# Extinction of the TeV Gamma-Ray Background by Sunlight

Abraham Loeb  Astronomy Department, Harvard University, 60 Garden St.,
Cambridge, MA 02138, USA

###### Abstract

I show that pair production on sunlight introduces a sizable anisotropy in the
cosmic background of TeV gamma-rays. The anisotropy amplitude in the direction
of the Sun exceeds the cosmic dipole anisotropy from the motion of the Sun
relative to the cosmic rest-frame.

##  1 Introduction

Given its effective surface temperature of $T_{\rm eff}=5.8\times
10^{3}~{}{\rm K}$, the Sun acts as a source of photons with a characteristic
energy, $E_{{\gamma}_{\odot}}\approx k_{B}T_{\rm eff}=0.5~{}{\rm eV}$. When
these photons collide with the cosmic background of TeV photons (Inoue &
Tanaka, 2016; Yan-kun & Hou-dun, 2022), they produce electron-positron pairs
through the process: $\gamma_{\odot}+\gamma\rightarrow e^{+}+e^{-}$. For pair
creation to occur, the energy of the background photon, $E_{\gamma}$, must
exceed the threshold value, $E_{\rm th}$ (Gould & Schréder, 1967a),

$E_{\gamma}>E_{\rm
th}=\frac{2m_{e}^{2}c^{4}}{E_{{\gamma}_{\odot}}\mu}\approx{0.5~{}{\rm
TeV}\over(\mu/2)},$ (1)

where $m_{e}=5.11\times 10^{5}~{}{\rm eV}$ is the electron rest-mass, $c$ is
the speed of light and $\mu=(1-\cos\theta)$ with $\theta$ being the angle
between the momenta of the two photons in the observer’s frame. The value of
$E_{\rm th}$ is minimized for a head-on collision ($\mu=2$) with a background
photon of energy $E_{\gamma}\gtrsim 0.5~{}{\rm TeV}$.

Above this threshold, the cross-section for pair production is given by,

$\sigma_{\gamma\gamma}=3.7\times 10^{-25}~{}{\rm
cm^{2}}\times(1-\beta^{2})\left[(1-{1\over
3}\beta^{4})\ln\left({{1+\beta}\over{1-\beta}}\right)-{2\over
3}\beta(2-\beta^{2})\right],$ (2)

where $\beta=(v/c)$, with $v$ being the speed of the electron or positron in
the center-of-mass system. The value of $\beta$ depends on $E_{\gamma}$ and
$\theta$ through the relation,

$\beta=\sqrt{1-{E_{\rm th}\over E_{\gamma}}}.$ (3)

The same pair-production process introduces a cosmic opacity to the TeV
radiation background as a result of photon-photon collisions with the
extragalactic background light (Gould & Schréder, 1967b; Singh et al., 2021;
Franceschini, 2021). Here we focus on the imprint of Solar photons on the
cosmic TeV background observable on Earth.

##  2 Results

The number density of Solar photons at a distance $R$ from the Sun is given
by,

$n_{{\gamma}_{\odot}}={L_{\odot}\over\left(4\pi
cR^{2}E_{{\gamma}_{\odot}}\right)}\approx 2.7\times 10^{12}~{}{\rm
cm^{-3}}\left({R\over R_{\odot}}\right)^{-2},$ (4)

where $L_{\odot}=4\times 10^{33}~{}{\rm erg~{}s^{-1}}$ is the Solar
luminosity. The Earth-Sun separation at $R=1.5\times 10^{13}~{}{\rm cm}$ is
$2.14\times 10^{2}$ times larger than the Solar radius $R_{\odot}=7\times
10^{10}~{}{\rm cm}$. This photon density yields a local optical-depth for
pair-production at an energy $E_{\gamma}\sim 1~{}{\rm TeV}$, of magnitude,

$\tau_{\gamma\gamma}(E_{\gamma}\sim 1~{}{\rm
TeV})=n_{{\gamma}_{\odot}}\sigma_{\gamma\gamma}R\sim 7\times
10^{-2}\left({R\over R_{\odot}}\right)^{-1}.$ (5)

The resulting absorption of background photons produces a deficit in the
gamma-ray background at the energy $\sim 1~{}{\rm TeV}$ with an anisotropy
amplitude $\sim 7\times 10^{-2}$ in the vicinity of the Sun and $\sim
3.3\times 10^{-4}$ at the Earth’s orbital radius, and an anisotropy angular
dependence that reflects the position of the Sun in the sky at the observing
time combined with the $\mu$ dependence of $\sigma_{\gamma\gamma}$ in
equations (2-3).

##  3 Implications

Interestingly, the expected anisotropy amplitude is significant relative to
the dipole associated with the motion of the Solar system relative to the
cosmic rest-frame, $\sim 1.2\times 10^{-3}$, as measured from the cosmic
microwave background (Kogut et al., 1993).

## Acknowledgements

This work was supported in part by Harvard’s Black Hole Initiative, which is
funded by grants from JFT and GBMF.

## References

  * Franceschini (2021) Franceschini, A. 2021, Universe, 7, 146, doi: 10.3390/universe7050146
  * Gould & Schréder (1967a) Gould, R. J., & Schréder, G. P. 1967a, Physical Review, 155, 1404, doi: 10.1103/PhysRev.155.1404
  * Gould & Schréder (1967b) —. 1967b, Physical Review, 155, 1408, doi: 10.1103/PhysRev.155.1408
  * Inoue & Tanaka (2016) Inoue, Y., & Tanaka, Y. T. 2016, ApJ, 818, 187, doi: 10.3847/0004-637X/818/2/187
  * Kogut et al. (1993) Kogut, A., Lineweaver, C., Smoot, G. F., et al. 1993, ApJ, 419, 1, doi: 10.1086/173453
  * Singh et al. (2021) Singh, K. K., Yadav, K. K., & Meintjes, P. J. 2021, Ap&SS, 366, 51, doi: 10.1007/s10509-021-03957-z
  * Yan-kun & Hou-dun (2022) Yan-kun, Q., & Hou-dun, Z. 2022, Chinese Astron. Astrophys., 46, 42, doi: 10.1016/j.chinastron.2022.05.003


# Non-unitary pairing, fractional vortices, and charge $4e$ superconductivity
in twisted bilayer graphene

Eslam Khalaf  Department of Physics, Harvard University, Cambridge, MA 02138
Patrick Ledwith  Department of Physics, Harvard University, Cambridge, MA
02138  Ashvin Vishwanath  Department of Physics, Harvard University,
Cambridge, MA 02138

# Symmetry constraints on superconductivity in twisted bilayer graphene:
Fractional vortices, $4e$ condensates or non-unitary pairing

Eslam Khalaf  Department of Physics, Harvard University, Cambridge, MA 02138
Patrick Ledwith  Department of Physics, Harvard University, Cambridge, MA
02138  Ashvin Vishwanath  Department of Physics, Harvard University,
Cambridge, MA 02138

###### Abstract

When two graphene sheets are twisted relative to each other by a small angle,
enhanced correlations lead to superconductivity whose origin remains under
debate. Here, we derive some general constraints on superconductivity in
twisted bilayer graphene (TBG), independent of its underlying mechanism.
Neglecting weak coupling between valleys, the global symmetry group of TBG
consists of independent spin rotations in each valley in addition to valley
charge rotations, ${\rm SU}(2)\times{\rm SU}(2)\times{\rm U}_{V}(1)$. This
symmetry is further enhanced to a full ${\rm SU}(4)$ in the idealized chiral
limit. In both cases, we show that any charge $2e$ pairing must break the
global symmetry. Additionally, if the pairing is unitary the resulting
superconductor admits fractional vortices. This leads to the following general
statement: Any superconducting condensate in either symmetry class has to
satisfy one of three possibilities: (i) the superconducting pairing is non-
unitary, (ii) the superconducting condensate has charge $2e$ but admits at
least half quantum vortices which carry a flux of $h/4e$, or (iii) the
superconducting condensate has charge $2me$, $m>1$, with vortices carrying
$h/2me$ flux. The latter possibility can be realized by a symmetric charge
$4e$ superconductor ($m=2$). Non-unitary pairing (i) is expected for
superconductors observed in the vicinity of flavor polarized states. On the
other hand, in the absence of flavor polarization, e.g. in the vicinity of
charge neutrality, one of the two exotic possibilities (ii) and (iii) is
expected. We sketch how all three scenarios can be realized in different
limits within a strong coupling theory of superconductivity based on
skyrmions. Finally we discuss the effect of symmetry lowering anisotropies and
experimental implications of these scenarios.

_Introduction_ — The discovery of superconductivity in magic angle twisted
bilayer graphene (TBG) Cao  _et al._ (2018a) has inspired immense research
activity to determine its origin Yankowitz  _et al._ (2019); Lu  _et al._
(2019); Stepanov  _et al._ (2019); Saito  _et al._ (2020); Xu and Balents
(2018); Lian  _et al._ (2018); Wu  _et al._ (2018); You and Vishwanath (2019);
Isobe  _et al._ (2018); Classen  _et al._ (2020); Chichinadze  _et al._
(2020); Fernandes and Venderbos (2020); Wang  _et al._ (2020); Kozii  _et al._
(2020); Khalaf  _et al._ (2020); Chatterjee  _et al._ (2020). Several
mechanisms for the superconductivity were proposed including phonon-induced
pairing Lian  _et al._ (2018); Wu  _et al._ (2018), bosonic order parameter
fluctuations You and Vishwanath (2019); Isobe  _et al._ (2018); Classen  _et
al._ (2020); Chichinadze  _et al._ (2020); Fernandes and Venderbos (2020);
Wang  _et al._ (2020); Kozii  _et al._ (2020), and topological skyrmion
superconductivity Khalaf  _et al._ (2020); Chatterjee  _et al._ (2020);
Christos  _et al._ (2020). At the same time, a few works have attempted to
connect band topology to superfluid density Lian  _et al._ (2020); Bernevig
_et al._ (2020) or categorized different paired states on the basis of
symmetry Scheurer and Samajdar (2020). TBG at small twist angles has a rather
large symmetry, particularly when we ignore weak Hunds coupling that couples
spins in opposite valleys. Here, we exploit this large symmetry group to
expose strong constraints on the nature of superconductivity regardless of the
mechanism responsible for it. Remarkably, these constraints imply that the
superconductivity has to be exotic in one of three possible ways: either (i)
the pairing is non-unitary Leggett (1975); Salomaa and Volovik (1987) which
means that the different flavors pair independently, (ii) the pairing breaks
the global symmetries leading to an order parameter manifold which admits
fractional vortices, (iii) the superconducting condensate has charge $2me$
with $m>1$ admitting fractional vortices with flux $h/2me$. Each of these
scenarios leads to measurable physical consequences.

A notable feature of magic angle TBG is that electrons appear with both spin
and valley flavors. The absence of spin-orbit coupling in graphene makes ${\rm
SU}(2)_{S}$ spin rotation an excellent symmetry. Similarly, the small angle
leads to an additional ${\rm U}(1)_{V}$ symmetry, implying conservation of the
valley quantum number, in addition to charge ${\rm U}(1)_{C}$ conservation.
Finally, again in the small angle limit, the independent spin rotation
symmetry in the two valleys is only weakly broken by a Hunds coupling, which
can be ignored for many purposes. Thus while the low energy symmetry group is
${\rm SU}(2)_{S}\times{\rm U}(1)_{C}\times{\rm U}(1)_{V}$, it is meaningful to
also consider the symmetry group ${\rm U}(2)_{K}\times{\rm U}(2)_{K^{\prime}}$
denoting independent ${\rm U}(2)$ symmetry groups in the opposite valleys.
Finally, several works Bultinck  _et al._ (2020); Kang and Vafek (2019); Seo
_et al._ (2019); Vafek and Kang (2020) have suggested that this symmetry is
enhanced to a full ${\rm U}(4)$ flavor symmetry to a good approximation. This
approximation becomes exact in the so-called chiral limit Tarnopolsky  _et
al._ (2019). In the following, we will omit the overall charge conservation
${\rm U}(1)_{C}$ which will always be assumed and denote the three possible
symmetry settings by $G_{\rm low}={\rm U}_{V}(1)\times{\rm SU}(2)_{S}$,
$G={\rm U}_{V}(1)\times{\rm SU}(2)_{K}\times{\rm SU}(2)_{K^{\prime}}$, and
$G_{\rm high}={\rm SU}(4)$. A hierarchy of energy scales lowers the symmetry
in steps $G_{\rm high}\rightarrow G\rightarrow G_{\rm low}$ Bultinck  _et al._
(2020). For the most part, we will ignore the weak intervalley Hunds coupling
and concern ourselves with either $G$ or the enlarged symmetry $G_{\rm high}$.
At the end we will discuss how the conclusions are modified on including the
weak Hunds coupling, which can be either ferromagnetic or antiferromagnetic,
and close with a discussion about the relevance to experiment.

_Summary of the results_ — Before presenting the details of our analysis, let
us summarize our main results. First, we note that the pairing function is a
matrix in the space of flat bands labelled by band, spin, and valley indices.
When considering pairing in a multi-orbital superconductor, it is important to
distinguish the cases of unitary and non-unitary pairing first discussed in
the context of Helium 3 Leggett (1975); Salomaa and Volovik (1987). Unitary
pairing corresponds to the case where the pairing matrix is proportional to a
unitary matrix which guarantees that its eigenvalues has the same magnitude
leading to a single pairing gap. In non-unitary pairing, this is not the case
and we can get different pairing gaps for different flavors e.g. different
gaps for the two spin species. Usually, unitary pairing is energetically
favored when the normal state neither breaks flavor symmetry Huy  _et al._
(2007); Aoki and Flouquet (2012) nor is close to a flavor-symmetry breaking
phase transition Nevidomskyy (2020a); Ran  _et al._ (2019a).

Our result can be phrased as a symmetry imposed constraint on the type of
unitary pairing possible in TBG. This follows from the simple observation that
there is no symmetric charge $2e$ superconductor either under $G_{\rm
high}={\rm SU}(4)$ or $G={\rm U}_{V}(1)\times{\rm SU}(2)_{K}\times{\rm
SU}(2)_{K^{\prime}}$ leaving one of two possibilities. First, a charge $2e$
superconductor which spontaneously breaks the global symmetry ($G$ or $G_{\rm
high}$) by transforming as a non-trivial representation. As we will show, such
superconductors always admit exotic vortices which carry a fractional flux.
These arise from the non-trivial flavor structure of the symmetry breaking
pairing function which allows us to compensate for a rotation in the phase of
the condensate with an internal rotation in the flavor space. This is in
complete analogy with the emergence of half-vortices in liquid Helium 3 or
spin-triplet superconductors Salomaa and Volovik (1987); Autti  _et al._
(2016); Mackenzie and Maeno (2003) where a $\pi$ phase rotation of the overall
phase is combined with an internal rotation in spin space. The second
possibility is a ($G$ or $G_{\rm high}$) charge $2me$ superconductor with
$m>1$. Such higher charge superconductor is allowed to be symmetric under the
global symmetry and it will automatically admit fractional vortices with flux
$h/2me$. Higher charge superconductivity can be easily detected experimentally
through the periodicity of the Fraunhofer pattern in Josephson junctions. In
this regard, we note an important distinction between the fractional vortices
of the $2e$ superconductor and the fractional vortices in higher charge
symmetric superconductors. While the former have logarithmically divergent
energies, even in the presence of a magnetic field, and can only be observed
in certain mesoscopic geometries Babaev (2002); Silaev (2011); Chung  _et al._
(2007), the latter has a finite energy and can be stabilized in the infinite
system. This difference can be understood by recalling the difference between
vortices in a superconductor whose logarithmic divergence is cancelled by the
coupling to the electromagnetic gauge field and a neutral superfluid where the
logarithmic divergence persists. The logarithmically divergent energy of the
$2e$ fractional vortices arise precisely because they combine a rotation of
the superconducting phase, which can be cancelled by an external flux, with an
internal flavor rotation which cannot. On the other hand, fractional vortices
in symmetric higher charge superconductors arise from pure phase winding which
can be completely cancelled with an external flux.

We discuss the implications of these results for finite temperature physics
and point out how, in certain cases, $2e$-superconductors with fractional
vortices are reduced by thermal fluctuations to higher charge superconductors.
We also demonstrate how these different scenarios for superconductivity are
explicitly realized in the recently proposed theory of skyrmion
superconductivity Khalaf  _et al._ (2020); Chatterjee  _et al._ (2020), by
constructing both $2e$ and $4e$ charged skyrmions near charge neutrality. We
note here a recent work Christos  _et al._ (2020) which also investigated
topological terms in TBG within a Dirac approximation and found level 1 WZW
terms for half-filling and level 2 WZW term for neutrality. These correspond
to cases (i) and (iii) in our analysis (see Fig. 1). Finally, we discuss the
effects of further symmetry lowering perturbations such as the intervalley
Hunds coupling.

Figure 1: Schematic illustration of the three possibilities for
superconducting pairing with the global symmetry $G={\rm U}(1)_{V}\times{\rm
SU}(2)_{K}\times{\rm SU}(2)_{K^{\prime}}$ or $G_{\rm high}={\rm SU}(4)$. Non-
unitary pairing (i) is expected in TBG for superconductivity in the vicinity
of flavor symmetry breaking whereas unitary charge $2e$ superconductivity with
fractional vortices (ii) or higher charge superconductivity (iii) is expected
in the absence of flavor symmetry breaking. The unitary charge $2e$
superconductor always breaks the global ($G$ or $G_{\rm high}$) symmetry and
is thus only possible at $T=0$. The possible order parameter manifolds for the
unitary charge $2e$ superconductor and its associated vortices are summarized
in Table 1.

To discuss the implications of these results for TBG, we need to distinguish
the cases of pairing in the presence or absence of flavor polarization. This
distinction can be made more clear by recalling some basic facts regarding
superconductivity in the presence of ferromagnetic or antiferromagnetic order.
Although both spontaneously break ${\rm SU}(2)$ rotation symmetry, only the
former has finite spin polarization, i.e. the state has a finite charge under
a ${\rm U}(1)$ subgroup of ${\rm SU}(2)$. This means that, at least in weak
coupling, we have a spin split Fermi surface where we expect pairing to take
place independently leading generically to non-unitary pairing. On the other
hand, an antiferromagnet breaks ${\rm SU}(2)$ but still retains some discrete
symmetries relating the two spin species which usually results in the same
pairing amplitude in the two spin species, leading generically to unitary
pairing. These ideas can be generalized to TBG by definition a flavor
polarized state to be any state which carries a finite charge under any ${\rm
U}(1)$ subgroup of $G$, indicating finite spin or valley polarization. Note
that many symmetry breaking state are not flavor polarized in analogy to the
antiferromagnetic case. For instance, the intervalley coherent states
considered in Ref. Bultinck  _et al._ (2020) or the
$C_{2}\mathcal{T}$-breaking valley Hall states Liu  _et al._ (2019); Xie and
MacDonald (2020); Bultinck  _et al._ (2020) are _not_ flavor polarized despite
breaking some symmetries since they still manage to retain some discrete
symmetries (the so-called Kramers time-reversal symmetry Bultinck  _et al._
(2020)).

This leads to the following conjectures about the nature of superconducting
pairing in TBG, summarized in Fig. 1. First, the superconductors obtained in
regions with flavor polarization – which can be identified experimentally
through the evolution of the Hall number or quantum oscillations – likely
correspond to non-unitary pairing, which is not constrained by our theory.
This corresponds to the majority of the superconductivity observed in
experiments close to half filling Cao  _et al._ (2018a, b); Yankowitz  _et
al._ (2019); Saito  _et al._ (2020); Arora  _et al._ (2020). However, in
certain isolated cases, superconductivity in the vicinity of charge neutrality
Lu  _et al._ (2019); Stepanov  _et al._ (2019), or reached from charge
neutrality in the absence of flavor polarization Stepanov  _et al._ (2019);
Arora  _et al._ (2020); Park  _et al._ (2020), has also been observed. We
conjecture that these cases present a fertile hunting ground for exotic forms
of superconductivity, either paired superconductors featuring fractional
vortices or even higher charge condensates such as $4e$ superconductors. We
hope future experiments will search for signatures of such novel condensates.

_Symmetry transformation of charge $2e$ superconductors_— Let us begin by
considering the transformation properties of an arbitrary charge $2e$
superconductor under a general unitary transformation. A charge $2e$
superconductor is generally described by an order parameter $\Delta$ which
transforms as $\Delta\mapsto\Delta e^{2i\phi}$ under global ${\rm U}_{C}(1)$
charge conjugation which acts on the electrons as $c\mapsto e^{i\phi}c$. This
breaks ${\rm U}_{C}(1)$ down to $\mathds{Z}_{2}$. We take the electron
operators $c_{a}$ to be labelled by an index $a$. For example, in a
translationally symmetric system, this index $a=(\alpha,{\bm{k}})$ includes
both orbital $\alpha$ and momentum ${\bm{k}}$ degrees of freedom. The pairing
function is thus taken to be a matrix of the form $\Delta_{ab}=\langle
c_{a}c_{b}\rangle$. Under a general unitary transformation $c\mapsto Uc$,
$\Delta$ transforms the same way as $cc^{T}$ leading to

$\Delta\mapsto U\Delta U^{T}$ (1)

Note that this relation holds as long as $\Delta$ transforms the same way as
$cc^{T}$ and does not make any addition assumption on its nature. In
particular, it holds well beyond the weak coupling BCS limit.

Throughout most of this paper, we will focus on global symmetries which only
act on the orbital index $\alpha$ of the electron operator. This orbital index
will be taken to label the 8 flat bands of TBG. We employ the Chern-
sublattice-spin basis defined in Refs. Bultinck  _et al._ (2020); Khalaf  _et
al._ (2020) with Pauli matrices $\gamma_{z}=\pm$ distinguishing the $\pm$
Chern sectors, $\eta_{z}=\pm=K/K^{\prime}$ distinguishing the two valleys and
$s_{z}=\pm=\uparrow/\downarrow$ distinguishing the two spins. The generators
of ${\rm U}(4)={\rm U}(1)_{C}\times G_{\rm high}$ are
$\gamma_{0}\eta_{\mu}s_{\nu}$ where $\mu,\nu=0,x,y,z$ whereas the generators
of ${\rm U}(2)\times{\rm U}(2)={\rm U}(1)_{C}\times G$ are
$\gamma_{0}\eta_{0,z}s_{\nu}$.

We first note that the generators for $G_{\rm high}$ (and as a result also the
generators of $G\subset G_{\rm high}$) are all proportional to $\gamma_{0}$.
This means that the pairing function can be decomposed into different pairing
channels in the Chern space which can be considered independently

$\Delta=\sum_{\mu=0,x,y,z}\Delta_{\mu}\gamma_{\mu},$ (2)

Here, $\Delta_{\mu}$ satisfies

$\Delta_{\mu}^{T}=\begin{cases}-\Delta_{\mu}&:\mu=0,x,z\\\
\Delta_{\mu}&:\mu=y\end{cases}$ (3)

Although the results in the remainder of this paper hold for arbitrary pairing
functions $\Delta_{\mu}$, it is instructive to consider the case of
translationally symmetric pairing between time-reversal related momenta
${\bm{k}}$ and ${\bm{k}}^{\prime}=-{\bm{k}}$ given by
$\Delta({\bm{k}},{\bm{k}}^{\prime})=\Delta({\bm{k}})\delta_{{\bm{k}},-{\bm{k}}^{\prime}}$.
This allows us to classify the pairing function according to their
transformation properties under spatial rotations into $s$, $p$, $d$, etc. We
then see from Eq. 3 that
$\Delta_{0,x,z}(-{\bm{k}})^{T}=-\Delta_{0,x,z}({\bm{k}})$ whereas
$\Delta_{y}(-{\bm{k}})^{T}=\Delta_{y}({\bm{k}})$. Thus, antisymmetric pairing
in orbital space $\Delta_{\mu}^{T}({\bm{k}})=-\Delta_{\mu}({\bm{k}})$ is
associated with even momentum pairing ($s$, $d$, etc) for
$\Delta_{0,x,z}({\bm{k}})$ and with odd momentum pairing ($p$, $f$, etc) for
$\Delta_{y}({\bm{k}})$. For symmetric pairing, the opposite is true.
Physically, we expect the pairing between Chern sectors $\Delta_{x,y}$ to be
favored compared to pairing within a Chern sector $\Delta_{0,z}$. However, in
the following, we will keep our discussion as general as possible.

Let us briefly comment on the role of spatial symmetries. First, they place
constraints on the ${\bm{k}}$-dependence of the pairing function
$\Delta_{\mu}({\bm{k}})$. In addition, as shown in the supplemental material,
symmetries which exchange valleys, such as mirror or two-fold rotation, relate
the two intra-Chern pairing channels $\gamma_{\pm}=\frac{1\pm\gamma_{z}}{2}$
which will then form a higher-dimensional representation of the full symmetry
group. These considerations, which are analyzed in detail in the supplemental
material, do not affect any of the conclusions we will derive below which only
rely on the transformation properties under global symmetries. For notational
simplicity, we will usually drop the ${\bm{k}}$ dependence of the pairing
function since we are mostly concerned with the symmetry transformation
properties under global symmteries which only act on the internal (orbital)
indices. We will also drop the $\mu$ subscript since the action of the global
symmetries do not mix the different Chern channels.

_Absence of symmetric charge $2e$ superconductors_— Let us first consider the
high-symmetry limit with $G_{\rm high}={\rm SU}(4)$ whose action on $\Delta$
is given by Eq. 1. One fundamental observation is that it is not possible to
choose a non-zero pairing matrix $\Delta$ which transforms as a singlet under
${\rm SU}(4)$. Instead, the action of ${\rm SU}(4)$ on $\Delta$ decomposes
into a sum of a 6-dimensional and a 10-dimensional representations. This can
be understood by noting that the action of ${\rm SU}(4)$ corresponds to the
tensor product of two copies of the fundamental representation $\bm{4}$ which
decomposes as ${\bm{4}}\otimes{\bm{4}}={\bm{6}}\oplus\bm{10}$. The ${\bm{6}}$
and $\bm{10}$ correspond to the antisymmetric and symmetric tensor
representations of ${\rm SU}(4)$, respectively.

Notice that ${\rm SU}(4)$ symmetry mixes inter-valley and intra-valley pairing
and there is no sense in distinguishing the two. This may seem at odds with
our physical expectation that pairing takes places between time-reversal
related states which live in opposite valleys. However, note that ${\rm
SU}(4)$ allows for unitary rotations which mix the two valleys. As a result,
we can define a modified time-reversal symmetry $\mathcal{T}^{\prime}$ that
acts within the same valley by combining such rotation with the standard time-
reversal $\mathcal{T}$. The existence of intra-valley time-reversal was noted
in Ref. Bultinck  _et al._ (2020) and more recently studied in Ref. Wang  _et
al._ (2020).

Next, we can consider what happens in the more realistic limit where $G_{\rm
high}$ is broken down to $G={\rm U}(1)\times{\rm SU}(2)\times{\rm SU}(2)$. Let
us first consider the ${\rm SU}(2)\times{\rm SU}(2)$ part. The antisymmetric
representation $\bm{6}$ splits into $\bm{4}\oplus\bm{1}\oplus\bm{1}$ under
${\rm SU}(2)\times{\rm SU}(2)$ whereas the symmetric representation splits
into $\bm{10}=\bm{4}\oplus\bm{3}\oplus\bm{3}$. These can be understood as
follows: first recall the representations of ${\rm SU}(2)$ are labelled by a
half-integer $S$ with dimension $2S+1$. Thus, the 1D irrep of ${\rm
SU}(2)\times{\rm SU}(2)$ is simply the singlet representation in both valleys
$(S_{K},S_{K^{\prime}})=(0,0)$. The 3D irreps above correspond to a singlet in
one valley and triplet in the other $(S_{K},S_{K^{\prime}})=(0,1)$ or $(1,0)$
whereas the 4D irreps correspond to $S=1/2$ in both valleys. Alternatively,
this 4D irrep can be understood as the fundamental of $\rm{Spin}(4)$ which is
isomorphic to ${\rm SU}(2)\times{\rm SU}(2)$. Note that, unlike the ${\rm
SU}(4)$ case, it is possible to have a ${\rm SU}(2)\times{\rm SU}(2)$ singlet.
These correspond to the intravelley pairing channels $\propto\eta_{0,z}$.
However, these pairing channels carry a charge under ${\rm U}_{V}(1)$. Thus,
similar to the $G_{\rm high}$ case, there is no symmetric pairing channel
under the full $G$ symmetry. We note that our statement for the symmetry group
$G$ is consistent with the symmetry analysis of Ref. Scheurer and Samajdar
(2020).

Although the above argument applies to all pairing channels, the most natural
expectation in the case of the symmetry group $G$ is pairing between time-
reversal related electrons in opposite valleys You and Vishwanath (2019);
Isobe  _et al._ (2018); Classen  _et al._ (2020); Chichinadze  _et al._
(2020). Restriction to intervalley pairing is equivalent to the condition
$\\{\Delta,\eta_{z}\\}=0$. This selects 8 of the 16 generators of ${\rm U}(4)$
given explicitly by $\eta_{x,y}s_{\nu}$, $\nu=0,x,y,z$. Among these
generators, four are symmetric and four are antisymmetric. Thus, the action of
${\rm SU}(2)\times{\rm SU}(2)$ on the inter-valley pairing (which is invariant
under ${\rm U}_{V}(1)$) splits into $\bm{4}\oplus\bm{4}$. For the remainder of
this work, we will focus on the physically relevant case of intervalley
pairing. The case of intravalley pairing is discussed in the supplemental
material for completeness.

_Charge $4e$ superconductors_— The previous discussion leaves open the more
intriguing possibility of a symmetric charge $4e$ condensate. The existence of
such condensate can be seen by first writing the charge $4e$ pairing function

$\Delta_{\alpha\beta\gamma\delta}=\langle
c_{\alpha}c_{\beta}c_{\gamma}c_{\delta}\rangle$ (4)

$\Delta_{\alpha\beta\gamma\delta}$ transforms as

$\Delta_{\alpha\beta\gamma\delta}\mapsto
U_{\alpha\alpha^{\prime}}U_{\beta\beta^{\prime}}U_{\gamma\gamma^{\prime}}U_{\delta\delta^{\prime}}\Delta_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}$
(5)

A singlet pairing channel is obtained by taking $\Delta$ to be proportional to
antisymmetric tensor
$\Delta_{\alpha\beta\gamma\delta}\propto\epsilon_{\alpha\beta\gamma\delta}$
symbol since

$U_{\alpha\alpha^{\prime}}U_{\beta\beta^{\prime}}U_{\gamma\gamma^{\prime}}U_{\delta\delta^{\prime}}\epsilon_{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}=\det
U\epsilon_{\alpha\beta\gamma\delta}=\epsilon_{\alpha\beta\gamma\delta}$ (6)

Thus, in contrast to the charge $2e$ superconductor, an ${\rm SU}(4)=G_{\rm
high}$ symmetric charge $4e$ superconductor is possible. This conclusion also
holds for the symmetry group $G\subset G_{\rm high}$.

_Unitary pairing and fractional vortices_ — We start by discussing the case of
unitary pairing Leggett (1975) which is typically energetically favored if
pairing takes place equally in all flavors. In TBG, we expect unitary pairing
in the vicinity of charge neutrality or in samples with no signature of flavor
symmetry breaking Lu  _et al._ (2019); Stepanov  _et al._ (2019) as will be
discussed later. For unitary pairing, $\Delta\Delta^{\dagger}$ is proportional
to the identity matrix so that we can rescale $\Delta$ to be unitary (away
from the points where it vanishes).

To study different types of vortices, we start by noting that the overall
phase for a unitary superconductor described by an $n\times n$ matrix order
parameter $\Delta$ is defined as

$\varphi=\frac{1}{n}\arg\det\Delta$ (7)

This definition guarantees that the phase changes as
$\varphi\mapsto\varphi+\phi$ whenever we multiply $\Delta$ by an overall phase
$\phi$. The phase $\varphi$ couples to the background gauge field with charge
$2e$ according to the Ginzburg-Landau (GL) functional

$\mathcal{L}\propto(\hbar\partial_{\mu}\varphi-2eA_{\mu})^{2}+\dots$ (8)

Variations with respect to $A$ leads to

$\int dl_{\mu}A_{\mu}=\frac{\hbar}{2e}\int dl_{\mu}\partial_{\mu}\varphi$ (9)

which is equivalent to the condition of vanishing current in the ground state.
Let us see the implications of this for the different symmetry settings.

Let us first discuss the more physical $G$ symmetry and focus on the
physically relevant intervalley pairing. This can be decomposed into a
symmetric and an antisymmetric component given by

$\Delta^{\rm S/A}=\left(\begin{array}[]{cc}0&\tilde{\Delta}\\\
\pm\tilde{\Delta}^{T}&0\end{array}\right)_{\eta}$ (10)

where $\tilde{\Delta}$ is unitary. Notice that due to the ${\rm
SU}(2)\times{\rm SU}(2)$ symmetry, the ${\rm SU}(2)$ spin singlet and triplet
components are joined in a single 4-dimensional irrep. As a consequence, we
cannot split $\Delta^{\rm S/A}$ further into a valley-singlet spin-triplet
channel and a valley-triplet spin-singlet channel Lee  _et al._ (2019);
Scheurer and Samajdar (2020). This is only possible in the presence of Hund’s
coupling which breaks ${\rm SU}(2)_{K}\times{\rm SU}(2)_{K^{\prime}}$ down to
${\rm SU}(2)_{S}$ as we will discuss later. For either the symmetric or
antisymmetric representation, we can write
$\varphi=\frac{1}{4}\arg\det\Delta=\frac{1}{2}\arg\det\tilde{\Delta}$. Using
(9), we see that if only one of the two eigenvalues of $\tilde{\Delta}$ winds
by $2\pi$, the phase $\varphi$ winds by $\pi$ leading to a vortex with a flux
of $h/4e$.

The case of ${\rm SU}(4)$ symmetry can be analyzed similarly. For the
antisymmetric representation with $\Delta^{T}=-\Delta$, the eigenvalues come
in pairs $\\{\lambda,-\lambda\\}$ which means that the winding of $\varphi$ is
necessarily a multiple of $\pi$ leading also to half-vortices. For the
symmetric representation, there is no constraint on the winding of individual
eigenvalues of $\Delta$, which means that the winding of $\varphi$ is a
multiple of $\pi/2$ leading to quarter vortices with flux $h/8e$ in this case.

Finally note that the case of intravalley pairing for the symmetry group $G$
can be obtained from the high symmetry case $G_{\rm high}$ as follows. Unitary
intravalley pairing splits into an antisymmetric representation corresponding
to spin-singlet in both valleys and a symmetric representation corresponding
to spin-triplet pairing in both valleys. These can be obtained by restricting
the ${\rm SU}(4)$ antisymmetric and symmetric representations using the
condition $[\Delta,\eta_{z}]=0$ which does not affect the flux quantization
condition leading to $h/4e$ vortices for the former and $h/8e$ vortices in the
latter. A more detailed analysis of these cases is given in the supplemental
material.

_Order parameter manifolds for unitary pairing_ — We will now present an
alternative and more abstract argument for the existence of fractional
vortices in the different unitary charge $2e$ superconductors. This argument
will not rely on any GL functional. Let us again start with the case of
intervalley pairing with $G$ symmetry. The symmetric/antisymmetric pairing
channels can be generated starting with an arbitrary symmetric/ antisymmetric
valley off-diagonal matrix and applying ${\rm U}(2)\times{\rm U}(2)$ symmetry
according to (1). The resulting pairing matrix is

$\Delta^{\rm S,A}=\left(\begin{array}[]{cc}0&U_{1}U_{2}^{T}\\\ \pm
U_{2}U_{1}^{T}&0\end{array}\right)_{\eta},\quad\implies\quad\tilde{\Delta}=U_{1}U_{2}^{T}$
(11)

Thus, $\tilde{\Delta}$ is parametrized by a pair of $2\times 2$ unitary
matrices $U_{1}$ and $U_{2}$ modulo the operation $U_{1}\mapsto U_{1}V$,
$U_{2}\mapsto U_{2}V^{*}$ for any $2\times 2$ unitary matrix $V$. This means
that $\tilde{\Delta}$ parametrizes the coset space ${\rm U}(2)\times{\rm
U}(2)/{\rm U}(2)\simeq{\rm U}(2)$. The existence of half-vortices can be
understood as follows. A full vortex is a vortex in the overall phase of
$\tilde{\Delta}$ which corresponds to a loop in the subgroup ${\rm U}_{C}(1)$
generated by $\tilde{\Delta}=e^{i\phi}$ with $\phi:0\rightarrow 2\pi$. On the
other hand, we can write the decomposition ${\rm U}(2)=\frac{{\rm
U}_{C}(1)\times{\rm SU}(2)}{\mathds{Z}_{2}}$ which is seen more explicitly by
writing $\tilde{\Delta}$ as $e^{i\varphi}V$ with $V\in{\rm SU}(2)$. The
identification of $(\varphi,V)$ and $(\varphi+\pi,-V)$ allows for a $\pi$
vortex in the superconducting phase $\varphi$ combined with a rotation which
sends $V$ to $-V$. This can be written as $\pi_{1}({\rm U}(2))=\mathds{Z}/2$,
where we identify the element $1\in\mathds{Z}$ with the fundamental loop in
${\rm U}_{C}(1)$ generated by $\tilde{\Delta}=e^{i\phi}$, $\phi:0\rightarrow
2\pi$. The minimal loop in this space $1/2\in\mathds{Z}/2$ thus corresponds to
a half quantum vortex with $h/4e$ flux.

The existence of half (quarter) vortices for the antisymmetric (symmetric)
repsenetations of ${\rm SU}(4)$ can also be understood by analyzing the order
parameter manifolds. For the antisymmetric representation $\Delta_{\mu}^{A}$,
the manifold is generated by acting with ${\rm U}(4)$ on any given unitary
antisymmetric matrix according to (1) leading to

$\Delta^{A}=U\eta_{y}s_{0}U^{T}$ (12)

where $U\in{\rm U}(4)$. We notice that $\Delta_{\mu}^{A}$ is invariant under
multiplying $U$ from the right by any unitary matrix $K$ satisfying

$K\eta_{y}s_{0}K^{T}=\eta_{y}s_{0}$ (13)

which is the definition of a $4\times 4$ symplectic matrix $K\in\rm{Sp}(4)$.
Thus, $\Delta_{\mu}^{A}$ parametrizes the coset space ${\rm U}(4)/\rm{Sp}(4)$.
Writing $U=e^{i\theta}V$ where $V\in{\rm SU}(4)$, we see that we can form a
closed path in ${\rm U}(4)$ by taking $V\mapsto-iV$ and
$\theta\mapsto\theta+\pi/2$. We note however that doing this twice yields a
regular vortex since the superconducting phase $\varphi$ is twice $\theta$ and
the path $V\mapsto-V$ can be deformed into a path $K\mapsto-K$ lying
completely in the symplectic group which is identified with the identity.
Thus, $\pi_{1}\left(\frac{{\rm U}(4)}{\rm{Sp}(4)}\right)=\mathds{Z}/2$. A
similar argument applies to the symmetric 10-dimensional representation. In
this case, the order parameter manifold $\frac{{\rm U}(4)}{{\rm O}(4)}$ which
admits half-vortices similar to the case of $\frac{{\rm U}(4)}{\rm{Sp}(4)}$.
However, due to the structure of the ${\rm O}(4)$ group, we can also form
_quarter_ vortices. These are obtained by taking a trajectory in ${\rm U}(4)$
which starts at an element $K_{+}\in O(4)$ with $\det K_{+}=1$ and ends at an
element $K_{-}$ with $\det K_{-}=-1$. Since $O(4)\subset U(4)$ and ${\rm
U}(4)$ is path connected, we can always find such trajectory. This trajectory
is a loop in the coset space $\frac{{\rm U}(4)}{{\rm O}(4)}$ which involves a
winding of the phase of $\theta$ by $(2l+1)\frac{\pi}{4}$ yielding a quarter
vortex for $l=0$ of the superconducting phase $\varphi=2\theta$. An explicit
example of such trajectory is given by the diagonal matrix $U={\rm
diag}(1,1,1,e^{i\theta})$, $\theta:0\mapsto\pi$ which connects the identity to
the matrix $U={\rm diag}(1,1,1,-1)$ which is contained in ${\rm O}(4)$.

In general, we can derive the flux quantization condition without a GL
function by employing the following two axioms: (a) the flux is identified as
a homomorphism from the first homotopy group of the order parameter manifold,
which we will denote by $\mathcal{M}_{\Delta}$, to the real numbers
$F:\pi_{1}(\mathcal{M}_{\Delta})\mapsto\mathbbm{R}$ and (b) the element of
$\pi_{1}(\mathcal{M}_{\Delta})$ corresponding to the fundamental loop in ${\rm
U}_{C}(1)$ generated by $e^{i\phi}$, $\phi:0\rightarrow 2\pi$, is assigned a
flux of $h/2e$. Using this definition, we can reproduce the half and quarter
vortices identified above. For example, the full vortices of the manifold
${\rm U}(2)$ were identified with the element $1\in\pi_{1}({\rm U}(2))$ and
the half vortices were identified with the element $1/2$. Since the flux $F$
is a homomorphism, we have $2F(1/2)=F(1)=h/2e\,\implies\,F(1/2)=h/4e$.

_Non-unitary pairing_ — Let us now see what happens if we lift the restriction
of unitary pairing. Non-unitary pairing is expected if the parent state has
broken flavor symmetry such that pairing takes place in each flavor
separately. Although not very common, there are a few known examples for non-
unitary pairing, particularly in ferromagnetic superconductors Aoki and
Flouquet (2012); Huy  _et al._ (2007); Aoki  _et al._ (2019a, b) where pairing
develops on top of a spin-split Fermi surface and has different amplitude for
the different spin species. Another example is the $A_{1}$ phase of He3
Salomaa and Volovik (1987). In the following, we will show that for non-
unitary pairing, it is possible to have no fractional vortices. For this
purpose, it suffices to focus on the most physically relevant case of
intervalley pairing with $G={\rm U}(1)\times{\rm SU}(2)\times{\rm SU}(2)$
symmetry. In general, we can specify the pairing function $\Delta$ using the
eigenvalues of $\Delta^{\dagger}\Delta$ which are invariant under the symmetry
transformation (1). For inter-valley pairing, the pairing function has the
form given in Eq. 10. Thus, the most general form of the pairing functions
$\tilde{\Delta}$ is

$\tilde{\Delta}=U_{1}\left(\begin{array}[]{cc}\lambda_{1}&0\\\
0&\lambda_{2}\end{array}\right)U_{2}^{T}$ (14)

For $|\lambda_{1}|\neq|\lambda_{2}|$, this corresponds to so-called two gap
superconductors with different amplitudes of the gap function for the
different electron species Babaev (2002). The GL theory in this case is given
by

$\mathcal{L}=\rho_{1}(\partial_{\mu}\varphi_{1}-\frac{2e}{\hbar}A_{\mu})^{2}+\rho_{2}(\partial_{\mu}\varphi_{2}-\frac{2e}{\hbar}A_{\mu})^{2}+\dots$
(15)

where the two ${\rm U}(1)$ phases $\varphi_{1,2}=\arg\lambda_{1,2}$ correspond
to the gap functions of the two different spin species. The vortex
quantization condition is

$\int dl_{\mu}A_{\mu}=\frac{\hbar}{2e}\int
dl_{\mu}\frac{\rho_{1}\partial_{\mu}\varphi_{1}+\rho_{2}\partial_{\mu}\varphi_{2}}{\rho_{1}+\rho_{2}}$
(16)

We see that this generically leads to fractional vortices which depend on the
relative stiffness of the two superconductors when both $\lambda_{1}$ and
$\lambda_{2}$ are non-zero Babaev (2002).

The existence of fractional vortices can also be understood from our more
abstract definition in the previous section. The order parameter manifold can
be understood as follows. Allowing $\lambda_{1}$ and $\lambda_{2}$ to be
complex, we can absorb the overall phases of $U_{1}$ and $U_{2}$ as well as
the effect of the transformations $U_{1,2}\mapsto
e^{i\phi_{1,2}\sigma_{z}}U_{1,2}$ in the phases of $\lambda_{1}$ and
$\lambda_{2}$. This means that $\tilde{\Delta}$ parametrizes the manifold
${\rm U}(1)\times{\rm U}(1)\times S^{2}\times S^{2}$ which can be understood
as follows. Due to the independent ${\rm SU}(2)$ spin rotation in each valley,
we can choose the spin quantization axis independently in the two valleys
leading to two unit vectors parametrizing $S^{2}\times S^{2}$. The two ${\rm
U}(1)$ phases describe the phase of the pairing function for the two spin
species. Note that $\pi_{1}({\rm U}(1)\times{\rm U}(1)\times S^{2}\times
S^{2})=\mathds{Z}\oplus\mathds{Z}$ where the fundemantal vortex in ${\rm
U}_{C}(1)$ corresponds to the element $(1,1)\in\mathds{Z}\oplus\mathds{Z}$
which represents $2\pi$ phase winding in both ${\rm U}(1)$ phases. The flux
quantization condition can then be written as
$F[(1,1)]=h/2e=F[(0,1)]+F[(1,0)]$. In the absence of any symmetry relating
$(0,1)$ and $(1,0)$, i.e. the two spin species, we can write $F[(0,1)]=xh/2e$
for some arbitrary $0\leq x\leq 1$ which implies $F[(1,0)]=(1-x)h/2e$.
Comparing with (16), we can identify $x$ with $\rho_{2}/(\rho_{1}+\rho_{2})$.
Note that whenever the two spin species are related by an unbroken symmetry,
$x=1/2$ and we recover half vortices as in the unitary case.

In the extreme non-unitary limit where pairing takes place in only one
species, i.e. one of $\lambda_{1,2}$ (and consequently $\rho_{1,2}$) is zero,
there are only regular $h/2e$ vortices. This can be seen by noting that the
order parameter manifold for this case reduces to ${\rm U}(1)\times
S^{2}\times S^{2}$ because phase rotations on the vanishing eigenvalue have no
effect. This manifold has a single ${\rm U}(1)$ with the GL functional (8)
leading to the same quantization condition (9) wheres the winding of $\varphi$
has to be a multiple of $2\pi$ leading to standard $h/2e$ flux. This serves to
illustrate that the non-unitary pairing allows for a conventional scenario
with a charge $2e$ superconductor that does not host fractional vortices.

_No-go theorem_ — In summary, we have proved the following general statement:
For any pairing in the ideal ${\rm SU}(4)$ symmetric model or the more
realistic ${\rm U}(1)\times{\rm SU}(2)\times{\rm SU}(2)$ model, one of three
possibilities should occur: (i) the superconducting pairing is non-unitary,
(ii) the superconducting condensate has charge $2e$ but breaks the global
symmetry and admits at least half quantum vortices with flux of $h/4e$ , or
(iii) the superconducting condensate has charge of at least $2me$ with $m>1$
with vortices carrying $h/2me$ flux.

Note an important difference between the cases (ii) and (iii). The fractional
vortices of the charge $2e$ superconductor combine a winding in the overall
phase with a winding in internal orbital indices. Whereas the former is
compensated by external magnetic flux, the latter is not, leading to
logarithmically divergent energy at long distances. This means that these
vortices are only relevant at sufficiently short distances e.g. in mesoscopic
geometries. Such type of vortices were already observed in mesoscopic samples
in liquid Helium 3 Autti  _et al._ (2016) and spin-triplet condensates Seo
_et al._ (2015). On the other hand, vortices of a charge $2me$ superconductor
with $m>1$ which carry a fractional flux of $h/2me$ have finite energy and can
be observed in the infinite system.

Symmetry | Irreps | generators | SC Manifold (unitary pairing) | Vortices
---|---|---|---|---
${\rm SU}(4)$ | $\bm{6}$ |  $\eta_{y}s_{0,x,z}$, $\eta_{0,x,z}s_{y}$ | $\frac{{\rm U}(4)}{\rm{Sp}(4)}$ | $\mathds{Z}/2$
$\bm{10}$ |  $\eta_{0,x,z}s_{0,x,z}$, $\eta_{y}s_{y}$ | $\frac{{\rm U}(4)}{{\rm O}(4)}$ | $\mathds{Z}/4$
${\rm SU}(2)\times{\rm SU}(2)\simeq\rm{Spin}(4)$ Inter-valley pairing | $\bm{4}$ |  $\eta_{y}s_{0,x,z}$, $\eta_{x}s_{y}$ | ${\rm U}(2)$ | $\mathds{Z}/2$
$\bm{4}$ |  $\eta_{y}s_{y}$, $\eta_{x}s_{0,x,z}$ | ${\rm U}(2)$ | $\mathds{Z}/2$
Table 1: Summary of the order parameter manifolds and the irreducible
representations for the unitary pairing charge $2e$ superconductors. For the
ideal ${\rm SU}(4)$ case, the symmetry action splits into a 6-dimensional
antisymmetric and a 10-dimensional symmetric irrep whose corresponding
manifolds admit half and quarter quantum vortices respectively. For the more
realistic ${\rm U}(1)_{V}\times{\rm SU}(2)\times{\rm SU}(2)$, we focus on the
physically relevant inter-valley pairing which is ${\rm U}(1)_{V}$ singlet and
splits into two 4-dimenstional irreps under ${\rm SU}(2)\times{\rm SU}(2)$
whose manifolds admit half vortices.

_Nonzero temperature and BKT transitions_ —Here we discuss how the different
scenarios we have discussed manifest at nonzero temperatures. At nonzero
temperature in two spatial dimensions fractional vortices in charge $2e$
superconductors manifest as pure phase windings of a higher charge
superconductor once a non-zero temperature is included; above a critical
temperature $T_{c}$ the superconductivity is lost due to a BKT transition
driven by the condensation of these fractional vortices Mukerjee  _et al._
(2006); Radzihovsky and Vishwanath (2009); Berg  _et al._ (2009); Jian and
Zhai (2011); Xu and Balents (2018).

We now describe this picture for the candidate superconductors allowed by our
statement, starting with the case of ${\rm U}(4)$ symmetry. The Mermin-Wagner
theorem forbids the continuous symmetry breaking of the charge $2e$
superconductors, but algebraic order in the ${\rm U}(1)$ phase may still
persist. In particular, the unitary charge $2e$ order parameters $\Delta^{\rm
S,A}$ that parameterize the manifolds ${\rm U}(4)/\rm{Sp}(4)$ and ${\rm
U}(4)/{\rm O}(4)$ respectively must be disordered, but the respective
submanifolds ${\rm U}(1)/\mathds{Z}_{4}$ and ${\rm U}(1)/\mathds{Z}_{8}$
retain XY order that melts for sufficiently high temperatures through a BKT
transition. The BKT transition is driven by the condensation of half quantum
vortices and quarter-quantum vortices for the ${\rm SU}(4)$ antisymmetric and
symmetric superconductors, respectively. We may interpet this in terms of
higher charge superconductors as follows. For $T>0$ the charge $2e$
superconductors are unstable: $\Delta^{\rm A,S}$ is disordered. However, for
the antisymmetric case,

$\Delta_{4e}=\varepsilon^{\alpha\beta\gamma\delta}\Delta^{A}_{\alpha\beta}\Delta^{A}_{\gamma\delta}$
(17)

is an ${\rm SU}(4)$ singlet and parameterizes ${\rm U}(1)/\mathds{Z}_{4}$. It
retains algebraic order and gives rise to a charge $4e$ superconductor for
$T>0$. The $h/4e$ vortices in $\Delta_{4e}$ are the finite temperature
remnants of the half quantum vortices in $\Delta^{A}_{2e}$. Similarly the
operator

$\Delta_{8e}=\varepsilon^{\alpha\beta\gamma\delta}\varepsilon^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}\delta^{\prime}}\Delta^{S}_{\alpha\alpha^{\prime}}\Delta^{S}_{\beta\beta^{\prime}}\Delta^{S}_{\gamma\gamma^{\prime}}\Delta^{S}_{\delta\delta^{\prime}}$
(18)

is an ${\rm SU}(4)$ singlet built out of $\Delta^{S}$ that parameterizes ${\rm
U}(1)/\mathds{Z}_{8}$. It can have algebraic order for $T>0$ and host $h/8e$
vortices.

For intervalley pairing in the ${\rm U}(1)\times{\rm SU}(2)\times{\rm SU}(2)$
symmetric case, the operator

$\tilde{\Delta}_{4e}=\varepsilon^{ab}\varepsilon^{a^{\prime}b^{\prime}}\tilde{\Delta}_{aa^{\prime}}\tilde{\Delta}_{bb^{\prime}}$
(19)

is a singlet with charge $4e$. Here we used (10) to write the order parameter
in terms of an arbitrary unitary matrix $\tilde{\Delta}$. The unprimed and
primed indices transform under the first and second copy of ${\rm SU}(2)$,
rotations in valley $K$ and $K^{\prime}$, respectively. For the antisymmetric
representation (17) and (19) coincide up to a constant.

It is possible that at $T=0$ these higher charge superconductors are the
ground state anyway and there is no charge $2e$ order at $T=0$. The $T>0$
phenomena of higher charge superconductivity are similar in both cases, though
the correlation length for charge $2e$ order will diverge exponentially as
$T\to 0$ if a charge $2e$ superconductor is the quantum ground state. For
Josephson junctions with linear extent smaller than the charge $2e$
correlation length the system will behave like a charge $2e$ superconductor.

Symmetry Lowering by Intervalley Hunds Coupling— Ultimately ${\rm
SU}(2)\times\ {\rm SU}(2)$ symmetry is broken down to its diagonal subgroup of
overall spin and charge rotations ${\rm SU}(2)$ by intervalley Hunds terms
Zhang  _et al._ (2019); Lee  _et al._ (2019); Scheurer and Samajdar (2020).
These terms are suppressed by the lattice to moiré scale $a/L\ll 1$, and have
magnitude $\absolutevalue{J_{H}}\approx 0.2-0.5\rm{meV}$. Nonzero $J_{H}$
splits the $\bf{4}$ representation of ${\rm SU}(2)\times{\rm SU}(2)$ to
$\bf{3}\oplus\bf{1}$ of ${\rm SU}(2)$, spin triplet and spin singlet
respectively Scheurer and Samajdar (2020). The sign of $J_{H}$ is unknown and
determines whether spin triplet or spin singlet states are favored.

If the singlet state is favored, then at zero temperature there will only be
full quantum vortices. The critical temperature for the spin-singlet pairing
will be smaller than the critical temperature for charge $4e$ order for
sufficiently small $J_{H}$. By analogy with the Heisenberg model with a small
easy plane bias Khokhlachev (1976); R.A. Pelcovits (1976); Hikami and Tsuneto
(1980); Cuccoli  _et al._ (1995), we expect the critical temperature to have
the following form as $J_{H}\to 0$

$T_{C\text{ singlet,\,}2e}=\frac{c_{1}\rho}{\log\frac{c_{2}\rho}{J_{H}}},$
(20)

where $c_{1,2}$ are numbers and $\rho$ is the phase stiffness. Thus even with
small but finite $J_{H}$ due to the slow variation of the logarithm, there may
not be a significant separation of scales between $T_{C\text{ singlet,\,}2e}$
and $T_{C\,4e}\propto\rho$. On the other hand, if the triplet state is favored
and the pairing is unitary, half quantum vortices persist. An extreme non-
unitary triplet state is worth mentioning as well, and was considered in Ref.
Cornfeld  _et al._ , 2020. Here the order parameter actually parameterizes
${\rm SO}(3)$ which has fundamental group $\mathds{Z}_{2}$ – a double vortex
is no longer a metastable configuration. Here there is no BKT transition
Scheurer and Samajdar (2020), however; the $\mathds{Z}_{2}$ nature of the
vortices implies that there is no composite operator built from the zero
temperature order parameter that retains XY order for nonzero temperatures.

Spin orbit coupling at very small scales may also break the overall ${\rm
SU}(2)$ spin rotation and stabilize a charge $2e$ state with no half quantum
vortices at nonzero temperature. However, spin orbit coupling is extremely
weak and so even with a logarithmic critical temperature dependence similar to
(20) we expect that there will be a large window where the charge $2e$ order
has melted but the charge $4e$ order is still present with observable half
quantum vortices.

_Skyrmions with charge $2e$ and $4e$ and superconductivity_— Let us now
discuss how the different pairing scenarios discussed in this work arise very
naturally from the topological mechanism of superconductivity proposed in Ref.
Khalaf  _et al._ (2020) which we review briefly below. In that work, the
authors proposed a topological mechanism for superconductivity based on
pairing topological skyrmion textures. They considered a simplified problem
where the spin degree of freedom is neglected. This is likely relevant to the
extreme non-unitary limit where superconductivity takes place in only one spin
species. In this limit, the system consists of two bands in each Chern sector
which are labelled by the pseudospin index $\eta$. The low energy states of
the system with vanishing Chern number correspond to ferromagnetic order of
the pseudospin vector in each Chern sector which are coupled
antiferromagnetically between the opposite Chern sectors. A pseudo-spin
skyrmion texture in the $C=\pm$ sector carries charge $\pm e$. The
antiferromagnetic coupling between the opposite Chern sectors leads to an
attractive interaction between a skyrmion in one sector and an antiskyrmion in
the opposite sector despite having the same charge. This attraction leads to
the formation of a charge $2e$ bound state which leads to superconductivity
upon condensation.

Let us now see how this picture changes when we consider the fully spinful
model in the absence of flavor symmetry breaking where unitary pairing is
expected. In this case, each Chern sector contains four bands labelled by spin
and pseudospin. The low energy states with zero total Chern number are
specified by filling 2 out of the 4 bands within each sector Bultinck  _et
al._ (2020). Such state is parametrized by the $4\times 4$ matrices $Q_{\pm}$
in the spin-pseudospin space satisfying $Q_{\pm}^{2}=1$ and
$\mathop{\mathrm{tr}}Q_{\pm}=0$. Within each Chern sector, $Q_{\pm}$
parametrizes the manifold $\frac{{\rm U}(4)}{{\rm U}(2)\times{\rm U}(2)}$
which, similar to the 2-sphere $S^{2}=\frac{{\rm U}(2)}{{\rm U}(1)\times{\rm
U}(1)}$, admits skyrmion textures due to $\pi_{2}\left(\frac{{\rm U}(4)}{{\rm
U}(2)\times{\rm U}(2)}\right)=\mathds{Z}$. A general skyrmion texture in this
manifold intertwines the pseudospin and spin degrees of freedom. We can
understand the different types of possible skyrmions by restricting ourselves
to some simple limiting cases. One possibility is to consider a pseudospin
skyrmion in only one of the spin species. Such a skyrmion can be written as

$Q_{+}({\bm{r}})=\left(\begin{array}[]{cc}{\bm{n}}_{0}\cdot{\bm{\eta}}&0\\\
0&{\bm{n}}_{\rm
sk}({\bm{r}})\cdot{\bm{\eta}}\end{array}\right)_{\uparrow/\downarrow}$ (21)

where ${\bm{n}}_{0}$ is some fixed vector whereas ${\bm{n}}_{\rm
sk}({\bm{r}})$ describes a skyrmion texture. This skyrmion is the same as the
spinless pseudospin skyrmion and will consequently carry an electric charge of
$\pm e$. Another more symmetric possibility is to take the same skyrmion in
both spin directions so that

$Q_{+}({\bm{r}})=s_{0}{\bm{n}}_{\rm sk}({\bm{r}})\cdot{\bm{\eta}}$ (22)

This skyrmion consists of two copies of the spinless charge $e$ skyrmion and
will thus carry a total charge of $2e$. Notice that these Chern skyrmions
cannot condense since they feel a net magnus force (effective magnetive
field). However, as in the spinless case, the two Chern sectors are
antiferromagnetically decoupled by a term of the form
$J\mathop{\mathrm{tr}}Q_{+}Q_{-}$ which favors $Q_{+}=-Q_{-}$ leading to an
attractive interaction between skyrmions in one Chern sector and antiskyrmions
with the same charge in the opposite sector. The resulting bound state feels
no net magnetic field and can condense leading to a charge $2e$ or charge $4e$
superconductor. The former is realized by pairing the single spin skyrmions in
Eq. 21 which is excepted to break the ${\rm SU}(4)$ symmetry while the latter
is realized by pairing the spin-singlet skyrmions in Eq. 22 which is expected
to yield an ${\rm SU}(4)$-singlet. A detailed analysis of the symmetry
transformation properties of the different skyrmion superconductors will be
provided in an upcoming work Khalaf and Vishwanath .

The discussion above can be substantiated by writing the field theory derived
in Ref. Khalaf  _et al._ (2020) (although that work focused on the spinless
limit, the field theory derived was valid for any number of flavors)

$\mathcal{L}[Q_{+},Q_{-}]=\mathcal{L}_{+}[Q_{+}]+\mathcal{L}_{-}[Q_{-}]+J\mathop{\mathrm{tr}}Q_{+}Q_{-}$
(23)

with $\mathcal{L}[Q]$ given by

$\displaystyle\mathcal{L}_{\pm}[Q]=\frac{\rho}{2}\mathop{\mathrm{tr}}(\partial_{i}Q)^{2}+ieA_{\mu}J_{\pm,\mu}[Q]+\mathcal{L}_{B}[Q]\pm\mathcal{L}_{\rm
WZW}[Q],$ $\displaystyle
J_{\pm,\mu}[Q]=\pm\frac{i\epsilon_{\mu\nu\lambda}}{16\pi}\mathop{\mathrm{tr}}Q\partial_{\nu}Q\partial_{\lambda}Q$
(24)

Here, $\mathcal{L}_{B}$ is the Berry phase term Moon  _et al._ (1995) whereas
$\mathcal{L}_{\rm WZW}$ is a Wess-Zumino-Wittern term which ensures that
charge $e$ skyrmions are fermions as expected. This term, which was not
included in Ref. Khalaf  _et al._ (2020), will not play a role in our
discussion but is generally important to obtain the right symmetry properties
as we will show in a subsequent paper Khalaf and Vishwanath . What is
important to our current discussion is the skyrmion charge given by
integrating $J_{\pm,0}[Q]$. We see that for the single spin skyrmion defined
in (21), the expression in (24) simplifies to $\frac{1}{4\pi}{\bm{n}}_{\rm
sk}\cdot(\partial_{x}{\bm{n}}_{\rm sk}\times\partial_{y}{\bm{n}}_{\rm sk})$
which corresponds to the charge density of a standard spin skyrmion that
integrates to 1, thus indicating it is indeed a charge $e$ skyrmion. On the
other hand, the spin singlet skyrmion defined in (22) has twice the
topological density leading a total electrical charge of $2e$. An effective
field theory describing the different insulators and skyrmion superconductors
analogous to the ${\rm CP}^{1}$ theory of Ref. Khalaf  _et al._ (2020) will be
presented in Ref. Khalaf and Vishwanath .

In conclusion, it is possible to realize the three possibilities – non-unitary
pairing, unitary charge $2e$ and charge $4e$ superconductivity – within the
skyrmion mechanism. The former corresponds to the spinless limit where pairing
takes place in a single spin species. The latter two possibilities are
realized in the full spinful model by pairing charge $e$ skyrmions (21)
leading to an ${\rm SU}(4)$-breaking superconductor or charge $2e$ objects
(22) leading to an ${\rm SU}(4)$-symmetric superconductor.

_Experimental consequences_ — To discuss the experimental implications of the
statement above for TBG, let us first review some experimental facts. In many
experiments Cao  _et al._ (2018a, 2020); Yankowitz  _et al._ (2019); Saito
_et al._ (2020); Stepanov  _et al._ (2019); Arora  _et al._ (2020),
superconductivity is observed in the vicinity of a flavor-symmetry-breaking
insulator at $\nu=-2$, which suggests non-unitary pairing. However, there are
a few samples Lu  _et al._ (2019); Stepanov  _et al._ (2019) where
superconductivity is observed close to charge neutrality with no signature of
flavor symmetry breaking nearby. This is likely to be unitary pairing where
cases (ii) or (iii) of our theory should apply. In addition, several groups
Stepanov  _et al._ (2019); Saito  _et al._ (2020); Arora  _et al._ (2020) have
observed superconductivity in the absence of an insulator close to $\nu=\pm
2$. The absence of insulator is not sufficient to rule out flavor symmetry
breaking since it does not exclude the possibility of a flavor-polarized
metal. A stronger evidence for flavor polarization is provided by the presence
of a Landau fan emanating for $\nu=\pm 2$ indicating a Fermi surface
reconstruction. Thus, our general criteria for unitary vs non-unitary pairing
can be translated respectively to the absence or presence of a Landau fan
emanating from an integer filling in the vicinity of the superconductor. This
can be taken as a proxy for flavor symmetry breaking in the parent state. We
leave open the unusual scenario where non-unitary pairing develops on top of a
flavor symmetric state. This was observed in $\text{UTe}_{\text{2}}$ Ran  _et
al._ (2019b) and was argued to occur due to proximity to a ferromagnetic
transition Nevidomskyy (2020b).

Having established the connection to the experimental setup in TBG, let us now
investigate the implications of our results. In the vicinity of $\nu=\pm 2$,
the most likely flavor polarization is spin which has two possibilities
depending on the sign of Hund’s coupling. For the ferromagnetic sign of Hund’s
coupling, the flavor polarized state is a spin ferromagnet which favors a
spin-triplet superconductor where pairing takes place in a single spin
species. This corresponds to the scenario discussed in Ref. Cornfeld  _et al._
(2020) with only $\mathds{Z}_{2}$ vortices. In principle, this implies the
superconductor cannot exist at finite temperature and has no BKT transition.
This appears hard to reconcile with the robust pairing state observed in
experiments. On the other hand, antiferromagnetic Hund’s coupling results a
non-unitary pairing between anti-aligned spin directions in the two valleys.
This state is a superposition of a spin singlet and the $L_{z}=0$ component of
spin-triplet. At finite temperature, ${\rm SU}(2)$ rotation symmetry is
restored and the spin-triplet component disorders leading to a spin-singlet
superconductor. In the absence of spin anisotropies, this appears to be the
likely candidate for the experimentally observed superconductivity.

For superconductors observed in the absence of flavor polarization, we expect
unitary pairing with fractional vortices or higher charge superconductivity.
Ordinary half quantum vortices are not present in large systems due to
logarithmically divergent energies associated with their windings in flavor
space. While the phase winding costs finite energy due to the presence of
magnetic flux in the vortex core, there is no physical gauge flux that can
cancel the flavor space winding. However, as discussed above, in two spatial
dimensions fractional vortices in charge $2e$ superconductors manifest as pure
phase windings of a higher charge superconductor once a non-zero temperature
is included Mukerjee  _et al._ (2006); Jian and Zhai (2011); Xu and Balents
(2018).

This has the following implications for the periodicity of the Fraunhofer
patterns in magnetic Josephson junctions or SQUIDS. If the ground state is a
charge $2me$ superconductor with $m>1$, this will be observed as a modified
periodicity of the Fraunhofer pattern with period $2\pi/m$ for any temperature
below the KT transition. On the other hand, if the ground state is a charge
$2e$ superconductor, we can only observe the modified periodicity for
sufficiently large junctions whose size exceed the charge $2e$ correlation
length. The latter decreases with increasing temperature making it likely to
observe the modified periodicity at sufficiently large temperatures provided
that they are still below the KT temperature.

In mesoscopic annular geometries it may be possible to directly observe half
quantum vortices as $h/4e$ jumps in the flux threaded through the hole of the
annulus. Mesoscopic geometries are ideal to avoid the logarithmically
divergent energy cost associated with the unscreened flavor space winding
Babaev (2002); Silaev (2011); Chung  _et al._ (2007) and were previously used
to observe half quantum vortices Jang  _et al._ (2011); Seo  _et al._ (2015);
Autti  _et al._ (2016).

As discussed above, a Hund’s coupling that favors spin singlet pairing will
stabilize a unitary charge $2e$ superconductor without half quantum vortices.
Because the critical temperature (20) goes to zero very slowly for small
$J_{H}$, it is likely that experiments done at cold temperatures would not
observe such vortices if the Hunds coupling favors singlet pairing. However,
the lack of a large separation of scales does not exclude a potentially large
temperature window where the singlet Hunds pairing has melted but the charge
$4e$ condensate has not. Within this window one would still observe fractional
vortices.

Fractional vortices have previously received much attention due to the promise
of Majorana zero modes Volovik (1999); Read and Green (2000); Ivanov (2001).
However, these zero modes are specific to $p$-wave pairing where winding in a
single spin mimics a full quantum vortex for a spinless $p+ip$ superconductor.
Full quantum vortices are not expected to host Majorana zero modes in spinless
TBG, and so half quantum vortices in spinful TBG also will not host them.
However, both full and half quantum vortices are expected to host multiple
protected fermionic zero modes if the pairing is between Chern sectors and
valleys. We will discuss these zero modes in more depth in a subsequent paper.

_Conclusion_ — In conclusion, we have proven a no-go theorem under mild
assumptions which constraints the properties of superconducting condensates in
TBG, particularly for regions in the phase diagram where flavor polarization
is absent. The main outcome of our analysis is that such regions present ideal
hunting grounds for unconventional and exotic types of superconductivity. Our
results are independent of the underlying mechanism of superconductivity and
are equally applicable to weak coupling BCS-type superconductor as well as
strong coupling superconductors. We hope our study will inspire future
experiments to look for signatures of fractional vortices and higher charge
consendensates in these superconducting regions.

_Acknowledgements_ — We thank Igor Babaev and Mathias Scheurer for comments on
an early draft of the manuscript. AV was supported by a Simons Investigator
award and by the Simons Collaboration on Ultra-Quantum Matter, which is a
grant from the Simons Foundation (651440, AV). EK was supported by a Simons
Investigator Fellowship, by NSF-DMR 1411343, and by the German National
Academy of Sciences Leopoldina through grant LPDS 2018-02 Leopoldina
fellowship.

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Supplemental material:
Symmetry constraints on superconducting pairing in twisted bilayer graphene

##  I Transformation under spatial symmetries

In the main text, we only considered the transformation properties of the
pairing function under global symmetries. Here, we consider the spatial
symmetries. We follow the notations of Refs. Bultinck  _et al._ (2020); Khalaf
_et al._ (2020) by writing the microscopic symmetries acting on the
microscopic annihilation operators $f_{\bm{k}}$ which is a vector in valley
($\tau$), sublattice ($\sigma$), spin ($s$), and layer ($\mu$):

$C_{3}:f_{\bm{k}}\mapsto
e^{i\frac{\pi}{3}\sigma_{z}\tau_{z}}f_{O_{3}{\bm{k}}},\qquad
C_{2}:f_{\bm{k}}\mapsto\sigma_{x}\tau_{x}f_{-{\bm{k}}},\qquad
M_{y}:f_{\bm{k}}\mapsto\sigma_{x}\mu_{x}f_{m_{y}{\bm{k}}}$ (S1)

where $O_{3}$ denotes the standard action of threefold rotation on a 2D vector
and $m_{y}(k_{x},k_{y})=(k_{x},-k_{y})$. Note that all these symmetries acts
trivially on the spin index due to the ${\rm SU}(2)$ rotation symmetry. Next,
we project these symmetries onto the flat bands. Due to the finite sublattice
polarization, we can label the wavefunctions with the sublattice $\sigma$
their weight is concentrated on Bultinck  _et al._ (2020), which means that
the flatband electron annihilation operators are labelled by valley ($\tau$),
sublattice $(\sigma)$, and spin ($s$). A more convenient basis is obtained by
replacing the valley-sublattice indices with the the Chern $(\gamma)$ \-
pseudospin $(\eta)$ indices defined via Khalaf  _et al._ (2020):

${\bm{\gamma}}=(\sigma_{x},\sigma_{y}\tau_{z},\sigma_{z}\tau_{z}),\qquad{\bm{\eta}}=(\sigma_{x}\tau_{x},\sigma_{x}\tau_{y},\tau_{z})$
(S2)

Due to the non-trivial band topology of the flat bands, some symmetries
acquire a ${\bm{k}}$-dependence when projected onto them. This dependence was
derived in detail in Refs. Bultinck  _et al._ (2020); Khalaf  _et al._ (2020)
leading to the results summarized in the table below

basis | $C_{2}$ | $C_{3}$ | $M_{y}$
---|---|---|---
microscopic | $\sigma_{x}\tau_{x}$ | $e^{i\frac{\pi}{3}\sigma_{z}\tau_{z}}$ | $\sigma_{x}\mu_{x}$
projected | $\sigma_{x}\tau_{x}e^{i\theta_{2}({\bm{k}})}$ | $e^{i\theta_{3}({\bm{k}})\sigma_{z}\tau_{z}}$ | $\sigma_{x}e^{i\theta_{y}({\bm{k}})\sigma_{z}\tau_{z}}$
projected $(\gamma,\eta)$ | $\eta_{x}e^{i\theta_{2}({\bm{k}})}$ | $e^{i\theta_{3}({\bm{k}})\gamma_{z}}$ | $\gamma_{x}e^{i\theta_{y}({\bm{k}})\gamma_{z}}$

Let us now consider the transformation properties of the pairing function. We
restrict ourselves to the physically relevant case of opposite momentum
pairing:

$\Delta_{\bm{k}}=c_{\bm{k}}c_{-{\bm{k}}}^{T}$ (S3)

which behaves under spatial symmetries as

$c_{\bm{k}}\mapsto
U_{\bm{k}}c_{O{\bm{k}}}\qquad\implies\qquad\Delta_{\bm{k}}\mapsto
U_{\bm{k}}\Delta_{O{\bm{k}}}U_{-{\bm{k}}}^{T}$ (S4)

As in the main text (Eq. 3), we can use the fact the global symmetries do not
have structure in the Chern space $\gamma$ to decompose the gap function into
different channels

$\Delta({\bm{k}})=\Delta_{x}({\bm{k}})\gamma_{x}+\Delta_{y}({\bm{k}})\gamma_{y}+\Delta_{+}({\bm{k}})\frac{1+\gamma_{z}}{2}+\Delta_{-}({\bm{k}})\frac{1-\gamma_{z}}{2}$
(S5)

The antisymmetry of the full pairing function
$\Delta({\bm{k}})^{T}=-\Delta(-{\bm{k}})$ means that

$\Delta_{x,\pm}({\bm{k}})^{T}=-\Delta_{x,\pm}(-{\bm{k}}),\qquad\Delta_{y}({\bm{k}})^{T}=\Delta_{y}(-{\bm{k}})$
(S6)

The transformation properties under $C_{2}$, $C_{3}$ and $M_{y}$ can now be
obtained:

$\displaystyle
C_{2}:\Delta_{x,y,\pm}({\bm{k}})\mapsto\eta_{x}\Delta_{x,y,\pm}(-{\bm{k}})\eta_{x},$
(S7) $\displaystyle
C_{3}:\Delta_{x,y}({\bm{k}})\mapsto\Delta_{x,y}(O_{3}{\bm{k}}),\quad\Delta_{\pm}({\bm{k}})\mapsto
e^{\pm 2i\theta_{3}({\bm{k}})}\Delta_{\pm}(O_{3}{\bm{k}})$ (S8) $\displaystyle
M_{y}:\Delta_{x}({\bm{k}})\mapsto\Delta_{x}(m_{y}{\bm{k}}),\quad\Delta_{y}({\bm{k}})\mapsto-\Delta_{y}(m_{y}{\bm{k}}),\quad\Delta_{\pm}({\bm{k}})\mapsto
e^{\pm 2i\theta_{y}({\bm{k}})}\Delta_{\mp}(m_{y}{\bm{k}}).$ (S9)

The last two equations imply that the $\Delta_{\pm}$ pairing channels are tied
together in a single irrep of the spatial symmetry group, whereas $\Delta_{x}$
and $\Delta_{y}$ transform as separate irreps. To understand the action of
$C_{2}$, we decompose each $\Delta_{\mu}$ into symmetric and antisymmetric
intervalley pairing and an intravalley pairing. For intervalley pairing
considered in the main text

$\Delta^{\rm S/A}({\bm{k}})=\left(\begin{array}[]{cc}0&\tilde{\Delta}^{\rm
S/A}({\bm{k}})\\\ \pm\tilde{\Delta}^{\rm
S/A}({\bm{k}})^{T}&0\end{array}\right)\quad\implies\quad\tilde{\Delta}^{\rm
S/A}({\bm{k}})=\pm\tilde{\Delta}^{\rm S/A}(-{\bm{k}})^{T}$ (S10)

Combining with Eq. S6 yields

$\tilde{\Delta}^{\rm S/A}_{x,\pm}({\bm{k}})^{T}=\mp\tilde{\Delta}^{\rm
S/A}_{x,\pm}({\bm{k}}),\quad\tilde{\Delta}^{\rm
S/A}_{y}({\bm{k}})^{T}=\pm\tilde{\Delta}^{\rm S/A}_{y}({\bm{k}})$ (S11)

This represents a restriction on the individual blocks $\tilde{\Delta}$ in the
different pairing channels $\pm,x,y$ for symmetric and antisymmetric pairing
functions. Finally, we consider intravalley pairing

$\Delta({\bm{k}})=\left(\begin{array}[]{cc}\Delta_{K}({\bm{k}})&0\\\
0&\Delta_{K^{\prime}}({\bm{k}})\end{array}\right),\quad\implies\quad\Delta_{K}({\bm{k}})=\Delta_{K^{\prime}}(-{\bm{k}})$
(S12)

which dictates that the pairing in both valleys has the same gap and symmetry
properties (singlet or triplet).

##  II Fractional vortices for intravalley pairing

Let us now consider the case of intra-valley pairing. This is not likely to be
the relevant case physically but we include it here for completeness. Assuming
unitary pairing, we can write the most general form for intra-valley pairing
as

$\Delta=\left(\begin{array}[]{cc}\Delta_{K}&0\\\
0&\Delta_{K^{\prime}}\end{array}\right),\qquad\Delta_{K}\Delta_{K}^{\dagger}=\Delta_{K^{\prime}}\Delta_{K^{\prime}}^{\dagger}\propto\mathbbm{1}$
(S13)

If both $\Delta_{K}$ and $\Delta_{K^{\prime}}$ are antisymmetric, which
corresponds to the singlet channel of ${\rm SU}(2)$ in each valley, then we
can use the same considerations as in the main text to determine the type of
vortices with the GL Lagrangian

$\mathcal{L}=\rho_{K}(\partial_{\mu}\varphi_{K}-\frac{2e}{\hbar}A_{\mu})^{2}+\rho_{K^{\prime}}(\partial_{\mu}\varphi_{K^{\prime}}-\frac{2e}{\hbar}A_{\mu})^{2}+\dots$
(S14)

leading to the vortex quantization condition

$\int dl_{\mu}A_{\mu}=\frac{\hbar}{2e}\int
dl_{\mu}\frac{\rho_{K}\partial_{\mu}\varphi_{K}+\rho_{K^{\prime}}\partial_{\mu}\varphi_{K^{\prime}}}{\rho_{K}+\rho_{K^{\prime}}}$
(S15)

Symmetries such as $C_{2}$ and time reversal exchange valleys and therefore
relate $\Delta_{K}$ and $\Delta_{K^{\prime}}$ as well as $\rho_{K}$ and
$\rho_{K^{\prime}}$. As long as the superconductivity develops in a state that
preserves one of $C_{2}$, time reversal or any combination of one of them with
an internal symmetry, we expect $\rho_{K}=\rho_{K^{\prime}}$ for unitary
pairing. We then obtain $h/4e$ vortices.

This conclusions can also be verified by considering the manifold of the
superconducting order parameter by writing
$\Delta=s_{y}e^{i(\phi\eta_{0}+\phi_{V}\eta_{z})}$. Thus, we can form a loop
in the space of $\Delta$ where both $\phi$ and $\phi_{V}$ go from 0 to $\pi$.
This will correspond to a half-vortex in the overall superconducting phase
$\phi$. This reflects the nature of the manifold of the superconducting order
parameter $\frac{{\rm U}_{C}(1)\times{\rm U}_{V}(1)}{\mathds{Z}_{2}}$ which
admits half-vortices. This case is similar to the case of non-unitary pairing
considered in the main text where $\pi_{1}$ of the superconducting manifold is
$\mathds{Z}\oplus\mathds{Z}$ and vortices with arbitrary flux $xh/2e$ are in
principle allowed. However, the presence of any symmetry relating the two
valleys would force $x=1/2$ leading to half quantum vortices.

If both $C_{2}$ and time reversal are broken in the parent state and there is
no other unbroken symmetry that relates the two valleys then $\rho_{K}$ and
$\rho_{K^{\prime}}$ may be different. In such a state we would expect
nonunitary pairing since there is also no symmetry relating $\Delta_{K}$ and
$\Delta_{K^{\prime}}$ which means unitary pairing should not be stable to RG
flows; it must be fine tuned. If we nonetheless fine tune to obtain unitary
pairing, and furthermore fine tune one of the stiffnesses to zero despite a
nonzero pairing potential, we obtain conventional $h/2e$ vortices. We do not
take this latter scenario into account for our main theorem due to the extreme
amounts of unphysical fine tuning, but detailed its existence here for
completeness.

Next, we consider the case where either $\Delta_{K}$ or $\Delta_{K^{\prime}}$
is symmetric, i.e. ${\rm SU}(2)$ triplet. The existence of fractional vortices
in this case follows immediately from the existence of half-vortices in the
spin-triplet case whose manifold is ${\rm U}(2)/{\rm O}(2)$ with $\pi_{1}({\rm
U}(2)/{\rm O}(2))=\mathds{Z}/2$. When included in the bigger order parameter
for both valleys, this yields fractional vortices. For instance, if pairing is
triplet in both valleys, this will be a quarter vortex similar to the
symmetric ${\rm SU}(4)$ case considered in the main text.