $2\uparrow$ and (ii), one
electron in band $1\uparrow$ and the other in band $2\downarrow$. Without
spin-splitting instead we have (iii), one electron in band 1 and the other in
band 2. The choice of the parameters $E_{c}$, $h$ and $\gamma$ determines the
size of the overlap between the different bands and the size of the
superconducting coupling constant $\lambda^{\mathrm{inter}}$ influences in
which of the three cases superconductivity is realized.

Qualitatively, in each case the maximum in the size of the overlap, and thus
in the superconducting critical temperature, occurs when the Fermi momenta of
the two bands involved are equal to each other. Therefore, equating the Fermi
momenta of the two bands for each of these cases, we can obtain, as a function
of the other parameters, the value of $h$ which determines the match of the
momenta, allowing for Cooper pair formation. In case (i) we have
$k_{F1\downarrow}=\sqrt{-2m(\mu+E_{c}-h)}$ and
$k_{F2\uparrow}=\sqrt{-2m_{1}/\gamma(\mu-E_{c}+h)}$, equating the two we
obtain:

$h=\frac{\gamma-1}{\gamma+1}\mu+E_{c}.$ (7)

In case (ii) we get $k_{F1\uparrow}=\sqrt{-2m_{1}(\mu+E_{c}+h)}$ and
$k_{F2\downarrow}=\sqrt{-2m_{1}/\gamma(\mu-E_{c}-h)}$, yielding:

$h=-\frac{\gamma-1}{\gamma+1}\mu-E_{c}.$ (8)

In case (iii) it is clear that $h=0$.

We note that the value of $h_{c}$ at which the interband superconducting
domains present a maximum in $T_{c}$, depends only on the band parameters,
$\mu$, $E_{c}$ and $\gamma$, and not on the superconducting coupling
constants. These instead, together with other factors, such as _e.g._ impurity
scattering, influence the value of $T_{c}$, and therefore, whether
superconductivity is realized.

Figure 4: Critical temperature of the interband superconducting domains as a
function of $\gamma$ and $h$, for $E_{c}=0.05\left|\mu\right|$,
$\lambda^{\mathrm{inter}}=0.2$, and $\omega_{c}=0.2\left|\mu\right|$.

Having determined the value of $h$ which produces a maximum of the overlap in
the different cases, with our choice of parameters we calculate numerically
the critical temperature of the interband domains as a function of $\gamma$
and $h$. The numerical results are plotted in Fig. 4 together with the
theoretical prediction given by Eqs. 7 and 8.

We observe that for a good range of $\gamma$ values
($0.95\lesssim\gamma\lesssim 1.05$ and $1.15\lesssim\gamma\lesssim 1.3$) the
numerical results follow the analytical results and superconductivity comes
from case (i) or case (ii), showing peaks in $T_{c}$ for non-zero values of
$h$. Instead, for $1.06<\gamma<1.14$, the superconducting domains are centered
in $h=0$, away from the analytical prediction, except for $\gamma\simeq 1.1$.
Therefore, for this choice of parameters, and this range of $\gamma$ values,
the system favors superconductivity in absence of spin-splitting. Considering
instead those points which do not exhibit superconductivity, for $\gamma<0.95$
and $\gamma>1.3$, we can state that the overlap between the bands is not large
enough for the chosen value of the interband superconducting coupling
constant. However, choosing a higher coupling would yield superconductivity
for values of $\gamma$ smaller than 0.95 and higher than 1.3.

Observing Fig. 4, we can see that peaks in $T_{c}$ can be found for rather
high values of the exchange field, up to $h\simeq 1.6E_{c}$. Therefore, when
the system has both intraband and interband superconducting pairing, it can
present two disconnected superconducting domains, one for low magnetic field
due to the intraband pairing, and one for high magnetic field coming
exclusively from interband pairing.

Finally, it is worth noting that the case of two electron-like bands would
yield qualitatively similar results. The difference would be in the value of
the band mass ratio $\gamma$ at which the band overlap occurs. The case where
there is a coupling between an electron-like and a hole-like band is more
complicated and would require a separate study.

##  IV Minimal Models for two-dimensional MgB2 and Ba0.6K0.4Fe2As2

Figure 5: Superconducting critical curves from a minimal model for (a) two-
dimensional MgB2 and (b) Ba0.6K0.4Fe2As2, with the highlighted high (dark
gray) and low (light) field superconducting domains.

The simple model presented in this work, for an ideal case, can be realized in
a 2D multiband superconductor with the application of an in-plane external
magnetic field. On the material side many superconductors have been shown to
have multiple hole-like bands close to the Fermi energy at the $\Gamma$ point.
For instance, band structure calculations in few monolayers (MLs) MgB2 in Ref.
[58] have shown that each ML contributes to the band structure with a pair of
$\sigma$ bands, $\sigma_{1}$ and $\sigma_{2}$ having different effective
masses and chemical potential $\mu\simeq 500$\mathrm{m}\mathrm{e}\mathrm{V}$$,
with each pair of bands separated by $E_{c}\simeq
37.5$\mathrm{m}\mathrm{e}\mathrm{V}$$. Therefore, we consider a minimal model
for 2ML two-dimensional MgB2, taking only a pair of $\sigma$ bands separated
by $37.5$\mathrm{m}\mathrm{e}\mathrm{V}$$. In this case we retain the
superconducting coupling constants representing scattering processes
connecting interband to intraband Cooper pairs and vice versa, which were
previously neglected. This means that in this case the intraband and interband
order parameters are coupled, the resulting set of linearized gap equations is
given in Appendix D. We reduce the indices of the
$\lambda_{\alpha\alpha^{\prime},\beta\beta^{\prime}}$ coupling constants in
the following way:

$\alpha\alpha^{\prime}\equiv\begin{cases}&\alpha\quad\mathrm{if}\quad\alpha=\alpha^{\prime},\\\
&3\quad\mathrm{if}\quad\alpha\neq\alpha^{\prime}.\end{cases}$

Therefore, the index 3 represents the interband pairing channel. Following
Ref.[13] we consider the following superconducting coupling matrix:

$\hat{\lambda}=\begin{pmatrix}0.275&0.032&\lambda_{13}\\\
0.032&0.274&\lambda_{23}\\\
\lambda_{31}&\lambda_{32}&\lambda_{33}\end{pmatrix},$ (9)

where the upper $2\times 2$ block represent the conventional superconducting
coupling for a two band system, the terms $\lambda_{\alpha 3}$ and
$\lambda_{3\alpha}$, $\alpha=1,2$, represent interband Cooper pairs scattering
to intraband Cooper pairs and vice versa. Finally $\lambda_{33}$ represents
purely interband scattering processes, and corresponds to the constant
$\lambda^{\mathrm{inter}}$. To obtain the correct value of the critical
temperature at zero field ($T_{c}\approx 39$\mathrm{K}$$), we set
$\lambda_{\alpha 3}=\lambda_{3\alpha}=0.045$ and $\lambda_{33}=0.2$. In Fig.
5(a) we report the superconducting domain delimited by the $h_{c}$-$T_{c}$
critical curves. As in the previous section we find two disconnected
superconducting domains for low and high magnetic field, with the high
magnetic field one centered at a value $B=\mu_{B}E_{c}\approx
650$\mathrm{T}$$.

Also, among FeBS, many materials in the iron-pnictides family, like
LaFeAsO$1-x$F$x$ [59, 60, 61] and Ba$1-x$K$x$Fe2As2 [62], exhibit multiple
hole-like bands at the $\Gamma$ points, mainly originating from the 3-d Fe
orbitals. Again, we consider a minimal model for optimally doped
Ba0.6K0.4Fe2As2, with the two hole-like $\alpha$ and $\beta$ bands at the
$\Gamma$ point, separated by $E_{c}=5$\mathrm{m}\mathrm{e}\mathrm{V}$$ [62].
Following Ref.[13] we take the superconducting coupling matrix to be:

$\hat{\lambda}=\begin{pmatrix}0.51&-0.005&\lambda_{13}\\\
-0.0025&0.39&\lambda_{23}\\\
\lambda_{31}&\lambda_{32}&\lambda_{33}\end{pmatrix},$ (10)

where the elements 12 and 21 are set to negative values as required by the
$s_{\pm}$ superconducting phase typical of FeBS. However, we note that the
critical temperature and field of the system are not sensitive to this
negative sign since the intraband gaps are effectively decoupled from the
interband gap for the parameters used. Once more, we set $\lambda_{\alpha
3}=\lambda_{3\alpha}=0.05$, $\alpha=1,2$, and $\lambda_{33}=0.25$ to obtain
the right critical temperature at zero field ($T_{c}\approx 36$\mathrm{K}$$)
and report the results in Fig. 5(b). Again we note two disconnected
superconducting domains with the high field one centered around $h_{c}\approx
90$\mathrm{T}$$.

##  V Conclusions

In summary, we have considered a superconducting system with two relevant low-
energy bands in the presence of a spin-splitting field. We have shown that the
inclusion of interband superconducting pairing processes, _i.e._ allowing for
the presence of Cooper pairs with the two electrons coming from two different
bands, can result in reentrant superconducting domains centered at a high
value of the spin-splitting field. These domains are exclusively due to the
interband pairing and generally do not overlap with the superconducting
domains originating from intraband pairing, which are always centered at zero
spin-splitting field.

###### Acknowledgements.

Computations have been performed on the SAGA supercomputer provided by UNINETT
Sigma2 - the National Infrastructure for High Performance Computing and Data
Storage in Norway. We acknowledge funding via the “Outstanding Academic
Fellows” programme at NTNU, the Research Council of Norway Grant No. 302315,
as well as through its Centres of Excellence funding scheme, Project No.
262633, “QuSpin.” Nordita is supported in part by NordForsk.

##  Appendix A Green’s function of the system.

The Green’s function of the system is obtained by inverting Eq. 3 in the main
text:

$\bm{\mathcal{G}}_{0}(\bm{k},\omega_{n})=\frac{1}{\mathrm{det}\bm{\mathcal{G}}_{0}^{-1}}\begin{pmatrix}\hat{\mathcal{A}}^{11}(\bm{k},\tilde{\omega}_{n})&\hat{\mathcal{A}}^{12}(\bm{k},\tilde{\omega}_{n})\\\
\hat{\mathcal{A}}^{21}(\bm{k},\tilde{\omega}_{n})&\hat{\mathcal{A}}^{22}(\bm{k},\tilde{\omega}_{n})\end{pmatrix},$
(11)

where
$\mathrm{det}\bm{\mathcal{G}}_{0}^{-1}=\left(\tilde{\omega}_{n}^{2}+E_{+}^{2}\right)\left(\tilde{\omega}_{n}^{2}+E_{-}^{2}\right)$
with $\tilde{\omega}_{n}=\omega_{n}-ih$. The two branches have the following
expression:

$E_{\pm}^{2}\\!=\\!\frac{1}{2}\\!\left\\{\\!\tilde{E}_{1}^{2}\\!+\\!\tilde{E}_{2}^{2}\\!+\\!2\left|\Delta_{12}\right|^{2}\pm\\!\left[\left(\tilde{E}_{1}^{2}\\!+\\!\tilde{E}_{2}^{2}\\!+\\!2\left|\Delta_{12}\right|^{2}\right)^{2}\\!-\\!4\left(\tilde{E}_{1}^{2}\tilde{E}_{2}^{2}\\!+\\!\left|\Delta_{12}\right|^{4}+\\!2\tilde{\xi_{1}}\tilde{\xi_{2}}\left|\Delta_{12}\right|^{2}\\!-\\!2\mathrm{Re}\left\\{\Delta_{11}\Delta_{22}{\Delta_{12}^{*}}^{2}\right\\}\right)\right]^{\frac{1}{2}}\right\\}.$
(12)

The the components of the $2\times 2$ matrices
$\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\omega_{n})$ are:

$\displaystyle\left[\hat{\mathcal{A}}^{11}(\bm{k},\tilde{\omega}_{n})\right]_{11}=$
$\displaystyle-\left(i\tilde{\omega}_{n}+\tilde{\xi_{1}}\right)\\!\left(\tilde{\omega}_{n}^{2}\\!+\\!\tilde{\xi_{2}}^{2}\\!+\\!\left|\Delta_{22}\right|^{2}\\!\right)$
$\displaystyle-\\!\left|\Delta_{12}\right|^{2}\left(i\tilde{\omega}_{n}+\tilde{\xi_{2}}\right),$
(13a)
$\displaystyle\left[\hat{\mathcal{A}}^{11}(\bm{k},\tilde{\omega}_{n})\right]_{12}=$
$\displaystyle\Delta_{11}\left(\tilde{\omega}_{n}^{2}+\tilde{\xi_{2}}^{2}+\left|\Delta_{22}\right|^{2}\\!\right)-\Delta_{12}^{2}\Delta_{22}^{*},$
(13b)
$\displaystyle\left[\hat{\mathcal{A}}^{12}(\bm{k},\tilde{\omega}_{n})\right]_{11}=$
$\displaystyle\left(i\tilde{\omega}_{n}+\tilde{\xi_{1}}\right)\Delta_{12}\Delta_{22}^{*}$
$\displaystyle+\Delta_{11}\Delta_{12}^{*}\left(i\tilde{\omega}_{n}+\tilde{\xi_{2}}\right),$
(13c)
$\displaystyle\left[\hat{\mathcal{A}}^{12}(\bm{k},\tilde{\omega}_{n})\right]_{12}=$
$\displaystyle\\!-\\!\left[\left(i\tilde{\omega}_{n}\\!+\\!\tilde{\xi_{1}}\right)\\!\left(i\tilde{\omega}_{n}\\!-\\!\tilde{\xi_{2}}\right)\\!-\\!\left|\Delta_{12}\right|^{2}\right]\Delta_{12}$
$\displaystyle-\Delta_{11}\Delta_{12}^{*}\Delta_{22},$ (13d)
$\displaystyle\left[\hat{\mathcal{A}}^{21}(\bm{k},\tilde{\omega}_{n})\right]_{11}=$
$\displaystyle\left[\hat{\mathcal{A}}^{12}(\bm{k},-\tilde{\omega}_{n}^{*})\right]_{11}^{*},$
(13e)
$\displaystyle\left[\hat{\mathcal{A}}^{21}(\bm{k},\tilde{\omega}_{n})\right]_{12}=$
$\displaystyle\left[\hat{\mathcal{A}}^{12}(\bm{k},-\tilde{\omega}_{n})\right]_{12},$
(13f)
$\displaystyle\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})\right]_{22}=$
$\displaystyle-\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},-\tilde{\omega}_{n})\right]_{11},$
(13g)
$\displaystyle\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})\right]_{21}=$
$\displaystyle\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\tilde{\omega}_{n}^{*})\right]_{12}^{*}.$
(13h)

where
$\tilde{E}_{\alpha}=\sqrt{\tilde{\xi}_{\alpha}^{2}+\left|\Delta_{\alpha\alpha}\right|^{2}},\,\,\alpha=1,2$,
with
$\tilde{\xi_{1}}=\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}-E_{c},\,\tilde{\xi_{2}}=\xi_{\bm{k}}^{\scaleto{(2)}{6pt}}+E_{c}$.

The elements of $\hat{\mathcal{A}}^{22}(\bm{k},\omega_{n})$ are obtained by
exchanging the indices 1 and 2 in the expressions for the elements of
$\hat{\mathcal{A}}^{11}(\bm{k},\omega_{n})$. The inter- and intra-band normal
and anomalous Green’s functions are given by:

$\displaystyle G^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})$
$\displaystyle=\frac{\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})\right]_{11}}{\left(\tilde{\omega}_{n}^{2}+E_{+}^{2}\right)\left(\tilde{\omega}_{n}^{2}+E_{-}^{2}\right)},$
(14a) $\displaystyle F^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})$
$\displaystyle=\frac{\left[\hat{\mathcal{A}}^{\alpha\beta}(\bm{k},\tilde{\omega}_{n})\right]_{12}}{\left(\tilde{\omega}_{n}^{2}+E_{+}^{2}\right)\left(\tilde{\omega}_{n}^{2}+E_{-}^{2}\right)}.$
(14b)

##  Appendix B Gap equation for the interband order parameter.

We derive the gap equation in the case of purely interband coupling, _i.e._
Cooper pairs formed exclusively by electrons in different bands. To do so we
consider the following form of the coupling matrix:

$g_{\alpha\alpha^{\prime}\beta\beta^{\prime}}=\begin{cases}g^{\mathrm{inter}},\quad\mathrm{if}\quad\alpha\neq\alpha^{\prime},\,\beta\neq\beta^{\prime}\\\
0,\quad\mathrm{otherwise}.\end{cases}$ (15)

Inserting this in Eq. 2 we obtain the following gap equation:

$\Delta_{12}=\frac{g^{\mathrm{inter}}T}{V}\sum_{\bm{k}\omega_{n}}\left[F^{12}(\bm{k},\omega_{n}-ih)+F^{21}(\bm{k},\omega_{n}-ih)\right].$
(16)

Setting $\Delta_{11}=\Delta_{22}=0$ in Eq. 12 we get:

$E_{\pm}=\frac{\tilde{\xi}_{1}-\tilde{\xi}_{2}}{2}\pm E_{12},$ (17)

with
$E_{12}=\sqrt{(\tilde{\xi}_{1}+\tilde{\xi}_{2})^{2}/4+\left|\Delta_{12}\right|^{2}}$.
Using the relation, defined in the main text, between the two bands
$\xi_{\bm{k}}^{\scaleto{(2)}{6pt}}=\gamma\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}+(\gamma-1)\mu$,
we can write:

$\displaystyle E_{\pm}$
$\displaystyle=\frac{(\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}+\mu)(1-\gamma)}{2}-E_{c}\pm
E_{12},$ (18a) $\displaystyle E_{12}$
$\displaystyle=\sqrt{\left(\frac{\xi_{\bm{k}}^{\scaleto{(1)}{6pt}}(1+\gamma)-\mu(1-\gamma)}{2}\right)^{2}+\left|\Delta_{12}\right|^{2}}.$
(18b)

The expression of the anomalous Green’s function is:

$\begin{split}F^{12}&(\bm{k},\omega_{n}-ih)=\\\
&-\frac{\left(i\tilde{\omega}_{n}+\tilde{\xi_{1}}+h\right)\left(i\tilde{\omega}_{n}-\tilde{\xi_{2}}+h\right)-\left|\Delta_{12}\right|^{2}}{\left[(\omega_{n}-ih)^{2}+E_{+}^{2}\right]\left[(\omega_{n}-ih)^{2}+E_{-}^{2}\right]}\Delta_{12}.\end{split}$
(19)

After summing over the Matsubara frequency the gap equation Eq. 16 takes the
following form:

$\begin{split}\frac{1}{g^{\mathrm{inter}}}=&-\frac{1}{V}\sum_{\bm{k}}\frac{1}{2E_{12}}\left[n_{F}(E_{+}-h)-n_{F}(-E_{+}-h)-n_{F}(E_{-}-h)+n_{F}(-E_{-}-h)\right]\\\
=&\frac{1}{V}\sum_{\bm{k}}\sum_{s=\pm}\frac{s}{4E_{12}}\left\\{\tanh\left[\frac{\beta}{2}\left(E_{s}+h\right)\right]+\tanh\left[\frac{\beta}{2}\left(E_{s}-h\right)\right]\right\\},\end{split}$
(20)

where $n_{F}(\varepsilon)=(e^{\beta\varepsilon}+1)^{-1}$ is the Fermi function
and we have used:

$\begin{split}n_{F}&(E_{\pm}-h)-n_{F}(-E_{\pm}-h)\\\ &=-\frac{\sinh\beta
E_{\pm}}{\cosh\beta E_{\pm}+\cosh\beta h}\\\
&=-\frac{1}{2}\left\\{\tanh\left[\frac{\beta}{2}\left(E_{\pm}+h\right)\right]+\tanh\left[\frac{\beta}{2}\left(E_{\pm}-h\right)\right]\right\\}.\end{split}$

We now switch from the summation over momenta to integral over the energy:
$V^{-1}\sum_{\bm{k}}(\cdot)=\int d\xi_{1}N_{1}(\xi_{1})(\cdot)$, where
$N_{1}(\xi_{1})$ is the density of state of band 1. We then approximate the
density of state with its value at the Fermi level $N_{1}(\xi_{1})\simeq
N_{1}(0)$ and, defining the dimensionless superconducting coupling constant as
done in the main text, we obtain the final expression for the interband gap
equation:

$\begin{split}\frac{1}{\lambda^{\mathrm{inter}}}&=\int_{-\omega_{c}}^{\omega_{c}}d\xi\sum_{s=\pm}\frac{s}{4E_{12}(\xi)}\\\
\times&\left\\{\tanh\left[\frac{\beta}{2}\left(E_{s}(\xi)+h\right)\right]+\tanh\left[\frac{\beta}{2}\left(E_{s}(\xi)-h\right)\right]\right\\}.\end{split}$
(21)

To obtain critical temperature and critical field, we linearize Eq. 21 with
respect to $\Delta_{12}$. The equation then takes the following simple form:

$\frac{1}{\lambda^{\mathrm{inter}}}=\int_{-\omega_{c}}^{\omega_{c}}\frac{d\xi}{2(\tilde{\xi}_{1}+\tilde{\xi}_{2})}\sum_{\alpha=1,2}\left\\{\tanh\left[\frac{\beta_{c}}{2}\left(\tilde{\xi}_{\alpha}+h_{c}\right)\right]+\tanh\left[\frac{\beta_{c}}{2}\left(\tilde{\xi}_{\alpha}-h_{c}\right)\right]\right\\}.$
(22)

We note that by setting $E_{c}=0$ and $\gamma=1$, we get
$\tilde{\xi}_{1}=\tilde{\xi}_{2}=\xi$, the system reduces to a single band
superconductor. Therefore, Eq. 22 takes the usual form for a single band spin-
split superconductor:

$\frac{1}{\lambda}\\!=\\!\int_{-\omega_{c}}^{\omega_{c}}\frac{d\xi}{2\xi}\left\\{\\!\tanh\\!\left[\\!\frac{\beta_{c}}{2}\\!\left(\xi+h_{c}\right)\\!\right]\\!+\\!\tanh\\!\left[\\!\frac{\beta_{c}}{2}\\!\left(\xi-
h_{c}\right)\\!\right]\\!\right\\}.$ (23)

##  Appendix C Gap equation for the intraband order parameters.

In order to consider purely intraband coupling we take the following form of
the coupling matrix:

$g_{\alpha\alpha^{\prime}\beta\beta^{\prime}}=\begin{cases}g_{\alpha\beta}^{\mathrm{intra}},\quad\mathrm{if}\quad\alpha=\alpha^{\prime},\,\beta=\beta^{\prime}\\\
0,\quad\mathrm{otherwise}\end{cases}$ (24)

Using this expression in Eq. 2 we get the following coupled gap equations:

$\displaystyle\Delta_{11}=$
$\displaystyle\frac{g_{11}^{\mathrm{intra}}T}{V}\sum_{\bm{k}\omega_{n}}F^{11}(\bm{k},\omega_{n}-ih)$
$\displaystyle+\frac{g_{12}^{\mathrm{intra}}T}{V}\sum_{\bm{k}\omega_{n}}F^{22}(\bm{k},\omega_{n}-ih),$
(25a) $\displaystyle\Delta_{22}=$
$\displaystyle\frac{g_{21}^{\mathrm{intra}}T}{V}\sum_{\bm{k}\omega_{n}}F^{11}(\bm{k},\omega_{n}-ih)$
$\displaystyle+\frac{g_{22}^{\mathrm{intra}}T}{V}\sum_{\bm{k}\omega_{n}}F^{22}(\bm{k},\omega_{n}-ih).$
(25b)

The two branches of the BCS quasiparticle spectrum are simply $E_{+}=E_{11}$
and $E_{-}=E_{22}$ and we get the following expression for the anomalous
Green’s function:

$F^{\alpha\alpha}(\bm{k},\omega_{n}-ih)=\frac{\Delta_{\alpha\alpha}}{\left(i\omega_{n}+E_{\alpha\alpha}+h\right)\left(i\omega_{n}-E_{\alpha\alpha}+h\right)}.$
(26)

Following the same steps as in the previous section we finally get the
following:

$\begin{split}\Delta_{\alpha\alpha}=\sum_{\beta=1,2}&\lambda_{\alpha\beta}\Delta_{\beta\beta}\int_{-\omega_{n}}^{\omega_{n}}\frac{d\xi_{\beta}}{4E_{\beta\beta}}\\\
&\times\left(\tanh\frac{E_{\beta\beta}+h}{2T}+\tanh\frac{E_{\beta\beta}-h}{2T}\right).\end{split}$
(27)

Linearizing Eq. 27 with respect to the order parameters, again allows to
obtain the superconducting critical temperature and critical field. We set
$\Delta_{\alpha\alpha}=\epsilon\delta_{\alpha}$, where $\epsilon(T=T_{c})=0$,
and obtain the following system:

$\begin{pmatrix}\delta_{1}\\\
\delta_{2}\end{pmatrix}\begin{pmatrix}\lambda_{11}I_{1}(T_{c},h_{c})-1&\lambda_{12}I_{2}(T_{c},h_{c})\\\
\lambda_{21}I_{1}(T_{c},h_{c})&\lambda_{22}I_{2}(T_{c},h_{c})-1\end{pmatrix}=0,$
(28)

where

$\begin{split}I_{\alpha}&(T,h)=\\\
\int_{-\omega_{c}}^{\omega_{c}}&\frac{d\xi}{4\tilde{\xi}_{\alpha}}\left\\{\tanh\left[\frac{\beta}{2}\left(\tilde{\xi}_{\alpha}+h\right)\right]+\tanh\left[\frac{\beta}{2}\left(\tilde{\xi}_{\alpha}-h\right)\right]\right\\},\end{split}$
(29)

with $\alpha=1,2$. The critical parameters are found by setting to zero the
determinant of the matrix in Eq. 28.

##  Appendix D Linearized gap equation for coupled interband and intraband
order parameters

When including the superconducting coupling constants representing scattering
processes connecting interband to intraband Cooper pairs, the gap equations
for the interband and intraband order parameters become coupled. In this case
the linearized gap equation yields the following system:

$\begin{pmatrix}\delta_{1}\\\ \delta_{2}\\\
\delta_{3}\end{pmatrix}\begin{pmatrix}\lambda_{11}I_{1}(T_{c},h_{c})-1&\lambda_{12}I_{2}(T_{c},h_{c})&\lambda_{13}I_{3}(T_{c},h_{c})\\\
\lambda_{21}I_{1}(T_{c},h_{c})&\lambda_{22}I_{2}(T_{c},h_{c})-1&\lambda_{23}I_{3}(T_{c},h_{c})\\\
\lambda_{31}I_{1}(T_{c},h_{c})&\lambda_{32}I_{2}(T_{c},h_{c})&\lambda_{33}I_{3}(T_{c},h_{c})-1\end{pmatrix}=0,$
(30)

where the constants $\lambda_{\alpha 3}$ and $\lambda_{3\alpha}$, with
$\alpha=1,2$, represent the scattering processes connecting interband to
intraband Cooper pairs, and $\lambda_{33}$ corresponds to
$\lambda^{\mathrm{inter}}$ in the main text. The terms $I_{\alpha}$,
$\alpha=1,2$ are given in Eq. 29, while $I_{3}$ corresponds to the energy
integral in Eq. 22. Again, the critical parameters are found by setting to
zero the determinant of the matrix in Eq. 30.

##  Appendix E Critical Temperature and Critical Field.

We presented the linearized equations Eqs. 22 and 28 allowing to obtain the
curve $h_{c}(T)$, for the interband and intraband domains, respectively. A
remark here is needed, since as is clear from Fig. 3 in the main text, in the
interband curves, to each temperature (except the maximum one), correspond two
solutions for $h$. This is a problem for the numerical solver. To address
this, first we find the value of the critical field corresponding to the
maximum temperature of the superconducting curve. This can be done through
analytical considerations as explained in the main text (see Eqs. 7 and 8).
Having obtained this value we then find separately a solution for the critical
field higher and lower than this value, as a function of the temperature.

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