# Altermagnetism from coincident Van Hove singularities: application to
$\kappa$-Cl

Yue Yu1, Han-Gyeol Suh1, Mercè Roig2, and Daniel F. Agterberg1 1Department of
Physics, University of Wisconsin–Milwaukee, Milwaukee, Wisconsin 53201, USA
2Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

###### Abstract

Realizing two-dimensional (2D) altermagnets is important for spintronics
applications. Here we propose a microscopic template for stabilizing 2D
altermagnetism through Van Hove singularities that are coincident in both
energy and momentum. These coincident Van Hove singularities are a generic
consequence of non-symmorphic symmetries in 8 2D space groups. They allow new
hopping interactions between the Van Hove singularities that do not appear in
analogous Van-Hove singularity based patch models for cuprates and graphene.
We show these new interactions can give rise to various weak coupling, and
BCS-based instabilities, including altermagnetism, nematicity, inter-band
d-wave superconductivity, and orbital altermagnetic order. We apply our
results to quasi-2D organic $\kappa$-Cl in which altermagnetism is known to
appear.

Altermagnetism, distinct from ferromagnetism and antiferromagnetism, preserves
translation symmetry and exhibits zero net magnetization with momentum-
dependent spin texturesKivelson et al. (2003); Šmejkal et al. (2022a, b); Noda
et al. (2016); Ahn et al. (2019); Hayami et al. (2019, 2020); Yuan et al.
(2021); Mazin et al. (2021); Berlijn et al. (2017); Zhu et al. (2019); López-
Moreno et al. (2016); Ishizaka et al. (2011). Analogous to unconventional
superconductors, it establishes a profound connection between magnetism and
topology, hosting nonzero Berry curvature for anomalous Hall transport Šmejkal
et al. (2020); Šmejkal et al. (2022c); Feng et al. (2022); Betancourt et al.
(2023); Naka et al. (2020); Nakatsuji et al. (2015); Samanta et al. (2020);
Sürgers et al. (2016); Ghimire et al. (2018). The momentum-dependent spin
splitting serves as a intrinsic platform for spin-current couplingJungwirth et
al. (2016), enabling practical control in spin devices through the application
of magnetic fields Šmejkal et al. (2022d); Hariki et al. (2023), electrical
currents Naka et al. (2019); Yuan et al. (2020); González-Hernández et al.
(2021); Naka et al. (2021); Shao et al. (2021); Bose et al. (2022), strain Ma
et al. (2021); Guo et al. (2023); Steward et al. (2023); Liu et al. (2018);
López-Moreno et al. (2012), torque Karube et al. (2022); Bai et al. (2022),
and heat Tomczak (2018); Zhou et al. (2023). When coupled with
superconductors, altermagnetism can induce intriguing phenomena Papaj (2023);
Beenakker and Vakhtel (2023); Zhu et al. (2023), such as the Fulde-Ferrell-
Larkin-Ovchinnikov (FFLO) state Zhang et al. (2023); Sumita et al. (2023) and
new platforms for enabling Majorana particles Ghorashi et al. (2023).

As demonstrated through density functional theory (DFT)-based Hartree-Fock
calculations Šmejkal et al. (2022a), the random phase approximation Roig et
al. (2024), and analysis of Fermi surface Pomeranchuk instabilities Wu et al.
(2007); Šmejkal et al. (2022a), altermagnetism is believed to be stabilized
through strong on-site Coulomb interactions. Searches for this strong coupling
instability form the basis for identifying new altermagnetic materials. In
principle, in 2D, Van Hove singularities offer a weak coupling route towards
stabilizing altermagnetism. Van Hove singularities are saddle points in the
energy dispersion and induce a logarithmic divergence in the 2D density of
states. Tuning the chemical potential across such singularities results in
divergent susceptibilities, signaling an instability into a variety of
possible competing orders that have been examined in cuprates, graphene, and
Kagome metals. Kiesel et al. (2012); Nandkishore et al. (2012); Wang et al.
(2012); Furukawa et al. (1998); Kampf and Katanin (2003); Honerkamp et al.
(2001); Le Hur and Rice (2009); Wang et al. (2013); Gonzalez (2008); Martin
and Batista (2008); Halboth and Metzner (2000); Honerkamp and Salmhofer
(2001); Schulz (1987); Kiesel and Thomale (2012); Yu and Li (2012). However,
in these applications, altermagnetism is not one of these competing orders,
suggesting that this route is not viable for stabilizing this state.

More recently, it has been found that $d$-wave altermagnetism can be an
instability of higher-order Van Hove singularities Classen et al. (2020).
However, higher-order Van Hove singularities require tuning the chemical
potential and also altering the band structure to eliminate the usual
logarithmic singularity. This degree of fine-tuning constraints the
practicality of experimentally exploiting higher-order Van Hove singularities
to stabilize altermagnetism.

Here we identify a mechanism based on usual 2D logarithmic Van-Hove
singularities that stabilize altermagnetism. Two ingredients are key to
realizing this mechanism. The first is the existence of a pair of Van Hove
singularities that are coincident in both energy and momentum. The second is
the existence of a new hopping interaction between these Van Hove
singularities. This hopping interaction does not appear in Van Hove-based
patch models for cuprates Furukawa et al. (1998); Kampf and Katanin (2003);
Honerkamp et al. (2001); Le Hur and Rice (2009) and graphene Kiesel et al.
(2012); Nandkishore et al. (2012); Wang et al. (2012); Classen et al. (2020)
where it is forbidden by translation invariance. Both ingredients are a
generic consequence of non-symmorphic symmetries that exist in 8 2D space
groups when spin-orbit coupling is neglected. Furthermore, unlike the Van Hove
driven spin density wave transitions that occur in cuprate Furukawa et al.
(1998); Kampf and Katanin (2003); Honerkamp et al. (2001); Le Hur and Rice
(2009), graphene Kiesel et al. (2012); Nandkishore et al. (2012); Wang et al.
(2012); Classen et al. (2020), and Kagome metal Wang et al. (2013) patch
models, our altermagnetic mechanism does not require nesting or near-nesting
of the bands. Instead, it is based on BCS instability, which typically yields
only a superconducting instability for the other Van Hove singularity
scenarios. Our analysis also stabilizes other orders including interband
d-wave superconductivity, nematicity, and orbital altermagnetism. To be
concrete, we apply our analysis to the organic material $\kappa$-Cl where an
altermagnetic state is believed to occur.

Coincident Van Hove singularities: The phase diagram of the quasi-2D
orthorhombic organic compound $\kappa$-(ET)2Cu[N(CN)2]Cl ($\kappa$-Cl) shares
similarities with cuprates McKenzie (1997), exhibiting unconventional
superconductivity adjacent to magnetism under pressure Williams et al. (1990);
Miyagawa et al. (2004); Kagawa et al. (2005) and anion substitution Kurosaki
et al. (2005); Miyagawa et al. (2002); Kanoda (1997); Mayaffre et al. (1995);
De Soto et al. (1995); Kanoda et al. (1996); Urayama et al. (1988); Kini et
al. (1990). Recently, considerable hole-doping has been achieved in
$\kappa$-Cl Kawasugi et al. (2016) allowing more control of the electronic
phases. In this context, DFT calculations reveal a Van Hove singularity at the
$S=(\pi,\pi)$ point that can be tuned to the chemical potential with an
achievable hole doping Koretsune and Hotta (2014). Here we show that the space
group symmetry requires that this Van Hove singularity consists of a pair of
Van Hove singularities that are coincident in both energy and momentum.

$\kappa$-Cl belongs to 2D space group L25 de la Flor et al. (2021); Kopskỳ and
Litvin (2002). With negligible spin-orbit coupling (SOC) Koretsune and Hotta
(2014); Naka et al. (2019), a detailed group theory analysis (provided in the
appendix) reveals its kp Hamiltonian at the $S=(\pi,\pi)$ point is:

$H(\pi+k_{x},\pi+k_{y})=t_{x}k_{x}^{2}+t_{y}k_{y}^{2}+k_{x}k_{y}(t_{4}\tau_{x}+t_{5}\tau_{z})-\widetilde{\mu},$
(1)

with Pauli matrices $\tau_{i}$. The resulting dispersions of the two bands
are:

$\begin{split}E_{\pm}=t_{x}k_{x}^{2}+t_{y}k_{y}^{2}\pm\sqrt{t_{4}^{2}+t_{5}^{2}}k_{x}k_{y}-\widetilde{\mu}.\end{split}$
(2)

The saddle point conditions for these two dispersions are the same:
$t_{4}^{2}+t_{5}^{2}+(t_{x}-t_{y})^{2}>(t_{x}+t_{y})^{2}$. Consequently, each
band hosts a Van Hove singularity at the band crossing point $S=(\pi,\pi)$.
This feature is a generic symmetry-required consequence of many 2D space
groups.

Figure 1: Crystal structure of $\kappa$-Cl. In each unit cell, there are two
ET molecule dimers (green ellipses).

The kp Hamiltonian in Eq.1 can also be obtained from a tight-binding model.
$\kappa$-Cl has four ET molecules per unit cell, forming two molecule-dimers
(Fig. 1) at positions: $(r_{1},r_{2})=\\{(0,0),(1/2,1/2)\\}$. As a minimal
model capturing the coincident Van Hove singularities, we consider a 2D tight-
binding model with one $s$-orbital on each dimer position. We use the Pauli
matrices $\tau_{z}=\pm$ to distinguish $(1,2)$ with different dimer (or
sublattice) positions and the Pauli matrices $\sigma_{i}$ for the spin. While
spin-orbit coupling (SOC) is negligible Koretsune and Hotta (2014); Naka et
al. (2019), the normal state tight-binding Hamiltonian is:

$\begin{split}H&=2t_{1}\cos{k_{x}}+2t_{2}\cos k_{y}+4t_{3}\cos k_{x}\cos
k_{y}\\\ &+4t_{4}\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\tau_{x}+t_{5}\sin
k_{x}\sin k_{y}\tau_{z}-\mu.\end{split}$ (3)

Each band is doubly degenerate, consisting of spin-up and down states. Due to
the nonsymmorphic symmetries, the band without SOC is 4-fold degenerate at the
entire Brillouin zone boundary $k_{x}=\pi$ and $k_{y}=\pi$ Suh et al. (2023).
Near the Brillouin zone corner S=$(\pi,\pi)$, we obtain the kp Hamiltonian in
Eq.1, with $\widetilde{\mu}=\mu+2t_{1}+2t_{2}-4t_{3}$, $t_{x}=(t_{1}-2t_{3})$,
and $t_{y}=(t_{2}-2t_{3})$.

A key property of the kp Hamiltonian in Eq.1 is that two sublattice operators,
$\tau_{x}$ and $\tau_{z}$, have the same symmetry at the $S$ point. The other
non-trivial sublattice operator $\tau_{y}$ is odd under time-reversal symmetry
and can only appear in the Hamiltonian multiplied by spin-1/2 operators Suh et
al. (2023). When this SOC is ignored, then symmetry requires the above kp
Hamiltonian to occur in the 2D orthorhombic space groups L21, L25, L44, and
the 2D tetragonal space groups L54, L56, L58, L60, and L63, where these groups
are in Refs. de la Flor et al., 2021; Kopskỳ and Litvin, 2002. Generically,
for all these space groups there is a 4-fold degeneracy at the zone boundary
and a coincident Van Hove point at the zone corner. When SOC is included, the
coincident Van Hove is unmodified in the 2D space groups orthorhombic L44 and
tetragonal L63. In the following, we ignore SOC and discuss its inclusion in
the appendix.

In the following, we work in the band basis for which the kp Hamiltonian, Eq.
1 is diagonal. To diagonalize the kp Hamiltonian, we take the unitary
transformation $u=\cos(\theta/2)I-i\sin(\theta/2)\sigma_{y}$ where
$\cos\theta=t_{5}/\sqrt{t_{4}^{2}+t_{5}^{2}}$. $\widetilde{\tau_{x}}$ and
$\widetilde{\tau_{z}}$ in the band basis are then linear combinations of
$\tau_{x}$ and $\tau_{z}$, and both have the same symmetry properties as
$k_{x}k_{y}$. $\widetilde{\tau_{y}}$ and $\tau_{y}$ are the same in the
sublattice and band bases and have the same symmetry properties as
$k_{x}k_{y}\sigma_{z}$ when spin and spatial rotations are coupled by SOC.

Patch model and interactions: The coincident Van Hove singularities provide a
novel platform to study strongly interacting fermions and a natural mechanism
for stabilizing 2D altermagnetism. To understand this, it is useful to
consider a patch model Furukawa et al. (1998). Here, the density of states of
each band is approximated as a patch situated solely at the Van Hove point.
With the coincident Van Hove singularities, we end up with two patches located
at the same point: $(\pi,\pi)$. This patch model has similarities with that
examined for cuprates. For cuprates, there are also two Van Hove
singularities, but these Van Hove singularities are not coincident and appear
at the two distinct momenta $(0,\pi)$ and $(\pi,0)$. The key difference
between patch models for coincident Van Hoves points and those for the
cuprates is in the interactions allowed by symmetry. Both models contain the
standard interactions $g_{1}$, $g_{2}$, $g_{3}$, and $g_{4}$ (see Fig. 2).
However, our coincident Van Hove patch model contains a new hopping
interaction, $g_{5}$, shown in Fig.2

Figure 2: Allowed interactions for two coincident patches: the exchange
interaction $g_{1}$, inter-patch Hubbard interaction $g_{2}$, pair-hopping
interaction $g_{3}$, intra-patch Hubbard interaction $g_{4}$, and the new
hopping interaction $g_{5}$. Here, upper propagators are for spin-up, while
lower ones are for spin-down. We choose the convention such that the
interaction with three solid and one dashed propagator is $g_{5}$, while that
with one solid and three dashed propagators is $-g_{5}$.

This $g_{5}$-interaction involves a hopping between the two patches which, due
to momentum conservation, is permissible only when two patches coincide. More
precisely, since $\widetilde{\tau_{x}}$ and $\widetilde{\tau_{z}}$ have the
same symmetry, it is possible to form local Cooper pairs
$\widetilde{\tau_{x}}i\sigma_{y}:\Delta_{xy,1}=c_{2\downarrow}c_{1\uparrow}+c_{1\downarrow}c_{2\uparrow}$
and
$\widetilde{\tau_{z}}i\sigma_{y}:\Delta_{xy,2}=c_{1\downarrow}c_{1\uparrow}-c_{2\downarrow}c_{2\uparrow}$
that also share the same symmetry (here 1 and 2 label the bands). The local
interaction $g_{5}\Delta_{xy,1}^{\dagger}\Delta_{xy,2}$ is then allowed by
symmetry.

This interaction is generated by an on-site Hubbard interaction after
transforming to the band basis, and its magnitude can be large. In particular,
for the interaction

$H_{\rm int}=U\sum_{m}n_{m\uparrow}n_{m\downarrow},$ (4)

where $m$ denotes the sublattice index, we find the bare interactions (denoted
by $g_{i}^{0}$): $g^{0}_{1}=g^{0}_{2}=g^{0}_{3}=(U/2)\sin^{2}\theta$,
$g_{3}^{0}+g^{0}_{4}=U$, $g^{0}_{4}-g^{0}_{3}=U\cos^{2}\theta$, and
$g^{0}_{5}=(U/2)\sin\theta\cos\theta$. The $g_{i}^{0}$ for the cuprate patch
model is recovered when $\theta=\pi/2$.

When the chemical potential is tuned close to the Van Hove singularity, the
enhancement in the density of states results in substantial corrections to
interaction strengths. These corrections originate through the intra- and
inter-band particle-particle ($\chi^{intra}_{pp}$, $\chi^{inter}_{pp}$ ) and
particle-hole ($\chi^{intra}_{ph}$, $\chi^{intra}_{ph}$ ) susceptibilities for
the free fermions Kiesel et al. (2012); Nandkishore et al. (2012); Wang et al.
(2012); Furukawa et al. (1998); Kampf and Katanin (2003); Honerkamp et al.
(2001); Le Hur and Rice (2009); Wang et al. (2013); Gonzalez (2008); Martin
and Batista (2008); Halboth and Metzner (2000); Honerkamp and Salmhofer
(2001); Schulz (1987); Kiesel and Thomale (2012); Yu and Li (2012). Here the
dominant corrections stem from $\chi^{intra}_{pp}$. This is because
$\chi^{intra}_{pp}$ exhibits the conventional BCS logarithmic divergence
multiplied by the logarithmic divergence in the density of states
($\chi^{intra}_{pp}$ diverges as $\ln^{2}\Lambda/T$) while other
susceptibilities only exhibit the latter and hence diverge as $\ln\Lambda/T$.
Keeping only the dominant interaction corrections, the one-loop BCS
corrections to the interactions are:

$\begin{split}&\Delta g_{1}=-2\chi_{pp}^{intra}g_{5}^{2}\\\ &\Delta
g_{2}=-2\chi_{pp}^{intra}g_{5}^{2}\\\ &\Delta
g_{3}=-2\chi_{pp}^{intra}g_{3}g_{4}\\\ &\Delta
g_{4}=-\chi_{pp}^{intra}(g_{3}^{2}+g_{4}^{2})\\\ &\Delta
g_{5}=-\chi_{pp}^{intra}(g_{4}-g_{3})g_{5}\\\
\rightarrow&\left[\begin{array}[]{l}\Delta(g_{4}+g_{3})=-\chi_{pp}^{intra}(g_{4}+g_{3})^{2}\\\
\Delta(g_{4}-g_{3})=-\chi_{pp}^{intra}(g_{4}-g_{3})^{2}\end{array}\right].\end{split}$
(5)

When $g_{5}=0$, these equations are the same as those found in cuprate patch
models Furukawa et al. (1998); Honerkamp et al. (2001). When $g_{5}\neq 0$,
these BCS corrections to $g_{3,4}$ do not involve $g_{5}$. Hence, a bare
attractive $g^{0}_{4}+g^{0}_{3}$ (or $g^{0}_{4}-g^{0}_{3}$) enhances itself
and gives an intra-band s-wave superconducting $\Delta=i\sigma_{y}$ (or d-wave
superconducting $\Delta=i\widetilde{\tau_{z}}\sigma_{y}$) instability. For
onsite Hubbard interactions, we have $g^{0}_{4}>g^{0}_{3}>0$, causing the BCS
correction to suppress these two instabilities.

Figure 3: $g_{5}$-interaction leads to BCS corrections to $g_{1}$ and $g_{2}$.

The $g_{5}$-interaction introduces new one-loop BCS corrections to the
exchange interaction $g_{1}$, and the inter-band Hubbard interaction $g_{2}$,
as shown in Fig.3. Notably, these corrections are always negative, regardless
of the signs of interactions. This can drive $g_{1,2}<0$. In the appendix, we
consider a patch renormalized group study including subleading corrections,
which are important as $g_{4}-g_{3}$ vanish Arovas et al. (2022). We consider
the small $g_{3,4}$ sector and find that BCS corrections from $g_{5}$ can push
the RG flow to stable fixed points with divergent $g_{1,2}<0$.

Instability | RPA eigenvalue
---|---
d-wave altermagnetism: $\widetilde{\tau_{z}}\sigma$ | $\chi_{ph}^{intra}|g_{2}|$
nematicity: $\widetilde{\tau_{x}}$ | $\chi_{ph}^{inter}|g_{1}|$
orbital altermagnetism: $\widetilde{\tau_{y}}$ | $\chi_{ph}^{inter}|g_{1}|$
d-wave superconductivity: $\widetilde{\tau_{x}}i\sigma_{y}$ | $\chi_{pp}^{inter}|g_{1}+g_{2}|$
Table 1: Orders stabilized by coincident Van Hove points and corresponding
eigenvalues.

Competing instabilities: The above discussion reveals that negative $g_{1}$
and $g_{2}$ can become the dominant interactions, and so we may neglect other
interactions. We find that $g_{1}$ and $g_{2}$ generically lead to competing
instabilities in the intra-band particle-hole, inter-band particle-hole, and
inter-band particle-particle channels (Table.1). These instabilities can be
captured by the self-consistent one-loop vertex corrections shown in Fig. 4.
These diagrams linearly couple the different vertices and the leading
instability is given by finding the largest eigenvalue of these coupled
vertices. The leading instabilities and their eigenvalues are given in Table.
1.

These instabilities include a conventional $d$-wave altermagnetic order (with
a momentum-dependent spin splitting that varies as $k_{x}k_{y}\sigma_{z}$),
inter-band $d_{xy}$ superconductivity, $\epsilon_{xy}$ nematicity, and an
orbital current loop order that has the same symmetry as the $d$-wave
altermagnetic order. Symmetry implies that this orbital altermagnetic order
will induce an anomalous Hall effect under the application of an
$\epsilon_{xy}$ strain, even without the presence of SOC (a detailed
explanation is in the appendix). When SOC is present, this orbital
altermagnetic order and the conventional $d$-wave altermagnetic order will
coexist since these two orders will then be bilinearly coupled. It is notable
that the d-wave superconductivity we find is inter-band, and cannot be
stabilized through a conventional BCS mechanism.

Figure 4: Self-consistent one-loop vertex corrections. (Top) Intra-band
particle-hole vertex correction for altermagnetism. (Middle) Inter-band
particle-hole vertex correction for nematicity and orbital altermagnetism.
(Bottom) Inter-band particle-particle vertex correction for inter-band
superconductivity. Figure 5: Bare susceptibilities $\chi_{pp}^{inter}$,
$\chi_{ph}^{inter}$ and $\chi_{ph}^{intra}$, normalized with respect to
$\chi_{ph}^{intra}$. The energy dispersion is in Eq.2, with $t_{x}=t_{y}$ and
$a\equiv\frac{\sqrt{t_{4}^{2}+t_{5}^{2}}}{2t_{x}}$. The same energy cutoff
$\Lambda_{i}$ is used for this computation. Van Hove singularities require
$a>1$. A factor of 2 is included for $\chi_{pp}^{inter}$ since the
corresponding SC vertex is contributed by both $g_{1}$ and $g_{2}$.

While the one-loop correction for the interaction comes from the dominant
intra-band particle-particle (BCS) channel, the one-loop correction for the
vertices all belong to the three subleading channels. The ultimate leading
instability can be determined by the magnitude of the three eigenvalues
$\chi_{pp}^{inter}|g_{1}+g_{2}|$, $\chi_{ph}^{inter}|g_{1}|$ and
$\chi_{ph}^{intra}|g_{2}|$. The three susceptibilities share the same scaling:
$\chi\propto\log\Lambda_{i}/T$, generically with different energy cutoff
$\Lambda_{i}$. To examine what parameter range enables the different
instabilities, we set the cutoffs in all three susceptibilities the same and
consider the limit $g_{1}=g_{2}$. This limit occurs when only BCS corrections
from on-site Coulomb interactions are included but generally $g_{1}\neq
g_{2}$. The results then depend upon the kp dispersion in Eq.2. Fig. 5 reveals
that all instability channels are stable for some choice of
$a\equiv\frac{\sqrt{t_{4}^{2}+t_{5}^{2}}}{2t_{x}}$. Furthermore, a smaller $a$
is beneficial for the conventional altermagnetic state. We note that once
$g_{1}>g_{2}$ (or $g_{1}<g_{2}$), the inter-band (or intra-band) altermagnetic
phase will expand.

Figure 6: (Top)Conventional RPA bubbles from $g_{1,2}$, for intra-band and
inter-band particle-hole channel. (Bottom) Contribution from BCS channel due
to the $g_{5}$ interaction. The two pairs of vertical propagators can be from
both patches.

It is worth noting that the above results are not captured by the standard
random phase approximation (RPA). The conventional RPA diagram is shown in the
top panels in Fig. 6. It gives the usual instability criteria
$g_{2}\chi_{ph}^{intra}>1$ (topleft) and $g_{1}\chi_{ph}^{inter}>1$
(topright). The BCS instability involving the $g_{5}$-interaction is
equivalent to a two-loop correction described by the bottom panel, with a
instability criteria $4|g_{5}|^{2}\chi_{pp}^{intra}\chi_{ph}^{intra}>1$. Since
$\chi_{pp}^{intra}\gg\chi_{ph}^{intra},\chi_{ph}^{inter}$ at low-temperature,
the BCS correction from $g_{5}$ describes a stronger weak-coupling
instability. While our argument emphasizes Van Hove singularities in 2D, in 3D
similar interactions appear for high symmetry momenta in some space groups,
including at the $R$-point of space group P42/nmm, which applies to RuO2.
While strictly speaking, in 3D, there is no Van Hove divergence, the BCS
instability still exists and will aid in stabilizing an altermagnetic state.

Discussion: Here we have identified coincident Van Hove singularities as a
platform to realize 2D altermagnetic states. These states are stabilized due
to a new interaction term, $g_{5}$, through a weak-coupling BCS mechanism that
leads to altermagnetism, nematic, d-wave superconducting, and orbital
altermagnetic orders. Our results apply to 8 2D space groups and we have
chosen a specific application in $\kappa$-Cl, where altermagnetism with the
same symmetry found here is observed in $\kappa$-Cl Miyagawa et al. (1995) and
d-wave superconductivity is reported under pressure Williams et al. (1990);
Miyagawa et al. (2004) and anion substitution Kurosaki et al. (2005); Miyagawa
et al. (2002); Kanoda (1997); Mayaffre et al. (1995); De Soto et al. (1995);
Kanoda et al. (1996); Urayama et al. (1988); Kini et al. (1990).

We thank Philip Brydon, Andrey Chubukov, Tatsuya Shishidou, Stephen Wilson,
and Michael Weinert for useful discussions. D.F.A. and Y.Y were supported by
the National Science Foundation Grant No. DMREF 2323857. H.G. Suh was
supported by the Department of Energy, Office of Basic Energy Science,
Division of Materials Sciences and Engineering under Award No. DE-SC0021971.
M. R. acknowledges support from the Novo Nordisk Foundation grant
NNF20OC0060019.

## References

  * Kivelson et al. (2003) S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. Tranquada, A. Kapitulnik, and C. Howald, Reviews of Modern Physics 75, 1201 (2003).
  * Šmejkal et al. (2022a) L. Šmejkal, J. Sinova, and T. Jungwirth, Physical Review X 12, 040501 (2022a).
  * Šmejkal et al. (2022b) L. Šmejkal, J. Sinova, and T. Jungwirth, Physical Review X 12, 031042 (2022b).
  * Noda et al. (2016) Y. Noda, K. Ohno, and S. Nakamura, Physical Chemistry Chemical Physics 18, 13294 (2016).
  * Ahn et al. (2019) K.-H. Ahn, A. Hariki, K.-W. Lee, and J. Kuneš, Physical Review B 99, 184432 (2019).
  * Hayami et al. (2019) S. Hayami, Y. Yanagi, and H. Kusunose, journal of the physical society of japan 88, 123702 (2019).
  * Hayami et al. (2020) S. Hayami, Y. Yanagi, and H. Kusunose, Physical Review B 102, 144441 (2020).
  * Yuan et al. (2021) L.-D. Yuan, Z. Wang, J.-W. Luo, and A. Zunger, Physical Review Materials 5, 014409 (2021).
  * Mazin et al. (2021) I. I. Mazin, K. Koepernik, M. D. Johannes, R. González-Hernández, and L. Šmejkal, Proceedings of the National Academy of Sciences 118, e2108924118 (2021).
  * Berlijn et al. (2017) T. Berlijn, P. C. Snijders, O. Delaire, H.-D. Zhou, T. A. Maier, H.-B. Cao, S.-X. Chi, M. Matsuda, Y. Wang, M. R. Koehler, et al., Physical review letters 118, 077201 (2017).
  * Zhu et al. (2019) Z. Zhu, J. Strempfer, R. Rao, C. Occhialini, J. Pelliciari, Y. Choi, T. Kawaguchi, H. You, J. Mitchell, Y. Shao-Horn, et al., Physical review letters 122, 017202 (2019).
  * López-Moreno et al. (2016) S. López-Moreno, A. Romero, J. Mejía-López, and A. Muñoz, Physical Chemistry Chemical Physics 18, 33250 (2016).
  * Ishizaka et al. (2011) K. Ishizaka, M. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miyahara, A. Kimura, et al., Nature materials 10, 521 (2011).
  * Šmejkal et al. (2020) L. Šmejkal, R. González-Hernández, T. Jungwirth, and J. Sinova, Science advances 6, eaaz8809 (2020).
  * Šmejkal et al. (2022c) L. Šmejkal, A. H. MacDonald, J. Sinova, S. Nakatsuji, and T. Jungwirth, Nature Reviews Materials 7, 482 (2022c).
  * Feng et al. (2022) Z. Feng, X. Zhou, L. Šmejkal, L. Wu, Z. Zhu, H. Guo, R. González-Hernández, X. Wang, H. Yan, P. Qin, et al., Nature Electronics 5, 735 (2022).
  * Betancourt et al. (2023) R. G. Betancourt, J. Zubáč, R. Gonzalez-Hernandez, K. Geishendorf, Z. Šobáň, G. Springholz, K. Olejník, L. Šmejkal, J. Sinova, T. Jungwirth, et al., Physical Review Letters 130, 036702 (2023).
  * Naka et al. (2020) M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Motome, and H. Seo, Physical Review B 102, 075112 (2020).
  * Nakatsuji et al. (2015) S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
  * Samanta et al. (2020) K. Samanta, M. Ležaić, M. Merte, F. Freimuth, S. Blügel, and Y. Mokrousov, Journal of applied physics 127 (2020).
  * Sürgers et al. (2016) C. Sürgers, W. Kittler, T. Wolf, and H. v. Löhneysen, AIP Advances 6 (2016).
  * Ghimire et al. (2018) N. J. Ghimire, A. Botana, J. Jiang, J. Zhang, Y.-S. Chen, and J. Mitchell, Nature communications 9, 3280 (2018).
  * Jungwirth et al. (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature nanotechnology 11, 231 (2016).
  * Šmejkal et al. (2022d) L. Šmejkal, A. B. Hellenes, R. González-Hernández, J. Sinova, and T. Jungwirth, Physical Review X 12, 011028 (2022d).
  * Hariki et al. (2023) A. Hariki, T. Yamaguchi, D. Kriegner, K. Edmonds, P. Wadley, S. Dhesi, G. Springholz, L. Šmejkal, K. Vỳbornỳ, T. Jungwirth, et al., arXiv preprint arXiv:2305.03588 (2023).
  * Naka et al. (2019) M. Naka, S. Hayami, H. Kusunose, Y. Yanagi, Y. Motome, and H. Seo, Nature communications 10, 4305 (2019).
  * Yuan et al. (2020) L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Physical Review B 102, 014422 (2020).
  * González-Hernández et al. (2021) R. González-Hernández, L. Šmejkal, K. Vỳbornỳ, Y. Yahagi, J. Sinova, T. Jungwirth, and J. Železnỳ, Physical Review Letters 126, 127701 (2021).
  * Naka et al. (2021) M. Naka, Y. Motome, and H. Seo, Physical Review B 103, 125114 (2021).
  * Shao et al. (2021) D.-F. Shao, S.-H. Zhang, M. Li, C.-B. Eom, and E. Y. Tsymbal, Nature Communications 12, 7061 (2021).
  * Bose et al. (2022) A. Bose, N. J. Schreiber, R. Jain, D.-F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal, D. G. Schlom, et al., Nature Electronics 5, 267 (2022).
  * Ma et al. (2021) H.-Y. Ma, M. Hu, N. Li, J. Liu, W. Yao, J.-F. Jia, and J. Liu, Nature communications 12, 2846 (2021).
  * Guo et al. (2023) S.-D. Guo, X.-S. Guo, K. Cheng, K. Wang, and Y. S. Ang, arXiv preprint arXiv:2306.04094 (2023).
  * Steward et al. (2023) C. R. Steward, R. M. Fernandes, and J. Schmalian, Physical Review B 108, 144418 (2023).
  * Liu et al. (2018) Z. Liu, H. Chen, J. Wang, J. Liu, K. Wang, Z. Feng, H. Yan, X. Wang, C. Jiang, J. Coey, et al., Nature Electronics 1, 172 (2018).
  * López-Moreno et al. (2012) S. López-Moreno, A. Romero, J. Mejía-López, A. Muñoz, and I. V. Roshchin, Physical Review B 85, 134110 (2012).
  * Karube et al. (2022) S. Karube, T. Tanaka, D. Sugawara, N. Kadoguchi, M. Kohda, and J. Nitta, Physical review letters 129, 137201 (2022).
  * Bai et al. (2022) H. Bai, L. Han, X. Feng, Y. Zhou, R. Su, Q. Wang, L. Liao, W. Zhu, X. Chen, F. Pan, et al., Physical Review Letters 128, 197202 (2022).
  * Tomczak (2018) J. M. Tomczak, Journal of Physics: Condensed Matter 30, 183001 (2018).
  * Zhou et al. (2023) X. Zhou, W. Feng, R.-W. Zhang, L. Smejkal, J. Sinova, Y. Mokrousov, and Y. Yao, arXiv preprint arXiv:2305.01410 (2023).
  * Papaj (2023) M. Papaj, arXiv preprint arXiv:2305.03856 (2023).
  * Beenakker and Vakhtel (2023) C. Beenakker and T. Vakhtel, Physical Review B 108, 075425 (2023).
  * Zhu et al. (2023) D. Zhu, Z.-Y. Zhuang, Z. Wu, and Z. Yan, arXiv preprint arXiv:2305.10479 (2023).
  * Zhang et al. (2023) S.-B. Zhang, L.-H. Hu, and T. Neupert, arXiv preprint arXiv:2302.13185 (2023).
  * Sumita et al. (2023) S. Sumita, M. Naka, and H. Seo, arXiv preprint arXiv:2308.14227 (2023).
  * Ghorashi et al. (2023) S. A. A. Ghorashi, T. L. Hughes, and J. Cano, arXiv preprint arXiv:2306.09413 (2023).
  * Roig et al. (2024) M. Roig, A. Kreisel, Y. Yu, B. Andersen, and D. Agterberg, arXiv preprint (2024).
  * Wu et al. (2007) C. Wu, K. Sun, E. Fradkin, and S.-C. Zhang, Phys. Rev. B 75, 115103 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.115103.
  * Kiesel et al. (2012) M. L. Kiesel, C. Platt, W. Hanke, D. A. Abanin, and R. Thomale, Physical Review B 86, 020507 (2012).
  * Nandkishore et al. (2012) R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Nature Physics 8, 158 (2012).
  * Wang et al. (2012) W.-S. Wang, Y.-Y. Xiang, Q.-H. Wang, F. Wang, F. Yang, and D.-H. Lee, Physical Review B 85, 035414 (2012).
  * Furukawa et al. (1998) N. Furukawa, T. Rice, and M. Salmhofer, Physical review letters 81, 3195 (1998).
  * Kampf and Katanin (2003) A. P. Kampf and A. Katanin, Physical Review B 67, 125104 (2003).
  * Honerkamp et al. (2001) C. Honerkamp, M. Salmhofer, N. Furukawa, and T. M. Rice, Physical Review B 63, 035109 (2001).
  * Le Hur and Rice (2009) K. Le Hur and T. M. Rice, Annals of Physics 324, 1452 (2009).
  * Wang et al. (2013) W.-S. Wang, Z.-Z. Li, Y.-Y. Xiang, and Q.-H. Wang, Physical Review B 87, 115135 (2013).
  * Gonzalez (2008) J. Gonzalez, Physical Review B 78, 205431 (2008).
  * Martin and Batista (2008) I. Martin and C. Batista, Physical review letters 101, 156402 (2008).
  * Halboth and Metzner (2000) C. J. Halboth and W. Metzner, Physical Review B 61, 7364 (2000).
  * Honerkamp and Salmhofer (2001) C. Honerkamp and M. Salmhofer, Physical Review B 64, 184516 (2001).
  * Schulz (1987) H. Schulz, Europhysics Letters 4, 609 (1987).
  * Kiesel and Thomale (2012) M. L. Kiesel and R. Thomale, Physical Review B 86, 121105 (2012).
  * Yu and Li (2012) S.-L. Yu and J.-X. Li, Physical Review B 85, 144402 (2012).
  * Classen et al. (2020) L. Classen, A. V. Chubukov, C. Honerkamp, and M. M. Scherer, Physical Review B 102, 125141 (2020).
  * McKenzie (1997) R. H. McKenzie, Science 278, 820 (1997).
  * Williams et al. (1990) J. M. Williams, A. M. Kini, H. H. Wang, K. D. Carlson, U. Geiser, L. K. Montgomery, G. J. Pyrka, D. M. Watkins, and J. M. Kommers, Inorganic Chemistry 29, 3272 (1990).
  * Miyagawa et al. (2004) K. Miyagawa, K. Kanoda, and A. Kawamoto, Chemical reviews 104, 5635 (2004).
  * Kagawa et al. (2005) F. Kagawa, K. Miyagawa, and K. Kanoda, Nature 436, 534 (2005).
  * Kurosaki et al. (2005) Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda, and G. Saito, Physical review letters 95, 177001 (2005).
  * Miyagawa et al. (2002) K. Miyagawa, A. Kawamoto, and K. Kanoda, Physical review letters 89, 017003 (2002).
  * Kanoda (1997) K. Kanoda, Physica C: Superconductivity 282, 299 (1997).
  * Mayaffre et al. (1995) H. Mayaffre, P. Wzietek, D. Jérome, C. Lenoir, and P. Batail, Physical review letters 75, 4122 (1995).
  * De Soto et al. (1995) S. M. De Soto, C. P. Slichter, A. M. Kini, H. Wang, U. Geiser, and J. Williams, Physical Review B 52, 10364 (1995).
  * Kanoda et al. (1996) K. Kanoda, K. Miyagawa, A. Kawamoto, and Y. Nakazawa, Physical Review B 54, 76 (1996).
  * Urayama et al. (1988) H. Urayama, H. Yamochi, G. Saito, K. Nozawa, T. Sugano, M. Kinoshita, S. Sato, K. Oshima, A. Kawamoto, and J. Tanaka, Chemistry Letters 17, 55 (1988).
  * Kini et al. (1990) A. M. Kini, U. Geiser, H. H. Wang, K. D. Carlson, J. M. Williams, W. Kwok, K. Vandervoort, J. E. Thompson, and D. L. Stupka, Inorganic Chemistry 29, 2555 (1990).
  * Kawasugi et al. (2016) Y. Kawasugi, K. Seki, Y. Edagawa, Y. Sato, J. Pu, T. Takenobu, S. Yunoki, H. M. Yamamoto, and R. Kato, Nature communications 7, 12356 (2016).
  * Koretsune and Hotta (2014) T. Koretsune and C. Hotta, Physical Review B 89, 045102 (2014).
  * de la Flor et al. (2021) G. de la Flor, B. Souvignier, G. Madariaga, and M. I. Aroyo, Acta Crystallographica Section A: Foundations and Advances 77, 559 (2021).
  * Kopskỳ and Litvin (2002) V. Kopskỳ and D. Litvin (2002).
  * Suh et al. (2023) H. G. Suh, Y. Yu, T. Shishidou, M. Weinert, P. Brydon, and D. F. Agterberg, Physical Review Research 5, 033204 (2023).
  * Arovas et al. (2022) D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu, Annual review of condensed matter physics 13, 239 (2022).
  * Miyagawa et al. (1995) K. Miyagawa, A. Kawamoto, Y. Nakazawa, and K. Kanoda, Physical review letters 75, 1174 (1995).

##  Appendix A symmetry arguments for the coincident Van Hove singularities

To derive the minimal kp Hamiltonian $H\propto k_{x}k_{y}\hat{O}$ with two
distinct operators $\hat{O}$ that share the same symmetry, it is first
necessary to demonstrate that the bands exhibit four-fold degeneracy along the
$k_{x}=\pi$ and $k_{y}=\pi$ lines. We adopt the notation
$\widetilde{M_{x}}=\\{M_{x}|t^{x}_{x},t^{y}_{x}\\}$ to describe the non-
symmorphic symmetry $\widetilde{M_{x}}$, followed by a translation vector
$(t^{x}_{x},t^{y}_{x})$. The inversion center is taken to be the origin.

###  A.1 With SOC

On the $k_{x}=\pi$ line, the symmetries that keep momenta invariant are
generated by $\\{TI,\widetilde{M_{x}},\widetilde{M_{z}}\\}$. The four-fold
degeneracy comes from the following two two-fold degeneracies. Firstly, when
$\\{\widetilde{M_{x}},\widetilde{M_{z}}\\}=0$, states $|\psi\rangle$ and
$M_{z}|\psi\rangle$ have opposite $\widetilde{M_{x}}$ eigenvalues, so they are
orthogonal degenerate eigenstates. Secondly, states $|\psi\rangle$ and
$TI|\psi\rangle$ are always orthogonal degenerate eigenstates due to the
Kramers’ degeneracy. To make $TI|\psi\rangle$ and $M_{z}|\psi\rangle$
orthogonal to each other, we require $|\psi\rangle$ and $TI|\psi\rangle$ share
the same $\widetilde{M_{x}}$ eigenvalue. Then $|\psi\rangle$,
$TI|\psi\rangle$, $M_{z}|\psi\rangle$ and $TIM_{z}|\psi\rangle$ are the
desired four mutually orthogonal degenerate eigenstates.

We now obtain the conditions for $|\psi\rangle$ and $TI|\psi\rangle$ to share
the same $\widetilde{M_{x}}$ eigenvalue. For
$\widetilde{M_{x}}=\\{M_{x}|t^{x}_{x},t^{y}_{x}\\}$, we have
$\widetilde{M_{x}}^{2}=(i\sigma_{x})^{2}\\{0,2t^{y}_{x}\\}=-\exp(2ik_{y}t_{x}^{y})$.
The eigenvalues of $\widetilde{M_{x}}$ are $\pm i\exp(ik_{y}t_{x}^{y})$. Let
$|\psi\rangle$ be an eigenstate of $\widetilde{M_{x}}$ with eigenvalue
$i\exp(ik_{y}t_{x}^{y})$, then its Kramers’ partner $TI|\psi\rangle$ has
eigenvalue:

$\begin{split}&\widetilde{M_{x}}TI|\psi\rangle=\\{2t_{x}^{x},2t_{x}^{y}\\}TI\widetilde{M_{x}}|\psi\rangle\\\
&=e^{2ik_{x}t_{x}^{x}+2ik_{y}t_{x}^{y}}TI(ie^{ik_{y}t_{x}^{y}}|\psi\rangle)\\\
&=(-e^{2ik_{x}t_{x}^{x}})ie^{ik_{y}t_{x}^{y}}TI|\psi\rangle.\end{split}$ (6)

To make $|\psi\rangle$ and $TI|\psi\rangle$ share the same $\widetilde{M_{x}}$
eigenvalue, we thus require $k_{x}=\pi$ and $t^{x}_{x}=1/2$.

We now obtain the conditions to make
$\\{\widetilde{M_{x}},\widetilde{M_{z}}\\}=0$ on the $k_{x}=\pi$ line. Their
spin part $\sigma_{x}$ and $\sigma_{z}$ already anticommute. The orbital parts
give $\exp(2ik_{x}t^{x}_{z})$. We thus require $k_{x}t^{x}_{z}=0$.

Now the four-fold degeneracy on the $k_{x}=\pi$ line requires $t^{x}_{x}=1/2$
and $t^{x}_{z}=0$. Similarly, the four-fold degeneracy on the $k_{y}=\pi$ line
requires $t^{y}_{y}=1/2$ and $t^{y}_{z}=0$. Using
$t^{x}_{x}+t^{x}_{y}+t^{x}_{z}=0$, we obtain the conditions for the kp
Hamiltonian:

$\begin{split}&\text{Centered at }(k_{x},k_{y})=(\pi,\pi)\\\
&\widetilde{M_{x}}=\\{M_{x}|1/2,1/2\\}\\\
&\widetilde{M_{y}}=\\{M_{y}|1/2,1/2\\}\\\
&\widetilde{M_{z}}=\\{M_{z}|0,0\\}\end{split}$ (7)

This condition applies to layer space groups L44 and L63. The explicit kp
Hamiltonians can be classified into the following three classes Suh et al.
(2023):

$\begin{split}D_{4h,2}:\;&A_{1g}+2B_{2g}+[A_{1g}]\\\
H=&k_{x}k_{y}\tau_{z}+k_{x}k_{y}\tau_{x}+k_{x}k_{y}(k_{x}^{2}-k_{y}^{2})\tau_{y}\sigma_{z}\\\
D_{4h,4}:\;&A_{1g}+A_{2g}+B_{2g}+[B_{1g}]\\\
H=&k_{x}k_{y}\tau_{z}+k_{x}k_{y}(k_{x}^{2}-k_{y}^{2})\tau_{x}+k_{x}k_{y}\tau_{y}\sigma_{z}\\\
D_{2h,2}:\;&A_{g}+2B_{1g}+[A_{g}]\\\
H=&k_{x}k_{y}\tau_{z}+k_{x}k_{y}\tau_{x}+k_{x}k_{y}\tau_{y}\sigma_{z},\\\
\end{split}$ (8)

up to proper coefficients. The corresponding point group symmetry at the high-
symmetry points is included. Space group L44 belong to the $D_{2h,2}$ class.
Space group L63 belong to either $D_{4h,2}$ or $D_{4h,4}$ class, depending on
the Wyckoff position.

###  A.2 Without SOC

When SOC is neglected, there is always a two-fold spin degeneracy. To get a
four-fold degeneracy, we need to obtain a two-fold orbital degeneracy. Spin
will be ignored in the following analysis.

On the $k_{x}=\pi$ line, the possible symmetries that keep momenta invariant
are generated by $\\{TI,\widetilde{M_{x}},\widetilde{M_{z}}\\}$. There are two
ways to obtain the two-fold orbital degeneracy. Firstly, if
$\\{\widetilde{M_{x}},\widetilde{M_{z}}\\}=0$, states $|\psi\rangle$ and
$M_{z}|\psi\rangle$ have opposite $\widetilde{M_{x}}$ eigenvalues, so they are
orthogonal degenerate eigenstates. For spinless fermions, this requires
$\exp(2ik_{x}t^{x}_{z})=-1$, i.e. $t^{x}_{z}=1/2$.

Another way involves orbital Kramers’ degeneracy from time-reversal operation:

$\begin{split}&(TIM_{x})^{2}=\\{2t_{x}^{x},0\\}=\exp(2ik_{x}t^{x}_{x})\\\
&(TIM_{z}M_{x})^{2}=\\{2t_{y}^{x},0\\}=\exp(2ik_{x}t^{x}_{y}).\end{split}$ (9)

The requirement is either $t^{x}_{x}=1/2$ or $t^{x}_{y}=1/2$. Here, inversion
symmetry $I$ does not have to be present. For example, if $C_{2x}=IM_{x}$ is a
symmetry and $(TIM_{x})^{2}=-1$, the desired degeneracy is guaranteed.

Similarly, to get the four-fold band degeneracy on the $k_{y}=\pi$ line, we
require either $t^{y}_{x,y,z}$ to be $1/2$. The relevant 2D layer groups are
L21, L25, L32, L33, L34, L39, L42, L43, L44, L45, L46, L54, L56, L58, L60,
L62, L63, and L64. By explicitly checking the kp Hamiltonian, we find that in
L21, L25, L44, L54, L56, L58, L60, and L63, $\tau_{x}$ and $\tau_{z}$ share
the same symmetry as $k_{x}k_{y}$, giving rise to the coincident Van Hove
singularity. For example, in layer group L25, the S=$(\pi,\pi)$ point is
invariant under the point group $C_{2v}$, which consists of the identity $E$,
two mirror symmetries $M_{x}$ and $M_{y}$, and their product $C_{2z}$. The
symmetry of the orbital operators are $A_{1}+2A_{2}+[A_{1}]$, which gives full
kp Hamiltonian:

$\begin{split}C_{2v}:\;&A_{1}+2A_{2}+[A_{1}]\\\
H=&k_{x}k_{y}\left(\tau_{z}+\tau_{x}+\tau_{y}\sigma_{z}\right)+k_{x}\tau_{y}\sigma_{y}+k_{y}\tau_{y}\sigma_{x}\\\
+&k_{x}\sigma_{y}+k_{y}\sigma_{x}\end{split}$ (10)

When SOC is negligible, the Hamiltonian reduces to Eq.1 in the main text.

##  Appendix B Details of Instability Analysis

For particle-hole vertices, we only consider the $(\uparrow,\uparrow)$ and
$(\downarrow,\downarrow)$ vertices due to the spin-rotational symmetry. The
top panel of Fig.4 shows a self-consistent one-loop correction in the intra-
band particle-hole channel. It describes the correction to the
$\Gamma_{1\uparrow,1\uparrow}$ vertex from the
$\Gamma_{2\downarrow,2\downarrow}$ vertex. The self-consistent equation can be
written as a matrix form:

$\begin{split}\vec{\Gamma}_{ph}^{intra}&\equiv\left[\Gamma_{1\uparrow,1\uparrow},\Gamma_{1\downarrow,1\downarrow},\Gamma_{2\uparrow,2\uparrow},\Gamma_{2\downarrow,2\downarrow}\right]\\\
\vec{\Gamma}_{ph}^{intra}&=\vec{\Gamma}_{ph,0}^{intra}+M_{ph}^{intra}\vec{\Gamma}_{ph}^{intra}\\\
M_{ph}^{intra}&=-\chi_{ph}^{intra}g_{2}\left[\begin{array}[]{cccc}&&&1\\\
&&1&\\\ &1&&\\\ 1&&&\end{array}\right]\end{split}$ (11)

The leading instability in this channel corresponds to the eigenvector of the
matrix $M_{ph}^{intra}$ with the largest positive eigenvalue. For a negative
$g_{2}$, such eigenvector is $[1,-1,-1,1]$ with eigenvalue
$\chi_{ph}^{intra}|g_{2}|$. The corresponding vertex,
$\Gamma_{1\uparrow,1\uparrow}-\Gamma_{1\downarrow,1\downarrow}-\Gamma_{2\uparrow,2\uparrow}+\Gamma_{2\downarrow,2\downarrow}$,
describes the altermagnetic state $M=\widetilde{\tau_{z}}\sigma_{z}$. Another
eigenvector $[1,1,1,1]$ has the same eigenvalue but does not correspond to any
orderings.

The middle panel of Fig.4 shows a self-consistent one-loop correction in the
inter-band particle-hole channel. It describes the correction to the
$\Gamma_{1\downarrow,2\downarrow}$ vertex from the
$\Gamma_{1\uparrow,2\uparrow}$ vertex. The self-consistent equations of this
channel have the following matrix form:

$\begin{split}\vec{\Gamma}_{ph}^{inter}&\equiv\left[\Gamma_{1\uparrow,2\uparrow},\Gamma_{1\downarrow,2\downarrow},\Gamma_{2\uparrow,1\uparrow},\Gamma_{2\downarrow,1\downarrow}\right]\\\
\vec{\Gamma}_{ph}^{inter}&=\vec{\Gamma}_{ph,0}^{inter}+M_{ph}^{inter}\vec{\Gamma}_{ph}^{inter}\\\
M_{ph}^{inter}&=-\chi_{ph}^{inter}g_{1}\left[\begin{array}[]{cccc}&1&&\\\
1&&&\\\ &&&1\\\ &&1&\end{array}\right]\end{split}$ (12)

There are two leading instabilities for $g_{1}<0$ with eigenvalue
$\chi_{ph}^{inter}|g_{1}|$. The first one has eigenvector $[1,1,1,1]$. The
corresponding vertex,
$\Gamma_{1\uparrow,2\uparrow}+\Gamma_{1\downarrow,2\downarrow}+\Gamma_{2\uparrow,1\uparrow}+\Gamma_{2\downarrow,1\downarrow}$,
describes the nematicity $O=\widetilde{\tau_{x}}$. In the lattice basis, it
has components $\tau_{x}$ and $\tau_{z}$ with the same symmetry as
$k_{x}k_{y}$, so it breaks mirror reflections symmetries for $x\rightarrow-x$
and $y\rightarrow-y$. The second instability has eigenvector $[1,1,-1,-1]$.
The corresponding vertex,
$\Gamma_{1\uparrow,2\uparrow}+\Gamma_{1\downarrow,2\downarrow}-\Gamma_{2\uparrow,1\uparrow}-\Gamma_{2\downarrow,1\downarrow}$
describes another unconventional magnetic state $O=\widetilde{\tau_{y}}$. In
the lattice basis, it is $\tau_{y}$, which is an even-parity current-loop
order. When SOC is present, it has the same symmetry as the altermagnetism
$k_{x}k_{y}\sigma_{z}$, so it is an orbital altermagnetic order.

The bottom panel of Fig.4 shows a self-consistent one-loop correction in the
inter-band particle-particle channel. The self-consistent equations in this
channel have a matrix form:

$\begin{split}\vec{\Gamma}_{pp}^{inter}&\equiv\left[\Gamma^{sc}_{1\uparrow,2\downarrow},\Gamma^{sc}_{2\uparrow,1\downarrow}\right]\\\
\vec{\Gamma}_{pp}^{inter}&=\vec{\Gamma}_{pp,0}^{inter}+M_{pp}^{inter}\vec{\Gamma}_{pp}^{inter}\\\
M_{pp}^{inter}&=-\chi_{pp}^{inter}\left[\begin{array}[]{cc}g_{2}&g_{1}\\\
g_{1}&g_{2}\end{array}\right]\end{split}$ (13)

For $g_{1},g_{2}<0$, the eigenvector with the largest positive eigenvalue is
$[1,1]$, with eigenvalue $\chi_{pp}^{inter}|g_{1}+g_{2}|$. The corresponding
vertex,
$\Gamma^{sc}_{1\uparrow,2\downarrow}+\Gamma^{sc}_{2\uparrow,1\downarrow}$,
describes the inter-band superconductivity
$\Delta=\widetilde{\tau_{x}}i\sigma_{y}$. In the lattice basis, it has
components $\tau_{x}i\sigma_{y}$ and $\tau_{z}i\sigma_{y}$, with the same
symmetry as a d-wave superconductor $k_{x}k_{y}i\sigma_{y}$. There is no
vertex correction in the intra-band particle-particle channel from $g_{1}$ or
$g_{2}$.

##  Appendix C Patch RG Calculation

We perform a one-loop patch renormalization group analysis. When only keeping
the leading (BCS) correction, the RG flow is

$\begin{split}&\dot{g_{1}}=-2\dot{\chi}_{pp}^{intra}g_{5}^{2}\\\
&\dot{g_{2}}=-2\dot{\chi}_{pp}^{intra}g_{5}^{2}\\\
&\dot{g_{4}}+\dot{g_{3}}=-\dot{\chi}_{pp}^{intra}(g_{4}+g_{3})^{2}\\\
&\dot{g_{4}}-\dot{g_{3}}=-\dot{\chi}_{pp}^{intra}(g_{4}-g_{3})^{2}\\\
&\dot{g_{5}}=\dot{\chi}_{pp}^{intra}g_{5}(g_{3}-g_{4})\end{split}$ (14)

Here $\dot{...}\equiv\frac{d(...)}{dt}$, with $t\equiv\log\Lambda/T$. Note
that $g_{5}$ does not introduce BCS correction to $g_{3,4}$. The RG flow for
$g_{3,4}$ has three fixed points. The first fixed point has divergent
$g_{4}+g_{3}<0$ while finite $g_{4}-g_{3}$, describing the standard BCS
instability for s-wave superconductivity $\Delta=i\sigma_{y}$. This fixed
point is not affected by $g_{5}$, as $g_{5}$ is finite. The second fixed point
has divergent $g_{4}-g_{3}<0$ while finite $g_{4}+g_{3}$, originally
describing the BCS instability for d-wave superconductivity
$\Delta=i\tau_{z}\sigma_{y}$. This fixed point has divergent $g_{5}$, and
consequently, divergent $g_{1,2}<0$. The presence of $g_{5}$ leads to other
subleading instabilities. In this work, we focus on the last fixed point
scenario where $g_{4}$ starts sufficiently repulsive, such that $g_{4}\pm
g_{3}$ are both reduced under the RG flow. For this case, subleading
corrections are important.

The full RG flows are :

$\begin{split}\dot{g_{1}}&=2g_{1}g_{4}+2ag_{1}(g_{2}-g_{1})-2bg_{1}g_{2}\\\
&-(2ct+2a)g_{5}^{2}\\\
\dot{g_{2}}&=2(g_{1}-g_{2})g_{4}+a(g_{2}^{2}+g_{3}^{2})-b(g_{1}^{2}+g_{2}^{2})\\\
&-(2ct-2)g_{5}^{2}\\\
\dot{g_{3}}&=2a(2g_{2}-g_{1})g_{3}-2ctg_{3}g_{4}+(2b+2)g_{5}^{2}\\\
\dot{g_{4}}&=g_{1}^{2}-2g_{2}^{2}+2g_{1}g_{2}+g_{4}^{2}-ct(g_{3}^{2}+g_{4}^{2})\\\
&+2(a-b)g_{5}^{2}\\\
\dot{g_{5}}&=g_{5}\left[g_{2}-2g_{1}+a(2g_{2}-g_{1})-b(g_{1}+g_{2})\right.\\\
&\left.+g_{4}+ag_{3}+ct(g_{3}-g_{4})\right]\end{split}$ (15)

For simplicity, we have defined $\dot{\chi}_{ph}^{intra}\equiv 1$,
$\dot{\chi}_{ph}^{inter}\equiv a$, $\dot{\chi}_{pp}^{inter}\equiv b$, and
$\dot{\chi}_{pp}^{intra}\equiv ct$.

Let us focus on the sector of fixed point, where $g_{3,4}$ becomes small due
to their BCS corrections. The RG flow can be simplified to:

$\begin{split}&\dot{g_{1}}=2ag_{1}(g_{2}-g_{1})-2bg_{1}g_{2}-2ctg_{5}^{2}\\\
&\dot{g_{2}}=ag_{2}^{2}-b(g_{1}^{2}+g_{2}^{2})-2ctg_{5}^{2}\\\
&\dot{g_{5}}=g_{5}\left[g_{2}-2g_{1}+a(2g_{2}-g_{1})-b(g_{1}+g_{2})\right]\end{split}$
(16)

For the $g_{5}^{2}$ corrections, we only keep the dominant intra-band
particle-particle channel. Near instabilities $t\rightarrow t_{c}$, the RG
flow for interaction strengths diverges. When studying weak-coupling
instabilities, the change in the interaction strengths predominantly happens
near $t\rightarrow t_{c}$. As an approximation, we will take $t=t_{c}$ in the
RG flow. Now the BCS corrections all have the form $ct_{c}g_{5}^{2}$. We can
define $\widetilde{g_{5}}=\sqrt{ct_{c}}g_{5}$ to absorb the unknown paramter
$t_{c}$. The RG flow becomes

$\begin{split}&\dot{g_{1}}=2ag_{1}(g_{2}-g_{1})-2bg_{1}g_{2}-2\widetilde{g_{5}}^{2}\\\
&\dot{g_{2}}=ag_{2}^{2}-b(g_{1}^{2}+g_{2}^{2})-2\widetilde{g_{5}}^{2}\\\
&\dot{\widetilde{g_{5}}}=\widetilde{g_{5}}\left[g_{2}-2g_{1}+a(2g_{2}-g_{1})-b(g_{1}+g_{2})\right]\end{split}$
(17)

Now the RG flow does not formally depend on $t_{c}$. At the end of the
calculation, we will take the $t_{c}\gg 1$ limit. For the above equations, the
generic solution for an interaction strength X is
$X\rightarrow\frac{G_{X}}{t_{c}-t}$. The equations for $G_{X}$ are:

$\begin{split}&G_{1}=2aG_{1}(G_{2}-G_{1})-2bG_{1}G_{2}-2\widetilde{G_{5}}^{2}\\\
&G_{2}=aG_{2}^{2}-b(G_{1}^{2}+G_{2}^{2})-2\widetilde{G_{5}}^{2}\\\
&\widetilde{G_{5}}=\widetilde{G_{5}}\left[G_{2}-2G_{1}+a(2G_{2}-G_{1})-b(G_{1}+G_{2})\right]\end{split}$
(18)

The fixed point behavior is described by $G_{X}$. To understand the dominant
instability at a given fixed point, we turn to the vertex RG. The RG flow for
a given vertex $\Gamma$ reads $\dot{\Gamma}\propto M\Gamma$, where $M$ is a
linear function of $X$ as described by Fig.4 in the main text. The generic
solution for the vertices is $\Gamma\propto(t_{c}-t)^{-\gamma_{i}}$. The
corresponding susceptiblity scales as
$\chi_{\Gamma}\propto(t_{c}-t)^{1-2\gamma_{i}}$. The leading instability thus
corresponds to the largest positive $\gamma_{i}$. These powers $\gamma_{i}$
are listed in Table. 2. Here, we have taken $t_{c}\gg 1$, such that
$\widetilde{G_{5}}$ is kept, while $\widetilde{G_{5}}/\sqrt{t_{c}}$ is
neglected.

O | $\sigma_{z}$ | $\widetilde{\tau_{z}}$ | $\widetilde{\tau_{z}}\sigma_{z}$ | $\widetilde{\tau_{x}}$ | $\widetilde{\tau_{x}}\sigma_{z}$
---|---|---|---|---|---
$\gamma$ | $G_{1}$ | $2G_{2}-G_{1}$ | $-G_{1}$ | $a(-2G_{1}+G_{2})$ | $aG_{2}$

$\Delta$ | $\widetilde{\tau_{x,z}}i\sigma_{y}$ | $\tau_{y}\sigma_{z}i\sigma_{y}$
---|---|---
$\gamma$ | $\frac{-b(G_{2}+G_{1})}{2}\pm\sqrt{(\frac{-b(G_{2}+G_{1})}{2})^{2}+4b\widetilde{G_{5}}^{2}}$ | $-b(G_{2}-G_{1})$

Table 2: (Top) Vertex powers in the particle-hole channels. (Bottom) Vertex
powers in the particle-particle channels. If $G_{1}+G_{2}>0$,
$\widetilde{\tau_{z}}$ takes the larger value; otherwise
$\widetilde{\tau_{x}}$ takes the larger value.

Fig. A1: (Left) Instabilities at the stable fixed point, as a function of the
bare susceptibility ratio. (Right) The corresponding fixed point solution with
$g_{1,2}<0$, when $g_{5}$ is not considered.

The stable fixed point solution is shown in the left panel of Fig.A1. We omit
the trivial fixed point, where all interactions vanish. The three possible
instabilities are inter-band SC $\Delta=\widetilde{\tau_{x}}i\sigma_{y}$
(yellow), nematicity $O=\widetilde{\tau_{x}}$ (lighter blue), and
altermagnetism $M=\widetilde{\tau_{z}}\sigma_{z}$ (darker blue). They become
dominant when $\chi_{pp}^{inter}$, $\chi_{ph}^{inter}$, and
$\chi_{ph}^{intra}$ are dominant, respectively. This is consistent with the
results in the main text. When $g_{3,4}$ are included, this fixed point leads
to a small repulsive $g_{4}\pm g_{3}$.

Fig. A2: RG flow for initial interactions $a=0.5$, $b=0.2$, $g_{1}=0.03$ and
$g_{2}=0.1$. (left) For $g_{5}=0$, $g_{1,2}$ flows to $+\infty$. (right) A
small $g_{5}\neq 0$ can drive $g_{1,2}$ to $-\infty$.

Qualitatively, $g_{5}$ does not introduce a new fixed point solution. The
fixed point $g_{1,2}<0$ exists even if $g_{5}$ is not considered (Right panel
of Fig.A1). However, $g_{5}$ leads to negative BCS corrections to $g_{1,2}$,
and makes this fixed point more preferable. An example is shown in Fig. A2.
When $g_{5}$-interaction is absent, $g_{1}$ and $g_{2}$ flow to positive
infinity for the given initial interactions. As $g_{2}$ approaches 1, the one-
loop RG flow becomes uncontrolled. The above fixed point is unstable to a non-
zero $g_{5}$-interaction. As shown in the right panel, $\widetilde{g_{5}}$ can
drive $g_{1}$ and $g_{2}$ to negative infinity, before the theory becomes
uncontrolled.

##  Appendix D Effect of SOC

We now consider the orthorhombic $D_{2h,2}$ class with SOC, which is relevant
for L44. It has the following kp Hamiltonian:

$\begin{split}D_{2h,2}:H=k_{x}k_{y}\tau_{z}+k_{x}k_{y}\tau_{x}+k_{x}k_{y}\tau_{y}\sigma_{z},\end{split}$
(19)

with proper coefficients. To diagonalize the kp Hamiltonian, we take the
unitary transformation $u\propto
c_{1}I+c_{2}\tau_{x}\sigma_{z}+c_{3}\tau_{z}\sigma_{z}+c_{4}\tau_{y}$, with
proper prefactors $c_{1,2,3,4}$, such that operators in band basis and lattice
basis have the same symmetry.

In this class, all three terms share the same symmetry as $k_{x}k_{y}$, so
there are more allowed interactions. Mixing the corresponding Cooper pairs:
$\widetilde{\tau_{x}i\sigma_{y}}:\Delta_{xy,1}=c_{2\downarrow}c_{1\uparrow}+c_{1\downarrow}c_{2\uparrow}$,
$\widetilde{\tau_{z}i\sigma_{y}}:\Delta_{xy,2}=c_{1\downarrow}c_{1\uparrow}-c_{2\downarrow}c_{2\uparrow}$,
and
$\widetilde{\tau_{y}\sigma_{z}i\sigma_{y}}:\Delta_{xy,3}=i(c_{1\downarrow}c_{2\uparrow}+c_{2\downarrow}c_{1\uparrow})$,
we have three interactions:

$\begin{split}&g_{5}(c^{\dagger}_{2\downarrow}c^{\dagger}_{1\uparrow}+c^{\dagger}_{1\downarrow}c^{\dagger}_{2\uparrow})(c_{1\downarrow}c_{1\uparrow}-c_{2\downarrow}c_{2\uparrow})+c.c.\\\
&ig_{6}(c^{\dagger}_{1\downarrow}c^{\dagger}_{2\uparrow}+c^{\dagger}_{2\downarrow}c^{\dagger}_{1\uparrow})(c_{1\downarrow}c_{1\uparrow}-c_{2\downarrow}c_{2\uparrow})+c.c.\\\
&ig_{7}(c_{1\downarrow}^{\dagger}c_{2\uparrow}^{\dagger}+c_{2\downarrow}^{\dagger}c_{1\uparrow}^{\dagger})(c_{2\downarrow}c_{1\uparrow}-c_{2\uparrow}c_{1\downarrow})+c.c..\end{split}$
(20)

Notably, $g_{7}$ is different from the exchange interaction. The absence of
spin-rotational symmetry also allows another inter-patch Hubbard interaction:
$g_{8}(n_{1\uparrow}n_{2\uparrow}+n_{1\downarrow}n_{2\downarrow})$, and the
spin-flip interactions:
$g_{9}c_{1\uparrow}^{\dagger}c_{2\uparrow}^{\dagger}c_{2\downarrow}c_{1\downarrow}+c.c.$.

In total, 9 interactions are needed under SOC for the orthorhombic class
$D_{2h,2}$. They are exchange interaction $g_{1}$, inter-band Hubbard
interaction $g_{2}n_{1\uparrow}n_{2\downarrow}$, pair-hopping interaction
$g_{3}$, intra-band Hubbard interaction $g_{4}$, real single-hopping
interaction $g_{5}$, imaginary single-hopping interaction $ig_{6}$, imaginary
exchange interaction $ig_{7}$, inter-band Hubbard interaction
$g_{8}n_{1\uparrow}n_{2\uparrow}$, and spin-flip interaction $g_{9}$.

The necessity of these nine interactions can also be understood from the point
group classification at the high-symmetry point: $A_{g}+2B_{1g}+[A_{g}]$.
These group elements correspond to operators $1$, $(\tau_{x},\tau_{z})$, and
$\tau_{y}$. When including the spin to the antisymmetric (time-reversal odd)
element $[A_{g}]$, the point groups elements are
$A_{g}+3B_{1g}+B_{2g}+B_{3g}$. This corresponds to operators $1$,
$(\tau_{x},\tau_{z},\tau_{y}\sigma_{z})$, $\tau_{y}\sigma_{x}$ and
$\tau_{y}\sigma_{y}$. Here, 3 elements have the same symmetry
$B_{1g}=k_{x}k_{y}$. Their couplings generate 6 interactions. The other three
independent elements generate the other 3 interactions.

In the main text, we neglect SOC and assume spin-rotational symmetry. Then
operators $1$, $(\tau_{x},\tau_{z})$ generate 4 interactions, while operators
$\tau_{y}\sigma_{i}$ generate interactions with the same strength. In total 5
interactions are needed.

##  Appendix E Anomalous Hall effect for orbital altermagnetic state

The Berry curvature for the following tight-binding Hamiltonian with the
orbital altermagnetic state and $\epsilon_{xy}$-strain is shown in Fig. A3.

$\begin{split}H&=4t_{4}\cos\frac{k_{x}}{2}\cos\frac{k_{y}}{2}\tau_{x}+t_{5}\sin
k_{x}\sin k_{y}\tau_{z}\\\
&+M\sin\frac{k_{x}}{2}\sin\frac{k_{y}}{2}\tau_{y}+\epsilon_{xy}\tau_{z}\end{split}$
(21)

The orbital altermagnetic state becomes $\tau_{y}$ at the $(\pi,\pi)$ point,
and the $\tau_{z}$ term captures the same symmetry as the
$\epsilon_{xy}$-strain.

Fig. A3: Berry curvature for the lower-band of Eq.21, with
$4t_{4}=t_{5}=M=\epsilon_{xy}=1$.

The orbital altermagnetic state breaks time-reversal symmetry, while the
$\epsilon_{xy}$-strain breaks mirror symmetry $\widetilde{M_{x,y}}$.
Consequently, the anomalous Hall conductivity of an orthorhombic Fermi surface
is generically non-zero. SOC is not required for the anomalous Hall effect
with such an orbital origin.