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Altermagnetic spin textures: Emergent electrodynamics, quantum geometry, and probes Constantin Schrade 1 and Mathias S. Scheurer 2 1 Emergent electrodynamics arising from spatially and temporally varying magnetic textures provides a framework for spin control in quantum materials. While this principle is established for ferromagnetic and antiferromagnetic textures, its consequences for altermagnets—magnetic orders with vanishing net magnetization but finite spin splitting—remain largely unexplored. In this work, we develop an effective low-energy theory of itinerant electrons coupled to smoothly varying altermagnetic spin textures. In the adiabatic regime, we show that altermagnetic textures generate additional emergent electromagnetic fields and quantum-geometric effects that are absent in conventional magnetic systems. These effects include emergent Zeeman fields that encode the structure of the altermagnetic order parameter, enabling local spin manipulation and a way to distinguish different altermagnetic orders. Moreover, we demonstrate a quantum-metric-induced, spin-dependent electron lensing effect that provides a mechanism for spin filtering, and discuss the local admixture of effective odd-parity magnetic components. Our results suggest that textured altermagnets could serve as a versatile resource for spintronics functionalities and a probe of altermagnetism.

I. INTRODUCTION

Spin textures play a central role in condensed matter systems, ranging from magnetic textures in spintronics [1, 2] to moiré lattices with spatially varying effective spin degrees of freedom [3–7]. A particularly interesting feature of such textures is the emergence of effective electrodynamics for itinerant electrons, which has been studied for ferromagnets [8–14] and antiferromagnets [15–18]. Intuitively, as an electron moves through a smoothly varying texture (with a characteristic length scale much larger than the Fermi wavelength), its spin must continuously align with the effective magnetic field, which generates additional forces and modifies its dynamics. These additional contributions can be mapped to emergent electromagnetic fields and a distortion of the underlying space in which the electrons propagate. From a lattice perspective, the same physics appears as a texture-dependent quenching of hopping amplitudes set by the relative orientation of neighboring spins. Emergent electrodynamics and quantum geometry of this kind, as well as current-induced torques [19–24], provide a powerful route to controlling spin degrees of freedom for spintronics applications [25–27].

Another exciting opportunity in spintronics that emerged more recently are altermagnets [28–32]. In the simplest collinear cases, these magnetically ordered states are characterized by two magnetic sublattices which are related by symmetries that are—as opposed to antiferromagnets—not translation or inversion. A direct consequence is a finite spin splitting of the electronic bands while retaining vanishing net magnetization, which makes altermagnets attractive for spintronics applications [27]. Although spatially (e.g., domain walls)

  Domain wall 

 arXiv:2602.20236v1 [cond-mat.mes-hall] 23 Feb 2026

Hearne Institute of Theoretical Physics, Department of Physics & Astronomy,

Louisiana State University, Baton Rouge LA 70803, USA 2 Institute for Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany  

 

 

 FIG. 1. Circular Néel domain wall separating two altermagnetic domains. (a) Real-space schematic of the Néel vector field, n(r), on the two sublattices (τ z = ±; orange and blue). Arrows indicate the local in-plane spin orientation. The shaded annulus denotes the domain wall. The solid circle marks the domain wall radius, R. (b) Radial rotation profile,

marks the domain wall radius, R. (b) Radial rotation profile,
ϕ(r), describing the π-rotation of n across the wall. The wall

center is at r = R. The illustrations are schematic and not to scale. In the paper, we assume a texture with a ≪ w ≪ R (a is the lattice constant and w is the wall width).

and possibly also temporally varying textures of the altermagnetic order parameter, n(r, t), are expected to be 2 ubiquitous in real samples, in line with recent experiments [33, 34], there are only very few theoretical works studying them [35–38]; these studies focus on the dynamics and form of the magnetic order parameter fields themselves, rather than on the impact on the electronic spectrum. Thus far, electronic properties of inhomogeneous altermagnets have primarily been studied in the context of atomic-scale defects [39–46], as well as in studies on strain-induced responses [47, 48].

In this work, we fill this gap and develop a theory for the impact of smoothly varying altermagnetic order parameters on the electronic properties. This is particularly relevant for altermagnets where the impact of the magnetic moments on the electronic bands are non-trivial and their key defining feature. We will contrast the emergent electromagnetic fields and curved spacetime with the ferromagnetic and antiferromagnetic cases, revealing that there are novel contributions unique to altermagnets. We show that this not only allows to identify altermagnetic order but also to distinguish between different forms (dversus g-wave) of altermagnetic order parameters. While our theory applies to generic textures, we primarily illustrate our findings using a circular domain wall, as illustrated in Fig. 1.

II. MINIMAL MODEL

As a starting point, we use the following minimal model for an altermagnetic spin texture,

ˆH = ε0 ,ˆpτ 0 + tx ,ˆpτ x + tz ,ˆpτ z + Jτz n(r, t) · s, (1)

motivated by earlier works on uniform altermagnets [39, 49, 50]. Here, the Pauli matrices (τ x , τ y , τ z ) and s = (s x , s y , s z ) T act in sublattice and spin-space, respectively. Moreover, ε 0,p̂ describes the sublattice-independent con-

Moreover, ε0 ,ˆp describes the sublattice-independent con- tribution to the Hamiltonian with ˆp = −iℏ(∂x , ∂y )T be-

ing the momentum operator, whereas t z,p̂ and t x,p̂ are sublattice asymmetric and intersublattice contributions. For an altermagnet, t z,p̂ transforms as a non-trivial, onedimensional irreducible representation of the crystallographic point group. In the last term, n(r, t) describes the spatially (r = (x, y) T ) and temporally (t) varying order parameter. When n(r, t) is spatially uniform and time-independent, Eq. (1) is readily diagonalized and yields the characteristic spin splitting of the bands with angular dependence determined by the functional form of t z,p (as will become more explicit below). In particular, the splitting will vanish for t z,p = 0, which corresponds to antiferromagnetism.

To describe the effects of the altermagnetic spin texture, it is useful to move to a frame in which the quantization axis follows the local orientation of the Néel vector. We therefore parametrize the Néel vector as n(r, t) =

We therefore parametrize the Néel vector as n(r, t) (sin θ(r, t) cos ϕ(r, t), sin θ(r, t) sin ϕ(r, t), cos θ(r, t))T

and define the unitary transformation U (r, t) =

and define the unitary transformation U (r, t) = exp[ i 2 θ(r, t) (sin ϕ(r, t)sx − cos ϕ(r, t)sy )]. Under this

transformation, the last term in Eq. (1) becomes uniform

transformation, the last term in Eq. (1) becomes uniform since U † (r, t) (n(r, t) · s) U (r, t) = sz . The information

on the spatial and temporal variation of the altermagnetic spin texture are now contained in spin-dependent gauge fields, A(r, t), that shift the momentum op-

gauge fields, A(r, t), that shift the momentum op- erators as, ˆπ ≡ U † (r, t)ˆpU (r, t) = ˆp − A(r, t) with A(r, t) ≡ −U † (r, t)[ˆpU (r, t)]. The time-dependence of

the order parameter further generates a temporal gauge

the order parameter further generates a temporal gauge field, A0 ( r, t) ≡ iℏ U † (r, t)[∂t U (r, t)]. The rotating frame Hamiltonian, ˆHr ot = U † (r, t) ˆ H (t)U (r, t)−A0 ( r, t),

is then given by

ˆHr ot = ε0 , ˆ π τ0 + tx , ˆ π τx + tz , ˆ π τz + Jτz sz − A0 . (2)

Here, we emphasize that the shifted momenta still com-

Here, we emphasize that the shifted momenta still com- mute, [ˆπi , ˆπj ] = U † (r, t)[ˆpi , ˆpj ]U (r, t) = 0. However, as

we will see in the next section, this commuting property will be violated upon projecting the rotating frame Hamiltonian onto its low-energy subspace, leading to emergent electromagnetic fields.

Finally, it is instructive to introduce two examples that will guide the rest of this work. The first example is that of a d-wave altermagnet. Only keeping the leading power in momentum for every term, we set

the leading power in momentum for every term, we set ε0 ,p = ϱ0 ( p2x + p2y ), tx ,p = tx , and tz ,p = ϱz (p2x − p2y ).

In the limit of a spatially uniform and time-independent n(r, t), this choice gives the usual dispersion of a conventional d-wave altermagnet. The second example is that

tional d-wave altermagnet. The second example is that of a g-wave altermagnet, where ε0 ,p and tx ,p are chosen

as before. Meanwhile, for t z,p we choose a g-wave form,

as before. Meanwhile, for tz ,p we choose a g-wave form, tz ,p = ¯ϱz px p y (p2x − p2y ). At the operator level, we will use the representation ˆtz ,ˆp = ¯ϱz W[ˆpx ˆ p y (ˆp2x − ˆp2y )] where

we have introduced the

fully-symmetrized “Weyl” order-

we have introduced the fully-symmetrized “Weyl” order- ing, W( ˆ O 1 ˆ O 2 ˆ O 3 ˆ O 4 ) = π∈S4 ( ˆ O π (1) ˆ O π (2) ˆ O π (3) ˆ O π (4)) /4!

with S 4 being the symmetric group on four elements. At this point, this representation is redundant, since both the momenta and shifted momenta commute. However, it will be useful later when discussing the semiclassical limit.

III. EMERGENT ELECTRODYNAMICS

Emergent electromagnetic fields are known to arise in ferromagnetic [8–14] and antiferromagnetic [15–18] spin textures. Here, we show that altermagnets support qualitatively new emergent fields. These fields provide ways to distinguish (i) altermagnets from ferro- and antiferromagnets, and (ii) different altermagnetic order parameters.

As a first step to derive the emergent electromagnetic fields, it will be useful to decompose the spin-dependent gauge fields into its longitudinal and transversal compo-

gauge fields into its longitudinal and transversal compo- nents, Aµ ( r, t) = A∥µ ( r, t) + A⊥µ (r, t) with A∥µ ( r, t) ∝ sz ,

3

A⊥µ (r, t) ∝ sx ,y , and µ ∈ {0, x, y}. This decomposi-

tion allows us introduce momentum operators shifted

tion allows us introduce momentum operators shifted by the longitudinal gauge field, ˆΠi = ˆpi − A∥i (r, t) so that ˆπi = ˆΠi − A⊥i (r, t). Also, we introduce ˆΠ0 = iℏ∂t − A∥0 (r, t). One important feature of these newly de-

fined momentum operators is that they do not generally

fined momentum operators is that they do not generally commute. Instead, we find that [ˆΠx , ˆΠy ] = iℏB(r, t) and [ˆΠ0 , ˆΠi ] = iℏEi ( r, t) with the effective spin-dependent or-

bital (out-of-plane) magnetic fields and (in-plane) electric field components,

B(r, t) = −sz ℏ 2 n(r, t) · [∂x n (r, t) × ∂y n(r, t)], Ei ( r, t) = −sz ℏ 2 n(r, t) · [∂t n (r, t) × ∂i n (r, t)].

Domain wall

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We will now show that the itinerant electrons in our altermagnetic spin texture will effectively be subject to these B(r, t) and E i (r, t) fields as well to an emergent Zeeman field. For this purpose, we will assume that J is a sufficiently large energy scale so that the low-energy states of our altermagnetic spin texture lie in the subspace where S ≡ τ z s z has eigenvalue 1, and define the associated projection operator P − = (1 − S)/2. We will next study our rotating frame Hamiltonian in Eq. (2) upon being projected onto the S = −1 while, for now, neglecting virtual transitions to the high-energy S = +1 subspace.

We initially focus on our first example of the d-wave altermagnet. In this case, we find that the effective Hamil-

termagnet. In this case, we find that the effective Hamil- tonian (to 0t h -order in 1/J), ˆH( 0) r ot = P− ˆ H r otP − , evaluates

ˆH( 0) r ot = ε 0, ˆ Π σ 0 + t z, ˆ Π σ z + ϱ0 V 0 σ 0 + ϱz Vz σz − σz a∥0 . (4)

Here, we already expressed the Hamiltonian as an effective 2-band model using the Pauli matrices σ j acting on the low-energy degrees of freedom with basis states

on the low-energy degrees of freedom with basis states (τ = +, s =↓) and (τ = −, s =↑). In this expression,

(τ = +, s =↓) and (τ = −, s =↑). In this expression, sz in ˆΠ is implicitly replaced by −σz . Furthermore, a0 ( r, t) = (ℏ/2)[ˆz · n × ∂t n ]/(1 + nz ) is the tempo-

ral component of the projected longitudinal gauge field. Moreover, we have defined the scalar potentials

V0 = ℏ2 δi j gi j (r, t), Vz = ℏ2 ηi j gi j (r, t). (5) Here, ηi j = diag(1, −1)i j = 1 2! 1 ϱz ∂2 tz ,p ∂pi ∂pj p=0 , gi j (r, t) =

Here, ηi j = diag(1, −1)i j = 1 2! 1 ϱz ∂2 tz ,p ∂pi ∂pj p=0 , gi j (r, t) = ∂i n (r, t) · ∂j n(r, t)/4 is the quantum metric associated 2

with the spin texture (i.e., of the eigenstates |ψ r ⟩ ∈ C 2 of n(r, t) · s), and summation over the repeated indices

of n(r, t) · s), and summatio i, j ∈ {x, y} is left implicit.

Two comments about this result for the d-wave altermagnetic texture are in order: First, the itinerant electrons experience emergent orbital magnetic and electric fields, B(r, t) and E i (r, t), as they move through the texture. These fields have opposite signs for the two sublattices due to the prefactor s z (which is equivalent to −τ z FIG. 2. Multipolar emergent Zeeman fields from altermagnetic domain walls. (a) Emergent Zeeman field, V z (r) (normalized by its maximum), for a circular domain wall in a d-wave altermagnet, showing a quadrupolar pattern localized at the wall. (b) Corresponding result for a g-wave altermagnet, showing an octupolar pattern. (c,d) Angular

altermagnet, showing an octupolar pattern. (c,d) Angular cuts, Vz (R, φ), at the wall radius, showing the cos(2φ) and

sin(4φ) dependence. The multipolar form of V z provides a novel probe for altermagnetic order.

in P − ) and require non-coplanar textures in space (B) or space-time (E i ). However, we note that both fields are not genuinely unique to altermagnets since they arise even in the antiferromagnet [15] limit, t z,p → 0, and as such not our main subject of interest here.

Second, the itinerant electrons also experience an emergent Zeeman field, V z . This Zeeman field is unique to the altermagnet, since it vanishes in the limit when ϱ z → 0. In particular, it is of geometric origin as it involves the quantum metric. To further explain the effects of this Zeeman field, we move to the principal axis frame of the quantum metric through a rotation,

axis frame of the quantum metric through a rotation, g = Rψ diag(λ1 , λ2 ) RTψ where Rψ is a 2d rotation by an

angle, ψ, and λ 1,2 ≥ 0 are the eigenvalues of the quantum

angle, ψ, and λ1 ,2 ≥ 0 are the eigenvalues of the quantum metric tensor. Then, Vz = (ℏ2 /4)(λ1 −λ2 ) cos 2ψ. Hence,

V z measures how the principal axes of the texture are oriented relative to d-wave altermagnetic order parameter.

ented relative to d-wave altermagnetic order parameter. Such an effect does not occur for V0 = (ℏ2 /4)(λ1 + λ2 ) ,

since it does not depend on ψ.

Another important feature of the emergent Zeeman field is that it produces a quadrupolar moment that inherits the the structure of the altermagnetic order through η ij . To explain this feature, we consider a

through η . To explain this feature, we consider a static radial domain wall with θ(r) = π/2 and ϕ(r) = (π/2) tanh((r − R)/w) where R is the domain radius,

w ≪ R the domain wall width, and r = |r|, see Fig. 1. In

w ≪ R the domain wall width, and r = |r|, see Fig. 1. In this case, λ1 = ϕ′ (r)2 , λ2 = 0, and ψ = φ is the polar an- gle. Hence, Vz = f (r) cos 2φ with f (r) = (ℏ2 /4)ϕ′ (r)2 re-

4 alizes a “four-lobe” quadrupolar pattern near the domain wall boundary, which we illustrate in Fig. 2(a,c). This

quadrupole moment leads to spin accumulation near the domain wall.

An interesting question is whether such multipolar moments allow us to distinguish between different altermagnetic order parameters. To answer this question, we consider the previously introduced example of a g-wave altermagnet. In this case, the effective Hamiltonian (again to 0 th -order in 1/J) takes on the form,

Domain wall

ˆH( 0) r ot = ε 0, ˆ Π σ 0 + (t z, ˆ Π + δt z, ˆ Π ) σz + ϱ0 V 0 τ 0 + ϱz Vz σz (6) Here, δtz ,ˆp = 6ℏ2 ¯ϱz T ijkl W[ˆpi ˆ p j gp l( r, t)] is a correction ijkl

term that is quadratic in the momenta, with T ijkl =

1 4! 1 ¯ϱz ∂4 tz ,p ∂ki ∂ kj ∂kk ∂ kl p=0 denoting a totally symmetric rank-

4 tensor and W is the previously introduced Weylordering. The emergent Zeeman field is given by

Vz = ℏ4 T ijkl gi j (r, t)gk l( r, t). (7)

To contrast this emergent Zeeman field with the d-wave example, let us again consider a radial domain wall. In

example, let us again consider a radial domain wall. In the g-wave case, we find that Vz = (ℏ4 /64)ϕ′ (r)4 sin 4φ,

which realizes an eight-lobe octupolar pattern localized near the domain-wall boundary, see Fig. 2(b,d). The multipolar pattern from the emergent Zeeman field thus indeed allows to distinguish between different altermagnetic order parameters. In general, we expect the octopolar pattern in the g-wave case to be weaker since V z ∝ ℏ 4 , not V z ∝ ℏ 2 as in the d-wave case. However, it should still be relevant at sharper domain walls where ϕ ′ (r) 4 is sizable.

IV. GEOMETRY AND ELECTRON DYNAMICS

So far, we have focused on the leading corrections in 1/J to describe the altermagnetic texture. At this zeroth order 1/J, the quantum metric, g ij (r, t), enters only through the scalar potentials, V 0 and V z . A natural question is if the metric can also modify the electron dynamics through the kinetic energy.

To address this question, we consider a timeindependent d-wave altermagnetic texture. Let us begin by assuming that the inter-sublattice hybridization is the smallest energy scale and set t x = 0, which will allow for a systematic presentation of the findings and discussion of the different contributions. The first order correction to the effective Hamiltonian in the rotating

correction to the effective Hamiltonian in the rotating frame is then ˆH( 1) r ot = −Λ† Λ/2J, where Λ = P− ˆ H r otP +

describes virtual transitions between the low-energy S = −1 and the high-energy S = +1 subspace. The full effective Hamiltonian up to first order in 1/J is given by

fective Hamiltonian up ˆH( eff) r ot = ˆH( 0) r ot + ˆH( 1) r ot .

FIG. 3. Texture-induced effective metric near a circular domain wall. (a,b) Spatial map of λ max − λ min for

cular domain wall. (a,b) Spatial map of λm ax − λm in for τz = ±, where λm ax / min( r) are the eigenvalues of the metric

tensor, g eff,τ (r), that enters the kinetic energy. At fixed r, the local dispersion forms an ellipse in momentum space and λ max − λ min quantifies its elongation. (c,d) Major axis of the ellipse (eigenvector of λ max ) evaluated along the wall, which sets the orientation of the local dispersion.

(1) We now evaluate Ĥ

rot semiclassically; the full quantum mechanical operator expression is given in the Supplemental Material. Here we use a Wigner-Weyl formulation [51], which maps operators to phase-space functions and replaces operator products by Moyal products. Keeping terms up to order ℏ 0 in the expansion of Moyal products corresponds to the semiclassical approximation. In this case, the correction to the Hamiltonian function is

H( 1) r ot = − 2ℏ2 J pi [ K ij gj ℓ( r) K ℓm ]pm .

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Here, K ij = ϱ0 δ i j + ϱz σz ηi j is the kinetic-energy tensor of the d-wave altermagnet.

This result has a geometric interpretation. For explaining this geometric interpretation, we focus on coplanar ∥ textures, so that the longitudinal gauge potential A

i (r) gives rise to a pure gauge and can be omitted. For a fixed sublattice τ = σ = ±, the semiclassical spectrum is then given by,

ετ (p, r) = pi [ gi j eff,τ (r)]pj + Vτ (r), ive metric gi j eff,τ (r) = K ij τ − 2ℏ2 J [Kτ

(9)

with effective metric gi j eff,τ (r) = K ij τ − 2ℏ2 J [Kτ g(r)Kτ ]i j , ij i j i j

with effective metric g eff,τ (r) = Kτ − J [Kτ g(r)Kτ ] , kinetic tensor K ij τ = ϱ0 δ i j + τ ϱz ηi j , and total scalar po-

kinetic tensor Kτ = ϱ0 δ + τ ϱz η , and total scalar po- tential Vτ = ϱ0 V 0 + τ ϱz Vz . Hence, the quantum metric

gives rise to a sublattice-dependent “deformation” of the kinetic energy. Equivalently, electrons propagate as if 5 moving in a curved space whose geometry is determined jointly by texture gradients (through g ij ) and by the altermagnetic order (through η ij ).

The same geometric picture follows from the equations of motion. Treat r i and p j as phase-space variables with Poisson brackets {r i , p j } = δ ij . Hamilton’s equations

Poisson brackets {ri , pj } = δi j . Hamilton’s equations are ˙ri = ∂ετ /∂pi and ˙pi ,τ = −∂ετ /∂ri . Eliminating the

canonical momenta leads to

¨ri + Γij k,τ (r) ˙ r j ˙rk = −2gi j eff,τ ∂j Vτ ,

Domain wall (10)

Here, the Christoffel symbols are Γ ijk,τ (r) =

Here, the Christoffel symbols are Γj k,τ (r) = 1 2 (ge ff,τ )i ℓ [∂j ge ff τ ,kℓ +∂k ge ff τ ,jℓ −∂ℓ g e ff τ ,jk ] with ge ff τ ,ij = (g− 1 eff,τ )i j .

Eq. (10) shows that a semiclassical electron wavepacket is subject to the following effects: First, a force from the total scalar potential, V τ . Second, a purely geometric contribution encoded in the Christoffel symbols of the effective metric. Third, for noncoplanar textures, an additional Lorentz-like force, ∝ ϵ jk B τ ṙ k with opposite signs

ditional Lorentz-like force, ∝ ϵj k Bτ ˙r with opposite signs for the two τ sectors, where Bτ = τ ℏ n · [∂x n × ∂y n]/2.

This term vanishes for the coplanar textures considered here, but becomes finite when the spin texture acquires a scalar spin chirality. All of these contributions differ significantly from their ferromagnetic analogue [14] and also lead to crucial differences between altermagnets and antiferromagnets, which we will next discuss in the context of a circular domain wall inhomogeneity.

V.

Electron trajectories ALTERMAGNETIC LENSING The transport of electrons through the altermagnetic texture will provide a bending of electron trajectories, similar to the “lensing effect” in optics. We will focus on static, coplanar textures (B τ = 0 and E i = 0), and discuss two cases: (1) lensing due to the scalar poten-

discuss two cases: (1) lensing due to the scalar poten- tial, Vτ , (2) lensing due to effective metric, ge ff τ ,ij . Here,

(1) corresponds to the situation when J is sufficiently large and electron speed is “slow”, while (2) corresponds to fast electrons and moderately large J so that metric corrections to the kinetic energy become relevant. To see this, suppose we effectively increase the speed of electrons by rescaling the time coordinate, t → t/α for some

trons by rescaling the time coordinate, t → t/α for some α ≥ 1. Then, ˙r → α˙r and ¨r → α2 ¨r. Hence, Eq. (10) reads ¨ri + Γij k,τ ˙rj ˙rk = −[2gi j eff,τ ∂j Vτ ]/α2 and the effects

of the scalar potential gets suppressed upon increasing α or, equivalently, the speed of the electrons [14].

First, let us consider (1), lensing due to the scalar potential, V τ , and focus on the previously introduced radial domain wall example. In this case, V z (r) = f (r) cos 2φ gives rise to a force with a tangential com-

f (r) cos 2φ gives rise to a force with a tangential com- ponent − 1 r ∂φ V z = 2f (r) r sin 2φ and opposite sign for the

two sublattices. This force field is localized near the domain wall where f (r) ̸ = 0. As a result an electron ray would propagate along a straight line away from the domain wall. At the wall, the ray receives a “kick” from FIG. 4. Altermagnetic domain wall as a sublatticeselective electron lens. Semiclassical trajectories incident from the left on a circular domain wall (gray disk), with τ = +1 (blue) and τ = −1 (red). Left: metric lensing from eff (r). Right: potential the texture-induced effective metric g

τ,ij lensing from the emergent scalar potential V τ (r). Potential lensing produces opposite deflections of the two τ sectors (focusing/defocusing). In contrast, metric lensing generates a transmission/reflection asymmetry.

the force field that leads to a change in its slope. This change in slope will depend on the sublattice/spin and the orientation of the ray relative to the crystal axis of the altermagnet; in particular, the “kick” vanishes along

the altermagnet; in particular, the “kick” vanishes along the crystal axis when φ = 0, π/2 and hence sin 2φ = 0.

To further illustrate this result, we can compute the momentum transfer by solving the equations of motion. Consider a ray injected with a group velocity

vi n ≃ ∇p ε 0 ,p p=pi n that hits the domain wall at a point r0 = Rˆr(φ0 ) where ˆr = (cos φ, sin φ) is the radial unit

vector and φ 0 is he impact angle. The radial velocity component at the point of impact is v r = v in · r̂(φ 0 ). To obtain a compact analytical expression, let us first study our equations of motion in the limit of a thin domain wall, where w is the shortest length scale. We find that the tangential momentum is deflected by an amount

∆pφ ,τ ≃ τ ℏ2 ϱz π2 6wR sin 2φ0 vr .

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The deflection of the radial component is negligible, ∆p r,± ≃ 0. This result shows, once more, that there is no deflection along the crystal axes, φ 0 = 0, π/2, and that it depends on the τ sector. What is more, we also see that the deflection gets negligible when v r becomes sizable. This corresponds to exactly the limit where effects of the quantum metric are of relevance. We verified that this behavior is also qualitatively found when solving the semiclassical equations numerically for finite w, see Fig. 4(b,d).

Let us continue with (2) and consider the metric6 induced lensing effects for the same radial domain wall example. In this case, the texture-induced quantum met-

example. In this case, the texture-induced quantum met- ric is gi j (r) = ϕ′ (r)2 ˆri ˆ r j /4 and the effective metric that i j

ric is gi j (r) = ϕ (r) ˆri ˆ r j /4 and the effective metric that the electrons feel is gi j eff,τ (r) = K ij τ − β(r)(Kτ ˆr)i (Kτ ˆr)j with β(r) = (ℏ2 /2J)ϕ′ (r)2 . The velocity, ˙r, of an itiner-

ant electron is determined by the conjugate momentum, p, through the Hamilton equation ṙ = 2g eff,τ (r)p. Projecting this equation onto the radial direction, r̂, gives,

˙r = 2v( 0) g ,r (φ)[1 − β(r)ˆrT Kτ ˆr] ≡ 2v( 0) g ,r Zτ (r, φ) (12) where v( 0) g ,r = ˆrT Kτ p is the radial component of the group

velocity of the uniform altermagnet and Z τ (r, φ) is a τ - dependent transmission factor. It is defined through the second relation in Eq. (12) and can be explicitly writ-

second relation in Eq. (12) and can be explicitly writ- ten as Zτ (r, φ) = 1 − β(r)(ϱ0 + τ ϱz cos 2φ). This finding

has interesting consequences by considering three different regimes:

nt regimes: If Zτ > 0, then the velocity ˙r has the same sign as v( 0) g ,r .

Hence, the ray will move in the direction of the radial momentum projection. If Z τ = 0, then ṙ = 0 and the ray will be tangent to the wall. If Z τ < 0, then ṙ has opposite

will be tangent to the wall. If Zτ < 0, then ˙r has opposite sign as v( 0) g ,r . Hence, the ray will be pushed away from the

domain wall, even if the momentum direction points into the domain wall. Interestingly, due to the τ -dependence of Z τ , a special situation can arise when, say, Z + > 0 but Z − < 0. In this situation, the domain wall will act like a geometric spin filter, which we illustrate numerically in Fig. 4(a,c). Only spins with one polarization are transmitted, while rays with the opposite polarization are scattered back. In terms of the microscopic system parameters the condition Z + > 0 but Z − < 0 is achieved

parameters the condition Z+ > 0 but Z− < 0 is achieved when (ϱ0 −ϱz cos 2φ)− 1 < β < (ϱ0 +ϱz cos 2φ)− 1 . We em-

phasize that both spin-dependent scattering effects discussed in this section would be absent in the antiferromagnetic limit (ϱ z → 0) where neither Z τ nor V τ depend on τ [also note that ∆p φ,τ ∝ ϱ z in Eq. (11)].

VI. EMERGENT SPIN-ORBIT COUPLING AND

ODD-PARITY MAGNETISM We finally allow for finite t x such that there is an additional, non-trivial second-order process: the transverse gauge-field components A ⊥ change the spin of the electron and, subsequently, the t x term swaps the sublattice, bringing us back to the low-energy subspace (with S = −1). This time, however, the component of the spin has changed, leading to an additional contribution

spin has changed, leading to an additional contribution ∆ ˆ H ( 1) r ot to ˆH( 1) r ot, which involves σx ,y . While the full ex-

pression can be found in the Supplemental Material, the correction reads in the semi-classical limit as

∆Hr ot = gp ( r) · σϕ (r), (13a)

with “emergent spin-orbit field”

gp = g0 pj sin(θ)∂j ϕ ∂j θ , g0 = − ℏϱ0 t x J ,

(13b)

and rotated Pauli matrices σϕ = Rϕ ( σx , σy )T where Rϕ

rotates two-component vectors by an angle ϕ. Note that the third second-order process, based on changing the sublattice twice with t x , does not lead to a non-trivial operator in the low-energy subspace and, hence, will not be discussed here.

Importantly, it holds g p = −g −p such that the term in Eq. (13a) preserves time-reversal symmetry but is odd under inversion (at fixed position r). As such, it can locally be thought of as effective spin-orbit coupling. However, this term does not arise from relativistic corrections but, instead, from non-trivial magnetic textures. Consequently, one might also think of Eq. (13) as representing some form of spontaneous generation of spinorbit coupling [52–54] or an admixed odd-parity (antialter)magnetic component [32, 55]. We emphasize, however, that Eq. (13b) is not unique to altermagnets as it persists in the limit ϱ z → 0, as opposed to the lensing effects of the previous section.

To discuss concrete examples, let us begin with coplanar textures, e.g., by setting θ = π/2 but allowing for ϕ = ϕ(r), which includes our domain wall example in Fig. 1 as a special case. From Eq. (13b), we find g p /g 0 = (p · ∇ϕ, 0) T . As can be read off from the corresponding local spectrum

E± ( p) = ε0 ,p + ϱ0 V 0 ± (tz ,p + ϱz Vz )2 + g2p (14) of our effective, semi-classical 2 × 2 low-energy Hamil- tonian H( eff) r ot (where we neglected the non-trivial met-

ric in Eq. (8) for simplicity), the spin-orbit splitting is |g p | and thus finite except for p perpendicular to ∇ϕ. In Fig. 5(a,b), we illustrate this behavior for a domain wall, where ∇ϕ = r̂ϕ ′ (r), by focusing on a single point at the boundary [indicated in Fig. 5(b)]. Note that a planar spiral, i.e., ϕ(r) = q · r, leads to the spatially constant spin-orbit field, g p = (p · q, 0) T , of the same form and, thus, the same type of splitting everywhere in space. We emphasize the similarity to the vanishing of the spin-splitting in one direction in the p-wave magnets discussed in Ref. 55.

Spatially varying altermagnets can also induce spinorbit splittings without nodal directions in g p . This, however, necessitates that sin(θ)∇ϕ and ∇θ are linearly independent, which in turn is equivalent to having a non-

independent, which in turn is equivalent to having a non- coplanar n(r). For instance, ϕ = q1 · r and θ = q2 · r

yields a finite |g p | for all momenta p ̸ = 0 as long as q 1 and q 2 are linearly independent and for points in space with sin q 2 · r ̸ = 0. Another important non-coplanar structure is a skyrmion, see Fig. 5(c), which also gives rise to extended regions in space with non-nodal spin-orbit splitting, see Fig. 5(d).

7

FIG. 5. Emergent spin-orbit coupling and odd-parity magnetism from altermagnetic textures. (a) Circular Néel domain wall. Faint arrows show the in-plane Néel vector field, n(r). The marked point lies at azimuthal angle φ = π/4, where the emergent Zeeman field, V z ∝ cos 2φ, vanishes. (b) Local constant-energy contours, E ± (p) = µ, evaluated at that point. Because V z = 0, the splitting arises purely from the metric-induced spin-orbit term, g p ∝ p·∇ϕ. The splitting vanishes for p ⊥ ∇ϕ (dashed line), producing a nodes. (c) Skyrmion texture. The color scale shows n z . Arrows indicate the in-plane components of n(r). (d) Local constant-energy contours at the point in (c). Here, both ∇θ and sin θ∇ϕ contribute to g p , which eliminates any nodes.

We finally point out that including an additional, subleading term in the intersublattice hopping, t x,p = t x + ϱ x p 2 , already generates a term of the same form as Eq. (13) at order 1/J 0 ; now with g 0 = ℏϱ 3 . Which one is more relevant depends on how large J is compared to t x ϱ 0 /ϱ x .

VII. CONCLUSION

Starting from the four-band model in Eq. (1), with a slowly varying altermagnetic order parameter n(r, t), we have derived an effective two-band model describing the electronic spectrum in a rotating reference frame and studied the physical consequences of the different corrections that emerge relative to the homogeneous limit n(r, t) = n 0 . First, to leading (zeroth) order 1/J, we find that non-coplanar textures induce the emergent magnetic and electric fields given in Eq. (3), which are opposite for the two sublattices and persist in the antiferromagnetic limit (t z → 0). We also uncover an emergent Zeeman field V z , which, in contrast, is unique to the altermagnet. It involves the quantum metric g ij of the magnetic texture, and as such does not require non-coplanar order, as well as the momentum-space structure of the altermagnetic order parameter [via η ij and T ijkl in Eqs. (5) and (7) for a d-wave and g-wave altermagnet, respectively]. Since the associated local spin polarization can be probed experimentally, e.g., with scanning SQUIDs or NV magnetometry, this not only allows to distinguish altermagnets from antiferromagnets but also to access the predominant angular dependence of the altermagnetic order parameter [cf. Fig. 2(a) and (b)]. We have further shown that the emergent Zeeman field also leads to a spin-dependent lensing effect at inhomogeneities such as domain walls, see Fig. 4(b,d).

At order 1/J, additional corrections appear, which we divide into two categories. The first type preserves the spin/sublattice degree of freedom and can be thought of as an inhomogeneous, anisotropic contribution to the effective mass tensor or emulating curved space for electronic motion; as with the Zeeman field, it is unique to the altermagnet, involves the metric g ij and the form of the altermagnetic order parameter, and also introduces a spin-dependent lensing and, depending on parameters, filtering effect, see Fig. 4(a,c). The second type of corrections to order 1/J mixes the spin species and can be interpreted as emergent spin-orbit coupling or the local admixture of an effectively odd-parity magnetic component. It can further gap out the nodes of the altermagnetic spin splitting of the bands.

Taken together, we have demonstrated that textures in altermagnets lead to particularly rich electronic properties. On the one hand, our findings show that taking into account such inhomogeneities, which are inevitable in experimental samples, is imperative for a proper understanding of the local and global electronic properties. On the other hand, altermagnetic textures also provide unprecedented opportunities for locally engineering electronic properties of interest, emulating dynamics in curved space and analogue gravity, creating spindependent electronic lenses and filters, and probing the altermagnetic order parameter symmetry experimentally.

Note added. At the final stages of the preparation of this manuscript, Ref. 56 appeared, which also discusses the impact of altermagnetic textures on electronic properties, albeit with a different focus.

ACKNOWLEDGMENTS M.S.S. thanks U. Seifert for insightful discussions and acknowledges funding by the European Union (ERC2021-STG, Project 101040651— SuperCorr). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the Euro8 pean Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. C.S. acknowledges support from the Louisiana Board of Regents.

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1 Supplemental Material 1 Constantin Schrade 1 and Mathias S. Scheurer 2 Hearne Institute of Theoretical Physics, Department of Physics & Astronomy, Louisiana State University, Baton

Rouge LA 70803, USA 2 Institute for Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany S1. ROTATING FRAME HAMILTONIAN In this section, we provide more details on the rotating frame Hamiltonian in which the quantization axis on each sublattice follows the local magnetization. For concreteness, we will focus on the case of a time-independent altermagnetic texture.

As a first step, we redisplay (for convenience) the Hamiltonian for the altermagnetic spin texture,

ˆH = ˆε0 ,ˆpτ 0 + ˆtx ,ˆpτ x + ˆtz ,ˆpτ z + Jτz n(r) · s. (S1)

as introduced in Eq. (1) of the main text. We then parameterize the unit vector field n(r) as,

n(r) = (sin θ(r) cos ϕ(r), sin θ(r) sin ϕ(r), cos θ(r)) ,

and introduce a position-dependent spin rotation that aligns n(r) with the spin quantization axis,

U (r) = exp − i 2 θ(r) (− sin ϕ(r)sx + cos ϕ(r)sy ) , U † (r) (s · n(r)) U (r) = sz .

(S2)

(S3)

As a second step, we define the rotating-frame Hamiltonian as,

ˆHr ot ≡ U † (r) ˆH U (r). (S4)

which evaluates to,

ˆHr ot = ˆε0 , ˆ π τ0 + ˆtx , ˆ π τx + ˆtz , ˆ π τz + Jτz sz , ˆπ = U † (r) ˆp U (r) = ˆp − A(r). (S5)

We note that the spatial variation of the altermagnetic texture gives rise to emergent SU(2) gauge fields that couple to the electron momentum,

Ai ( r) = iℏ U † (r)∂i U (r), A(r) = (Ax ( r), Ay (r), Az (r)). (S6)

For the parameterization introduced above, these SU(2) gauge fields take the explicit form, As a third step, it is useful to separate the SU(2) gauge fields into components that are parallel and perpendicular to the quantization axis,

Ai ( r) = A∥i (r) + A⊥i (r), A∥i ∝ sz , A⊥i ∝ sx , sy ,

and thus commute and anticommute with τ z s z , respectively. These components are explicitly given by

A∥i (r) = −ℏ ∂i ϕ (r) sin2 θ(r) 2 sz A⊥i (r) = ℏ 2 [(∂i θ (r)) eϕ ( r) − sin θ(r) (∂i ϕ (r)) er (r)] · (sx , sy ), with er (r) = (cos ϕ(r), sin ϕ(r))T and eϕ ( r) = (− sin ϕ(r), cos ϕ(r))T .

(S7)

(S8)

Using this decomposition, we also introduce a momentum operator shifted by the longitudinal SU(2) gauge field,

ˆΠi = ˆpi − A∥i , (S9) ˆπi = ˆΠi − A⊥i . (S10)

so that the we can write This formulation will be the starting point for our subsequent derivation of the low-energy effective Hamiltonian.

2 S2. EFFECTIVE HAMILTONIAN In this section, we provide details on the derivations of the effective Hamiltonians shown in the main text.

S2.1. Schrieffer-Wolff Transformation We begin by redisplaying (for convenience) the rotating-frame Hamiltonian given in Eq. (2) of the main text,

ˆHr ot = ˆε0 , ˆ π τ0 + ˆtx , ˆ π τx + ˆtz , ˆ π τz + JS (S11) S ≡ τz sz , (S12)

We then introduce the operator, whose eigenvalues s = ±1 label the two subspaces separated by the large energy scale, J. In the rotating frame, S is a good quantum number for distinguishing the low-energy (s = −1) and high-energy (s = +1) subspaces. The projection operators onto these subspaces are given by,

P± = 1 2 (1 ± S).

(S13)

It will now be convenient to decompose Ĥ rot into diagonal and off-diagonal blocks with respect to the low- and high-energy subspaces,

H1 ,2 ≡ P∓ ˆ H r otP ∓ , Λ ≡ P− ˆ H r otP + , (S14)

With these definitions, we find that the low-energy effective Schrieffer-Wolff Hamiltonian is (to first order in 1/J) given by,

He ff = H1 − 1 2J ΛΛ†

(S15)

We will now evaluate this effective Hamiltonian for the d-wave and g-wave altermagnet.

S2.2. Effective Hamiltonian for the d-wave altermagnet (0 th order in 1/J)

For our first example of the d-wave altermagnet, we have ˆε0 ,ˆp = ϱ0 ( ˆp2x + ˆp2y ), ˆtx ,ˆp = tx , and ˆtz ,ˆp = ϱz (ˆp2x − ˆp2y ). In

this case, the kinetic term in our texture Hamiltonian can be written compactly as,

ˆε0 , ˆ π τ0 + ˆtz , ˆ π τz = ˆπi K ij ˆπj , with K ij = ϱ0 δ i j + ϱz τz ηi j and η = diag(1, −1). (S16)

We initially decompose the SU(2) gauge fields as ˆπi = ˆΠi − A⊥i with ˆΠi = ˆki − A∥i and rewrite the first

contribution to the kinetic term as,

ˆπi K ij ˆπj = (ˆΠi − A⊥i )K ij (ˆΠj − A⊥j ) = ˆΠi K ij ˆΠj − (ˆΠi K ij A⊥j + A⊥i K ij ˆΠj ) + A⊥i K ij A⊥j = ˆΠi K ij ˆΠj − (ˆΠi K ij A⊥j + K ij A⊥j ˆΠi ) + A⊥i K ij A⊥j

(S17)

j j i = ˆΠi K ij ˆΠj − { ˆ Π i , K ij A⊥j } + A⊥i K ij A⊥j ,

Using this result, the full rotating-frame Hamiltonian takes on the form,

Hr ot = ˆΠi K ij ˆΠj + A⊥i K ij A⊥j − { ˆ Π i , K ij A⊥j } + tx τ x + JS. (S18)

3 With this rewriting, we are now in the position to evaluate the lowest-order contribution to our effective Hamiltonian,

H (0) r ot = H1 = −J + P− [ ˆΠi K ij ˆΠj + A⊥i K ij A⊥j ]P− = −J + P− [ ϱ0 ( ˆΠ2x + ˆΠ2y ) + ϱz τz (ˆΠ2x − ˆΠ2y ) + ϱ0 V 1 + ϱz τz Vz ]P− ,

where we have defined the quantities,

V0 = ℏ2 4 [(∂x θ )2 + (∂y θ)2 + sin2 θ (∂x ϕ )2 + (∂y ϕ)2 ] = ℏ2 δi j gi j (r), Vz = ℏ2 4 [(∂x θ )2 − (∂y θ)2 + sin2 θ (∂x ϕ )2 − (∂y ϕ)2 ] = ℏ2 ηi j gi j (r),

(S20)

with η = diag(1, −1) and the quantum metric gi j (r) = ∂i n (r)·∂j n(r)/4. Introducing the Pauli matrices σj that act in

the low-energy subspace, with basis (τ = +, s =↓) and (τ = −, s =↑), we obtain the effective low-energy Hamiltonian given by (upon dropping an irrelevant constant)

H (0) r ot = ϱ0 σ 0 ( ˆΠ2x + ˆΠ2y ) + ϱz σz (ˆΠ2x − ˆΠ2y ) + ϱ0 V 1 + ϱz σz Vz . (S21)

This concludes our derivation for the effective Hamiltonian of the d-wave altermagnet system.

S2.3. Effective Hamiltonian for the g-wave altermagnet (0 th order in 1/J)

For our second example of the g-wave altermagnet, we again set ˆε0 ,ˆp = ϱ0 ( ˆp2x + ˆp2y ), ˆtx ,ˆp = tx . Moreover, we

conveniently write,

ˆtz , ˆ π = ¯ϱz W[ˆπx ˆ π y (ˆπ2x − ˆπ2y )] = ¯ϱz T ijkl W[ˆπi ˆ π j ˆπk ˆπl ] . (S22)

where we have defined the totally symmetric rank-4 tensor T ijkl as,

T ijkl = 1 4! 1 ¯ϱz ∂4 tz ,p ∂pi ∂pj ∂pk ∂pl p=0 .

(S23)

In particular, we remark that its non-zero components are given by,

T xxxy = T xxyx = T xyxx = T yxxx = 1 4 , T xyyy = T yxyy = T yyxy = T yyyx = − 1 4 ,

(S24)

As a next step, we insert our decomposition of the shifted momenta in terms of the longitudinal and transversal SU(2)

As a next step, we insert our decomposition of the shifted momenta in terms of the longitudinal and gauge fields, ˆπi = ˆΠi − ˆA⊥i and ˆΠi = ˆki − ˆA∥i . Projected onto the S = −1 subspace, we find that

P− ˆ t z , ˆ π P− = ¯ϱz P− T ijkl W[ˆΠi ˆ Π j ˆΠk ˆΠl ] + 6 W[ˆΠi ˆ Π j ˆA⊥k ˆA⊥l ] + W[ ˆ A ⊥i ˆA⊥j ˆA⊥k ˆA⊥l ] P− (S25)

where we have used that only products involving an even number of transversal gauge field factors are non-zero upon projecting with P − .

To further simplify the above expression, we note that

ˆA⊥k ˆA⊥l = 1 2 { ˆ A ⊥k , ˆA⊥l } + 1 2 [ ˆ A ⊥k , ˆA⊥l ] = ℏ2 (gi j + i χi j sz ) (S26)

where χ ij = n · (∂ i n × ∂ j n)/4 is the scalar spin chirality. This result can be seen from the explicit

where χi j = n · (∂i n × ∂j n)/4 is the scalar spin chirality. This result can be seen from the explicit form of the transverse gauge potential, ˆA⊥i = ℏ 2 ((∂i θ )eϕ − sin θ (∂i ϕ )er ) · (sx , sy ), and the quantum metric, gj k (r) = [(∂j θ)(∂k θ) + sin2 θ (∂j ϕ)(∂k ϕ)]/4 = (∂i n ) · (∂j n)/4.

4 We can now write the second term in our expansion of P − t̂ z, π̂ P − as,

T ijkl W[ˆΠi ˆ Π j ˆA⊥k ˆA⊥l ] = 1 2 T ijkl W[ˆΠi ˆ Π j [ ˆ A ⊥k , ˆA⊥l ]] + 1 2 T ijkl W[ˆΠi ˆ Π j { ˆ A ⊥k , ˆA⊥l }] = 1 2 T ijkl W[ˆΠi ˆ Π j { ˆ A ⊥k , ˆA⊥l }] = ℏ2 T ijkl W[ˆΠi ˆ Π j gk l( r)].

(S27)

Here, we have noted that T ijkl and { ˆ A ⊥k , ˆA⊥l } are both even under k ↔ l, while [ ˆ A ⊥k , ˆA⊥l ] is odd under k ↔ l and

thus vanishes upon carrying out the summation.

In a similar way, we find for the last term in the expansion of P − t̂ z, π̂ P − that,

T ijkl ˆA⊥i ˆA⊥j ˆA⊥k ˆA⊥l = ℏ4 T ijkl [(gi j gk l − χi j χk l) + i(gi j χk l + χi j gk l) sz ] = ℏ4 T ijkl gi j gk l, (S28)

where the last equality follows because T ijkl is totally symmetric and χ ij is antisymmetric under i ↔ j.

We can now combine these results to arrive at,

ˆH( 0) r ot = P− [ ˆε 0, ˆ Π τ 0 + (ˆt z, ˆ Π + δˆt z, ˆ Π ) τz + ϱ0 V 0 τ 0 + ¯ϱz Vz τz ]P− , (S29)

with

δˆt z ,ˆp = 6ℏ2 ¯ϱz T ijkl W[ˆpi ˆ p j gk l( r, t)]

(S30)

Vz = ℏ4 T ijkl gi j (r, t)gk l( r, t),

This concludes our derivation of the effective Hamiltonian to 0 th order in 1/J for the g-wave altermagnet example.

S2.4. Effective Hamiltonian for the d-wave altermagnet (1 st order in 1/J)

As a next step, we return to our first first example of the d-wave altermagnet and compute the correction to the effective Hamiltonian in 1 st order in 1/J.

For this purpose, it is initially useful to formulate a few definitions and identities. The first identity is

A⊥i A⊥j = ℏ2 (gi j + iχi j sz ) with {A⊥i , A⊥j } = 2ℏ2 gi j and [A⊥i , A⊥j ] = 2iℏ2 χi j sz (S31) with χi j = n · (∂i n × ∂j n)/4. The second identity is gi j = a=x,y ei ae j a, (S32)

where,

A⊥i = ℏ a=x,y ei as a , with ei x = − 1 2 [(∂i θ ) sin ϕ + sin θ(∂i ϕ ) cos ϕ] ,

(S33)

ei y = 1 2 [(∂i θ )(cos ϕ) − sin θ(∂i ϕ ) sin ϕ] .

The third identity is,

{ ˆ Π i , A⊥j } = {ˆpi , A⊥j } = ℏ a=x,y {ˆpi , ej a} sa (S34)

5 With the help of these identities, we can now compute the 1 st order correction in 1/J to the effective Hamiltonian. For this purpose, we write the off-diagonal blocks as,

Λ = P− H r otP + = P− tx τ x − { ˆ Π i , K ij A⊥j } P+ (S35)

and define,

Λ = Λ∆ + Λπ , Λ∆ ≡ P− ( tx τ x ) P+ , Λπ ≡ −P− { ˆ Π i , K ij A⊥j }P+ (S36)

We then decompose the following product,

ΛΛ† = Λ∆ Λ †∆ + Λπ Λ†π + Λ∆ Λ †π + Λπ Λ†∆ . (S37)

as it appears in our effective Hamiltonian and evaluate this expression term by term:

First, we consider the contribution originating from the sublattice hybridization. This term takes the form,

Λ∆ Λ †∆ = t2x P − , (S38)

which gives a constant energy shift within the low-energy (S = −1) subspace.

Second, we consider the product Λ π Λ † π . Here, we first write,

Λπ = −P− { ˆ Π i , K ij A⊥j }P+ = −ℏK ij a=x,y {ˆpi , ej a} (P− s a P + ) Λ†π = −ℏK kℓ {ˆpk , eℓ a′ } (P+ s a ′ P− ) .

(S39)

Λ†π = −ℏK kℓ a′ =x,y {ˆpk , eℓ a′ } (P+ s a ′ P− ) .

We then find that the product of these operators can be written as,

Λπ Λ†π = ℏ2 K ij K kℓ a,a′ =x,y {ˆpi , ej a} {ˆpk , eℓ a′ } (P− s a P + s a ′ P− ) = ℏ2 K ij K kℓ a,a′ =x,y {ˆpi , ej a} {ˆpk , eℓ a′ }P− ( sa s a ′ )P− = ℏ2 K ij K kℓ a=x,y {ˆpi , ej a} {ˆpk , eℓ a} P− + i a,a′ =x,y εa a′ {ˆpi , ej a} {ˆpk , eℓ a′ } (P− s z P− ) (S40) where we have used that P− s a P + s a ′ P− = P− s a s a ′ P− and, for a, a′ = x, y, we have sa s a ′ = δa a′ + iεa a′ sz with the

totally antisymmetric tensor ε aa ′ that satisfies ε xy = 1.

In the semiclassical limit, we find for the first term in the above expression,

Λπ Λ†π → 4ℏ2 (pi K ij ) gj ℓ( K ℓm pm ) . (S41)

Finally, we evaluate the cross terms involving both Λ ∆ and Λ π .

Λ∆ Λ †π + Λπ Λ†∆ = −tx P − τx , { ˆ Π i , K ij A⊥j } P− . (S42)

where we have used that for any pair of operators, A and B, that anti-commute with S, we have

where we have used that for any pair of operators, A and B, that anti-commute with S, we have (P− A P+ ) (P+ B P− ) + (P− B P+ ) (P+ A P− ) = P− { A, B}P− with O1 = tx τ x and O2 = −{ ˆ Π i , K ij A⊥j }. To further simplify this, we note the general anti-commutator identity {A, {B, C}} = {B, {A, C}} if [A, B] = 0.

To further simplify this, we note the general anti-commutator identity {A, {B, C}} = {B, {A, C}} if [A, B] = 0. Using this identity combined with [τx , ˆΠi ] = 0 and [τx , A⊥j ] = 0 (since ˆΠi and A⊥j are comprised of sx ,y,z Pauli

6 matrices), we find that,

Λ∆ Λ †π + Λπ Λ†∆ = −tx P − τx , { ˆ Π i , K ij A⊥j } P− = −tx P − ˆΠi , {τx , K ij A⊥j } P− = −tx P − ˆΠi , {τx , K ij }A⊥j P− = −tx P − ˆΠi , {τx , K ij }A⊥j = −tx P − ˆΠi , {τx , K ij }A⊥j P− = −2ϱ0 t x P− { ˆ Π i , τx A ⊥i }P−

(S43)

0 x i=x,y − i x i − = −2ϱ0 t x i=x,y P− τ x { ˆ Π i , A⊥i }P− = −2ϱ0 t x i=x,y P− τ x { ˆp, A⊥i }P− = −2ℏϱ0 t x i=x,y a=x,y { ˆ Π i , ei a( r)}P− ( τx s a ) P− Here, we also used {A∥i , A⊥i } = 0 and {τx , K ij } = {τx , ϱ0 δ i j + ϱz τz ηi j } = 2ϱ0 δ i j τx . In summary, with this we see that

the full quantum mechanical correction is given by,

ˆH( 1) r ot = − 1 2J t2x P − + ℏ2 K ij K kℓ a=x,y {ˆpi , ej a( r)}{ˆpk , eℓ a( r)}P− + i a,a′ =x,y εa a′ {ˆpi , ej a( r}{ˆpk , eℓ a′ (r)} (P− s z P− ) − 2ℏϱ0 t x i=x,y a=x,y {ˆpi , ei a( r)}P− ( τx s a ) P− (S44)

We will now discuss the term ∝ ϱ 0 t x in more detail.

Defining, as before, Pauli matrices σ j acting in the low-energy subspace (P − projects on) with basis choice (τ = +, s =↓) and (τ = −, s =↑), the associated contribution to the Hamiltonian reads as

∆ ˆ H r ot = ℏϱ0 t x J i=x,y ({ˆpi , ei x( r)}σx − {ˆpi , ei y (r)}σy ) , (S45)

where we explicitly indicated that e ia (r), as defined in Eq. (S33), depend in general on position. Again in the semi-classical limit, we can effectively replace {p̂ i , e ia (r)} → 2p i e ia leading to

∆Hr ot = 2ℏϱ0 t x J i=x,y pi (ei x( r)σx − ei y (r)σy ) = gp ( r) · σϕ (r), (S46)

where we introduced the spatially varying spin-orbit field and rotated Pauli matrices given by (suppressing position dependence for notational simplicity)

gp = − ℏϱ0 t x J sin(θ) p · ∇ϕ p · ∇θ , and σϕ = Rϕ σx σy , Rϕ = cos ϕ − sin ϕ sin ϕ cos ϕ , (S47)

respectively.

To derive the condition stated in the main text, we first note that

g2p = ℏ2 ϱ20 t 2x J 2 sin2 θ(p · ∇ϕ)2 + (p · ∇θ)2 (S48) is non-vanishing for all p ̸= 0 if and only if sin θ ̸= 0 and the two vectors ∇ϕ and ∇θ are linearly independent. At the same time, the texture is (locally) non-collinear if and only if n · (∂x n × ∂y n) = sin θ [(∂x θ )(∂y ϕ) − (∂y θ)(∂x ϕ )] (S49)

7

= sin θ det[∇θ ∇ϕ] ̸= 0 ⇔ sin θ ̸= 0 and ∇ϕ, ∇θ linearly independent. (S50)

Finally, we point out that extending the interlayer term to depend on momentum via t x,p = t x + ϱ x p 2 already induces a term that couples (τ = +, s =↓) and (τ = −, s =↑) on the level of the first term in Eq. (S15). This leads to the additional contribution to Eq. (S21) given by

∆H = −ϱ3 i=x,y ({ˆpi , ei x( r)}σx − {ˆpi , ei y (r)}σy ) , (S51)

which is identical to Eq. (S45) except for the different prefactor.

S3. LENSING In this section, we provide more details on the derivation of the lensing effects of an incoming semiclassical ray from the potential term.

As a starting point, we define a circular domain wall texture as

θ(r) = π 2 , ϕ(r) = ϕ(r) = π 2 tanh r − R w , n(r, φ) = (cos ϕ(r), sin ϕ(r), 0). (S52)

where w is the domain wall width and R is its radius. We note that,

n · (∂x n × ∂y n) = 0 (S53)

for this texture and, as a result, the orbital magnetic field contribution vanishes.

We want to evaluate the potentials, V z and V 0 , for this domain wall texture. For this purpose, we first need to compute the metric,

gi j = 1 4 (∂i ϕ )(∂j ϕ) so that gx x = 1 4 ϕ′ (r)2 cos2 φ, and gy y = 1 4 ϕ′ (r)2 sin2 φ.

(S54)

From this form of the metric, we find that,

Vz (r) = ℏ2 4 ηi j gi j = ℏ2 4 (gx x − gy y ) = ℏ2 4 ϕ′ (r)2 cos 2φ = f (r) cos 2φ with f (r) = ℏ2 4 ϕ′ (r)2 V0 ( r) = ℏ2 4 δi j gi j = ℏ2 4 (gx x + gy y ) = ℏ2 4 ϕ′ (r)2 (S55)

As a next step, we write down the semiclassical equations of motion,

˙r = 1 ℏ ∇k ε ± , and ℏ ˙ k = −∇r ε± . (S56) ε± ( k, r) = ε0 ( k) + ϱ0 V 0 ( r) ± ϱz Vz (r). (S57)

where we have defined the energies, We will primarily be interested in the equation of motion,

ℏ ˙ k = −ˆr[ϱ0 f ′ (r) ± ϱz f ′ (r) cos 2φ] ± ˆφϱz 2f (r) r sin 2φ

(S58)

to understand the change in momentum of an incoming ray due to the domain wall. Here, we have defined the

to understand the change in momentum of an incoming ray due to the domain wall. Here, we have defined the unit vectors in the radial and angular directions ˆr = (cos φ, sin φ) and ˆφ = (− sin φ, cos φ). Moreover, we have used

the polar representation of the gradient, ∇ r = r̂∂ r + φ̂ 1 r ∂ φ . Lastly, it will be useful to separately write down the components of this equation in the radial and angular direction,

ℏ ˙ k r = −[ϱ0 f ′ (r) ± ϱz f ′ (r) cos 2φ] and ℏ ˙ k φ = ±ϱz 2f (r) r sin 2φ

(S59)

8 Let us now assume that we have some incoming ray with velocity,

vi n ≃ 1 ℏ ∇k ε 0 ( k) k=ki n . (S60) r0 = Rˆr(φ0 ) , (S61)

The ray hits the domain wall at, which defines an angle, φ 0 . The velocity components at this point of impact are,

vr = vi n · ˆr(φ0 ) and vφ = vi n · ˆφ(φ0 ) . (S62)

We assume now that the domain wall, i.e. the region where the potentials V 0 and V z are finite, is very thin, wr ≪ R. In this case, we will assume that φ(t) and r(t) are unchanged for the moment that the ray transverses through the domain wall,

φ(t) ≃ φ0 , r(t) ≃ R, dt ≃ dr vr .

(S63)

We then first compute the change in the angular component of the momentum by integrating the semiclassical equation in the angular direction,

∆kφ ,± = ± ϱz ℏ dt 2f (r(t)) r(t) sin 2φ(t) ≃± ϱz ℏ 2 sin 2φ0 R 1 vr ∞ −∞ drf (r) = ± ϱz ℏ 2 sin 2φ0 R 1 vr ℏ2 π2 12w = ± ℏϱz π2 6wR sin 2φ0 vr .

(S64)

We can perform a similar calculation for the radial component of the momentum,

∆kr ,± = ∞ −∞ dt˙kr = − 1 ℏ ∞ −∞ dt[ϱ0 f ′ (r(t)) ± ϱz f ′ (r(t)) cos 2φ(t)] ≃− 1 ℏvr [ϱ0 ± ϱz cos 2φ0 ] ∞ −∞ drf ′ (r) = − 1 ℏvr [ϱ0 ± ϱz cos 2φ0 ] [f (∞) − f (−∞)] = 0,

where in the last equality we have used that f (∞) = f (−∞) ≃ 0.

Hence, we see that the ray experiences primarily a momentum deflection along the angular coordinate.

S4. SIMULATION DETAILS In this section, we provide technical details on our simulations for the figures of the main text.

(S65)

9 S4.1. Simulation details for Fig. 2 We first summarize our approach for obtaining the results shown in Fig. 2 of the main text.

In all panels, we consider a static, coplanar circular Néel domain wall with,

θ(r) = π 2 , ϕ(r) = π 2 tanh r − R w , and ϕ′ (r) = π 2w sech2 r − R w ,

(S66)

where we are using polar coordinates, r = (r, φ). Since the texture is coplanar and time-independent, the emergent orbital fields vanish, B τ = E i,τ = 0. We then evaluate the metric tensor of the texture,

gi j (r) = 1 4 ∂i n (r) · ∂j n(r) = ϕ′ (r)2 4 ˆri ˆ r j , ˆr = (cos φ, sin φ). (S67) with K ij τ = ϱ0 δ i j + τ ϱz ηi j . For the d-wave altermagnetic texture in panel (a), we plot, Vz (r) = ℏ2 ηi j gi j (r) = ℏ2 [gx x( r) − gy y (r)] = (ℏ2 /4)ϕ′ (r)2 cos(2φ). (S68)

For the g-wave altermagnetic texture in panel (b), we plot,

V (g) z (r) = ℏ4 T ijkl gi j (r)gk l( r) = ℏ4 gx y (gx x − gy y ) = (ℏ4 /64)ϕ′ (r)4 sin(4φ). (S69)

The parameters used in the simulations are given in Table S1. We set ℏ = 1.

R w
4.0 0.55

TABLE S1. Simulation parameters for Fig. 2.

S4.2. Simulation details for Fig. 3 We next summarize our approach for obtaining the results shown in Fig. 3 of the main text.

In all panels, we consider again a static, coplanar circular Néel domain wall with the same parameterization as in Fig. 3. The effective metric is given by,

gi j eff,τ (r) = K ij τ − 2ℏ2 J Kτ g(r)Kτ i j , (S70) with K ij τ = ϱ0 δ i j + τ ϱz ηi j

For panels (a) and (b), we compute the eigenvalues λm ax( r) ≥ λm in( r) of gi j eff,τ (r) and plot the anisotropy, ∆λ(r) = λm ax( r) − λm in( r), (S71)

which we normalize by its maximum value across both τ sectors.

For panels (d) and (d), we evaluate the eigenvector associated with λ max along the domain-wall circle r = R. We then sample angles φ on the circle and draw short line segments in the direction of that eigenvector to show the local principal axis orientation. The parameters used in our simulations of Fig. 3 are shown in Table S2, where we have again set ℏ = 1.

10

ϱ0 ϱz J R w 1.0 0.30 10.0 4.0 0.55

TABLE S2. Simulation parameters for Fig. 3.

S4.3. Simulation details for Fig. 4 We next summarize our approach for obtaining the results shown in Fig. 4 of the main text.

In all panels, we consider again a static, coplanar circular Néel domain wall with the same parameterization as in Fig. 2 and Fig. 3. The semiclassical dispersion for each τ = ± sector is,

ετ (p, r) = pT ge ff τ (r)p + Vτ (r) with Vτ (r) = ϱ0 V 0 ( r) + τ ϱz Vz (r). (S72)

We obtain the ray trajectories from Hamilton’s equations,

˙ri = ∂ετ ∂pi , ˙pi = − ∂ετ ∂ri .

(S73)

For the simulations, we again set ℏ = 1 and use the initial conditions r in = (−x 0 , b) and p in = (p 0 , 0), with several impact parameters b ∈ [−2.2, 2.2]. While we fix the incoming momentum p 0 to be the same for both τ = ±1, we

impact parameters b ∈ [−2.2, 2.2]. While we fix the incoming momentum p0 to be the same for both τ = ±1, we remark that the group velocity along the x-direction, vτ = ∂ετ ∂px = 2(ϱ0 + τ ϱz )p0 , will still be τ -dependent. With our simulation parameters, ϱ0 = 1, ϱz = 0.35, we obtain v+ / v− ≃ 2.08. So the τ = + rays are more than twice as fast

as the τ = − rays. Because the group velocity is larger for τ = +, the corresponding potential-induced deflection is suppressed.

The parameters used in Fig. 4 are summarized in Table S3.

ϱ0 ϱz w J Π0 R Fig. 4(a,c) 1.0 0.35 0.50 5.5 3.0 4.0 Fig. 4(b,d) 1.0 0.35 0.60 30.0 2.0 4.0

TABLE S3. Simulation parameters for Fig. 4.

S4.4. Simulation details for Figure 5 of the main text We lastly summarize our approach for obtaining the results shown in Fig. 5 of the main text.

In panel (a), we show a circular Néel domain wall with the same parameterization as in Figs. 2-4.

In panel (b), we obtain the constant-energy contour from the local dispersion, E ± (p; r) = µ, with,

E± ( p; r) = ε0 ,p + ϱ0 V 0 ( r) ± (tz ,p + ϱz Vz (r))2 + g 2 p (r), (S74) evaluated at the marked point, (r, φ) = (R, φ0 ) . Here, tz ,p = ϱz (p2x − p2y ) and the spin-orbit field is given by, gp ( r) = g0 ( sin θ(r)p · ∇ϕ(r), p · ∇θ(r))T

In panel (c), we plot a skyrmion texture defined by

θ(r) = 2 arctan a2 r2 , ϕ(r) = φ (S75)

In panel (d), the constant-energy contours are computed as in panel (b), evaluated at the marked point (r, φ) = (r 0 , φ 0 ).

The parameters used in our simulations are given in Table S4.

11

ϱ0 ϱz g0 µ (r0 , φ0 ) R w a Fig. 5(a,b) 1.0 0.3 −0.4 1.0 (R, π/4) 3.6 0.7 – Fig. 5(c,d) 1.0 0.3 −0.4 1.0 (1.0, π/4) – – 2.0

TABLE S4. Simulation parameters for Fig. 5.