⬇ Download Markdown  ·  ⬇ Download LaTeX  ·  ⬇ original  ·  view original  ·  CSV: table 1

Angular momentum splitter effect of d-wave axial phonons in orbital altermagnets Dimos Chatzichrysafis Institute of Physics, Johannes Gutenberg University Mainz, 55128 Mainz, Germany Alexander Mook Institute of Solid State Theory, University of Münster, 48149 Münster, Germany We theoretically demonstrate that axial phonons, lattice vibration quanta carrying finite angular momentum, can host a d-wave angular momentum texture in orbital altermagnets in the absence of spin-orbit coupling. We consider a minimal electronic tight-binding model with d-wave loop-current order that breaks time-reversal symmetry. Within the Born-Oppenheimer approximation, we incorporate electron-phonon coupling via the molecular Berry curvature and show that the underlying d-wave orbital magnetic moment texture of the electronic state is transferred to the phonons without requiring the relativistic spin-orbit coupling. Our results expand the range of platforms available for engineering axial phonons and point to functionality unique to d-wave textures, including angular-momentum Seebeck and splitter effects, corresponding to longitudinal and transverse angular-momentum currents driven by a temperature gradient.

arXiv:2607.13923v1 [cond-mat.str-el] 15 Jul 2026

I. INTRODUCTION

Axial (or chiral) phonons are quantized lattice vibrations that carry finite angular momentum [1]. They have attracted significant attention not only for fundamental reasons but also for their application potential [2–5]. Furthermore, they have emerged as a unifying theme across diverse areas of condensed matter physics, including ultra-fast dynamics [6–8], magnetism [9–12], orbitronics [13, 14], and strongly correlated systems [15–17]. Consequently, substantial effort has been devoted to understanding their microscopic origin. While external stimuli such as magnetic fields or laser pulses can induce rotational motion in phonons [18–21], it is particularly compelling to identify intrinsic, material specific mechanisms that generate such angular momentum.

To understand how axial phonons can arise intrinsically, we recall that a finite angular momentum J k of a quasiparticle requires the breaking of either inversion or time-reversal symmetry [22]. In non-centrosymmetric but time-reversal symmetric systems, angular momentum is odd in momentum J k = −J −k . Such phonons have been predicted to occur in kagome [23] and hexagonal lattices [24] and were subsequently observed in the latter through transient infrared spectroscopy [25]. They also exhibit experimental signatures in thermal transport measurements, such as those performed in quartz [14, 26]. By contrast, realizing axial phonons with even parity, characterized by J k = J −k , requires broken time-reversal symmetry while preserving inversion symmetry. This scenario is more subtle because phonons cannot intrinsically break time-reversal symmetry; instead, they acquire effective time-reversal symmetry breaking through coupling to a suitable magnetic background.

One route to breaking time-reversal symmetry is through coupling the lattice to electronic degrees of freedom. In the context of axial phonons, a prominent framework is based on the molecular Berry curvature (MBC) [27–30]. Within the BornOppenheimer approximation, the electronic wave function depends parametrically on the lattice coordinates. Adiabatically evolving the set of the lattice coordinates leads to geometric effects, whose gauge-invariant description is provided by the MBC. Acting as an effective real-space gauge field, the MBC modifies the phonon dynamics and can generate even-parity axial phonons when the electronic system breaks time-reversal symmetry [22]. This mechanism has been explored in both model studies [30–36] and first-principles calculations [37–41]. Notably, with the exception of Ref. [42, 43] existing studies have focused on electronic systems compatible with ferromagnetic point groups, where the induced axial phonons exhibit an s-wave angular-momentum texture in momentum space and can therefore possess a finite angular momentum at the Brillouin-zone center [31, 37, 41].

Beyond s-wave symmetry, higher-order even-parity angularmomentum textures have recently emerged as a hallmark of altermagnetism [44, 45]. Altermagnets are magnetically compensated materials whose electronic bands exhibit non-relativistic spin-splitting with a d-, g-, or i-wave form factor [45]. They have attracted considerable interest for spintronic applications because they combine key advantages of ferromagnets and antiferromagnets [45]. In particular, their spin-polarized band structure enables non-relativistic transport phenomena such as spin-dependent Seebeck and spin-splitter effects [46] , corresponding to longitudinal and transverse spin currents driven by an electric field.

Notably, many established and candidate altermagnets are insulating [45, 47], placing their magnetic excitations— magnons—at the center of thermally driven spin transport. These magnons inherit the unconventional spin-splitting textures of the underlying electronic structure [45, 48–51]. Recent theoretical work [42, 43] further demonstrated that relativistic spin-orbit coupling can transfer these textures to phonons, generating corresponding angular-momentum textures. This finding naturally raises the question: can phonons, as the primary heat carriers in insulating materials, acquire beyond s-wave even-parity angular momentum textures through mechanisms that are not limited by the typically weak scale of relativistic effects?

In this work, we demonstrate that d-wave axial phonons, characterized by a d-wave angular momentum texture in reciprocal space, can arise in orbital altermagnets without relying on spin-orbit coupling. Orbital altermagnets realize beyond-swave even-parity symmetries through orbital rather than spin 2 degrees of freedom. In particular, they break time-reversal symmetry in real space via loop-current order, where individual lattice plaquettes carry orbital magnetic moments whose spatial arrangement encodes the underlying altermagnetic symmetry [52–56]. Motivated by this emerging material class, we consider a minimal tight-binding model on the checkerboard lattice that hosts loop currents and exhibits an altermagnetic d-wave orbital magnetic moment texture in the electronic spectrum. Because time-reversal symmetry is broken at the realspace level it is directly transferred to the phonons through the molecular Berry curvature without the need for relativistic interactions. Consequently, both optical and acoustic phonon branches acquire a d-wave angular momentum texture in reciprocal space. To detect these axial phonons, we propose angular-momentum Seebeck and splitter effects, corresponding to longitudinal and transverse angular-momentum currents driven by a temperature gradient. Our results establish orbital altermagnets as a robust platform for engineering axial phonons beyond s-wave symmetry and reveal a direct link between altermagnetic order and phonon angular momentum.

The rest of the paper is organized as follows. In Sec. II, we present a minimal tight-binding model of an orbital altermagnet with a d-wave loop-current order on the checkerboard lattice. We compute the momentum texture of the electronic orbital magnetic moments, demonstrating their altermagnetic character. In Sec. III, we review the concept of the molecular Berry curvature and specify it to our orbital altermagnet system. In Sec. IV, we discuss how the MBC modifies the phonon dynamics of the checkerboard lattice and calculate the phonon angular momentum that also exhibits d-wave symmetry. Finally, in Sec. V, we propose the phonon angular-momentum Seebeck and splitter effects as characteristic thermal transport signatures of d-wave axial phonons. We conclude in Sec. VI and provide details in several appendices.

II.

FIG. 1: Orbital altermagnet on the checkerboard lattice illustrating the tight-binding Hamiltonian in Eq. (1). Nearest-neighbor hopping amplitudes are complex, e iϕ t, where the arrows indicate the direction corresponding to a positive phase ϕ. Next-nearest-neighbor hopping amplitudes on the A and B sublattices are denoted by t A and t B , respectively, with t A , t B . The circulating currents can be thought to generate out-of-plane orbital magnetic moments µ A and µ B , which are guides to the eye and make the d-wave symmetry apparent.

the nearest-neighbor hopping, encodes the loop current order. From here on, we choose for simplicity t A = −t B that renders the system particle-hole symmetric. The numerical values of the parameters used throughout the text are summarized in Table I.

We Fourier transform the fermionic operators,

ORBITAL d-WAVE ALTERMAGNET: A MINIMAL

TIGHT BINDING MODEL We begin by introducing the electronic system and considering a tight-binding Hamiltonian on the checkerboard lattice that captures the orbital altermagnet with d-wave loop-current order shown in Fig. 1. In real space, the Hamiltonian takes the form

where N is the total number of unit cells, and R 0 l the equilibrium position of the lth unit cell. We obtain the following electronic Hamiltonian in momentum space,

k l,κ

(1) where c † lκ and c lκ are spinless canonical fermionic creation and annihilation operators, respectively. Here, l, l ′ are unit cell indices, and κ, κ ′ = A, B refer to the two sublattices of the system. We choose the basis vectors of the two sublattices within the unit cell as τ A = 0 and τ B = a ( x̂ + ŷ) /2, where a is the lattice constant, which is set to unity unless stated otherwise. Here, x̂ and ŷ are unit vectors along the x- and ydirection, respectively. Furthermore, t is the nearest-neighbor hopping amplitude, t κ that of the second neighbors, with t A , t B , and V A = −V B = V is a staggered potential. The phase ′ ′ ′ ′ ′ ′ ϕ l lκ κ = η l

lκ κ ϕ, whose sign η l

lκ κ = ±1 depends on the direction of (2)

  T where Ψ

k = c kA c kB is the two-dimensional sublattice basis vector. The Bloch Hamilton kernel

contains

FIG. 2: (a) Electron spectrum plotted in reciprocal space along high-symmetry directions. Dashed lines indicate the electron energies in Eq. (6) and color indicates the orbital magnetic moment m z k . (b) Momentum-resolved orbital magnetic moment m z k , with m 0 = et 0 a 20 /(2cℏ) being the unit magnetic moment.

Diagonalizing H k provides two electronic bands, with energies

and corresponding eigenstates

factors.

For V = 0, the spectrum is gapless, with band-touching points at the X and Y high-symmetry points. Since we focus on the insulating regime, we set V , 0 from here on, as shown in Fig. 2.

The magnetic point group of H el is 4 ′ /mm ′ m in HermannMauguin notation. The following symmetries, which map red sites to red sites and blue sites to blue sites in Fig. 1, are important:

(a) T C 4z,+ : time-reversal symmetry T followed by a 90 ◦

clockwise rotation around the out-of-plane z-axis perpendicular to the xy plane,

(b) T M x : time-reversal symmetry T followed by reflection

about the x-axis, (c) M xy : reflection against the x = y-axis.

Note in particular that time-reversal T by itself is not a symmetry. Nor is PT symmetry (where P denotes inversion symmetry) as long as the two sublattices are different: t A , t B or V , 0.

These symmetries are compatible with a d-wave magnetic order. Following Refs. [54, 57] we make the d-wave texture explicit in momentum space by computing the orbital magnetic moments m kn of the two bands within the modern theory of orbital magnetization [58–60]. We assume an insulating zerotemperature limit at half filling, where the lower band with energy E k− is fully occupied. In two dimensions, only the out-of-plane component of the orbital moment, m z kn = − 1 e X h x y i

Im v nn ′ v n ′ n , 2cℏ n ′

,n E kn − E kn ′ (8)

is nonzero. Here, e > 0 is the elementary charge, and c is the speed of light. Additionally, we have defined the ve-

ℏ states of the tight binding Hamiltonian kernel H k that is related to the one defined in Eq. (4) by a unitary transformation,

(9)

This transformation is required because orbital magnetic moment in Eq. (8) is formulated in terms of the lattice-periodic part of the Bloch wave function. Within the tight-binding formalism, this quantity is obtained by diagonalizing the Bloch Hamiltonian constructed using the Fourier transform in Eq. (2) only when the basis-atom positions τ κ are included in the Fourier phase.

We plot m z k+ = m z k− = m z k within the first Brillouin zone in Fig. 2(a) along high-symmetry lines. The d-wave texture becomes fully apparent in the full momentum dependence of m z k shown in Fig. 2(b).

4 TABLE I: Numerical values of all physical parameters used in this work.

Physical quantity Numerical value Electron energy scale t 0 = 1 eV

Ionic mass scale M 0 = 10 −25 kg force constant scale n 0 = 0.1 N/m a 0 = 1 Å Lattice constant scale

Phase angle ϕ = π/4

Ionic masses M B /M 0 = 2, M A /M 0 = 1 Nearest-neighbor force constants n 11 /n 0 = −10 2 , n 12 /n 0 = −50 Second-neighbor force constants γ 11 /n 0 = −10, γ 22 /n 0 = −30

Electron-phonon coupling III.

t ′ /t 0 ′ = 1 MOLECULAR BERRY CURVATURE IN D-WAVE

ORBITAL ALTERMAGNETS To describe phonon dynamics in an interacting electronnuclear system, one typically employs the Born-Oppenheimer (BO) approximation. More specifically, one assumes that due to the large difference in the energy scales of the two subsystems, the electrons follow changes brought to the lattice configuration {R l } instantaneously, where R l = R 0 lκ + u lκ with R 0 lκ = R 0 l + τ κ , and u lκ being the displacement of the lattice site κ in the l-th unit cell. Consequently, one can decouple the two systems and consider the dynamics of the many-body electron ground state Φ 0el (r, {R l }) independently of the lattice by treating the set of all ion positions, {R l }, as a parameter. Therefore, one considers the parameter space spanned by {R l } and evolve the electronic wavefunction adiabatically by slowly varying the value of {R l }. If one performs the above procedure in a closed loop, Φ 0el (r, {R l }) can only change by a phase, known as the molecular Berry phase (MBP) [27]

Here, A ({R l }) is the molecular Berry connection. The component of A ({R l }) corresponding to the unit cell with equilibrium position R 0 l , the κth sublattice and the αth component of u lκ is given by [30–32, 35, 36]

(MBC), defined as [31, 32, 35, 36]

(12)

Since the MBC appears in real space as an effective gauge field, a finite value requires the electronic system to break time-reversal symmetry in real space.

To facilitate the calculation of the MBC, we express it in terms of single-particle wave functions obtained by diagonalizing the tight-binding Hamiltonian in Eq. (3). The derivation follows Ref. [31]; here we briefly summarize the main steps. We assume that we are dealing with an insulating system at zero temperature. Firstly, we Fourier transform the curvature in Eq. (12) and introduce a complete basis of states denoted by {Φ n } with corresponding energies {E n }, where n > 0 corresponds to excited states, to get

 0 0 1 X κα  0 G

κ ′ β R l − R

0 l ′ e −i ( R lκ −R l ′ κ ′ ) ·k N l,l ′

G κα κ ′ β (k) = where

and

To obtain Eq. (13) we made use of the identity

where Φ 0el is the electronic many-body ground state wave function. To declutter notation, we will call it Φ 0 from here on. From the molecular Berry connection one can then construct a gauge-independent quantity, the molecular Berry curvature (15)

(16)

By Fourier transforming the fermion operators using Eq. (2) we obtain

XM kκα = Ψ † q M kκα Ψ q+k ,

q(11)

(14)

M −kκ ′ β =

X Ψ †

q+k M −kκ ′ β Ψ q , (17)

q

  T where Ψ

q+k = c q+kκ 1 , ...., c q+kκ n , with n the being the total number of sublattices and M kκα and M −kκ ′ β are (nonHermitian) electron-phonon coupling kernels.

5 To compute the analytical form of M −kκ ′ β and M kκα for H el in Eq. (1), we assume that the spatial dependence of the tight-binding Hamiltonian enters through the nearest-neighbor hopping amplitude t. Accordingly, the electron-phonon coupling parameter t ′ (see also Tab. I) is identified with the spatial modulation of t. With the technical details delegated to App. A, we arrive at ! [M kκα ] 12 , M

kκα = [M

kκα ] 21 0 0 (18)

with the correspoding elements for κ = A, B and α = x, y being given by

 −iϕ −i(q+k)·(a 2 +a 1 ) , −e e

 iϕ −i(q+k)·a 1 +e e − e −iϕ e −i(q+k)·(a 2 +a 1 ) ,

(19) and

 +e −iϕ e −iq·(a 2 +a 1 ) e −ik·τ B ,

 iϕ i(q+k)·(a 2 +a 1 ) −ik·τ B +e e e ,

 +e −iϕ e −iq·(a 2 +a 1 ) e −ik·τ B ,

 iϕ i(q+k)·(a 2 +a 1 ) −ik·τ B e e e .

The elements of M −kκ ′ β are obtained by exchanging k + q ↔ q in the expression above.

We continue by further simplifying the expression for G κα κ ′ β (k) in Eq. (13). We do so by taking Φ 0 and Φ n to be Slater determinants and by using the Slater-Condon rules to write the matrix element products involving the many-body states in terms of single-particle states. The final expression for computing the MBC is [31] (see App. A for technical details)

 u † q+km − M −kκ ′ β u qm + u † qm + M kκα u q+km −   , −   2 

E qm + − E q+km −

(21) where E km ± is the energy of the single-electron state u km ± . We use m − to label the occupied states, and m + to label the unoccupied states. We readily see from Eq. (21) that since the matrix elements of Eq. (19) and Eq. (20) are proportional to t ′ , the MBC scales quadratically with the electron-phonon coupling strength.

On a technical note, the single-electron states u km ± appearing in Eq. (21) are those used to construct the Slater determinants Φ 0 and Φ n and correspond thus to the full Bloch function, rather than their periodic parts alone. Accordingly, obtaining these single-particle states requires the Fourier transform of the fermionic operators in the gauge that excludes the basis-atom positions τ κ , as defined in Eq. (2).

For a two-band electronic system with particle-hole symmetry, E k+ = −E k− , the MBC simplifies considerably. This applies, in particular, to the checkerboard loop-current model in Eq. (4) for the choice t A = −t B adopted here

h i h i

In the following, we will use Eq. (22) when computing the MBC. We continue by defining the rescaled MBC as

(23)

which has units of frequency and—as it will become clear in Sec. IV—is the relevant quantity entering the phonon dynamics. κα The elements of G κ ′ β (k) derived using Eq. (22) for κ, κ ′ = A, B and α, β = x, y can be grouped up in a matrix reading

(24)

The MBC matrix in Eq. (24) is anti-Hermitian and is constrained by the symmetries of the electronic system. Here, we highlight how inversion symmetry P, time-reversal with rotation C 4z,+ T , and the mirror M xy constrain G k . For the checkerboard lattice where every site is a center of inversion, P enforces

6

FIG. 3: Real part of all elements of the MBC for checkerboard-lattice model in Eq.

(1).

Each element is plotted as a function of k within

ordering of the elements corresponds to the ordering of the MBC matrix in Eq. (24).

rendering the MBC matrix real and antisymmetric. Moreover, T C 4z,+ leads to the constraint

and C 4z,+ k being the rotated k vector. Finally, the reflection M xy enforces κα G κ ′ β (k) =

X h i

M xy αα h i  κα  M xy G κ ′ β M xy k , ββ (28) αβ h i where M

xy αα (30b)

(30d)

(30a)

Equation (30a) follows from inversion symmetry forcing the matrix to be real and thus antisymmetric, leading to vanishing diagonal elements. The symmetry constrains in Eq. (26) and Eq. (28) lead to Eq. (30b). However, Eqs. (30c) and (30d) are not a product of symmetries. They rather derive from the simple nature of the two-band electronic system that we consider and the approximations made in the construction of the M kκα and M −kκ ′ β kernels in Eqs. (19) and (20), where only the spatial modulation of the nearest-neighbor hopping is taken into account. Figure 3 shows all elements of G k across the Brillouin zone, making the symmetry constraints manifest. is an element of the reflection matrix M xy = ! 0 1 . 1 0 IV. (29)

We provide a detailed proof of these symmetry constrains in App. A 2. The simple form of the above relations relies on the definition of the MBC Fourier transform in Eq. (13), which includes the atomic basis positions.

Considering the aforementioned symmetry relations, we Ax identify only two independent elements of

G k , namely G Ay (k) Ax

and G By (k). The other elements are either zero or related to D-WAVE AXIAL PHONONS FROM MOLECULAR

BERRY CURVATURE The effects of electron-phonon coupling can be incorporated in the kinetic part of the phonon Hamiltonian by means of minimal coupling [30–36, 38]

where M κ is the mass of the ion at the basis site κ = A, B. By employing the harmonic approximation for the lattice potential, 7 the full phonon Hamiltonian is given by H ph = 1 X 1 (p

lκα − ℏA lκα )

2 2 l,κ,α M κ   1 X X X 0 0 + u

lκα Φ

κα κ ′ β R l − R l ′ u l ′ κ ′ β , 2

l,l ′ κκ ′ αβ (32)

  0 0 where Φ κα

κ ′ β R l − R l ′ is the element of the force constant matrix (FCM) generated by the displacements u lκα and u l ′ κ ′ β . The molecular Berry connection can be expressed in terms of the MBC using an antisymmetric gauge that is valid in equilibrium as [15, 31, 32]

(33)

By Fourier transforming the momenta and the displacements,

0 1 X p kκα e ik·R lκ , p lκα = √

N k 0 1 X u kκα e ik·R lκ , u lκα = √

N k and γ 22 , for longitudinal and transverse second-neighbor interactions. Their numerical values are given in Tab. I. After a Fourier transformation [see Eq. (36)], we obtain the dynamical matrix as



  By By By By D Ax (k) D Ay (k) D Bx (k) D By (k)

with the corresponding elements being

 +e −ik·(a 1 +a 2 ) + 1 e i ( k x +k y ) /2 ,

 +e −ik·(a 1 +a 2 ) + 1 e i ( k x +k y ) /2 , (34)

κα

for the nearest-neighbor couplings, and

 1   2 γ 11 cos k x + γ 22 cos k y M

A−2 (γ 22 + γ 11 ) − 4n 11 ) ,  1   Ay D Ay (k) = 2 γ 22 cos k x + γ 11 cos k y M

A−2 (γ 22 + γ 11 ) − 4n 11 ) ,  1   Bx 2 γ 11 cos k x + γ 22 cos k y D Bx (k) = M

B−2 (γ 22 + γ 11 ) − 4n 11 ) ,  1   By D By (k) = 2 γ 22 cos k x + γ 11 cos k y M

B−2 (γ 22 + γ 11 ) − 4n 11 ) , Ax D Ax (k) =

(35)

where G k is the rescaled MBC matrix defined in Eq. (22), D k is the phonon dynamical matrix with elements D κ ′ β (k) =

one can write H ph in compact form

(39)

X   0 0 1 1 0 0

e −ik· ( R lκ −R l ′ κ ′ ) Φ κα p ′ β R l − R l ′ , (36) κ N M κ M κ ′ ll ′ and I is an identity matrix of the same dimension as G k and D k . For the checkerboard lattice, the vectors p k and u k are given by

(37) With G k from Eq. (22), we now construct the dynamical matrix for the checkerboard lattice to solve the harmonic phonon problem coupled to the electrons.

We consider up to

 second-neighbor couplings in the FCM,  0 0 whose elements Φ κα R − R have units of spring constants.

′ ′ l l κ β To deduce the number of independent elements of FCM we use the crystallographic symmetries of the checkerboard lattice, as shown in detail in App. C. For simplicity we assume B that the blocks Φ A

A and Φ B of the FCM (corresponding to the second-neighbor couplings that describe intra-sublattice force constants) are identical. Within this approximation, a total of four independent spring constants are required; two spring constants, namely n 11 , and n 12 , corresponding to longitudinal and transverse nearest-neighbor coupling, and two more, γ 11 , (40)

for the second-neighbor coupling. The remaining elements of D k in Eq. (39) are given by Hermitian conjugation.

With H ph fully constructed, we continue by promoting p k and u k to operators and determining their time evolution. We do so by assuming plane-wave solutions of the form p k = µ kσ e −iω kσ t and u k = ϵ kσ e −iω kσ t and derive the eigenvalue problem

with

   G (k) −D (k) − G † (k) G (k)   .  H eff = i  k I G (k) (42) Here, σ is the phonon band index, ω kσ is the phonon frequency, µ kσ and ϵ kσ are the momentum and displacement polarization

FIG. 4: (a) Phonon band structure of the in-plane phonons on the checkerboard lattice, including the effects of the MBC in the phonon Hamiltonian, with ε 0 = ℏω 0 = 1 meV being the characteristic phonon energy scale. Dashed lines indicate the harmonic phonon energies and color shows the out-of-plane phonon angular momentum l kσ . (b) d-wave texture of l kσ for all four phonon bands.

vectors, respectively, while µ̃ kσ , and ϵ̃ kσ are their normalized counterparts. The technical steps to derive and diagonalize H eff , and to proceed with the second quantization of the u k and k p k are described in detail in Ref. [18] and reviewed also for completeness in App. B.

Eventually, we can proceed as in Ref. [19] and compute the angular momentum of the phonons. Starting from the semiclassical definition of the kinetic angular momentum

l,κ

and using the canonical momentum for the derivative of the displacements, we obtain the following relation for the thermal average of the out-of-plane component of J,

In accordance with Ref. [19], we have defined the angular momentum of a phonon in band σ with momentum k as

For the orbital altermagnet on the checkerboard lattice, we show l kσ in Fig. 4(a) along high-symmetry directions, and in Fig. 4(b) across the full Brillouin zone for all four phonon bands. The resulting d-wave phonon angular momentum texture caused by the electron-phonon coupling is clearly visible.

Interestingly, the acoustic modes also exhibit finite angular momentum, albeit of smaller magnitude compared to the optical modes. This is in contrast to the s-wave axial phonons in the Haldane model studied in Ref. [31], where the obtained acoustic modes, compared to the optical ones, have a negligibly small angular momentum over the full Brillouin zone. Moreover, the s-wave axial phonons in Ref. [31] come with lifted degeneracy of the optical phonon modes at the Γ point. In contrast, the MBC of our d-wave system does not permit a splitting of the optical phonon modes. Specifically, the combined constraints of M xy and C 4z,+ T [Eqs. (26) and (28)] restrict the form of G k such that the optical modes remain degenerate. As shown in Appendix A 3 b, this result follows solely from the underlying symmetries and is independent of the complexity of the toy model. We therefore conclude that the magnetic point group 4 ′ /mm ′ m of the orbital altermagnet introduced in Sec. II, which is incompatible with ferromagnetism, forbids a finite time-reversal-odd axial moment at the Γ point.

The magnitude of l kσ is directly determined by the strength of the MBC, which in turn is governed by two key factors. The first factor is the microscopic origin of the MBC, namely its relation to electron-phonon coupling. Since the MBC depends quadratically on t ′ , its characteristic scale is set by   2 g 0 = ℏ/ M 0 a 0 . For a representative mass scale of M 0 = 10

−25 kg and a lattice constant of a 0 = 1 Å, we obtain g 0 = 10 11 Hz. While electron-phonon coupling determines this fundamental scale, the actual magnitude of the MBC is highly sensitive to the underlying electronic band structure. In particular, Eq. (22) shows that the MBC at the Γ point (k = 0) scales inversely with the square of the direct electronic band gap E q+ − E q− . Therefore, systems with small direct electronic gaps are expected to exhibit strongly enhanced MBC effects. In our model, however, the MBC vanishes at the Γ point due to the numerator in Eq. (22). Instead, it is finite only at k , 0 (and largest at the X and Y points of the Brillouin zone), where it grows with the inverse square of an indirect gap E q+ − E q+k− . As a result in our model, for t ′ /t 0 ′ = 1, the magnitude of the MBC is restricted to G 0 = 10 −2 g 0 = 10 9 Hz. For a fixed phonon bandwidth of ∆ε = 25 meV, obtained by choosing the dominant force constant as n 11 = −100 N/m, the corresponding MBC energy scale is four orders of magnitude smaller than ∆ε. This hierarchy of energy scales is directly reflected in the 9 relatively small values of l kσ , which remain in the range of ∼ 10 −3 ℏ. We expect, however, that the magnitude of d-wave angular momentum l kσ can vary substantially across materials, as it is controlled by the underlying electronic band structure, in particular by the relevant indirect electronic gaps.

V. PHONON ANGULAR MOMENTUM TRANSPORT

Axial phonons with a d-wave angular momentum texture can exhibit characteristic thermal transport responses such as an angular momentum splitter effect, as pointed out in Ref. [42]. To see so, consider the constitutive equation

where j z µ is the µth component of a non-equilibrium angular momentum current (polarized along the z direction), ∂ ν T is a temperature gradient along the νth direction, and the response tensor χ µν is the thermal angular momentum conductivity. To compute χ µν from microscopics, we define the angular momentum polarized current as

j z = 1

X

l kσ v kσ f kσ , V k,σ (47)

where l kσ is the phonon angular momentum given in Eq. (45), v kσ = 1 ℏ ∂ε ∂k kσ is the phonon group velocity, and f kσ is the steadystate distribution function of the phonons induced by the temperature gradient. We assume that the temperature gradient is applied in the νth direction and utilize Boltzmann’s semiclassical kinetic equation which upon linearization in the temperature gradient reduces to

  −1 0 where f

kσ = e

βε kσ − 1 is the equilibrium Bose-Einstein distribution function, and τ is a constant phonon relaxation time. We solve Eq. (48) for f kσ and compare with Eq. (46), to extract the response tensor

The symmetries of the d-wave restrict the shape of the response tensor as follows:

The off-diagonal elements are zero because the product v x kσ v y kσ l kσ is an odd function in k and thus vanishes upon integration. The diagonal elements are finite and related by a factor of −1 due to C 4 T symmetry.

Since, according to Eq. (50), a temperature gradient generates a longitudinal angular-momentum current, we refer to this response as the phonon angular-momentum Seebeck effect.

This effect is depicted in Fig. 5(a). Its key signature is the pronounced directional dependence: rotating the temperature gradient by 90 ◦ reverses the sign of the response, reflecting the d-wave character. We note that this effect is necessarily accompanied by a phonon heat current parallel to the temperature gradient, in accordance with Fourier’s law.

To produce a pure angular momentum response not accompanied by a heat current, we consider transverse responses and apply −∇T along the nodal lines of l kσ that are located along the two diagonals k x = k y and k x = −k y , as seen in Fig. 5(b). Rotating the diagonal response tensor χ by 45 ◦ (with R indicating the corresponding rotation matrix) we obtain

This purely transverse but symmetric response is reminiscent of the spin splitter effect in d-wave spin altermagnets [46] and can be referred to as phonon angular momentum splitter effect. (Note that this is not the same as the phonon angular momentum Nernst effect discussed in Ref. [61], which enters the antisymmetric part of χ, is not linear in τ, and a pure interband response.)

The angular momentum splitter effect, or likewise the sign change of the corresponding angular momentum Seebeck effect upon rotating the temperature gradient by 90 degrees, is a characteristic hallmark of d-wave axial phonons. Using Eq. (49), the quantitative value of χ xx as a function of temperature and relaxation time (between τ = 1 ps to τ = 1 ns) is depicted in Fig. 5(c). Relaxation times of the order of ns are realistic for long-lived acoustic phonons in crystalline solids at sufficiently low or intermediate temperatures, where Umklapp scattering due to lattice anharmonicity is suppressed [62]. Assuming τ = 1 ns and the parameters for which we have calculated the phonon angular momentum in Fig. 4 of Sec. IV, we obtain a peak phonon angular momentum conductivity of χ xx ∼ 10 −4 k B .

VI. DISCUSSION AND CONCLUSION

We have shown that orbital altermagnets can host axial phonons with a d-wave angular-momentum texture in reciprocal space. This texture originates from the ubiquitous nonrelativistic electron-phonon coupling present in all solids. The resulting phonon bands exhibit a pronounced d-wave angularmomentum structure in both acoustic and optical branches. In addition, we predict a phonon angular-momentum splitter effect, which provides a promising route for experimental detection.

Regarding candidate materials, orbital altermagnets— systems with d-wave loop-current order as considered here— offer a natural platform. Besides cuprates [63], recent theoretical work identifies kagome metals of the family AV 3 Sb 5 [53] as promising candidates, where A denotes alkali metals such as K, Rb, or Cs. Other studies suggest a possible coexistence of orbital altermagnetism with (conventional or unconventional) spin magnetism in several materials, such as CuBr 2 , VS 2 , MoO, and CrO [54]. Orbital altermagnetism may also be engineered 10 y

1 −

5 10

15 k B T/ 0 FIG. 5: (a,b) Schematic of the angular-momentum splitter effect, with the red and blue lobes representing the d-wave phonon angular momentum texture in reciprocal space. The red color denotes the positive part of the texture and the blue the negative one. The nodal lines of the texture along the diagonals are colored in gray. We distinguish the cases where −∇T is applied (a) along the x-axis, resulting in a longitudinal angular momentum current response, and (b) along the nodal lines of the texture, thus generating a transverse angular momentum current response. (c) Response tensor element χ xx (angular momentum conductivity) as a function of temperature T and constant phonon relaxation time τ.

via adatoms, as demonstrated in Ref. [52]. Beyond orbital systems, analogous physics should also arise in spin altermagnets when spin-orbit coupling is included [42]. Our results therefore establish a general principle: both orbital and spin altermagnets can host d-wave axial phonons as a robust consequence of electron-phonon coupling.

The phonon angular-momentum splitter effect leads to an accumulation of angular momentum at sample boundaries, which may be detected via an inverse spin Hall voltage, similar to experiments on quartz [14, 26]. While our analysis demonstrates that the effect is symmetry-allowed and generically expected, a quantitative description calls for further refinement. In particular, a more complete treatment of the angular-momentum current in Eq. (47) should account for the non-commutativity of velocity and angular-momentum operators, and for the nonconservation of phonon angular momentum. In addition, a realistic description of phonon transport requires incorporating momentum- and temperature-dependent phonon lifetimes arising from anharmonic scattering. These additions will refine the temperature dependence of χ xx , which in the present work is governed by Bose-Einstein statistics alone.

The peak value of the estimated response tensor element corresponds to χ xx ∼ 10 −4 k B for the simple toy model considered here. To benchmark the magnitude of our result, we compare it with previously reported transverse and longitudinal angular momentum current responses. Reference [64] predicts an orbital angular momentum Nernst conductivity of magnons of the order of 10 −2 k B in collinear antiferromagnets, while the magnon spin Nernst conductivity for the same type of system has been computed to be of the order of 10 −4 k B in Ref. [65]. Regarding spin altermagnets, Ref. [66] obtained a value of ∼ 0.1k B for the magnon spin Seebeck and splitter coefficients, albeit for an exceptionally long magnon relaxation time, corresponding to systems of high magnetic quality. While the phonon angular momentum splitter conductivity in our toy model does not reach the large values predicted for the orbital magnon Nernst transport in Ref. [64], its magnitude is comparable to established spincaloritronic responses, and may be further enhanced in materials with smaller indirect electronic band gaps, as discussed at the end of Sec. IV. We 11 have thus demonstrated that d-wave phonon angular momentum transport can generate sizable longitudinal and transverse signals in orbital altermagnets.

More broadly, our work opens several directions for future research. Extensions to g- and i-wave axial phonons, as recently proposed in Ref. [43], in (orbital) altermagnets are natural next steps, as is the exploration of their dynamical and transport signatures. It will be particularly interesting to investigate the interplay between odd-partiy wave angularmomentum textures, such as p-wave textures [57], and axial phonon responses. Together, these developments point toward a rich landscape of symmetry-controlled phonon phenomena [1] D. M. Juraschek, R. M. Geilhufe, H. Zhu, M. Basini, P. Baum,

A. Baydin, S. Chaudhary, M. Fechner, B. Flebus, G. Grissonnanche, A. I. Kirilyuk, M. Lemeshko, S. F. Maehrlein, M. Mignolet, S. Murakami, Q. Niu, U. Nowak, C. P. Romao, H. Rostami, T. Satoh, N. A. Spaldin, H. Ueda, and L. Zhang, Chiral phonons, Nature Physics 21, 1532–1540 (2025). [2] T. Zhang, Y. Liu, H. Miao, and S. Murakami, Advances in

phonons: From band topology to phonon chirality (2025), arXiv:2505.06179 [cond-mat.mtrl-sci]. [3] T. Zhang, S. Murakami, and H. Miao, Weyl phonons: the con-

nection of topology and chirality, Nature Communications 16, 10.1038/s41467-025-58913-0 (2025). [4] T. Wang, H. Sun, X. Li, and L. Zhang, Chiral phonons: Predic-

tion, verification, and application, Nano Letters 24, 4311 (2024), pMID: 38587210, https://doi.org/10.1021/acs.nanolett.4c00606. [5] Z. Chen, T. Zhang, X. Wang, J. Li, Z.-K. Lin, F. Gao, L.-

W. Wang, Y. Liu, Q. Wang, X. Zhang, G. Ma, X. Chen, M. Lu, Y. Chen, and J.-H. Jiang, Topological phononics (2026), arXiv:2605.20900 [cond-mat.mtrl-sci]. [6] M. S. Mrudul, M. Weißenhofer, and P. M. Oppeneer, Generation

of phonons with angular momentum during ultrafast demagnetization, Physical Review B 112, 10.1103/nt8w-47hb (2025). [7] S. R. Tauchert, M. Volkov, D. Ehberger, D. Kazenwadel,

M. Evers, H. Lange, A. Donges, A. Book, W. Kreuzpaintner, U. Nowak, and P. Baum, Polarized phonons carry angular momentum in ultrafast demagnetization, Nature 602, 73–77 (2022). [8] C. S. Davies, F. G. N. Fennema, A. Tsukamoto, I. Razdolski,

A. V. Kimel, and A. Kirilyuk, Phononic switching of magnetization by the ultrafast barnett effect, Nature 628, 540–544 (2024). [9] J. Luo, T. Lin, J. Zhang, X. Chen, E. R. Blackert, R. Xu, B. I.

Yakobson, and H. Zhu, Large effective magnetic fields from chiral phonons in rare-earth halides, Science 382, 698–702 (2023). [10] D. M. Juraschek and N. A. Spaldin, Orbital magnetic moments

of phonons, Phys. Rev. Mater. 3, 064405 (2019). [11] S. Chaudhary, D. M. Juraschek, M. Rodriguez-Vega, and G. A.

Fiete, Giant effective magnetic moments of chiral phonons from orbit-lattice coupling, Phys. Rev. B 110, 094401 (2024). [12] D. Yao and S. Murakami, Theory of spin magnetization driven

by chiral phonons, Phys. Rev. B 111, 134414 (2025). [13] D. Yao, D. Go, Y. Mokrousov, and S. Murakami, Dynami-

cal orbital angular momentum induced by circularly polarized phonons (2026), arXiv:2511.09271 [cond-mat.mes-hall]. [14] Y. Nabei, C. Yang, H. Sun, H. Jones, T. Mai, T. Wang, R. Bodin,

B. Pandey, Z. Wang, Y. Xiong, A. H. Comstock, B. Ewing, J. Bingen, R. Sun, D. Smirnov, W. Zhang, A. Hoffmann, R. Rao, M. Hu, Z. V. Vardeny, B. Yan, X. Li, J. Zhou, J. Liu, and D. Sun, in quantum materials with unconventional magnetic order.

VII. ACKNOWLEDGMENTS

We would like to thank Yafei Ren, Verena Brehm, Dominik Juraschek and Robin R. Neumann for helpful discussions. This work was funded by the German Research Foundation (DFG) as part of Project No. 504261060 (Emmy Noether Programme). We acknowledge support by the Dynamics and Topology Center (TopDyn) funded by the State of Rhineland-Palatinate.

Orbital seebeck effect induced by chiral phonons, Nature Physics 22, 245 (2026). [15] Y. Chen, W. Qin, S. Zhang, P. Cui, Q. Niu, and Z. Zhang, Emer-

gence of chiral phonons in two-dimensional kagome lattices harboring electronic chirality, Phys. Rev. Lett. 135, 126608 (2025). [16] G. Grissonnanche, S. Thériault, A. Gourgout, M.-E. Boulanger,

E. Lefrançois, A. Ataei, F. Laliberté, M. Dion, J.-S. Zhou, S. Pyon, T. Takayama, H. Takagi, N. Doiron-Leyraud, and

L. Taillefer, Chiral phonons in the pseudogap phase of cuprates,

Nature Physics 16, 1108–1111 (2020). [17] M.-E. Boulanger, L. Chen, V. Oliviero, D. Vignolles, G. Grisson-

nanche, K.-J. Xu, Z.-X. Shen, C. Proust, J. Baglo, and L. Taillefer, Thermal hall conductivity of electron-doped cuprates: Electrons and phonons, Phys. Rev. B 113, 094511 (2026). [18] J.-S. W. Lifa Zhang, Jie Ren and B. Li, The phonon hall effect:

theory and application, Journal of Physics: Condensed Matter, Volume 23, Number 30 23, 10.1088/0953-8984/23/30/305402 (2011). [19] L. Zhang and Q. Niu, Angular momentum of phonons and the

einstein–de haas effect, Phys. Rev. Lett. 112, 085503 (2014). [20] H. Komiyama and S. Murakami, Universal features of canonical

phonon angular momentum without time-reversal symmetry, Phys. Rev. B 103, 214302 (2021). [21] C. P. Romao and D. M. Juraschek, Phonon-induced geometric

chirality, ACS Nano 18, 29550–29557 (2024). [22] S. Coh, Classification of materials with phonon angular momen-

tum and microscopic origin of angular momentum, Phys. Rev. B 108, 134307 (2023). [23] H. Chen, W. Wu, S. A. Yang, X. Li, and L. Zhang, Chiral

phonons in kagome lattices, Phys. Rev. B 100, 094303 (2019). [24] L. Zhang and Q. Niu, Chiral phonons at high-symmetry points

in monolayer hexagonal lattices, Phys. Rev. Lett. 115, 115502 (2015). [25] H. Zhu, J. Yi, M.-Y. Li, J. Xiao, L. Zhang, C.-W. Yang, R. A.

Kaindl, L.-J. Li, Y. Wang, and X. Zhang, Observation of chiral phonons, Science 359, 579 (2018). [26] K. Ohe, H. Shishido, M. Kato, S. Utsumi, H. Matsuura, and

Y. Togawa, Chirality-induced selectivity of phonon angular momenta in chiral quartz crystals, Phys. Rev. Lett. 132, 056302 (2024). [27] C. A. Mead, The geometric phase in molecular systems, Rev.

Mod. Phys. 64, 51 (1992). [28] S. K. Min, A. Abedi, K. S. Kim, and E. K. U. Gross, Is the

molecular berry phase an artifact of the born-oppenheimer approximation?, Phys. Rev. Lett. 113, 263004 (2014).

12 [29] O. Bistoni, F. Mauri, and M. Calandra, Intrinsic vibrational

angular momentum from nonadiabatic effects in noncollinear magnetic molecules, Phys. Rev. Lett. 126, 225703 (2021). [30] T. Oh and N. Nagaosa, Phonon thermal hall effect in mott insu-

lators via skew scattering by the scalar spin chirality, Phys. Rev. X 15, 011036 (2025). [31] D. Saparov, B. Xiong, Y. Ren, and Q. Niu, Lattice dynamics

with molecular berry curvature: Chiral optical phonons, Phys. Rev. B 105, 064303 (2022). [32] T. Saito, K. Misaki, H. Ishizuka, and N. Nagaosa, Berry phase

of phonons and thermal hall effect in nonmagnetic insulators, Phys. Rev. Lett. 123, 255901 (2019). [33] D. Li, G. Yang, T. Qin, J. Zhou, and Y. Yao, Phonon dichro-

isms revealing unusual electronic quantum geometry (2025), arXiv:2511.16141 [cond-mat.mes-hall]. [34] D. Liu and J. Shi, Circular phonon dichroism in weyl semimetals,

Phys. Rev. Lett. 119, 075301 (2017). [35] Y. Ren, D. Saparov, and Q. Niu, Nonreciprocal phonons in

PT -symmetric antiferromagnets, Phys. Rev. Lett. 134, 206701 (2025). [36] S. K. Das, R. Yeh, and Y. Ren, Antiferro-chiral phonons in PT -

symmetric antiferromagnets (2026), arXiv:2605.08490 [condmat.mes-hall]. [37] F. Wang, X. Liu, H. Sun, H. Wang, S. Murakami, L. Zhang,

H. Zhang, and D. Xing, Ab initio theory of phonon magnetic moment induced by electron-phonon coupling in magnetic materials, Phys. Rev. Lett. 135, 256701 (2025). [38] R. Dhakal, D. A. S. Kaib, K. Choi, S. Biswas, R. Valenti, and

S. M. Winter, Theory of intrinsic phonon thermal hall effect in α-rucl 3 (2025), arXiv:2407.00660 [cond-mat.str-el]. [39] J. Bonini, S. Ren, D. Vanderbilt, M. Stengel, C. E. Dreyer, and

S. Coh, Frequency splitting of chiral phonons from broken timereversal symmetry in cri 3 , Phys. Rev. Lett. 130, 086701 (2023). [40] S. Ren, J. Bonini, M. Stengel, C. E. Dreyer, and D. Vanderbilt,

Adiabatic dynamics of coupled spins and phonons in magnetic insulators, Phys. Rev. X 14, 011041 (2024). [41] S. Zhang, M. Wang, and T. Zhang, General ab initio frame-

work for chiral phonons induced by electronic order (2025), arXiv:2509.09253 [cond-mat.mtrl-sci]. [42] H. Bendin, A. Mook, I. Mertig, and R. R. Neumann, D-wave

phonon angular momentum texture in altermagnets by magnonphonon-hybridization (2026), arXiv:2511.08357 [cond-mat.meshall]. [43] F. Wang, J. Xu, X. Liu, H. Wang, L. Zhang, and H. Zhang, Al-

teraxial phonons in collinear magnets (2026), arXiv:2512.07518 [cond-mat.mtrl-sci]. [44] L. Šmejkal, J. Sinova, and T. Jungwirth, Emerging research

landscape of altermagnetism, Phys. Rev. X 12, 040501 (2022). [45] L. Šmejkal, J. Sinova, and T. Jungwirth, Beyond conventional

ferromagnetism and antiferromagnetism: A phase with nonrelativistic spin and crystal rotation symmetry, Phys. Rev. X 12, 031042 (2022). [46] R. González-Hernández, L. Šmejkal, K. Výborný, Y. Yahagi,

J. Sinova, T. c. v. Jungwirth, and J. Železný, Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021). [47] A. Sufyan, B. Marfoua, J. A. Larsson, E. v. Loon, and

R. Armiento, High-throughput quantification of altermagnetic band splitting, Phys. Rev. Mater. , (2026). [48] R. Hoyer, P. P. Stavropoulos, A. Razpopov, R. Valentı́,

L. Šmejkal, and A. Mook, Altermagnetic splitting of magnons

in hematite α−fe 2 o 3 , Phys. Rev. B 112, 064425 (2025). [49] L. Šmejkal, A. Marmodoro, K.-H. Ahn, R. González-Hernández,

I. Turek, S. Mankovsky, H. Ebert, S. W. D’Souza, O. c. v. Šipr,

J. Sinova, and T. c. v. Jungwirth, Chiral magnons in altermagnetic ruo 2 , Phys. Rev. Lett. 131, 256703 (2023). [50] M. Gohlke, A. Corticelli, R. Moessner, P. A. McClarty, and

A. Mook, Spurious symmetry enhancement in linear spin wave theory and interaction-induced topology in magnons, Phys. Rev. Lett. 131, 186702 (2023). [51] Z. Liu, M. Ozeki, S. Asai, S. Itoh, and T. Masuda, Chiral split

magnon in altermagnetic mnte, Phys. Rev. Lett. 133, 156702 (2024). [52] L. V. Pupim and M. S. Scheurer, Adatom engineering magnetic

order in superconductors: Applications to altermagnetic superconductivity, Phys. Rev. Lett. 134, 146001 (2025). [53] A. R. Chakraborty, F. Yang, T. Birol, and R. M. Fernandes,

Orbital altermagnetism on the kagome lattice and possible application to av 3 sb 5 (2025), arXiv:2509.26596 [cond-mat.str-el]. [54] M. Pan, F. Liu, and H. Huang, Orbital altermagnetism (2026),

arXiv:2510.00509 [cond-mat.mtrl-sci]. [55] Y. Yu, H. G. Suh, M. Roig, and D. F. Agterberg, Altermag-

netism from coincident van hove singularities: application to κ-cl, Nature Communications 16, 10.1038/s41467-025-57970-9 (2025). [56] V. Leeb and J. Knolle, Collinear p-wave magnetism and hidden

orbital ferrimagnetism (2026), arXiv:2601.07418 [cond-mat.strel]. [57] Y. Li and P. Sukhachov, p-wave orbital magnetism (2026),

arXiv:2604.18695 [cond-mat.mes-hall]. [58] T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Orbital

magnetization in periodic insulators, Phys. Rev. Lett. 95, 137205 (2005). [59] D. Xiao, J. Shi, and Q. Niu, Berry phase correction to electron

density of states in solids, Phys. Rev. Lett. 95, 137204 (2005). [60] T. THONHAUSER, Theory of orbital magnetization in solids, In-

ternational Journal of Modern Physics B 25, 1429–1458 (2011). [61] S. Park and B.-J. Yang, Phonon angular momentum hall

effect, Nano Letters 20, 7694 (2020), pMID: 32955897, https://doi.org/10.1021/acs.nanolett.0c03220. [62] H. Zhao and J. B. Freund, Full-spectrum phonon relaxation times

in crystalline si from molecular dynamics simulations, Journal of Applied Physics 104, 10.1063/1.2963721 (2008). [63] C. M. Varma, Non-fermi-liquid states and pairing instability of

a general model of copper oxide metals, Phys. Rev. B 55, 14554 (1997). [64] G. Go, D. An, H.-W. Lee, and S. K. Kim, Intrinsic magnon

orbital hall effect in honeycomb antiferromagnets (2023), arXiv:2303.11687 [cond-mat.mes-hall]. [65] R. Cheng, S. Okamoto, and D. Xiao, Spin nernst effect of

magnons in collinear antiferromagnets, Phys. Rev. Lett. 117, 217202 (2016). [66] Q. Cui, B. Zeng, T. Yu, H. Yang, and P. Cui, Efficient spin

seebeck and spin nernst effects of magnons in altermagnets (2023), arXiv:2306.08976 [cond-mat.mes-hall]. [67] D. Saparov, Effect of Molecular Berry Curvature on the Dynam-

ics of Phonons, Ph.D. thesis, The University of Texas at Austin, Austin, Texas, USA (2022). [68] J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, and B. A.

Bernevig, Non-trivial quantum geometry and the strength of electron–phonon coupling, Nature Physics 20, 1262–1268 (2024). [69] A. A. Maradudin, E. W. Montroll, G. H. Weiss, and I. P. Ipatova,

Theory of Lattice Dynamics in the Harmonic Approximation, 2nd ed. (Academic Press, New York, 1971). [70] K. Leraand, K. Mæland, and A. Sudbø, Phonon-mediated spin-

polarized superconductivity in altermagnets, Phys. Rev. B 112, 104510 (2025).

13 Appendix A: Molecular Berry Curvature This section reviews the molecular Berry curvature (MBC). We first follow Ref. [31] to derive an expression for the MBC in terms of single-particle eigenstates and the microscopic electron-phonon coupling. We then examine how the symmetries of the checkerboard model constrain the MBC and prove that the MBC matrix is anti-Hermitian. Finally, for the checkerboard model, we derive the explicit electron-phonon coupling matrices and prove that the magnetic point group of our electronic system does not allow for a finite phonon angular momentum at the Γ point.

1. Molecular Berry Curvature as a Gauge Field: General Derivation

Our starting point is the MBC in Eq. (13) reading

P ∂H el −iR 0 ·k P 0 where we recall that M

kκα = l ∂u e lκ and M −kκ ′ β = l ′ ∂u

∂H

l ′ κ el ′ β e

iR l ′ κ ′ ·k are the Fourier transform of the electron-phonon lκα coupling operators. To make analytical progress we assume that the electronic Hamiltonian is a non-interacting tight-binding Hamiltonian containing hopping terms of electrons between different sites, which generally assumes the form

Here, t ll ′ are, in general complex, hopping amplitudes between the l-th and l ′ -th unit cell, and κ, κ ′ are internal indices which, without loss of generality, are assumed to be sublattice indices. We note that t ll ′ are chosen such so that H el = H el † applies. Later on, we furthermore assume the t ll ′ to carry distance dependence that ultimately gives rise to electron-phonon coupling, as shown in Sec. A 3 for the case of the checkerboard lattice. By Fourier transforming the fermion operators as

(A3)

one can write M kκα and M −kκ ′ β as

M kκα =

X Ψ †

q M kκα Ψ q+k , q M −kκ ′ β =

X Ψ †

q+k M −kκ ′ β Ψ q , (A4)

q

  T where Ψ

q+k = c q+kκ 1 , ...., c q+kκ N , and M kκα and M −kκ ′ β are the kernels of the electron-phonon coupling. In reciprocal space, the Hamiltonian in Eq. (A2) can be written as

k

Denoting the normalized column eigenvectors of H k by u km (with m the band index) we can construct a unitary matrix

whose action on the vector Ψ k is

To compute matrix elements such as ⟨Φ 0 |M kκα |Φ n ⟩ appearing in Eq. (A1) we assume that the many-body states Φ n , Φ 0 are Slater determinants [67]. This allows us to apply the Slater-Condon rules, according to which the matrix element of a one-body operator between two Slater determinants is nonzero only if the determinants differ by a single occupied single-particle state. In that case, the many-body matrix element reduces to the matrix element of the one-body operator between the two single-particle states in which the Slater determinants differ, rather than requiring an evaluation between the many-body states themselves.

14 Assuming that our electronic model is insulating with a well-defined gap separating the occupied from the unoccupied states at T = 0, we can write [67]

E E (A8) |Φ

where the minus sign in the superscript of the band index denotes that all the states in Eq. (A8) are occupied states below the Fermi level of the system at T = 0, and A is an operator anti-symmetrizing the state product to the right of it. An excited state Φ n is then defined as

E E E (A9) |Φ

E where in contrast to Φ

0 the electron originally in the single-particle state u k r m − r has been excited to the state u k n m + n , whose D E energy lies above the Fermi level. Thus, we find for the product of the elements P ≡ Φ

0 M k,κα Φ n Φ n M −k,κ ′ β Φ 0 that

qq ′ (A10)

To the fourth to the fifth D go from

Eline we noticed that one of the two terms in the expression of P, namely P 1 = † u k r m − r Ψ̃ q U q

† M

kκα U q+k Ψ̃ q+k u k n m + n , can be written as

qm c q+km ′ u k n m + n (A11)

where we expanded the product of the four operators as

(A12)

using Wick’s theorem. Using the result for P 1 one can simplify in a similar same manner the expression for P 2 = D E † u

k n m + n Ψ̃ q ′ +k U q

† ′ +k M −kκ ′ β U q ′ Ψ̃ q ′ u k r m − r and reach the final line in Eq. (A10). Finally, following the same logic for the second term in Eq. (A1), the MBC can be obtained as

  X X  u † qm − M kκα u q+km + u † q+km + M −kκ ′ β u qm − u † q+km − M −kκ ′ β u qm + u † qm + M kκα u q+km −  i   , G κα − (A13)   2   2 κ ′ β (k) =

 N m − ,m + q  − + + − E qm − E q+km E qm − E q+km

where E km ± is the energy of the single-particle state u km ± . Moreover, m − labels the occupied states and m + the unoccupied ones.

15

2. Properties and Symmetries of the Molecular Berry Curvature

In this section, we discuss how the symmetries of the crystal constrain the MBC and subsequently show that the MBC is anti-Hermitian. Specifically, we consider the following symmetry operations:

(a) time-reversal (T ),

(b) inversion (P),

(c) clockwise 90 ◦ rotation around the z-axis followed by time-reversal (C 4z,+ T ),

(d) reflection about the x = y-line (M xy ).

In the following, we consider the checkerboard lattice introduced in the main text and analyze the action of all crystal symmetry operations. Among the symmetries discussed above, only (b)-(d) are symmetries of the altermagnetic loop-current model defined in Eq. (1). They are of particular relevance because the MBC originates from the coupling between the electronic degrees of freedom and the crystal lattice. By contrast, time-reversal symmetry T by itself is not a symmetry of the electronic model considered here. Nevertheless, analyzing its action provides valuable insight into the physical origin and symmetry properties of the MBC.

First, recall that the MBC enters the phonon Hamiltonian as a magnetic Raman-like coupling, such that the relevant contribution in reciprocal space is

We first determine how the relevant symmetry operations act on the momentum and displacement operators entering H RM . We specify their action on the corresponding real-space operators p lκα and u lκα , where l labels the unit cells. The transformation properties of the reciprocal-space operators, p kκα and u kκα , then follow directly from the Fourier transformations

1 X −ik· ( R 0

l +τ κ ) e u lκα , N l 1 X −ik· ( R 0 l +τ κ ) e p kκα = p lκα .

N l u kκα =

(A15)

We recall that, for the checkerboard lattice considered in the main text, the basis vectors are chosen as τ A = 0 and τ B = ( x̂ + ŷ) /2, while the primitive lattice vectors are a 1 = x̂ and a 2 = ŷ. Throughout, the lattice constant is set to unity.

Before analyzing the symmetry constraints, we briefly comment on the choice of Fourier convention. The atomic gauge introduced in Eq. (A15) is particularly convenient for lattices with a basis, as it leads to considerably simpler symmetry transformation laws than the convention in which the basis vectors are omitted from the Fourier transform.

(a) Time-reversal Symmetry (T )

The action of the time-reversal operator T on the momentum and displacement operators in real space is defined as [68]

T p lκα T −1 = −p lκα , T u lκα T −1 = u lκα .

(A16)

Since T is anti-unitary, its action on a scalar c is complex conjugation, T cT −1 = c ∗ . It then follows for the displacement that

1 X −ik·(R l +τ κ ) T

u kκα T −1 = √ T e u lκα T −1

N l

16 and similarly for the momentum that

Using Eqs. (A17) and (A18) in Eq. (A14), one can show that

(A19)

and similarly

  X X 1 X  X X X 1  ′ κ ′ β †  T  u † kκα G κ κα β (k) p kκ ′ β . u kκα G κα (k) p kκ ′ β  T −1 = − M κ ′ M κ ′ k κκ ′ βα k κκ ′ (A20) βα

Therefore, if T is a symmetry of the system, that is, if T H RM T −1 = H RM , all elements of the MBC matrix must vanish identically:

(b) Inversion Symmetry (P)

Inversion symmetry acts on displacement and momentum operators in real space as

(A22)

where l ′ is the index of the inverted unit cell. Unlike time-reversal symmetry, inversion symmetry is unitary. Since the checkerboard lattice is non-Bravais, the lattice vectors associated with unit cells l and l ′ in Eq. (A22) are not simply related by inversion (R 0 l , −R 0 l ′ ). Instead, they satisfy

We emphasize that, since every lattice site of the checkerboard lattice is an inversion center, the basis index κ remains unchanged under inversion. This situation may differ for other lattices, such as the honeycomb lattice, where inversion symmetry interchanges the two sublattices. Using Eq. (A23) as well as the Fourier transformation in Eq. (A15), we determine the action of P on u kκα and p kκα as

0 1 X Pu kκα P −1 = √ Pu lκα P −1 e −ik·R l e −ik·τ κ

and

17 Using Eqs. (A24) and (A25) in Eq. (A14), the two terms comprising H RM can be seen to transform as follows

  X X 1 X   X X 1 X † −1 κα p kκα G κ ′ β (k) u kκ ′ β  P −1 = Pp † kκα G κα P  κ ′ β (k) u kκ ′ β P

M M κ κ βα βα k κκ ′ k κκ ′

(A26)

and

(A27)

Therefore, in centrosymmetric lattices with PH RM P −1 = H RM , where each lattice site is an inversion center, we find that

h ′ i ∗ ′ G κ

κα β (k) = G κ

κα β (k) ∈ Real. (A28)

We emphasize that the above result applies only within the Fourier gauge defined in Eq. (A15), in analogy to the well-known case of the dynamical phonon matrix [69].

(c) 90 ◦ rotation followed by time-reversal (C 4z,+ T )

We start off with the effect of the clockwise 90 ◦ rotation in real space:

a

(A29)

a

  where l is the transformed unit cell index under C

4z,+ and C 4z,+ αα are the elements of the 90

◦ clockwise rotation matrix

′ We recall that we use calligraphic letters for the symmetry operators, while upright letters denote their representations. For the checkerboard lattice, we find

(A31)

with C 4z,+ R 0 l being the rotated unit cell vector. For displacements of ions belonging to the A sublattice, we obtain

0
1 X C 4z,+ u kAα C −1 = C 4z,+ u lAα C 4z,+ −1 e −ik·R l √ 4z,+

N l

α (A32)

18 and, similarly, for displacements of the B sublattice,

0
1 X C 4z,+ u lBα C 4z,+ −1 e −ik·R l e −ik·τ B C 4z,+ u kBα C −1 4z,+ = √

N l