# Altermagnetic superconducting diode effect

Sayan Banerjee  Institute for Theoretical Physics III, University of
Stuttgart, 70550 Stuttgart, Germany  Mathias S. Scheurer  Institute for
Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany

###### Abstract

Non-reciprocal superconductivity, also known as the superconducting diode
effect, has been extensively studied in the presence of a magnetic field or
some form of ferromagnetic order breaking time-reversal symmetry. We here show
that another class of magnetic order known as altermagnetism, which also
breaks time-reversal symmetry but does not exhibit a finite net magnetic
moment, can also give rise to a superconducting diode effect. Whether this is
the case depends on the combination of the system’s point group and
altermagnetic order parameter which we explore systematically for two-
dimensional crystalline systems. If the superconducting electrons are in a
centrosymmetric crystalline environment, an electric field $E_{z}$ (or other
sources of inversion symmetry breaking) can be used to turn on and tune the
non-reciprocity, yielding an electric-field tunable diode effect; there are
also non-centrosymmetric point groups, which are not reached by applying
$E_{z}\neq 0$ in a centrosymmetric crystal, but still allow for an
altermagnetic order parameter with non-reciprocal superconductivity. Depending
on the residual magnetic point group, the zeros of the critical current
asymmetry, $J_{c}(\hat{\boldsymbol{n}})-J_{c}(-\hat{\boldsymbol{n}})$, are
pinned along high-symmetry crystalline directions
$\hat{\boldsymbol{n}}=\hat{\boldsymbol{e}}_{j}$ or are free to rotate in the
plane of the system. In some cases, the zeros can be rotated by tuning the
electric field $E_{z}$. We discuss all of these phenomena both on the general
level using exact symmetry arguments and more explicitly by constructing and
solving minimal lattice models. We provide experimental setups to realize the
altermagnetic superconducting diode effect.

##  I Introduction

The broad class of magnetically ordered states referred to as “altermagnets”
(AMs) has recently become a subject of notable interest and has motivated a
significant amount of theoretical Yuan  _et al._ (2020); Šmejkal  _et al._
(2022a, b); Ahn  _et al._ (2019); Bhowal and Spaldin (2022); Cuono  _et al._
(2023); Guo  _et al._ (2023); Maier and Okamoto (2023); Mazin (2023);
Oganesyan  _et al._ (2001); Sato  _et al._ (2023); Steward  _et al._ (2023);
Turek (2022); Ouassou  _et al._ (2023); Mazin  _et al._ (2021); Hayami  _et
al._ (2019); Šmejkal  _et al._ (2020); Brekke  _et al._ (2023); Das  _et al._
(2023); Fakhredine  _et al._ (2023); Fang  _et al._ (2023); Leeb  _et al._
(2023); Mazin  _et al._ (2023); Šmejkal  _et al._ (2023); Sun and Linder
(2023); Gao  _et al._ (2023); Beenakker and Vakhtel (2023); Zhou  _et al._
(2024); Fernandes  _et al._ (2023); Sun  _et al._ (2023); Zhang  _et al._
(2023a); Brekke  _et al._ (2023); Giil and Linder (2023); Zhu  _et al._
(2023); Li and Liu (2023); Ghorashi  _et al._ (2023); Cheng and Sun (2024);
Antonenko  _et al._ (2024); Yu  _et al._ (2024); Wei  _et al._ (2023);
McClarty and Rau (2023) and experimental Aoyama and Ohgushi (2023); Bai  _et
al._ (2023); Krempaský  _et al._ (2023); Lee  _et al._ (2023); Reimers  _et
al._ (2023); Feng  _et al._ (2022) studies, probing into its nature and
emergence. Just as with any form of magnetic order, time-reversal symmetry is
broken in an AM. However, as opposed to ferromagnets, AMs do not exhibit a
finite net magnetic moment, which is guaranteed by a magnetic point symmetry.
This means that the state is invariant under the product $\Theta g_{m}$ of
time-reversal $\Theta$ and a unitary lattice symmetry $g_{m}$. Unlike a
collinear antiferromagnet, $g_{m}$ is not lattice translation for an AM, where
the reversal of all local magnet moments resulting from application of
$\Theta$ cannot be undone by a lattice translation $T_{\boldsymbol{R}}$.
Instead, $g_{m}$ is a point group symmetry such as a rotation or reflection
symmetry of the underlying crystalline lattice.

While a Pomeranchuk instability I. (1958); Wu  _et al._ (2007) in the $l=2$
(or higher $l$) spin-triplet channel of a Fermi liquid falls into the category
of AMs, a lot of interesting novel consequences have recently been uncovered.
For instance, since $g_{m}$ is not translation, AMs can exhibit an anomalous
Hall effect, even without net magnetic moment, unlike antiferromagnets where
$\Theta T_{\boldsymbol{R}}$ does not allow for a finite Hall response Feng
_et al._ (2022); Sato  _et al._ (2023). Furthermore, the spin texture of the
Bloch states in the presence of an AM can lead to a Zeeman splitting with
protected nodal lines Fernandes  _et al._ (2023). Further consequences have
been explored in thermal transport Zhou  _et al._ (2024), in Josephson
junctions Beenakker and Vakhtel (2023); Ouassou  _et al._ (2023), and other
tunnel junction setups Chi  _et al._ (2023). What is more, the effect of AMs
on other coexisting interaction-induced orders is very rich; in particular,
superconductivity Chakraborty and Black-Schaffer (2023); Zhang  _et al._
(2023a); Brekke  _et al._ (2023); Giil and Linder (2023); Zhu  _et al._
(2023); Li and Liu (2023); Ghorashi  _et al._ (2023); Cheng and Sun (2024);
Wei  _et al._ (2023) has been explored, sparking off broader questions
regarding the nature of strongly correlated materials exhibiting
superconductivity and magnetism beyond the conventional paradigm of
ferromagnetism and antiferromagnetism.

Concurrently, non-reciprocal transport in superconductors and superconducting
junctions, particularly the superconducting diode effect (SDE), has garnered
various experimental Jiang  _et al._ (1994); Ando  _et al._ (2020); Lyu  _et
al._ (2021); Du  _et al._ (2023); Sundaresh  _et al._ (2023); Kealhofer  _et
al._ (2023); Hou  _et al._ (2023); Chen  _et al._ (2023); Gupta  _et al._
(2022); Baumgartner  _et al._ (2022); Banerjee  _et al._ (2023); Pal  _et al._
(2022); Kim  _et al._ (2023); Gupta  _et al._ (2022); Turini  _et al._ (2022);
Bauriedl  _et al._ (2022); Wu  _et al._ (2022); Diez-Merida  _et al._ (2021);
Golod and Krasnov (2022); Chiles  _et al._ (2022); Zhang  _et al._ (2023b);
Shin  _et al._ (2021); Jeon  _et al._ (2022); Kealhofer  _et al._ (2023); Hou
_et al._ (2023); Lin  _et al._ (2022); Anwar  _et al._ (2022); Narita  _et
al._ (2022); Gutfreund  _et al._ (2023); Zhao  _et al._ (2023) and theoretical
Daido and Yanase (2022); Daido  _et al._ (2022); Yuan and Fu (2022); He  _et
al._ (2022a); Ilić and Bergeret (2022); Scammell  _et al._ (2022); Zinkl  _et
al._ (2022); Zhang  _et al._ (2022); He  _et al._ (2022b); Zhai  _et al._
(2022); Jiang  _et al._ (2022); Kokkeler  _et al._ (2022); Debnath and Dutta
(2024); Chazono  _et al._ (2022); Vodolazov and Peeters (2005); de Picoli  _et
al._ (2023); Kochan  _et al._ (2023); Ikeda  _et al._ (2022); Tanaka  _et al._
(2022); Wang  _et al._ (2022); Haenel and Can (2022); Legg  _et al._ (2023);
Cuozzo  _et al._ (2023); Souto  _et al._ (2022); Cheng and Sun (2023); Steiner
_et al._ (2022); Costa  _et al._ (2023); Wei  _et al._ (2022); Legg  _et al._
(2022); Karabassov  _et al._ (2022); Hu  _et al._ (2023); Wu  _et al._ (2023);
Nunchot and Yanase (2024); Daido and Yanase (2023); Souto  _et al._ (2023)
attention. The SDE refers to a difference in the critical current
$J_{c}(\hat{\boldsymbol{n}})$ in opposite directions, $\hat{\boldsymbol{n}}$
and $-\hat{\boldsymbol{n}}$. In that case, applying a current $J$ with
$\min(J_{c}(\pm\hat{\boldsymbol{n}}))<J<\max(J_{c}(\pm\hat{\boldsymbol{n}}))$,
leads to superconducting transport along one and resistive behavior along the
opposite direction, yielding a superconducting analogue of a semi-conducting
diode. On top of fundamental theoretical questions, the rich space of
potential applicability of such non-reciprocal responses on modern-day
electronics and devices renders the SDE an exciting topic to delve into from
an experimental and technological standpoint.

Importantly, time-reversal symmetry $\Theta$ implies
$J_{c}(\hat{\boldsymbol{n}})=J_{c}(-\hat{\boldsymbol{n}})$ which therefore
needs to be broken to obtain a finite SDE. So far, external magnetic fields
Ando  _et al._ (2020); Du  _et al._ (2023); Hou  _et al._ (2023); Chen  _et
al._ (2023); Gupta  _et al._ (2022); Baumgartner  _et al._ (2022); Banerjee
_et al._ (2023); Pal  _et al._ (2022); Kim  _et al._ (2023); Turini  _et al._
(2022); Bauriedl  _et al._ (2022); Lyu  _et al._ (2021); Sundaresh  _et al._
(2023); Kealhofer  _et al._ (2023); Diez-Merida  _et al._ (2021); Daido and
Yanase (2022); Daido  _et al._ (2022); Yuan and Fu (2022); He  _et al._
(2022a); Ilić and Bergeret (2022); He  _et al._ (2022b); de Picoli  _et al._
(2023); Kochan  _et al._ (2023); Ikeda  _et al._ (2022); Cheng and Sun (2023);
Legg  _et al._ (2022); Wu  _et al._ (2023); Nunchot and Yanase (2024),
proximity-induced magnetism Shin  _et al._ (2021); Jeon  _et al._ (2022);
Narita  _et al._ (2022), and interaction-induced order parameters in the same
electron liquid Lin  _et al._ (2022); Scammell  _et al._ (2022); Banerjee and
Scheurer (2024); Hu  _et al._ (2023) have been discussed as possible causes
for the broken time-reversal symmetry inducing the SDE. All of these
$\Theta$-breaking orders fall under the category of (spin and/or orbital)
ferromagnetism and are associated with a non-zero net magnetic moment. Due to
its residual magnetic symmetry $\Theta T_{\boldsymbol{e}_{x,y}}$, with
nearest-neighbor vectors $\boldsymbol{e}_{x,y}$, collinear antiferromagnetism
as the only cause of symmetry reduction cannot stabilize the SDE. This
therefore begs the question if and under which conditions AMs can induce a SDE
in an otherwise symmetry-unbroken superconductor and what the consequences are
of the non-trivial magnetic point symmetries of the AM on the critical current
$J_{c}(\hat{\boldsymbol{n}})$. This is the scope of our present work.

To this end, we here systematically analyze, using a combination of symmetry
arguments and explicit model calculations, which of the possible AMs in two-
dimensional (2D) crystalline systems can yield a critical current asymmetry
$J_{c}(\hat{\boldsymbol{n}})-J_{c}(-\hat{\boldsymbol{n}})\neq 0$. Indeed,
there are several candidate AMs where a SDE is possible despite the lack of
net magnetization. We will show that, depending on the residual magnetic point
group, the zeros of the current asymmetry
$J_{c}(\hat{\boldsymbol{n}})-J_{c}(-\hat{\boldsymbol{n}})$ are either aligned
with high-symmetry crystalline directions, can be rotated by an electric
field, or are not at all pinned to the crystalline axes.

The remainder of the manuscript is organized as follows. In Sec. II, we
describe the method of calculating $J_{c}(\hat{\boldsymbol{n}})$ and discuss
the general constraints on its directional dependence and on its asymmetry
resulting from magnetic point symmetries. We further systematically go through
all AM order parameters in 2D single-band models, study whether they induce an
SDE and, if so, how the current asymmetry
$J_{c}(\hat{\boldsymbol{n}})-J_{c}(-\hat{\boldsymbol{n}})$ depends on
$\hat{\boldsymbol{n}}$. In Sec. III, we demonstrate the AM-induced SDE on a
case-by-case basis by constructing and solving explicit lattice models,
detailing the distinctive characters in each of the cases. Finally, Sec. IV
summarizes our findings.

##  II General considerations

Before presenting explicit calculations, we start by discussing general
constraints on the AM-induced SDE, resulting from symmetries. We will see
under which conditions the SDE is forced to vanish along high-symmetry
directions and systematically go through all possible AM order parameters of
2D systems.

###  II.1 Diode effect efficiency

To set the stage for our symmetry discussion, we begin by first outlining the
basic procedure for the calculation of the critical current. Denoting the
superconducting order parameter for Cooper pairs with center-of-mass momentum
$\boldsymbol{q}$ by $\Delta_{\boldsymbol{q}}$, we expand the change of the
free-energy as a result of superconductivity
$\delta\mathcal{F}_{\text{S}}:=\mathcal{F}[\Delta_{\boldsymbol{q}}]-\mathcal{F}[0]$
up to quadratic order in $\Delta_{\boldsymbol{q}}$,

$\delta\mathcal{F}_{\text{S}}\sim\sum_{\boldsymbol{q}}a^{\text{S}}_{\boldsymbol{q}}\,|\Delta_{\boldsymbol{q}}|^{2}+b^{\text{S}}\sum_{\boldsymbol{q}_{i}}\Delta^{*}_{\boldsymbol{q}_{1}}\Delta^{*}_{\boldsymbol{q}_{2}}\Delta^{\phantom{*}}_{\boldsymbol{q}_{3}}\Delta^{\phantom{*}}_{\boldsymbol{q}_{4}}\delta_{\boldsymbol{q}_{1}+\boldsymbol{q}_{2},\boldsymbol{q}_{3}+\boldsymbol{q}_{4}},$
(1)

where we neglected the momentum dependence of the quartic term. The
equilibrium superconducting state is found by minimizing
$\delta\mathcal{F}_{\text{S}}$. Restricting the analysis to
single-$\boldsymbol{q}$ states,
$\Delta_{\boldsymbol{q}}\propto\delta_{\boldsymbol{q},\boldsymbol{q}_{0}}$,
the value of $\boldsymbol{q}_{0}$ is determined by the minimum of
$a^{\text{S}}_{\boldsymbol{q}}$. We define the critical current
$J_{c}(\hat{\boldsymbol{n}})$ along $\hat{\boldsymbol{n}}$ as the maximal
magnitude of
$\boldsymbol{J}(\boldsymbol{q})=2e\Delta_{\boldsymbol{q}}\boldsymbol{\nabla}a^{\text{S}}_{\boldsymbol{q}}$
oriented along $\hat{\boldsymbol{n}}$. Postponing a microscopic evaluation of
$a^{\text{S}}_{\boldsymbol{q}}$ to Sec. III below, we here only note that a
unitary symmetry with vector representation $g$ implies
$a^{\text{S}}_{\boldsymbol{q}}=a^{\text{S}}_{g\boldsymbol{q}}$ and, thus,
$\boldsymbol{J}(g\boldsymbol{q})=g\boldsymbol{J}(\boldsymbol{q})$ as well as
$J_{c}(\hat{\boldsymbol{n}})=J_{c}(g\hat{\boldsymbol{n}})$.

To quantify the ‘degree’ or amount of diode effect, it will be convenient for
us to first introduce a signed efficiency parameter

$\eta_{s}(\hat{\boldsymbol{n}}):=\frac{J_{c}(\hat{\boldsymbol{n}})-J_{c}(-\hat{\boldsymbol{n}})}{J_{c}(\hat{\boldsymbol{n}})+J_{c}(-\hat{\boldsymbol{n}})}$
(2)

such that the angle-resolved diode-effect efficiency is
$\eta(\hat{\boldsymbol{n}})=|\eta_{s}(\hat{\boldsymbol{n}})|\geq 0$. We next
discuss symmetry constraints on $\eta(\hat{\boldsymbol{n}})$. From the
discussion above, it follows that a unitary symmetry with vector
representation $g$ enforces
$\eta_{s}(g\hat{\boldsymbol{n}})=\eta_{s}(\hat{\boldsymbol{n}})$. Since
$\eta_{s}(\hat{\boldsymbol{n}})=-\eta_{s}(-\hat{\boldsymbol{n}})$, which
immediately follows from the definition (2), the SDE efficiency is required to
vanish along any direction $\hat{\boldsymbol{n}}_{0}$ obeying
$g\hat{\boldsymbol{n}}_{0}=-\hat{\boldsymbol{n}}_{0}$; in short:

$g\hat{\boldsymbol{n}}_{0}=-\hat{\boldsymbol{n}}_{0}\quad\Rightarrow\quad\eta(\hat{\boldsymbol{n}}_{0})=0.$
(3)

Clearly, Eq. (3) is obeyed for all $\hat{\boldsymbol{n}}$ if $g$ is inversion
($I$) or two-fold rotation along the out-of-plane ($\hat{\boldsymbol{z}}$)
direction $C_{2z}$ for a 2D system; these symmetries are thus required to be
broken to allow for a finite SDE. While the following constraints are readily
generalized to three spatial dimensions, we next focus for notational
simplicity on a 2D system in the $xy$ plane. If $g$ is a two-fold rotational
symmetry, $C_{2\hat{\boldsymbol{m}}}$, along an in-plane direction
$\hat{\boldsymbol{m}}$ or a mirror-plane symmetry,
$\sigma_{\hat{\boldsymbol{m}}}$, spanned by $\hat{\boldsymbol{z}}$ and
$\hat{\boldsymbol{m}}$, Eq. (3) implies that $\eta(\hat{\boldsymbol{n}})=0$
for $\hat{\boldsymbol{n}}$ perpendicular to $\hat{\boldsymbol{m}}$. In other
words, the zeros of $\eta(\hat{\boldsymbol{n}})$ are pinned along high-
symmetry directions if the point group contains two-fold rotations along in-
plane directions or mirror planes perpendicular to the system plane. Note that
rotations with degrees higher than two do not pin the zeros.

Similar constraints arise from magnetic point symmetries, i.e., non-unitary
symmetries, $\Theta g_{m}$, given by the product of time-reversal $\Theta$ and
some unitary symmetry $g_{m}$. Such a magnetic symmetry implies
$a^{\text{S}}_{\boldsymbol{q}}=a^{\text{S}}_{-g_{m}\boldsymbol{q}}$ and
consequently $J_{c}(\hat{\boldsymbol{n}})=J_{c}(-g_{m}\hat{\boldsymbol{n}})$;
this, in turn, leads to
$\eta_{s}(g_{m}\hat{\boldsymbol{n}})=\eta_{s}(-\hat{\boldsymbol{n}})=-\eta_{s}(\hat{\boldsymbol{n}})$.
As such, the presence of the magnetic symmetry $\Theta g_{m}$ implies that

$g_{m}\hat{\boldsymbol{n}}_{0}=\hat{\boldsymbol{n}}_{0}\quad\Rightarrow\quad\eta(\hat{\boldsymbol{n}}_{0})=0.$
(4)

Equation (4) contains the special case of preserved time-reversal symmetry,
$g=\mathbbm{1}$ for which $g\hat{\boldsymbol{n}}=\hat{\boldsymbol{n}}$ always
holds, implying $\eta(\hat{\boldsymbol{n}})=0$ for all $\hat{\boldsymbol{n}}$;
this just recovers the often-stated fact that $\Theta$ needs to be broken to
get a SDE. However, Eq. (4) also shows that zeros of the angle-resolved SDE
efficiency are pinned along high-symmetry directions if the system exhibits a
magnetic reflection symmetry, $\Theta\sigma_{\hat{\boldsymbol{m}}}$, or
magnetic rotational symmetry, $\Theta C_{2\hat{\boldsymbol{m}}}$; in both
cases, it holds $\eta(\hat{\boldsymbol{m}})=0$. We will see examples of all of
these constraints in the explicit model calculations of Sec. III below.

Figure 1: Schematic of the possible realizations of the considered models by
(a) proximity-induced AM in a superconductor, (b) proximity-induced
superconductivity on an AM metal, and (c) both orders coexisting in a single
material. Our analysis also applies to the case where the proximitizing
material, the AM in (a) and the superconductor in (b), is only present on one
side. The advantage of the symmetric design is that an external electric field
$E_{z}$ can be used to reduce the symmetries of the systems in a controllable
way.

###  II.2 AM-induced SDE

With these symmetry constraints at hand, we are now in position to analyze
under which conditions an AM moment can induce a SDE and, if non-zero, whether
the angle-resolved SDE efficiency $\eta(\hat{\boldsymbol{n}})$ has to have
zeros along certain directions. To be precise, in this work, we use the
following definition of an AM order parameter. We define an AM order parameter
as a time-reversal-odd order parameter (i.e., some form of magnetic order),
which unlike a ferromagnet does not lead to a finite net magnetic moment or
spin-polarization; in addition, we require that it preserve lattice
translation, unlike an antiferromagnet or more complex translational-symmetry-
breaking orders (such as an incommensurate spiral or a tetrahedral magnet).

For concreteness, we study the prototypical case of a minimal model for the
normal state—one that only contains a single, spinful band,

$\mathcal{H}_{0}=\sum_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k},s}\left[s_{0}\epsilon_{\boldsymbol{k}}+\boldsymbol{g}_{\boldsymbol{k}}\cdot\boldsymbol{s}+\boldsymbol{N}_{\boldsymbol{k}}\cdot\boldsymbol{s}\right]_{s,s^{\prime}}c^{\phantom{\dagger}}_{\boldsymbol{k},s^{\prime}},$
(5)

where $c^{\dagger}_{\boldsymbol{k},s}$ are electron creation operators of spin
$s$ and momentum $\boldsymbol{k}$, $\boldsymbol{s}=(s_{x},s_{y},s_{z})$,
$s_{0}=\mathbbm{1}$, are spin Pauli matrices. Furthermore,
$\epsilon_{\boldsymbol{k}}$ is the bare dispersion,
$\boldsymbol{g}_{\boldsymbol{k}}$ the spin-orbit vector, and
$\boldsymbol{N}_{\boldsymbol{k}}$ the AM order parameter. Since these three
quantities are, by design, even, even, and odd under $\Theta$, respectively,
they have to obey

$\epsilon_{\boldsymbol{k}}=\epsilon_{-\boldsymbol{k}},\qquad\boldsymbol{g}_{\boldsymbol{k}}=-\boldsymbol{g}_{-\boldsymbol{k}},\qquad\boldsymbol{N}_{\boldsymbol{k}}=\boldsymbol{N}_{-\boldsymbol{k}}.$
(6)

We, therefore, conclude that $\epsilon_{\boldsymbol{k}}$ always preserves
inversion, $\boldsymbol{g}_{\boldsymbol{k}}$ can only be non-zero if
inversion, $I$, is broken (as usual), and the AM order parameter cannot break
$I$ (and will always be odd under $\Theta I$) in the class of models in Eq.
(5). To describe superconductivity, we assume that Cooper pairs form in the
spin-singlet channel, the superconducting order parameter transforms trivially
under all point symmetries and preserves time-reversal. Therefore,
$\boldsymbol{N}_{\boldsymbol{k}}$ is the only term breaking time-reversal and
$\boldsymbol{g}_{\boldsymbol{k}}$ the only source of broken inversion symmetry
in Eq. (5); this is why both are required to obtain a finite SDE, $\eta\neq
0$.

The different possible physical realizations of this class of models are
illustrated in Fig. 1. Either the AM term $\boldsymbol{N}_{\boldsymbol{k}}$
will be proximity-induced in superconducting systems, see Fig. 1(a), or vice
versa—a proximitized superconductor induces pairing in a metallic AM material
[Fig. 1(b)]. Finally, our analysis also applies if both AM and
superconductivity arise spontaneously in the same electron liquid, see (c).
Although not essential for the realization of the following effects, we here
assumed that the proximitizing materials are placed symmetrically on both
sides such that an electric field, $E_{z}$, can be used to further reduce the
symmetries in situ. Our symmetry analysis and minimal-model study here still
applies when the proximitizing material is only present on one side. In that
case, the terms induced by $E_{z}\neq 0$ in the models below will already be
present for $E_{z}=0$.

To address the question which AM order parameters can induce a SDE, we start
by considering all possible point groups $\mathcal{G}_{0}$ of 2D crystalline
systems. For each of them, we go through all of its irreducible
representations and check whether a generic time-reversal-odd
$\boldsymbol{N}_{\boldsymbol{k}}\cdot\boldsymbol{s}$ in Eq. (5) transforming
under it is an AM; this requires the presence of a residual symmetry that
ensures that the expectation value of the total spin,

$\boldsymbol{S}=\sum_{\boldsymbol{k}}c^{\dagger}_{\boldsymbol{k},s}\boldsymbol{s}_{s,s^{\prime}}c^{\phantom{\dagger}}_{\boldsymbol{k},s^{\prime}},$
(7)

has to vanish. For instance, an AM order parameter transforming under $A_{g}$
of $\mathcal{G}_{0}=D_{2h}$ will lead to a residual $C_{2z}$ rotation
symmetry, implying that $\braket{S_{x}}=\braket{S_{y}}=0$. The additional
$\sigma_{v}$ mirror symmetry further implies that also $\braket{S_{z}}$
vanishes. In the first three columns of Table 1, we list all such combinations
of $\mathcal{G}_{0}$ and AM order parameters, some of which were already
tabulated in Ref. Fernandes  _et al._ , 2023. To organize the list, the first
11 lines refer to the AM order parameters for centrosymmetric point groups
$\mathcal{G}_{0}$; interestingly, we find that, in all cases, these AM order
parameters will continue to be proper AMs with $\braket{\boldsymbol{S}}=0$
when an electric field is applied, $E_{z}\neq 0$, which reduces the point
groups to $\mathcal{G}_{E_{z}}\leq\mathcal{G}_{0}$. Combing back to our
previous example, $\mathcal{G}_{0}=D_{2h}$ will be reduced to
$\mathcal{G}_{E_{z}}=C_{2v}$. It still contains the two symmetries used above
to show that there is no finite magnetization. The remaining lines of Table 1
refer to the AM order parameters in the 2D non-centrosymmetric point groups
that are not reached by applying $E_{z}$ in centrosymmetric point groups. In
all those cases, reducing the point group further by application of $E_{z}$
leads to a finite magnetization.

For all of these AMs and point groups, we investigate whether a
superconducting order parameter in the singlet channel and transforming under
the trivial representation of $\mathcal{G}_{0}$ can give rise to a SDE.
Clearly, due to inversion symmetry, there cannot be a SDE for the
centrosymmetric point groups at $E_{z}=0$. However, applying an electric field
does allow for finite $\eta$ in a few cases, see penultimate column in Table
1. This is remarkable since it shows that finite magnetization is not required
to obtain a SDE. Also some of the AMs in certain non-centrosymmetric point
groups allow for a SDE without finite moment. In Table 1 we also list whether
and, if so, how the residual symmetries pin the zeros of
$\eta(\hat{\boldsymbol{n}})$ to high-symmetry lines. These aspects will be
further discussed on a case-by-case basis in the following section.

Table 1: Summary of possible AM order parameters $\boldsymbol{N}_{\boldsymbol{k}}$ in 2D models of the form of Eq. (5). For each point group $\mathcal{G}_{0}$, we list all the irreducible representations (IR) that can give rise to an AM; the respective form of $\boldsymbol{N}_{\boldsymbol{k}}$ for small $\boldsymbol{k}$ is shown in the third column, where $\tau$ and $R_{\varphi}$ are, respectively, a real-valued constant and a rotation matrix along the third direction not fixed by symmetry. Applying an electric field $E_{z}$ reduces the point group to $\mathcal{G}_{E_{z}}$. In case of the centrosymmetric $\mathcal{G}_{0}$ (first 11 lines), all order parameters remain AMs (as indicated in the fourth column), characterized by zero net magnetization; this is not the case for the non-centrosymmetric $\mathcal{G}_{0}$ (line 12-20). In the column SDE, we indicate whether the respective AM will give rise to an SDE (in the centrosymmetryic point groups, only possible for $E_{z}\neq 0$). We also indicate whether the angle-resolved SDE efficiency $\eta(\hat{\boldsymbol{n}})$ has zeros along generic directions $\hat{\boldsymbol{n}}_{0}$ (and their $C_{3z}$-related partners) or along crystalline axes, $\hat{\boldsymbol{e}}_{x,y}$. For $D_{3}$ and $D_{6}$, we introduced $\hat{\boldsymbol{n}}^{i}_{E_{z}}$ with $\hat{\boldsymbol{n}}^{i}_{E_{z}=0}=\hat{\boldsymbol{e}}_{i}$, $i=x,y$, and pointing along a generic direction for $E_{z}\neq 0$. The last column shows candidate materials, based on their conduction type—metallic (M) or insulating (I)—focusing on those cases where an SDE is present.  $\mathcal{G}_{0}$ | IR of AM | $\boldsymbol{N}_{\boldsymbol{k}}$ | $\mathcal{G}_{E_{z}}$ | SDE | Materials
---|---|---|---|---|---
$C_{4h}$ | $B_{g}$ | $(0,0,k_{x}^{2}-k_{y}^{2}+\tau k_{x}k_{y})$ |  $C_{4}$; AM | ✗ |
$C_{6h}$ |  | $B_{g}$
---
| $R_{\varphi}(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},0)$
---
|  $C_{6}$; AM
---
| $\eta(\pm C_{3z}^{j}\hat{\boldsymbol{n}}_{0})=0$
---
$D_{2h}$ | $A_{g}$ | $(0,0,k_{x}k_{y})$ |  $C_{2v}$; AM | ✗ |
$D_{4h}$ |  | $A_{1g}$
---
$B_{1g}$
$B_{2g}$
| $(0,0,k_{x}k_{y}(k_{x}^{2}-k_{y}^{2}))$
---
$(0,0,k_{x}k_{y})$
$(0,0,k_{x}^{2}-k_{y}^{2})$
$C_{4v}$; AM |  | ✗
---
✗
✗
$D_{6h}$ |  | $A_{1g}$
---
$B_{1g}$
$B_{2g}$
$E_{2g}$
| $(0,0,k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(3k_{x}^{2}-k_{y}^{2}))$
---
$(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},0)$
$(2k_{x}k_{y},k_{x}^{2}-k_{y}^{2},0)$
$(0,0,2k_{x}k_{y}),(0,0,k_{x}^{2}-k_{y}^{2})$
$C_{6v}$; AM |  | ✗
---
$\eta(\pm C_{3z}^{j}\hat{\boldsymbol{e}}_{x})=0$
$\eta(\pm C_{3z}^{j}\hat{\boldsymbol{e}}_{y})=0$
✗
| CrSb (M)Šmejkal  _et al._ (2022a), MnTe (I)Nie  _et al._ (2023); Gonzalez
Betancourt  _et al._ (2023)
---
$D_{3d}$ |  | $A_{1g}$
---
| $(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},\tau
k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2}))$
---
$C_{3v}$; AM |  | $\eta(\pm C_{3z}^{j}\hat{\boldsymbol{e}}_{y})=0$
---
Fe2O3, CoF3 (I) Šmejkal  _et al._ (2022a)
$D_{2}$ | $A$ | $(0,0,k_{x}k_{y})$ |  $C_{2}$; no AM | ✗ |
$D_{3}$ |  | $A_{1}$
---
| $(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},\tau
k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2}))$
---
$C_{3}$; no AM | $\eta(\pm C_{3z}^{j}\hat{\boldsymbol{n}}_{E_{z}}^{y})=0$ |
$D_{4}$ |  | $A_{1}$
---
$B_{1}$
$B_{2}$
| $(0,0,k_{x}k_{y}(k_{x}^{2}-k_{y}^{2}))$
---
$(0,0,k_{x}k_{y})$
$(0,0,k_{x}^{2}-k_{y}^{2})$
$C_{4}$; no AM | ✗ |
$D_{6}$ |  | $A_{1}$
---
$B_{1}$
$B_{2}$
$E_{2}$
| $(0,0,k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(3k_{x}^{2}-k_{y}^{2}))$
---
$(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},0)$
$(2k_{x}k_{y},k_{x}^{2}-k_{y}^{2},0)$
$(0,0,2k_{x}k_{y}),(0,0,k_{x}^{2}-k_{y}^{2})$
$C_{6}$; no AM |  | ✗
---
$\eta(\pm C_{3z}^{j}\hat{\boldsymbol{n}}^{y}_{E_{z}})=0$
$\eta(\pm C_{3z}^{j}\hat{\boldsymbol{n}}_{E_{z}}^{x})=0$
✗
| VNb3S6 (M)Šmejkal  _et al._ (2022a); Inoshita  _et al._ (2019)
---

##  III Model studies

In order to demonstrate the features we have deduced by symmetry analysis
above more explicitly and to discuss additional aspects, we here construct and
solve concrete lattice models; to this end, we use three different point
groups and AM order parameters that can induce an SDE as examples.

###  III.1 General formalism

In all of these lattice models, superconductivity will be described in the
same way. We start with an attractive ($g>0$) interaction in the spin-singlet
Cooper channel,

$\mathcal{H}_{I}=-g\sum_{\boldsymbol{q}}C^{\dagger}_{\boldsymbol{q}}C^{\phantom{\dagger}}_{\boldsymbol{q}},\quad
C^{\phantom{\dagger}}_{\boldsymbol{q}}=\sum_{\boldsymbol{k}}c_{\boldsymbol{k}+\boldsymbol{q}/2}is_{y}c_{-\boldsymbol{k}+\boldsymbol{q}/2},$
(8)

and add it to the Hamiltonian in Eq. (5),
$\mathcal{H}_{0}\rightarrow\mathcal{H}_{0}+\mathcal{H}_{I}$. We then perform a
mean-field decoupling in the Cooper channel and introduce
$\Delta_{\boldsymbol{q}}=g\braket{C_{\boldsymbol{q}}}$, leading to the
following Hamiltonian

$\displaystyle\begin{split}\mathcal{H}_{\text{MF}}=&\sum_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\dagger}h_{\boldsymbol{k}}c_{\boldsymbol{k}}^{\phantom{\dagger}}+\sum_{\boldsymbol{k},\boldsymbol{q}}\left[\Delta_{\boldsymbol{q}}c^{\dagger}_{\boldsymbol{k}+\boldsymbol{q}/2}is_{y}c^{\dagger}_{-\boldsymbol{k}+\boldsymbol{q}/2}+\text{H.c.}\right]\\\
&+\sum_{\boldsymbol{q}}\frac{|\Delta_{\boldsymbol{q}}|^{2}}{g},\quad
h_{\boldsymbol{k}}=s_{0}\epsilon_{\boldsymbol{k}}+\boldsymbol{g}_{\boldsymbol{k}}\cdot\boldsymbol{s}+\boldsymbol{N}_{\boldsymbol{k}}\cdot\boldsymbol{s}.\end{split}$
(9)

Using a redundant Nambu basis, defined by
$\Psi_{\boldsymbol{k}}=(c_{\boldsymbol{k}+\boldsymbol{q}/2},is_{y}c^{\dagger}_{-\boldsymbol{k}+\boldsymbol{q}/2})^{T}$,
we obtain a Bogoliubov-de-Gennes Hamiltonian,

$H_{\text{BdG}}=\begin{pmatrix}h_{\boldsymbol{k}+\boldsymbol{q}/2}&\Delta_{\boldsymbol{q}}\\\
\Delta^{*}_{\boldsymbol{q}}&-s_{y}h^{*}_{-\boldsymbol{k}+\boldsymbol{q}/2}s_{y}\end{pmatrix}$
(10)

which allows us to rewrite the Hamiltonian (9) for a given $\boldsymbol{q}$
as,

$\mathcal{H}_{\text{MF}}(\boldsymbol{q})=\sum_{\boldsymbol{k}}\Psi_{\boldsymbol{k}}^{\dagger}H_{\text{BdG}}\Psi_{\boldsymbol{k}}^{\phantom{\dagger}}+\frac{|\Delta_{\boldsymbol{q}}|^{2}}{g}.$
(11)

Next, we integrate out the electrons in this quadratic mean-field Hamiltonian
and recover the Ginzburg-Landau expression in Eq. (1). The key quantity
$a^{\text{S}}_{\boldsymbol{q}}$ is related to the particle-particle bubble
$\Gamma(\boldsymbol{q})$ at finite $\boldsymbol{q}$ in the usual way,
$a^{\text{S}}_{\boldsymbol{q}}=g^{-1}-\Gamma(\boldsymbol{q})$. Keeping both
intra- and inter-band contributions, the particle-particle bubble is of the
form

$\Gamma(\boldsymbol{q})=\frac{1}{4}\sum_{\boldsymbol{k},\nu=p,m,\eta=\pm}\Lambda_{\boldsymbol{k},\boldsymbol{q},\nu,\eta}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},\nu,\eta}}{2T}},$
(12)

where $\Lambda_{\boldsymbol{k},\boldsymbol{q},\nu,\eta}$ contains the
coherence factors and $\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},\nu,\eta}$
are Bogoliubov energies (see Appendix B for their explicit form); the indices
$\nu$ and $\mu$ label the four different states associated with spin and
particle-hole space in our redundant Nambu basis, cf. Eq. (10). From the
resulting $a^{\text{S}}_{\boldsymbol{q}}$, we compute the directional
dependence of the critical current and hence $\eta(\hat{\boldsymbol{n}})$ as
described in Sec. II.1. We will next apply this formalism using a few
different normal-state Hamiltonians, $\mathcal{H}_{0}$, to elucidate the
nature of the SDE in various point groups.

###  III.2 Pinned zeros and $E_{z}$ tunability in $D_{6h}$

We start from the centrosymmetric point group $\mathcal{G}_{0}=D_{6h}$, where
$\epsilon_{\boldsymbol{k}}$ obeys
$\epsilon_{\boldsymbol{k}}=\epsilon_{-\boldsymbol{k}}=\epsilon_{C_{3z}\boldsymbol{k}}=\epsilon_{\sigma_{v}\boldsymbol{k}}$.
To model such a dispersion, we take a nearest-neighbor hopping model on the
triangular lattice,
$\epsilon_{\boldsymbol{k}}=-\sum_{j=1}^{3}t\cos(\boldsymbol{a}_{j}\cdot\boldsymbol{k})$,
where $\boldsymbol{a}_{j}$ are three $C_{3z}$-related primitive vectors. For
notational simplicity, we measure all energies in units of $t$ in the
following. The presence of $I$ in $D_{6h}$ implies
$\boldsymbol{g}_{\boldsymbol{k}}=0$. We can see in Table 1 that $D_{6h}$ has
four possible AM order parameters and only two of them—those transforming
under $B_{1g}$ and $B_{2g}$—can give rise to a SDE when also applying an
electric field $E_{z}$. For concreteness, we will here focus on $B_{1g}$,
which reduces $D_{6h}$ to the magnetic point group generated by $C_{3z}$,
$\Theta C_{2z}$, $\Theta\sigma_{v}$, and $I$. We construct the lowest-order,
Brillouin-zone-periodic form of $\boldsymbol{N}_{\boldsymbol{k}}$ consistent
with $B_{1g}$, which corresponds to first-nearest-neighbor-hopping processes
on the triangular lattice, with explicit form presented in Appendix B; it
behaves as

$\boldsymbol{N}_{\boldsymbol{k}}\sim
N_{0}(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},0)^{T}$ (13)

around the $\Gamma$ point, i.e., for $|\boldsymbol{k}|\rightarrow 0$. Due to
$\boldsymbol{N}_{\boldsymbol{k}}=\boldsymbol{N}_{-\boldsymbol{k}}$, see Eq.
(6), and $B_{1g}$ being odd under $C_{2z}$, it holds
$(\boldsymbol{N}_{\boldsymbol{k}})_{z}=0$ to arbitrary order in
$\boldsymbol{k}$. Note that this order parameter obeys all of the properties
in our definition of an AM: by design [see Eq. (9)] it is translationally
invariant and breaks time-reversal symmetry (due to
$\boldsymbol{N}_{\boldsymbol{k}}=\boldsymbol{N}_{-\boldsymbol{k}}$). The
absence of finite spin polarization immediately follows by noting that the
residual $C_{3z}$ and $C_{2z}\Theta$ symmetries (also present for $E_{z}\neq
0$) imply, respectively, that $\braket{S_{x,y}}$ and $\braket{S_{z}}$ have to
vanish, which we have also checked explicitly.

To understand the salient features of this model, let us start first with
$E_{z}=0$, with results displayed in the first column (a) of Fig. 2. One can
see in Fig. 2(ai) and (aii) that the maximum of $\Gamma(\boldsymbol{q})$ is
pinned to $\boldsymbol{q}=0$ as long as $N_{0}$ in Eq. (13) is smaller than a
critical value. This pinning is a consequence of $C_{3z}$ symmetry. It means
that, in the current model, a sufficiently strong AM order is required to
induce finite-momentum pairing. While finite-momentum pairing can be induced
by AM order, as pointed out in recent publications in the context of different
models without $C_{3z}$ symmetry Zhang  _et al._ (2023a); Chakraborty and
Black-Schaffer (2023), it does not necessarily follow immediately, as we can
see here. Moreover, since we have inversion symmetry and unbroken $C_{3z}$, we
observe that $\Gamma(\boldsymbol{q})$ is six-fold symmetric. This symmetry is
inherited by the critical current $J_{c}(\hat{\boldsymbol{n}})$ shown in Fig.
2(aiii); it thus obeys
$J_{c}(\hat{\boldsymbol{n}})=J_{c}(-\hat{\boldsymbol{n}})$ and there is no
SDE, $\eta=0$, see Fig. 2(aiv).

As anticipated in Table 1, we therefore have to apply an electric field
$E_{z}$, which reduces $D_{6h}$ to $C_{6v}$, to get an SDE. On the level of
the Hamiltonian, the electric field induces a non-zero spin-orbit vector
$\boldsymbol{g}_{\boldsymbol{k}}$. We again refer to Appendix B for its
explicit form throughout the Brillouin zone and here only note that
$\boldsymbol{g}_{\boldsymbol{k}}\sim\alpha(k_{y},-k_{x},0)^{T}$ as
$|\boldsymbol{k}|\rightarrow 0$. Similar to
$(\boldsymbol{N}_{\boldsymbol{k}})_{z}=0$ noted above, the last component
$(\boldsymbol{g}_{\boldsymbol{k}})_{z}$ has to vanish to all orders in
$\boldsymbol{k}$ as a consequence of $C_{2z}$ and Eq. (6).

The corresponding results for finite $E_{z}$ are shown in the second column,
(b), of Fig. 2. We observe that, even in the presence of $\alpha\neq 0$, the
maximum of $\Gamma(\boldsymbol{q})$ is pinned to zero momentum, see Fig.
2(bi), due to the still unbroken $C_{3z}$ symmetry. However, as a consequence
of inversion symmetry breaking, $\Gamma(\boldsymbol{q})$ is not an even
function anymore. As soon as a critical value of $N_{0}$ is surpassed, the
maxima of $\Gamma(\boldsymbol{q})$ move from zero $\boldsymbol{q}$ to three
non-zero, $C_{3z}$-related momenta, as indicated by the orange circles in Fig.
2(bii). The number of degenerate maxima is reduced from six to three [compared
with Fig. 2(aii)] when applying $E_{z}\neq 0$ as $I$ is broken.

By the same token, the critical current $J_{c}(\hat{\boldsymbol{n}})$ is not
an even function anymore, see Fig. 2(biii), and we obtain a SDE, $\eta\neq 0$,
as can be clearly seen in Fig. 2(biv). This demonstrates explicitly that a SDE
is also possible when the only source of broken $\Theta$ is an AM, i.e., in
the absence of a net magnetic moment. As the SDE requires broken $I$, it is
tunable by an electric field. What is more, we observe that
$\eta(\hat{\boldsymbol{n}})$ has zeros along the high-symmetry directions
$\hat{\boldsymbol{e}}_{x}$, $C_{3z}\hat{\boldsymbol{e}}_{x}$, and
$C_{3z}^{2}\hat{\boldsymbol{e}}_{x}$ as follows from the magnetic reflection
symmetry $\Theta\sigma_{v}$ and Eq. (4) [or $\sigma_{d}$ and Eq. (3) for that
matter]. For an AM order parameter transforming under $B_{2g}$, the behavior
is very similar. The main difference is that the aforementioned zeros of
$\eta(\hat{\boldsymbol{n}})$ are pinned along the orthogonal directions
$\hat{\boldsymbol{e}}_{y}$, $C_{3z}\hat{\boldsymbol{e}}_{y}$, and
$C_{3z}^{2}\hat{\boldsymbol{e}}_{y}$, since, instead of $\Theta\sigma_{v}$, we
have the magnetic reflection symmetry $\Theta\sigma_{d}$.

Figure 2: SDE in point group $D_{6h}$. Column (a) and (b) compare the cases
for zero and non-zero electric field $E_{z}$. Row (i) and (ii) show the
particle-particle bubble $\Gamma(\boldsymbol{q})$ in the Brillouin zone below
and above the critical value of the strength, $N_{0}$, of the AM order; the
orange circles denote the location of the maxima. Row (iii) shows the critical
current $J_{c}(\hat{\boldsymbol{n}})$ as a function of direction
$\hat{\boldsymbol{n}}$. The resulting angle-resolved SDE efficiency (2) is
plotted in the last row (iv) and is only finite for $E_{z}\neq 0$. The
magnetic point group dictates the zeros of $\eta$ to be confined along high-
symmetry directions, as discussed in the main text. Explicit parameters are
given in Cap .

###  III.3 Generic zeros in $C_{6h}$

To demonstrate explicitly that the zeros of $\eta(\hat{\boldsymbol{n}})$ are
not always pinned along high-symmetry directions for the SDE induced by an AM,
we next consider the case $\mathcal{G}_{0}=C_{6h}$. It exhibits no rotations
$C_{2\hat{\boldsymbol{m}}}$ or reflections $\sigma_{\hat{\boldsymbol{m}}}$
with in-plane $\hat{\boldsymbol{m}}$, such that neither Eq. (3) nor Eq. (4)
can apply (irrespective of the form of broken time-reversal symmetry). One
consequence of the reduced symmetries is that the bare dispersion exhibits
more terms compared to $D_{6h}$ discussed above, which we describe with the
leading-order contribution, $\propto t^{\prime}$, in

$\epsilon_{\boldsymbol{k}}=-\sum_{j=1}^{3}\left(t\cos{(\boldsymbol{a}_{j}\cdot\boldsymbol{k})}+t^{\prime}\cos{(\boldsymbol{a}^{\prime}_{j}\cdot\boldsymbol{k})}\right).$
(14)

Here, $\boldsymbol{a}^{\prime}_{j}$ are three $C_{3z}$-related next-nearest-
neighbor vectors on the triangular lattice
($\boldsymbol{a}^{\prime}_{1}=2\boldsymbol{a}_{2}-\boldsymbol{a}_{1}$,
$\boldsymbol{a}^{\prime}_{j}=C_{3z}^{j-1}\boldsymbol{a}^{\prime}_{1}$). As can
be seen in Table 1, only one of the two possible AM order parameters—the one
transforming under $B_{g}$ of $C_{6h}$—can give rise to a SDE. As a result of
the reduced symmetry, the two different AM order parameters associated with
$B_{1g}$ and $B_{2g}$ in $D_{6h}$ now both transform under $B_{g}$ of $C_{6h}$
and can, hence, mix. Introducing a $3\times 3$ matrix $R_{\varphi}$ describing
rotations by $\varphi$ along the third direction, this admixture can be
parametrized as

$\boldsymbol{N}_{\boldsymbol{k}}=R_{\varphi}\boldsymbol{N}^{B_{1g}}_{\boldsymbol{k}}\sim
N_{0}R_{\varphi}(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},0)^{T},$ (15)

where $\boldsymbol{N}^{B_{1g}}_{\boldsymbol{k}}$ is the $B_{1g}$ AM order
parameter of $D_{6h}$ used above in Sec. III.2 and the asymptotic relation
again refers to the vicinity of the $\Gamma$ point, $\boldsymbol{k}\rightarrow
0$.

To obtain a finite SDE, $\eta\neq 0$, inversion symmetry $I$ still needs to be
broken by application of an electric field $E_{z}\neq 0$. This induces a spin-
orbit coupling term $\boldsymbol{g}_{\boldsymbol{k}}$. For the same reason as
above, the spin-orbit vector is less constrained than in $D_{6h}$ and can be
written in the form $R_{\varphi^{\prime}}\boldsymbol{g}_{\boldsymbol{k}}$,
where $\boldsymbol{g}_{\boldsymbol{k}}$ is identical to the one used for
$D_{6h}$ above, obeying
$\boldsymbol{g}_{\boldsymbol{k}}\sim\alpha(k_{y},-k_{x},0)^{T}$. For
$\boldsymbol{N}_{\boldsymbol{k}},\,\boldsymbol{g}_{\boldsymbol{k}}\neq 0$, the
residual magnetic point group of the system is generated by $C_{3z}$ and
$C_{2z}\Theta$. As such, the only constraints on the particle-particle bubble
and the critical current are
$\Gamma(\boldsymbol{q})=\Gamma(C_{3z}\boldsymbol{q})$ and
$J_{c}(\hat{\boldsymbol{n}})=J_{c}(C_{3z}\hat{\boldsymbol{n}})$, which are
clearly visible in our numerical results in Fig. 3(a) and (b), respectively.
As expected, the maxima and minima of these two quantities are not pinned
along high-symmetry directions and, therefore, the zeros of
$\eta(\hat{\boldsymbol{n}})$, shown in Fig. 3(c), are not pinned either.

Figure 3: SDE in the point group $C_{6h}$ at $E_{z}\neq 0$. (a) Particle-
particle bubble $\Gamma(\boldsymbol{q})$ in the Brillouin zone; (b) shows the
critical current $J_{c}(\hat{\boldsymbol{n}})$ as a function of direction
$\hat{\boldsymbol{n}}$; (c) displays the angle-resolved SDE efficiency
$\eta(\hat{\boldsymbol{n}})$. Due to the reduced symmetry compared to $D_{6h}$
[cf. Fig. 2b(iv)], the zeros (and also maxima) of $\eta$ are rotated away from
high-symmetry directions. We choose $N_{0}=-0.21,\alpha=0.4,t^{\prime}=0.05t$,
$g^{-1}=0.5\Gamma(\boldsymbol{q}=0)$, $\varphi=-0.25\pi$,
$\varphi^{\prime}=0$.

###  III.4 $E_{z}$ tunable zeros and polarization in $D_{3}$

We finally discuss the AM-induced SDE for the case where $\mathcal{G}_{0}$ is
already a non-centrosymmetric point group. As such, an electric field will not
be required to get a SDE. Nonetheless, as we will see, $E_{z}$ can still be
used to tune the properties of the system in a non-trivial way. For
concreteness, let us choose $\mathcal{G}_{0}=D_{3}$. As can be seen in Table
1, there is only a single AM order parameter, transforming under the trivial
representation of the point group. While, to leading (quadratic) order in
$\boldsymbol{k}$, it has the same form as $\boldsymbol{N}_{\boldsymbol{k}}$ in
Eq. (13), the subleading corrections are different. In particular, the reduced
symmetries allow for a finite $(\boldsymbol{N}_{\boldsymbol{k}})_{z}$
component;

$\boldsymbol{N}_{\boldsymbol{k}}\sim
N_{0}(k_{x}^{2}-k_{y}^{2},-2k_{x}k_{y},\tau
k_{x}k_{y}(k_{x}^{2}-3k_{y}^{2})(k_{y}^{2}-3k_{x}^{2})),$ (16)

where we only kept the leading term in every component and $\tau$ is a real
parameter that is not determined by symmetry. The total spin polarization
still vanishes, $\braket{\boldsymbol{S}}=0$, as required for an AM, which
follows from the preserved $C_{3z}$ ($\braket{S_{x,y}}=0$) and $C_{2x}$
($\braket{S_{z}}=0$) symmetries.

Interestingly, the spin-orbit vector has two different contributions,
$\boldsymbol{g}_{\boldsymbol{k}}=\boldsymbol{g}_{\boldsymbol{k}}^{\text{I}}+\boldsymbol{g}_{\boldsymbol{k}}^{\text{II}}$
where
$\boldsymbol{g}_{\boldsymbol{k}}^{\text{I}}\sim\beta(k_{x},k_{y},k_{y}(3k_{x}^{2}-k_{y}^{2}))$
is already allowed by the point group $D_{3}$ without the need of an external
electric field, and
$\boldsymbol{g}_{\boldsymbol{k}}^{\text{II}}\sim\alpha(k_{y},-k_{x},k_{x}(k_{x}^{2}-3k_{y}^{2}))$
is the spin-orbit coupling term induced by a perpendicular electric field
$E_{z}$, i.e., $\alpha\propto E_{z}+\mathcal{O}(E_{z}^{3})$. The full
expressions for $\boldsymbol{N}_{\boldsymbol{k}}$,
$\boldsymbol{g}_{\boldsymbol{k}}^{\text{I}}$, and
$\boldsymbol{g}_{\boldsymbol{k}}^{\text{II}}$ we use are provided in Appendix
B. Finally, for the dispersion, we restrict ourselves just to a nearest-
neighbor hopping term on the triangular lattice,
$\epsilon_{\boldsymbol{k}}=-\sum_{j=1}^{3}t\cos(\boldsymbol{a}_{j}\cdot\boldsymbol{k})$.

Let us first focus on $E_{z}=0$ ($\alpha=0$). Since $C_{2z}$, $I$, and
$\Theta$ are already broken, we find a finite SDE, see Fig. 4(a,b).
Furthermore, we observe that the extrema of the critical current rotated by
$90^{\circ}$ in comparison to $D_{6h}$ [cf. Fig. 2(biii)]. This is closely
related to the discussion in Sec. II.1; since $D_{3}$ has two-fold rotational
symmetry $C_{2x}$ about the in-plane direction $x$, the current has to obey
$J_{c}(C_{2x}\hat{\boldsymbol{n}})=J_{c}(\hat{\boldsymbol{n}})$ which pins its
extrema along $\hat{\boldsymbol{e}}_{x}$ and $C_{3z}$-related directions.
Similarly, Eq. (3) implies that the angle-resolved efficiency
$\eta(\hat{\boldsymbol{n}})$ has to go to zero for
$\hat{\boldsymbol{n}}=\hat{\boldsymbol{e}}_{y}$,
$C_{3z}\hat{\boldsymbol{e}}_{y}$ and $C_{3z}^{2}\hat{\boldsymbol{e}}_{y}$,
which is also observed in our numerics [see Fig. 4(b)].

To demonstrate the effect of a finite $E_{z}$, we repeat the analysis now in
the presence of the induced spin-orbit coupling term
$\boldsymbol{g}_{\boldsymbol{k}}^{\text{II}}$. The electric field transforms
as $A_{2}$ of $D_{3}$ and reduces the point group to $C_{3}$. Altogether,
since time-reversal and inversion are still broken, the system still shows a
SDE, see Fig. 4(c,d), only now that there is no two-fold in-plane rotation
($C_{2x}$) or a mirror-plane $\sigma_{\hat{\boldsymbol{m}}}$ as a symmetry.
Naturally, it follows that the zeros of $\eta(\hat{\boldsymbol{n}})$ are not
pinned to the high symmetry directions, but are now free to move in-plane, as
is clearly visible in Fig. 4(d). An electric field can thus be used to tune
the orientation of the zeros of the SDE efficiency
$\eta(\hat{\boldsymbol{n}})$.

Figure 4: SDE in the non-centrosymmetric point group $D_{3}$. (a) and (c) show
the directional-dependence of the critical current
$J_{c}(\hat{\boldsymbol{n}})$ for zero and finite electric field,
respectively. The corresponding SDE efficiencies $\eta(\hat{\boldsymbol{n}})$
are displayed in (b) and (c). Since $E_{z}\neq 0$ breaks all in-plane rotation
symmetries $C_{2\hat{\boldsymbol{m}}}$, it can be used to rotate the symmetry-
imposed zeros of $\eta(\hat{\boldsymbol{n}})$ away from high-symmetry
directions. It can also be used, to control the net magnetization, see (e),
where the spin polarization as a function of $E_{z}$ is shown; the inset is a
zoomed-in version of the respective contribution of the two bands (in orange
and green) to $\braket{S_{z}}$. (f) reveals that both Fermi surfaces
(corresponding to the two bands) are present for $\alpha=0.05$ and, for
$\alpha=0.1$, only one of the bands contribute. Explicit parameters are given
in d (3).

By the same token, we also find that the $E_{z}$-induced reduction of the
point group leads to the order parameter lose its AM nature—since all in-plane
rotations are broken, we expect $\braket{S_{z}}\neq 0$ for $\alpha\neq 0$;
meanwhile we still have $\braket{S_{x,y}}=0$ since $C_{3z}$ remains unbroken.
This is indeed what we find when computing the expectation value of the spin
operators in Eq. (7) explicitly within our model, with the result displayed in
Fig. 4(e). We can further see that the orientation of the spin polarization
can be swapped by changing the sign of $\alpha$, as expected by symmetry, and
also by increasing $|\alpha|$ beyond a certain magnitude. To understand the
latter observation better, we consider the inset in Fig. 4(e), which shows the
contributions from the bands $\sigma=\pm 1$ to the total polarization. As we
start increasing $\alpha$ in the positive direction, for very small values of
$\alpha$, the $\sigma=-1$ band (shown in green) starts contributing more than
its counterpart $\sigma=1$ (shown in orange) to $\braket{S_{z}}$. This happens
until a certain value of $\alpha$ is reached from whence the $\sigma=1$ starts
taking over and ends up contributing the bulk of the total polarization,
thereby flipping the value of $\braket{S_{z}}$. This flipping of polarization
on the same side can also be well understood by noting that at $\alpha=0.05t$,
both Fermi surfaces [see Fig. 4(f)] corresponding to $\sigma=\pm 1$, shown in
orange and green respectively contribute. Meanwhile, when $\alpha=0.1t$, only
the green one contributes to the spin polarization. Taken together, we see
that an AM order parameter in the non-certrosymmetric point group $D_{3}$
(same applies to $D_{6}$, see Table 1) can induce a SDE, with both the zeros
in the efficiency $\eta(\hat{\boldsymbol{n}})$ and the spin polarization being
tunable by an applied electric field $E_{z}$.

##  IV Conclusion

In this work, we have studied under which conditions a conventional two-
dimensional superconducting phase can exhibit a SDE, $\eta_{s}\neq 0$ in Eq.
(2), if the only source of broken time-reversal symmetry is an AM order
parameter. To approach this question systematically, we started by listing all
AM order parameters $\boldsymbol{N}_{\boldsymbol{k}}$, coupling to the
electrons as given by Eq. (5), for all possible 2D crystalline point groups,
see Table 1. For each of them, we checked whether there are still residual
symmetries, such as inversion symmetry or two-fold rotation along a direction
perpendicular to the plane of the system, that would enforce the critical
currents along opposite directions to be the same,
$J_{c}(\hat{\boldsymbol{n}})=J_{c}(-\hat{\boldsymbol{n}})$. We found that
there are indeed several point groups and associated AM order parameters
($C_{6h}$ with AM in $B_{g}$; $D_{6h}$ with AM in $B_{1g}$ and $B_{2g}$;
$D_{3d}$ with $A_{1g}$; $D_{3}$ with AM in $A_{1}$; $D_{6}$ with $B_{1}$ and
$B_{2}$) that give rise to an SDE, as we have also checked by explicit model
calculations. Since, by design, the total magnetic moment vanishes when an AM
is the only source of broken time-reversal symmetry, this demonstrates that
the SDE does not require finite net magnetization.

In the context of the SDE, the AM order parameters can be further categorized
in the following way: the AMs transforming under $B_{g}$ of $C_{6h}$, $B_{1g}$
or $B_{2g}$ of $D_{6h}$, and $A_{1g}$ of $D_{3d}$ will only give rise to an
SDE if also an electric field, $E_{z}$, perpendicular to the plane of the
system is applied. This breaks inversion symmetry, a necessary requirement for
the SDE. Note, however, that in all cases listed here, there are still enough
symmetries present to guarantee a vanishing net magnetic moment. This shows
that AMs allow for electric-field-tunable (rather than the frequently
discussed magnetic-field-tunable) SDEs. In turn, the AMs transforming under
$A_{1}$ and $B_{1,2}$ for the non-centrosymmetric point groups $D_{3}$ and
$D_{6}$, respectively, will immediately give rise to a SDE, without
application of an electric field; moreover, for $E_{z}\neq 0$, the symmetries
will be lowered in such a way that the order parameters cease to be AMs. Here
the electric field can be used to control the out-of-plane magnetization of
the system.

Another important distinction of the different scenarios for AM-induced SDEs
can be made based on the directions $\hat{\boldsymbol{n}}_{j}$ where the SDE
efficiency vanishes, $\eta(\hat{\boldsymbol{n}}_{j})=0$. While
$\hat{\boldsymbol{n}}_{j}$ are pinned along high-symmetry directions for
$D_{6h}$ (reduced to $C_{6v}$ by $E_{z}\neq 0$) and $D_{3d}$ (reduced to
$C_{3v}$), this is not the case for the SDEs induced by the $B_{g}$ AM in
$C_{6h}$ (reduced to $C_{6}$), where $\hat{\boldsymbol{n}}_{j}$ point along
generic ($C_{3z}$-related) directions. Finally, for the AMs of non-
centrosymmetric point groups $D_{3}$ and $D_{6}$ that can induce an SDE,
applying an electric field $E_{z}\neq 0$ allows to rotate the
$\hat{\boldsymbol{n}}_{j}$ with $\eta(\hat{\boldsymbol{n}}_{j})=0$ away from
crystalline to generic directions.

These findings, that can be deduced from symmetry considerations, are
summarized in Table 1. We have further constructed and solved minimal lattice
models for the different classes of and phenomena associated with AM-induced
SDEs discussed above. This further illustrates and explicitly demonstrates the
complex interplay of AM and superconductivity. We finally point out that there
are several setups that can be used to realize and probe the proposed AM-
induced SDE experimentally, see Fig. 1. Given that there are already candidate
materials with the right symmetries (see last column in Table 1), the rapidly
growing number of AM candidate materials Gao  _et al._ (2023), and the fact
that heterostructures are routinely grown or stacked, this seems to be well
within current experimental reach.

###### Acknowledgements.

S.B. and M.S.S. acknowledge funding by the European Union (ERC-2021-STG,
Project 101040651—SuperCorr). Views and opinions expressed are however those
of the authors only and do not necessarily reflect those of the European Union
or the European Research Council Executive Agency. Neither the European Union
nor the granting authority can be held responsible for them. The authors thank
P. McClarty, R. Fernandes, L. Pupim, A. Rastogi and J. Sobral for discussions.

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  * (117) For a(i) weak altermagnetism $N_{0}=-0.1,\alpha=0$; a(ii) strong altermagnetism $N_{0}=-0.2,\alpha=0$; (b)(ii-iv) in presence of electric field $E_{z}$, $N_{0}=-0.23,\alpha=0.4$ and $g^{-1}=0.5\Gamma(\boldsymbol{q}=0)$.
  * d (3) For $E_{z}=0$ we take $N_{0}=-0.1,\beta=0.7,\alpha=0$ and $g^{-1}=0.5\Gamma(\boldsymbol{q}=0)$; for $E_{z}\neq 0$ in second row, $N_{0}=-1.4,\tau=1,\beta=1.2,\alpha=1.0$ and $g^{-1}=0.8\Gamma(\boldsymbol{q}=0)$; (e), (f) $N_{0}=-1.4,\beta=0.8$.

##  Appendix A Expression for the particle-particle bubble

In this section, we state the expression for the particle-particle bubble in
presence of both inter- and intraband pairings. Given the Hamiltonian, defined
in Eq.(11), we begin by defining a few quantities,

$l_{i,\boldsymbol{k},\boldsymbol{q}}=g_{i,\boldsymbol{k},\boldsymbol{q}}+N_{i,\boldsymbol{k},\boldsymbol{q}},\quad
i=x,y,z\quad
g_{p,\boldsymbol{k},\boldsymbol{q}}^{\pm}=l_{x,\boldsymbol{k}+\boldsymbol{q}/2}\pm
il_{y,\boldsymbol{k}+\boldsymbol{q}/2},\quad
g_{m,\boldsymbol{k},\boldsymbol{q}}^{\pm}=l_{x,-\boldsymbol{k}+\boldsymbol{q}/2}\pm
il_{y,-\boldsymbol{k}+\boldsymbol{q}/2}.$ (17)

In addition the renormalized energies are given as

$\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,\pm}=\epsilon_{\boldsymbol{k}+\boldsymbol{q}/2}\pm
g_{p},\quad\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,\pm}=\epsilon_{-\boldsymbol{k}+\boldsymbol{q}/2}\pm
g_{m},\quad$ (18)

where
$g_{p}^{2}=\sum\limits_{i=x,y,z}l_{i,\boldsymbol{k}+\boldsymbol{q}/2}^{2}$ and
$g_{m}^{2}=\sum\limits_{i=x,y,z}l_{i,-\boldsymbol{k}+\boldsymbol{q}/2}^{2}$.
Armed with these definitions, we can write down the various coherence factors
$\Lambda_{\boldsymbol{k},\boldsymbol{q},\nu,\eta}$ for the spinful model as

$\Lambda_{\boldsymbol{k},\boldsymbol{q},m,\mp}=\mp\frac{G_{\boldsymbol{k},\boldsymbol{q}}}{g_{m}(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,\mp}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,+})(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,\mp}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,-})}+\frac{2}{(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,\mp}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,\mp})};$
(19)
$\Lambda_{\boldsymbol{k},\boldsymbol{q},p,\mp}=\mp\frac{G_{\boldsymbol{k},\boldsymbol{q}}}{g_{p}(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,-}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,\mp})(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,+}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,\mp})}+\frac{2}{(\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,\mp}+\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,\mp})};$
(20)

where
$G_{\boldsymbol{k},\boldsymbol{q}}=g_{m,\boldsymbol{k},\boldsymbol{q}}^{+}g_{p,\boldsymbol{k},\boldsymbol{q}}^{-}+g_{m,\boldsymbol{k},\boldsymbol{q}}^{-}g_{p,\boldsymbol{k},\boldsymbol{q}}^{+}+2g_{p}g_{m}+2l_{z,\boldsymbol{k}+\boldsymbol{q}/2}l_{z,-\boldsymbol{k}+\boldsymbol{q}/2}$.
The final expression for the particle-particle bubble $\Gamma(\boldsymbol{q})$
then reads

$\Gamma(\boldsymbol{q})=\frac{1}{4}\sum_{\boldsymbol{k}}\left(\Lambda_{\boldsymbol{k},\boldsymbol{q},p,+}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,+}}{2T}}+\Lambda_{\boldsymbol{k},\boldsymbol{q},p,-}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},p,-}}{2T}}+\Lambda_{\boldsymbol{k},\boldsymbol{q},m,+}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,+}}{2T}}+\Lambda_{\boldsymbol{k},\boldsymbol{q},m,-}\tanh{\frac{\mathcal{E}_{\boldsymbol{k},\boldsymbol{q},m,-}}{2T}}\right).$
(21)

##  Appendix B Details of the models

We here state the basis functions used to characterise the terms
$\boldsymbol{N}_{\boldsymbol{k}}$ and $\boldsymbol{g}_{\boldsymbol{k}}$ of the
Hamiltonian in Eq. (9) of the main text for various point groups. Applying the
convention, where one of the primitive lattice vectors of the underlying
triangular lattice is given by $\boldsymbol{a}_{1}=(1,0)^{T}$, the leading
basis functions explicitly read as

Figure 5: Multi-component momentum dependence of (a) the altermagnet term
$\boldsymbol{N}_{\boldsymbol{k}}$; (b) inversion-symmetry breaking term
$\boldsymbol{g}_{\boldsymbol{k}}^{I}$ and (c) shows the spin-orbit coupling
term $\boldsymbol{g}_{\boldsymbol{k}}^{II}$ induced by $E_{z}$. For point
groups $D_{6h}$ and $C_{6h}$, $\nu=0$ and $\beta=0$, while for $D_{3}$ $\nu=1$
and $\beta\neq 0$.

$\boldsymbol{N}_{\boldsymbol{k}}=N_{0}\left(\frac{-8}{3}(\cos{k_{x}}-\cos{\frac{k_{x}}{2}}\cos{\frac{\sqrt{3}k_{y}}{2}}),\frac{-8\sin{\frac{k_{x}}{2}}\sin{\frac{\sqrt{3}k_{y}}{2}}}{\sqrt{3}},64\nu\frac{(\sin{\frac{5k_{x}}{2}}\sin{\frac{3\sqrt{3}k_{y}}{2}}+\sin{\frac{k_{x}}{2}}\sin{\frac{\sqrt{3}k_{y}}{2}}-\sin{2k_{x}}\sin{\sqrt{3}k_{y}})}{3\sqrt{3}}\right)$
(22)
$\boldsymbol{g}^{\text{I}}_{\boldsymbol{k}}=\beta\left(\frac{2}{3}(2\cos{\frac{k_{x}}{2}}+\cos{\frac{\sqrt{3}k_{y}}{2}})\sin{\frac{k_{x}}{2}}\right.,\left.{}\frac{2}{\sqrt{3}}\cos{\frac{k_{x}}{2}}\sin{\frac{\sqrt{3}k_{y}}{2}},\frac{8(-2\cos{\frac{3k_{x}}{2}}\sin{\frac{\sqrt{3}k_{y}}{2}}+\sin{\sqrt{3}k_{y}})}{3\sqrt{3}}\right),$
(23)
$\boldsymbol{g}^{\text{II}}_{\boldsymbol{k}}=\alpha\left(\frac{2}{\sqrt{3}}\cos{\frac{k_{x}}{2}}\sin{\frac{\sqrt{3}k_{y}}{2}}\right.,\left.{}\frac{-2}{3}(2\cos{\frac{k_{x}}{2}}+\cos{\frac{\sqrt{3}k_{y}}{2}})\sin{\frac{k_{x}}{2}},16\nu\left(-\cos{\frac{k_{x}}{2}}+\cos{\frac{\sqrt{3}k_{y}}{2}}\right)\sin{\frac{k_{x}}{2}}\right).$
(24)

These are the functions with the least number of nodes/nodal lines that
satisfy the required symmetry constraints while being periodic on the
Brillouin zone. As explained in the main text, $\nu=0$ for point groups
$D_{6h}$ and $C_{6h}$ and therefore Eq. (22) reduces to Eq. (13) and Eq. (15)
respectively when expanded for small $\boldsymbol{k}$. As the point group
$D_{3}$ permits the AM order parameter to also have a third component, it is
given fully by Eq. (22) with $\nu=1$ resulting in Eq. (16) for
$\boldsymbol{k}\rightarrow 0$. On the same token, the leading basis functions
for the electric-field induced spin-orbit coupling
$\boldsymbol{g}^{\text{II}}_{\boldsymbol{k}}$ are given by Eq. (24), with
$\nu=0$ for $\mathcal{G}=D_{6h}$ and $C_{6h}$ and $\nu=1$ for
$\mathcal{G}=D_{3}$. Finally, the leading basis function for
$\boldsymbol{g}^{\text{I}}_{\boldsymbol{k}}\sim\beta(k_{x},k_{y},k_{y}(3k_{x}^{2}-k_{y}^{2}))$
is given by Eq. (23).