A unified symmetry framework for spin–ferroelectric coupling in altermagnetic multiferroics

Domain: chemistry
Published: 2026-02-24
Editorial outcome: major_revision

Abstract
Altermagnetic multiferroics, hosting coexisting spin-splitting bands and ferroelectric polarization, offer a promising route to magnetoelectric coupling beyond conventional relativistic spin–orbit mechanism. However, the lack of a unified principle connecting ferroelectric switching symmetry to spin-band topology has impeded rational material design. Here, we establish a universal symmetry-based framework that classifies all possible spin–ferroelectric couplings in altermagnets into three fundamental types: decoupling, pseudo-time-reversal coupling, and asymmetric momentum mapping. This classification stems directly from the relation between ferroelectric switching operators and the spin Laue group, creating a decisive symmetry-to-function paradigm. First-principles calculations on bilayer MnPS3 confirm the framework, showing that 3 distinct ferroelectric switching paths produce characteristic spin-band reconstructions and discriminable electrical transport signatures. The universality of the framework is further validated in BiFeO3. Our work provides a predictive design principle for voltage-programmable spintronics, effectively transforming ferroelectric symmetry from a structural descriptor into a dynamic functional control knob for altermagnetic spin states.

Introduction
Altermagnetic multiferroics—materials that simultaneously host altermagnetic and ferroelectric orders—represent a new frontier in the design of strongly coupled magnetoelectric systems1–4. Conventional multiferroics are often limited by the inherent separation between the coexisting ferroelectric and magnetic orders, resulting in weak magnetoelectric coupling that typically depends on spin-orbit coupling (SOC) as an indirect mediator5,6. In contrast, altermagnetic multiferroics feature opposite spin sublattices within the crystal lattice, which are connected by rotation or mirror symmetries, while prohibiting connection through translational or inversion symmetries7–15. This gives rise to momentum-space spin splitting in the compensated magnetic ordering, which is naturally compatible with the inversion symmetry breaking induced by the ferroelectric. Central to this is the spatial symmetry, which serves as a key parameter for geometrically coupling the spin-split electronic bands of altermagnets with the switchable ferroelectric polarization, bypassing the reliance on SOC. This intrinsic symmetry coupling offers a direct route to achieving robust magnetoelectric responses, fulfilling the critical requirements for next-generation spintronic memory and logic devices16–27.

Recent theoretical studies have shown that in various low-dimensional and three-dimensional altermagnetic multiferroic materials, ferroelectric switching can reversibly modulate the spin texture without altering the global magnetic ordering28–32. This offers an ideal platform for electric-field control of spin states and highlight promising prospects for low-power, nonvolatile spintronic applications. However, it is important to note that this coupling is subject to strict symmetry constraints. For example, in some altermagnetic multiferroics, it has been found that the switching of ferroelectric polarization does not affect the spin electronic bands of the altermagnetic29,31. This suggests that the coexistence of ferroelectric and altermagnetic does not necessarily lead to their coupling, and the symmetry conditions that govern whether such coupling occurs remain unclear. More fundamentally, a unified theoretical framework is still lacking to systematically describe how ferroelectric switching modulates altermagnetic spin splitting. In particular, symmetry-based classification criteria that can predict spin responses under different switching operations remain underdeveloped, hindering rational material design and functional optimization.

To address this gap, we develop a group-theoretic classification of spin–ferroelectric coupling in altermagnets. Our analysis identifies three fundamental response types—Type I: spin–ferroelectric decoupling, and Type II: pseudo-time-reversal (pseudo-spin-flip) coupling, and Type III: non-symmetric momentum mapping—each determined by the relationship between the ferroelectric switching operator and the crystal’s spin Laue group. These theoretical predictions are first confirmed through first-principles calculations on bilayer MnPS3 and subsequently validated in BiFeO3 as a prototypical 3D system, in which distinct ferroelectric switching paths are shown to produce characteristic spin-band reconstructions.

This work establishes a unified symmetry-guided framework for understanding and designing altermagnetic multiferroics with tailored magnetoelectric functionality. The distinct spin-band reconstructions further give rise to discriminable electrical transport signatures, enabling simple electrical measurements to serve as direct experimental fingerprints of the three symmetry-determined coupling types. Thus, the framework not only offers fundamental theoretical insight but also provides a practical material platform for the development of voltage-programmable, low-power spintronic devices.

Results
Group-theoretical analysis of spin–ferroelectric locking
In crystals, symmetry imposes fundamental constraints on the electronic band structure. For any non-relativistic spin space operation U ^ =[g ||g], the crystal Hamiltonian obeys the conjugation relation U ^ H(k)U ^-1 =H(k0). Consequently, the energy eigenvalues transform as:
U ^ E(s,k)=[g ||g] E(s,k)=E(s0,k0) (1)

Where, g (g) acts in spin (real) space, with s’=gs and k’=gk. In conventional ferromagnetic or antiferromagnetic systems, this relation ensures common features such as spin splitting or degeneracy. In non-relativistic altermagnets, however, specific spatial operations exchange opposite-spin sublattices, corresponding to a nontrivial spin Laue group7:
RAM =[E||H]+[C ||A] [E||H]=[E||H]+[C ||G-H] (2)
2 2

Here, H G is a halving subgroup of the crystallographic Laue group, representing operations that preserve the same spin sublattice. The coset G−H=AH is generated by an element A, which is a real-space rotation or mirroring (excluding pure inversion or translation; hence, the inversion operation in G is contained within H). The action of A connects the two opposite-spin sublattices. Thus, the band structure of altermagnets satisfies:
E(s,k)=[E||H] E(s,k)=E(s,Hk) (3)

E(s,k)=[C ||AH] E(s,k)=E(-s,Ak) (4)
2

In this formulation, H ensures k=hk for all h H, whereas for k Ak, the relations correspond to an alternating sign pattern of spin splitting in momentum space.

Furthermore, non-relativistic collinear magnets possess a fundamental symmetry [C ||T], which implies:
2
E(s,k)=[C ||T] E(s,k)=E(s, -k)=[E||I] E(s,k) (5)
2

This indicates that the band structure always preserves inversion symmetry I in momentum space (k=−k), even if the crystal lacks real-space inversion symmetry.

We now consider the ferroelectric switching operation O acting in real space, represented as a symmetry operation U ^ =[E||O ], which reverses the ferroelectric polarization P while preserving the system’s magnetic configuration, i.e., [E||O ]P=P0. Note that the operation U ^ is independent of its own symmetry; rather, it describes a transformation that maps one system to another with different polarization. During this process, the spins remain unchanged in real space, but exhibit distinct responses in momentum space. By matching the real-space ferroelectric switching operation with the Laue group in momentum space, three distinct types of effects on the spin-resolved band structure E(s,k) of an altermagnet are observed, as illustrated in Fig. 1.

(1) Type I: Spin-ferroelectric decoupling described by O H
[E||O ] E(s,k)=E(s,O k)=E(s,k) (6)
Here, the energy eigenvalues for each spin channel remain unchanged at all k-points. Ferroelectric switching in this case leaves the spin-split spectrum unmodified, representing a decoupled regime with no direct spectral signature.

(2) Type II: Pseudo-time-reversal coupling described by O AH
Since H is a halving subgroup of G and the generator A satisfies A2 H, it follows that AH=A-1H. Thus, the action of ferroelectric switching in momentum space satisfies O k=A-1k, and the corresponding energy eigenvalue mapping becomes:
[E||O ] E(s,k)=[E||O ] [C ||AH] E(s,k)=E(-s,k)=E(-s, -k) (7)
2
This operation effectively flips the spin label, acting as a pseudo-time-reversal (or pseudo-spin-flip). Ferroelectric switching thus induces a global spin spectrum reversal, representing strong spin–ferroelectric coupling.

(3) Type III: General non-symmetric mapping described by O G
[E||O ] E(s,k)=E(s,O k), k O k (8)
In this case, ferroelectric switching does not correspond to any symmetry element that preserves or exchanges spin sublattices. Instead, it geometrically maps the spin texture in momentum space, leading to momentum-dependent spin reconstructions and an anisotropic response that varies across the Brillouin zone.

In summary, the effect of ferroelectric switching on spin is classified as:
[E||O ] E(s,k)= E(s,k), O 2H (Type I)
E(-s,k), O 2AH (Type II) (9)
E(s,O k), O 2=G (Type III)

We further obtained numerical results from the tight-binding model constructed under the 2/m Laue group (Supplementary Fig. S1), which are in perfect agreement with the aforementioned group-theoretical classification. This further validates the impact of different ferroelectric switching symmetries on the three characteristic types of spin bands in altermagnetism.

Notably, in altermagnets, the set {E, I} always belong to H. Thus, any ferroelectric switching operation involving only inversion leaves the spin spectrum unaffected and is inherently of Type I. This is particularly relevant in 2D systems, where the z-component is a pseudoscalar, rendering the set {C , M} effectively equivalent to {E, I}. Therefore, operations O {I, C , M } in 2D systems are invariably classified as Type I, remaining decoupled from spin in any Laue group.

First-principles validation in 2D bilayer MnPS3
First-principles calculations offer a realistic platform for validating the theoretical classification of spin–ferroelectric locking. We selected bilayer MnPS3 as a candidate system, where sliding ferroelectricity in van der Waals bilayers exhibits exceptional tunability. Unlike conventional ferroelectrics with a single, fixed switching path, polarization reversal in this system can be realized through multiple layer-sliding trajectories defined by distinct symmetry operations. This intrinsic flexibility makes sliding ferroelectrics an ideal platform for systematically exploring different ferroelectric switching symmetries within a single material family.

Starting from a monolayer of MnPS3 with Néel-type antiferromagnetic ordering (its relevant structural and magnetic properties are presented in Supplementary Fig. S2), we construct the bilayer by applying the mapping {M z |τ o } together with an out-of-plane translation τ, resulting in a crystallographic point group of m and a crystallographic Laue group of 2/m, with the full set of symmetry operations G={E, C , I, M}. The halving subgroup H={E, I} preserves each spin sublattice. Consequently, the system is governed by the altermagnetic group R AM =[E||{E, I}]+[C ||{C , M }], which generates the characteristic alternating sign pattern of spin splitting in the Brillouin zone. In this construction, the sliding ferroelectric polarization P is controlled by the in-plane translation vector τ o of the top layer relative to the bottom layer. DFT total-energy scans identify a global maximum for the paraelectric phase at τ =(0,0), along with six degenerate global minima for the ferroelectric phase at τ {(va,0), (0,vb), (va,vb) | v=±1/3}, as shown in Fig. 2b, leading to uncompensated interfacial charge transfer and a combined polarization of 0.92 μC/m (in-plane) and 0.26 μC/m (out-of-plane), as demonstrated by the charge difference between the paraelectric and ferroelectric phases in Supplementary Fig. S3. We identify three transition paths: path 1 maintains the crystal’s mirror symmetry orientation and reverses only the out-of-plane component with an energy barrier of 11.8 meV/f.u., while paths 2 and 3 rotate the crystal’s mirror symmetry orientation by π/3, switch both in-plane and out-of-plane components. The energy barriers for paths 2 and 3 are degenerate, both at 9.2 meV/f.u. (Fig. 2c). The theory of bilayer stacking ferroelectricity (BSF) provides a method for selectively choosing sliding paths using an external electric field33, allowing for controllable polarization switching, as shown in Supplementary Fig. S4.

To comprehensively investigate possible ferroelectric switching symmetries in the system, we employ two distinct interlayer antiferromagnetic configurations as analysis platforms. The two configurations are nearly degenerate in energy (difference ∼0.22 meV), indicating weak interlayer exchange coupling. For compact notation, we denote the polarization as P± L , where the superscript ± denotes the sign of the out-of-plane component, the subscript n=1, 2, 3 labels the in-plane polarization directions [100], [010], [110], and m=1, 2 distinguishes the two magnetic configurations, with L corresponding to the ground-state magnetic ordering. Under these conventions, we obtain four representative ferroelectric switching operations, as summarized in Fig. 2d. The detailed structure mapping process, including magnetic ordering, is presented in Supplementary Fig. S5.
[E||M ] P+ L =P- L (11)
z 1 1 1 1
[E||M ] C P+ L =P- L (12)
z 3z 1 1 2 1
[E||M ] M P+ L =P- L (13)
z y 1 2 1 2
[E||M ] M C P+ L =P- L (14)
z y 3z 1 2 2 2

Equations (11, 13) correspond to ferroelectric transition path 1, while (12) and (14) correspond to path 2. Their effects on the altermagnetic spin spectrum are as follows. In Equation (11), M acts in the 2D Brillouin zone as equivalent to the identity element E H, realizing Type I—spin–ferroelectric decoupling. Since the M operation reduces to the identity in the 2D momentum space, it imposes no additional constraints and is therefore not considered further in Equations (12)–(14). Equation (12) involves the C operation. Since C =G, it induces a Type III — non-symmetric mapping, geometrically rotating the momentum-space spin texture by π/3. In Equation (13), M y AH enacts Type II—pseudo-time-reversal coupling, producing a global reversal of spin labels. Finally, Equation (14), the composite M C , combines Type II and Type III: it applies a pseudo-time-reversal–like inversion of the spin spectrum, followed by a π/3 rotation in k-space. The resulting geometric and spectral reconstructions are illustrated in Fig. 3. The complete electronic structures, including the conduction bands for the L and L magnetic configurations, are presented in Supplementary Fig. S6. In addition, the ferroelectric switching along path 3 exhibits a similar effect to that of path 2, as shown in Supplementary Fig. S7. This result further confirms the three characteristic effects of ferroelectric switching on the spin-resolved band structure, providing a concrete material realization and a physical basis for the theoretical classification of spin-ferroelectric coupling.

Within the Boltzmann transport framework, the band structure—by governing the nonequilibrium distribution function—directly translates into electrical signals, providing a direct electrical response measurement to probe the aforementioned band mapping relationship. We accordingly calculated the spin-resolved charge conductivity σ , as shown in Fig. 4. When a current-driving electric field is applied along the mirror-symmetric x direction, spin-up and spin-down channels generate transverse Hall currents of opposite signs along the y direction. During ferroelectric switching along path 1, the spin Hall current follows the symmetry-locked characteristics of Type I or Type II coupling, either remaining unchanged in the L magnetic ordering according to Equation (11), or reversing its overall sign in the L magnetic ordering according to Equation (13). In contrast, along path 2, the involvement of an additional C =G symmetry modifies the anisotropic characteristics of the spin-band, thereby giving rise to a Type III strongly direction-asymmetric transverse current response in the L and L magnetic orderings, as described by Equations (12) and (14). This behavior is likewise manifested during the ferroelectric transition along path 3, as shown in Supplementary Fig. S8. Such distinct transport behavior provides a functional verification of the theoretical classification, clearly distinguishing the three representative coupling types identified in the symmetric analysis, and establishes an experimentally accessible electrical criterion for identifying different ferroelectric switching pathways.

Proof of universality using BiFeO3 as a 3D exemplar
To further demonstrate the generality of our symmetry-based framework beyond 2D systems, we extend our analysis to a classic 3D multiferroic material, G-type antiferromagnetic BiFeO3 multiferroics with R3c space group, the altermagnetism was confirmed recently32. From a symmetry perspective, the corresponding crystallographic Laue group of this system is 3m, and its complete set of symmetry operations is G={E, I, C 3[111] , C2 3[111] , S 3[111] , S2 3[111] , C 2[-110] , C 2[01-1] , C 2[10-1] , M[-110] , M[01-1] , M[10-1] }. The halving subgroup H={E, I, C 3[111] , C2 3[111] , S 3[111] , S2 3[111] } preserves each spin sublattice. Therefore, the system is described by the altermagnetic group R AM =[E||{E, I, C 3[111] , C2 3[111] , S 3[111] , S2 3[111] }]+[C 2 ||{C 2[-110] , C 2[01-1] , C 2[10-1] , M[-110] , M[01-1] , M[10-1] }], representing a g-wave altermagnetic system, as shown in Fig. 5a, b.

In bulk BiFeO3, the R3c crystal structure arises from Pm3 m by the condensation of two primary order parameters: a polar distortion along the [111] direction and a rotation of the oxygen octahedra along the [111] direction, described by the R+ 4 irreducible representation. For polarization switching achieved by independent displacement of Bi3+ ions (referred to as path 1), the overall effect is equivalent to an inversion operation O =I, i.e., [E||I]P=P’ (Fig. 5b). However, in certain conditions, polarization switching maybe accompanied by the rotation of the oxygen octahedra (referred to as path 2)34. In this case, inversion symmetry cannot map the lattice structure while preserving the system’s magnetic ordering; instead, path 2 affects the system in an O =C 2[-110] manner, i.e., [E||C 2[-110]]P=P’ (Fig. 5b). According to Equation 9, the path 1, where O H, does not alter the spin splitting during polarization reversal, and is classified as Type I (decoupled). Path 2, where O AH, maps the opposite spin channels and is classified as Type II (pseudo-time-reversal coupling). The band structure calculations shown in Fig. 5 clearly confirm these two distinct responses.

This analysis not only emphasizes that the spin splitting control in BiFeO3 is directly related to octahedral rotation rather than polarization switching, providing an intuitive symmetry perspective for electric control of spin splitting in BiFeO3, but also clarifies the universal applicability of this symmetry mechanism in various multiferroic materials. By identifying the symmetry of the ferroelectric switching operation, one can predict and classify its magnetoelectric coupling functionality, independent of the specific system under study.

Methods
The atomic properties and electronic structure of the materials were calculated using first-principles simulations within density functional theory (DFT)35,36. The projected augmented wave pseudopotentials method, as implemented in the Vienna Abinitio Simulation Package (VASP)37,38 was employed. The exchange-correlation energy was calculated using the generalized gradient approximation (GGA) of the Perdew-Burke-Ernzerhof form39, and the plane wave cutoff energy was set to 500 eV. A Hubbard U =4 eV with the Dudarev parametrization was applied to properly describe the localization of Mn 3d orbitals40. The semiempirical DFT-D3 method was used to include the van der Waals interaction41. For MnPS3 calculations, a centered 9×9×1 Monkhorst-Pack k-point mesh was used42. To eliminate periodic boundary effects, the vacuum space between adjacent slabs was set to exceed 15 Å along the z direction. Using the conjugate gradient method, the plane lattice constant and atomic coordinates were fully relaxed until the energy and force converged to 10−6 and 10−2 eV/Å, respectively. The Berry-phase method was employed to evaluate polarization magnitude43, and the ferroelectric transition switching pathway was obtained using the climbing image nudged elastic band (CI-NEB) method44. The spin-resolved charge conductivity was calculated using Boltzmann transport theory within the constant-relaxation-time approximation as implemented in the Wannier90 package45, with the electronic temperature and relaxation time set to 300 K and 1 ps, respectively. The lattice structure was visualized using VESTA46.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "I have carefully read manuscript titled “A unified Symmetry Framework for Spin Ferroelectric Coupling in Altermagnetic Multiferroics”. Authors have described concept of altermagnetic multiferroelectric which would represent both altermagnetism and ferroelectricity, both individually holding interesting and useful properties by its own. Coexistence is often debated, while coupling, especially concepts like the coupling factor is even more controversial within the community. Authors have tried to come up with theory to explain all possible spin–ferroelectric couplings in altermagnets, again a topic of discussion and interest. Authors have taken into consideration the example of MnPS3 as example system showing sliding ferroelectricity and tried explaining possible ferroelectric switching states, possible pathways and resultant spin-band splitting and possible electric signature. Hypothesis of the manuscript and structure is sound, even though not groundbreaking, but useful for the community considering its systematic representation of possible mechanisms. Ferroelectric materials are getting lot of attention in recent months for transistors and MRAMs and coupled properties with altermagnetism would be an interesting avenue for research community. The referee provides an overall positive evaluation of this work, while also highlighting notable concerns regarding its possible similarity to recent comparable publications. The following comments are provided to help enhance its quality, depth, and accessibility for members of the ferroic research community, with particular attention to those working in ferroelectricity.\n1) Can authors explain clear distinctions between recently published PRB: (cid:1)-type antiferromagnetic BiFeO3 is a multiferroic (cid:2)-wave altermagnet (https://journals.aps.org/prb/abstract/10.1103/v3fg-6smc) with this manuscript. Is this manuscript a subset of the previously mentioned publication? The publication of this work in a renowned journal such as Nature Communications is highly questionable without adequate justification.\n2) Authors are advised to clearly differentiate between “coexistence” and “coupling” of ferroelectric/polar and magnetic/spin properties and explain a bit more in the introduction. Current format is only accessible to highly advanced researchers in the community.\n3) Why does the \"Method\" section (Section IV) appear immediately after the introduction? It is essential to conduct thorough proofreading prior to submission.\n4) In Group-Theoretical Analysis of Spin–Ferroelectric Locking section, can authors comments on general observations for standard ferroelectric systems (so called 3D, other than 2D) which may have similar altermagnetic coupled ferroelectric properties (if any) and if not why not?\n5) Figure 2a looks incomplete without the descriptors of spin-texture in momentum space. Only Figure 2b has shown a schematic in the inset. Figure 2c and 2d should also have similar schematic to match the commentary in main text.\n6) SI information: Part 2. Differential charge distribution of MnPS3 bilayer should be extended further to show baseline structure with electronic density and, switched to path1, path2 and path3 for clear representation of ferroelectric switching for both in-plane and out-of-plane switching mechanisms. Presenting just one state does not provide enough information for a general reader to fully understand the topic. Good Potential reference, Figure 1 of Ultrafast high-endurance memory based on sliding ferroelectrics (DOI: 10.1126/science.adp3575).\n7) What kind of energy distribution is mentioned in figure 3B? Can authors explain why Path1 has higher barrier energy for switching compared to path2 or 3? Authors should add clear schematics to show L1 and L2 magnetic order atleast in SI for clear representation.\n8) What stimulus is applied to achieve this sliding ferroelectric switching? Is it mechanical or electrical? Is it out of plane or in-plane application of stimulus?\n9) Can authors careful map each of equation 11,12,13 and 14 to show its corresponding structure in figure 4?\n10) Authors should add xyz coordinates in figure 5 to indicate coordinates system.\n11) Similar comments for Figure 5. Authors should try to correlate equations and corresponding figure 4 and figure 5 to clearly show connections between different sections of the manuscript.\n12) Can authors add comments about all the presented theory for its applicability or limitations for prototypical multiferroic (possible altermagnetic) 3D ferroelectric such as BiFeO3 or any other relevant standard system in ferroelectric world in last portion of the manuscript.", "reviewer_2": "Sun and co-workers present a symmetry analysis enabling the prediction of couplings between spins and ferroelectric displacements in altermagnets. They identify three different types of coupling: a first mechanism in which spins and ferroelectric displacements are decoupled, and two additional paths in which polarization reversal can reverse the spin texture. Using these couplings, they further demonstrate that, in one of these mechanisms, the spin texture can generate a finite charge current. This could be highly desirable for spintronic applications.\nOverall, I find the paper interesting and relevant for the spintronics community. Nevertheless, I also find it rather technical in places and perhaps too specialized for a broad audience. In addition, I miss a clear recipe for identifying compounds that satisfy the optimal criteria — the reader is left somewhat “starved” at the end. Therefore, I feel the manuscript falls a bit short for publication in Nature Communications. I would instead recommend submission to a more specialized journal such as Physical Review Letters or a similar outlet. The paper also appears to be a modified manuscript transferred from another journal, with “Section IV (Methods)” unexpectedly placed in the middle of the text.\nBeyond these comments, I also find the duplication between the tight-binding and density-functional-theory approaches somewhat puzzling. Wouldn’t it be simpler to present a single test case based solely on DFT? I list below a few additional remarks that the authors may find relevant:\n(I) Please clarify the antiferromagnetic orders used in the calculations.\n(II) The units used for the spontaneous polarization are unusual and likely incorrect (µC·cm-2 are standard).\n(III) Energy barriers are discussed in meV per formula unit, whereas Fig. 3(c) uses a different normalization.\n(IV) Sketches for the different pathways are highly recommended, as these are not easy to infer from Fig. 3(b) alone."}, "author_response": "We sincerely thank the reviewers for their valuable feedback and positive evaluation of our work,\nwhich have played a crucial role in enhancing the quality of the manuscript. Below, we provide a\ndetailed point-by-point response to the reviewers’ comments.\nReviewer #1 (Remarks to the Author):\nI have carefully read manuscript titled “A unified Symmetry Framework for Spin Ferroelectric\nCoupling in Altermagnetic Multiferroics”. Authors have described concept of altermagnetic\nmultiferroelectric which would represent both altermagnetism and ferroelectricity, both\nindividually holding interesting and useful properties by its own. Coexistence is often debated,\nwhile coupling, especially concepts like the coupling factor is even more controversial within the\ncommunity. Authors have tried to come up with theory to explain all possible spin–ferroelectric\ncouplings in altermagnets, again a topic of discussion and interest. Authors have taken into\nconsideration the example of MnPS3 as example system showing sliding ferroelectricity and tried\nexplaining possible ferroelectric switching states, possible pathways and resultant spin-band\nsplitting and possible electric signature. Hypothesis of the manuscript and structure is sound, even\nthough not groundbreaking, but useful for the community considering its systematic representation\nof possible mechanisms. Ferroelectric materials are getting lot of attention in recent months for\ntransistors and MRAMs and coupled properties with altermagnetism would be an interesting\navenue for research community. The referee provides an overall positive evaluation of this work,\nwhile also highlighting notable concerns regarding its possible similarity to recent comparable\npublications. The following comments are provided to help enhance its quality, depth, and\naccessibility for members of the ferroic research community, with particular attention to those\nworking in ferroelectricity.\nReviewer #1: 1) Can authors explain clear distinctions between recently published PRB: -type\nantiferromagnetic BiFeO3 is a multiferroic -wave altermagnet\n𝐺\n(https://journals.aps.org/prb/abstract/10.1103/v3fg-6smc) with this manuscript. Is this manuscript\n𝑔\na subset of the previously mentioned publication? The publication of this work in a renowned\njournal such as Nature Communications is highly questionable without adequate justification.\nReply: We sincerely thank the reviewer for raising this important point and for providing us with\nthe opportunity to clarify the fundamental distinctions between our work and the recent publication\nin Physical Review B titled “G-type antiferromagnetic BiFeO is a multiferroic -wave\n3\naltermagnet” (PHYSICAL REVIEW B 112, 104411, 2025). We respectfully believe that our work\ng\nis not a subset of this study, but rather establishes a completely new symmetry-based classification\nframework focused on the dynamic coupling mechanisms between ferroelectricity and\naltermagnetism. This contrasts sharply with the core focus of the PRB paper, which is primarily\nconcerned with the static classification of altermagnetic states themselves (i.e., distinguishing d-,\ng-, and i-wave states).\nSpecifically, the PRB paper uses BiFeO as a model system, with the main goal being the\n3\ndetermination of its intrinsic altermagnetic wave state through the spin Laue group, which is\nultimately identified as a g-wave state. While the influence of structural distortions on spin splitting\nis discussed, the paper’s main starting point and focus lie in determining “what symmetries the\nmaterial possesses and which type of altermagnetic state it belongs to,” providing crucial\nclassification criteria for understanding the static electronic structure and topology of\naltermagnetism.\nIn contrast, our work poses a different, more function-oriented core question: How does the\nreversal of ferroelectric polarization symmetrically control and reconstruct the spin splitting of\naltermagnetism? To address this, we have established a universal theoretical framework that\nsystematically predicts all possible coupling outcomes by analyzing the relationship between\nferroelectric reversal operations and the system’s spin Laue group. The three fundamental\nmagnetoelectric coupling types we identify—decoupled, pseudo-time-reversal coupling, and\nasymmetric momentum mapping—precisely describe the dynamic process of spin-electronic state\nevolution triggered by the switching between two ferroelectric polarization states.\nTherefore, while both studies rely on symmetry analysis, the scientific goals, the starting points of\ntheir theoretical frameworks, and their final applications are fundamentally different. The PRB\nwork is an excellent study of state classification, while our work is a pioneering investigation into\nthe coupling and control mechanisms.\nIn the second paragraph of the introduction of the revised manuscript, we emphasize the\nfoundational premise of our work:\nHowever, it is important to note that this coupling is subject to strict symmetry constraints. For\nexample, in some altermagnetic multiferroics, it has been found that the switching of ferroelectric\npolarization does not affect the spin electronic bands of the altermagnetic.29,31 This suggests that\nthe coexistence of ferroelectric and altermagnetic does not necessarily lead to their coupling, and\nthe symmetry conditions that govern whether such coupling occurs remain unclear. More\nfundamentally, a unified theoretical framework is still lacking to systematically describe how\nferroelectric switching modulates altermagnetic spin splitting. In particular, symmetry-based\nclassification criteria that can predict spin responses under different switching operations remain\nunderdeveloped, hindering rational material design and functional optimization.\nReviewer #1: 2) Authors are advised to clearly differentiate between “coexistence” and “coupling”\nof ferroelectric/polar and magnetic/spin properties and explain a bit more in the introduction.\nCurrent format is only accessible to highly advanced researchers in the community.\nReply: We sincerely thank the reviewer for this valuable suggestion. We have clearly distinguished\nthe concepts of “coexistence” and “coupling” of ferroelectric and magnetic properties in our\nmanuscript. In the revised version, we explicitly define “coexistence” as the simultaneous presence\nof ferroelectric and magnetic properties in a material without implying any direct interaction, while\n“coupling” refers to the mutual influence between these properties. This distinction is now more\nclearly emphasized in the Introduction part of the revised manuscript to ensure better\nunderstanding by a broader audience. (Introduction, Paragraphs 1-2):\nAltermagnetic multiferroics—materials that simultaneously host altermagnetic and\nferroelectric orders—represent a new frontier in the design of strongly coupled magnetoelectric\nsystems.1-4 Conventional multiferroics are often limited by the inherent separation between the\ncoexisting ferroelectric and magnetic orders, resulting in weak magnetoelectric coupling that\ntypically depends on spin-orbit coupling (SOC) as an indirect mediator.5,6 In contrast,\naltermagnetic multiferroics feature opposite spin sublattices within the crystal lattice, which are\nconnected by rotation or mirror symmetries, while prohibiting connection through translational or\ninversion symmetries.7-15 This gives rise to momentum-space spin splitting in the compensated\nmagnetic ordering, which is naturally compatible with the inversion symmetry breaking induced\nby the ferroelectric. Central to this is the spatial symmetry, which serves as a key parameter for\ngeometrically coupling the spin-split electronic bands of altermagnets with the switchable\nferroelectric polarization, bypassing the reliance on SOC. This intrinsic symmetry coupling offers\na direct route to achieving robust magnetoelectric responses, fulfilling the critical requirements\nfor next-generation spintronic memory and logic devices.16-27\nRecent theoretical studies have shown that in various low-dimensional and three-dimensional\naltermagnetic multiferroic materials, ferroelectric switching can reversibly modulate the spin\ntexture without altering the global magnetic ordering.28-32 This offers an ideal platform for electric-\nfield control of spin states and highlights promising prospects for low-power, nonvolatile spintronic\napplications. However, it is important to note that this coupling is subject to strict symmetry\nconstraints. For example, in some altermagnetic multiferroics, it has been found that the switching of\nferroelectric polarization does not affect the spin electronic bands of the altermagnetic.29,31 This\nsuggests that the coexistence of ferroelectric and altermagnetic does not necessarily lead to their\ncoupling, and the symmetry conditions that govern whether such coupling occurs remain unclear.\nReviewer #1: 3) Why does the \"Method\" section (Section IV) appear immediately after the\nintroduction? It is essential to conduct thorough proofreading prior to submission.\nReply: We thank the reviewer for pointing this out. We have carefully proofread the manuscript\nand have removed the “Method” section from its current position. It has now been placed after the\nconclusion, as it should be.\nReviewer #1: 4) In Group-Theoretical Analysis of Spin–Ferroelectric Locking section, can authors\ncomments on general observations for standard ferroelectric systems (so called 3D, other than 2D)\nwhich may have similar altermagnetic coupled ferroelectric properties (if any) and if not why not?\nReply: We appreciate the reviewer’s insightful comment. We would like to clarify that although\nwe focus on a 2D system in the subsequent material analysis, the group-theoretical analysis section\ndoes not impose any limitations or assumptions regarding dimensionality. The analysis applies to\nboth 2D and 3D systems in general. The special consideration of the z-direction in 2D systems,\ndue to the pseudodistance, is highlighted only towards the end of the paper. To strengthen this point,\nwe have now included BiFeO as an example of a 3D system to demonstrate the applicability of\n3\nour framework to 3D materials as well. (Results, Section 3)\n3. Proof of Universality Using BiFeO as a 3D Exemplar\nwe extend our analysis to a classic 3D multiferroic material, G-type antiferromagnetic BiFeO\n3\nmultiferroics with R3c space group, the altermagnetism was confirmed recently. From a symmetry\nperspective, the corresponding crystallographic Laue group of this system is 3m, and its complete\nset of symmetry operations is G = {E, I, C , , S , , , , ,\n3[111] 3[111]\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅]\n, , }. The halving subgroup H = {E, I, C , , S , }\n3[111] 3[111]\n2 2\n𝑀pr [ e 1̅ s 1 e 0 r ] ves𝑀 e [ a 0 c 1 h 1̅] spi𝑀n [ s 1 u 0 b 1̅] lattice. Therefore, the system is described by the a𝐶l3t [ e1r1m1] agnetic gro𝑆u3p [1 1R1]\nAM\n= [E||{E, I, C , , S , }] + [C ||{ , , , , ,\n3[111] 3[111] 2\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅] 𝑀 [1̅10] 𝑀[011̅]\n}], representing a g-wave altermagnetic system, as shown in Fig. 5a-b.\n𝑀[10\nI\n1̅\nn\n]\nBiFeO , the switching of ferroelectric polarization can occur through two possible\n3\nmicroscopic paths. The path 1 is mainly realized by the displacement of Bi3+ ions along the [111]\ndirection, with the overall effect equivalent to an O = I inversion operation, i.e., [E||I]P = P’\nFE\n(Fig. 5b). The path 2 involves the displacement of Bi3+ ions accompanied by the rotation of the\noxygen octahedra. In this case, inversion symmetry cannot map the lattice structure while\npreserving the system’s magnetic ordering. Instead, path 2 must map the system in a O =\nFE\nmanner, i.e., [E|| ]P = P’ (Fig. 5b). According to Eq. 9, the path 1, where O\nFE\n, do\n𝐶\ne2s[ 1̅n1o0]t\nalter the spin spl\n𝐶\nit2ti[n1̅1g0 d]uring polarization reversal, and is classified as Type I (de\n∈\nc\nH\noupled). The\npath 2, where O , maps the opposite spin channels and is classified as Type II (pseudo-\nFE\ntime-reversal coupling). The band structure calculations shown in Fig. 5 clearly confirm these two\n∈ AH\ndistinct responses.\nThis analysis not only provides an intuitive symmetry explanation for the recently proposed\nidea that spin splitting in BiFeO requires oxygen octahedral rotation for electric control, but also\n3\nclarifies the universal applicability of this symmetry mechanism in various multiferroic materials.\nBy identifying the symmetry of the ferroelectric switching operation, one can predict and classify\nits magnetoelectric coupling functionality, independent of the specific system under study.\nFig. 5. The crystal structure of BiFeO and its corresponding Brillouin zone of the rhombohedral\n3\nlattice. Red and blue represent Fe ions with opposite spins, with k (k’) located at the midpoint\nbetween Γ and P (P ’) in the Brillouin zone. (b) Spin-splitting band structure with alternating\n2 2\nmagnetism. (c) and (d) represent two ferroelectric switching paths, where the spin band structure\nremains unchanged under O = I, but reverses under O = .\nFE FE\n𝐶2[1̅10]\nReviewer #1: 5) Figure 2a looks incomplete without the descriptors of spin-texture in momentum\nspace. Only Figure 2b has shown a schematic in the inset. Figure 2c and 2d should also have similar\nschematic to match the commentary in main text.\nReply: We sincerely thank the reviewer for the helpful suggestion. Since Figure 2b already\nincludes the identity operation E, which represents the description of the structure shown in Figure\n2a in momentum space, we have added clarifying annotations to enhance clarity. Additionally, we\nhave included schematic representations of spin splitting in Figures 2c and 2d to match the\ncommentary in the main text. Following Reviewer 2’s suggestion, we have moved this detailed\ndiscussion to the Supplementary Information for better readability. As a result, the figure\nnumbering in the subsequent sections has been shifted forward accordingly.\nFig. S1. (a) Schematic of a minimal altermagnetic tight-binding model with the crystallographic\nLaue group 2/m. (b) Spin-split spectra under symmetry transformations , since the\nidentity operation E is included, this also represents the intrinsic spin spectrum of the model. (c)\n𝑂 = {𝐸,𝐼}\nand (d) Spin-split spectra under and , respectively, showing a global\nspin reversal and a non-symme tric𝑂 g=eo{m𝐶e2𝑦tr,i𝑀c 𝑦m}apping𝑂 o=f th𝐶e3 𝑧spin spectrum. Parameters used:\nJ S/t = 1, and t /t = 0.5.\nsd inter\nReviewer #1: 6) SI information: Part 2. Differential charge distribution of MnPS3 bilayer should\nbe extended further to show baseline structure with electronic density and, switched to path1, path2\nand path3 for clear representation of ferroelectric switching for both in-plane and out-of-plane\nswitching mechanisms. Presenting just one state does not provide enough information for a\ngeneral reader to fully understand the topic. Good Potential reference, Figure 1 of Ultrafast high-endurance\nmemory based on sliding ferroelectrics (DOI: 10.1126/science.adp3575).\nReply: We sincerely thank the reviewer for this constructive suggestion. In response, we have\nextended the differential charge density comparison to include both out-of-plane polarization states\nas well as three in-plane polarization states. Additionally, we have introduced the paraelectric\nphase as a comparison to provide a more comprehensive view of the ferroelectric switching\nmechanisms. These modifications should help the general reader better understand the topic.\n(Results, Section 2, Paragraph 2, Lines 9-14; Supplementary Information, Part 3)\nDFT total-energy scans identify a global maximum for the paraelectric phase at τ (0,0), along\no\nwith six degenerate global minima for the ferroelectric phase at τ {(va,0), (0,vb), (va,vb) | v =\no =\n±1/3}, as shown in Fig. 2b, leading to uncompensated interfacial charge transfer and a combined\n∈\npolarization of 0.92 μC/m (in-plane) and 0.26 μC/m (out-of-plane), as demonstrated by the charge\ndifference between the paraelectric and ferroelectric phases in Fig. S3.\nPart 3. Differential charge distribution of MnPS bilayer.\n3\ninversion symmetry due to the stacking, resulting in a combination of in-plane and out-of-plane\npolarization. The polarization direction is denoted as , where the superscript ± denotes the\n±\nsign of the out-of-plane component, the subscript n =\n𝑷\n1𝑛, 2, 3 labels the in-plane polarization\ndirections [100], [010], [110]. For comparison, we also constructed the paraelectric phase P at\n0\nτ = (0,0), where the charge redistribution maintains mirror symmetry in the out-of-plane direction,\no\nwhile the in-plane charge transfer follows the C rotational symmetry, thereby prohibiting any\n3z\nform of polarization. The out-of-plane ferroelectricity of MnPS is demonstrated using the\n3\n±\nstates at τ {(a/3, b/3), (-a/3, -b/3)}, where charge accumulates at the interface S atoms a𝑷n3d\no\ndepletes around the P atoms, generating out-of-plane polarization that can be reversed via sliding,\n∈\nas shown in Fig. S3a. Additionally, the three in-plane polarization directions are demonstrated by\nstates at τ {(-a/3, -b/3), (a/3, 0), (0, b/3)}, where these directions are controlled by the\no\n+\n𝑷𝑛 ∈\nsliding paths. The different paths determine the final orientation of the mirror symmetry. Due to\nthe presence of mirror symmetry, any polarization component perpendicular to the mirror plane is\nforbidden, meaning the polarization is controlled by the mirror plane direction and can only occur\nalong its orientation, as shown in Fig. S3b.\nFig. S3. Differential charge distribution of MnPS bilayer. (a) The side view clearly shows electron\n3\naccumulation at the interfacial S atoms and depletion around the P atoms, establishing a vertical\nout-of-plane dipole moment. (b) In the in-plane direction, mirror symmetry M enforces a symmetric\ndistribution on both sides, thereby aligning the polarization direction with the mirror plane. The\nblack solid line represents the unit cell lattice, and the blue dashed line highlights the crystal's\nmirror symmetry.\nReviewer #1: 7) What kind of energy distribution is mentioned in figure 3B? Can authors explain\nwhy Path1 has higher barrier energy for switching compared to path2 or 3? Authors should add\nclear schematics to show L1 and L2 magnetic order atleast in SI for clear representation.\nReply: We sincerely thank the reviewer for the valuable suggestion. In Figure 2B, the energy\ndistribution represents the relative energy of the top layer with respect to the bottom layer, defined\nby translation vectors. This highlights the stability of the system for different stacking\nconfigurations defined by these translation vectors. Paths 2 and 3 are degenerate because both\nresult in a mirror rotation of 120°, leading to the switching of polarization both in-plane and out-\nof-plane, and are symmetric to each other along their paths. In contrast, Path 1 preserves the mirror\nsymmetry and only switches the out-of-plane polarization, distinguishing it from Paths 2 and 3.\nThe energy difference reflects the non-degeneracy (asymmetry) between these paths.\nWe have made these clarifications in the manuscript and added the requested schematics of the L1\nand L2 magnetic orders in the Supplementary Information for clearer representation. (Results,\nSection 2, Paragraph 2, Lines 7-11 and 14-18; Results, Section 2, Paragraph 3, Lines 9-10;\nSupplementary Information, Part 5)\nIn this construction, the sliding ferroelectric polarization P is controlled by the in-plane translation\nvector τ of the top layer relative to the bottom layer. DFT total-energy scans identify a global\no\nmaximum for the paraelectric phase at τ (0,0), along with six degenerate global minima for the\no\nferroelectric phase at τ {(va,0), (0,vb), (va,vb) | v = ±1/3}, as shown in Fig. 2b\no =\n∈\nWe identify three transition paths: path 1 maintains the crystal’s mirror symmetry orientation and\nreverses only the out-of-plane component with an energy barrier of 11.8 meV/f.u., while paths 2\nand 3 rotate the crystal’s mirror symmetry orientation by π/3, switch both in-plane and out-of-\nplane components. The energy barriers for paths 2 and 3 are degenerate, both at 9.2 meV/f.u. (Fig.\n2c).\nThe detailed structure mapping process including magnetic ordering is presented in Fig. S5.\nFig. S5. (a) Top and side views of the L and L antiferromagnetic ordering configurations in the\n1 2\nstate. Red and blue represent opposite spins. The inset on the right highlights the spin positions,\n+\n𝑷\ncl1early illustrating the spin and stacking order. The out-of-plane polarization direction is indicated\nby a black arrow, while the in-plane polarization direction aligns along the x-axis and is parallel\nto the mirror plane (cyan plane).\nReviewer #1: 8) What stimulus is applied to achieve this sliding ferroelectric switching? Is it\nmechanical or electrical? Is it out of plane or in-plane application of stimulus?\nReply: We appreciate the reviewer’s comment. The theory of bilayer stacking ferroelectricity (BSF)\nprovides a controlled, electrically-driven approach for the manipulation of polarization. We have\nadded a detailed description of this electrical control mechanism in the Supplementary Information\nto clarify the stimulus involved in the ferroelectric switching process. (Results, Section 2,\nParagraph 2, Lines 18-20; Supplementary Information, Part 4)\nThe theory of bilayer stacking ferroelectricity (BSF) provides a method for selectively choosing\nsliding paths using an external electric field,33 allowing for controllable polarization switching, as\nshown in Fig. S4.\nPart 4. Controllable switching of the six states achieved by the external electric field.\nSince each ferroelectric ground state corresponds to three distinct translational directions, it\nis not possible to achieve stable switching between all six states using only an out-of-plane electric\nfield. We demonstrate the controllable polarization switching method introduced by the bilayer\nstacking ferroelectricity (BSF) theory through an external electric field using the state. In the\n+\ncase of path 1, where only out-of-plane polarization switching is involved, th\n𝑷\ne1 transition is\nachieved by applying an out-of-plane electric field while keeping the in-plane electric field fixed\nalong the direction of in-plane polarization, as shown in Fig. S4a. For paths 2 and 3, which involve\nthe transition between in-plane and out-of-plane polarization, the coupling of these two\npolarization components is exploited. In this approach, an in-plane electric field is applied\nperpendicular to the switching path, aligning with the direction of in-plane polarization change\n(the [ ] direction for path 2 or the [2 0] direction for path 3). This approach reduces the\nintera\n2\nct 1̅io\n0\nn energy in the form of −E·P, thus\n1\nproviding the optimal switching pathway, as shown in\nFig. S4b-c.\nFig. S4. Switching paths selected by different external electric fields, enabling the switching of six\npolarization states. The simplified schematic of the bilayer stacking uses black spheres to represent\nMn atoms, highlighting the interlayer configuration. Panels (a), (b), and (c) represent paths 1, 2,\nand 3, respectively. The green arrows represent the direction of the applied electric field, where\nfor path 1, a combination of in-plane and out-of-plane fields is required, while only an in-plane\nelectric field is needed for paths 2 and 3. The red arrows indicate the direction of ionic\ndisplacement required to complete the polarization switching. The cyan plane illustrates the\ncrystal’s mirror symmetry, which changes depending on the sliding path.\nReviewer #1: 9) Can authors careful map each of equation 11,12,13 and 14 to show its\ncorresponding structure in figure 4?\nReply: We thank the reviewer for this helpful suggestion. We have added the corresponding\nmappings of equations (11), (12), (13), and (14) to the structures shown in Figure S5 to provide a\nclearer connection between the equations and their graphical representations. (Results, Section 2,\nParagraph 3, Lines 9-10; Supplementary Information, Part 5)\nThe detailed structure mapping process including magnetic ordering is presented in Fig. S5.\nPart 5. Spin Structure Mapping for Polarization Switching.\nFigure S5a shows the magnetic ordering configurations of L and L , both of which exhibit\n1 2\nthe same intralayer Néel-type magnetic ordering, but with different interlayer magnetic couplings.\nFor clarity in illustrating the magnetic ordering and related symmetry operations, we simplify\nthese configurations into stacking models that only include the magnetic sites. Based on the sliding\npaths shown in Fig. S4, the sliding direction of path 1 is aligned with the mirror plane, causing the\ntop layer to slide relative to the bottom layer in-plane, thus achieving electric-field controlled\npolarization switching. For the L magnetic ordering, by applying the [E||M ] mapping to the\n1 z\n+\nstate, we obtain a structure with the same position and magnetic ordering as the sta\n𝑷\nte1.\n−\nHowever, for the L\n2\nmagnetic ordering, the [E||M\nz\n] mapping only transforms the lattice\n𝑷\nst1ructure;\nan additional M operation is required to restore the original magnetic ordering, specifically an\ny\n[E||M M ] operation, as shown in Fig. S5b.\nz y\nNext, we consider paths 2 and 3, as shown in Fig. S5c, another state is generated from the\nsame polarization state by following three different sliding paths. These states have the same out-\nof-plane polarization but a 3/π angle between their in-plane polarizations. The stacking structure\nreveals that the three states can be mapped onto each other via the C symmetry. Therefore, the\n3z\npolarization switching achieved through these paths can be viewed as the result of applying an\nadditional [E||C ] operation to the structure from path 1, yielding [E||M C ] for the L magnetic\n3z z 3z 1\nordering and [E||M M C ] for the L magnetic ordering.\nz y 3z 2\nFig. S5. (a) Top and side views of the L and L antiferromagnetic ordering configurations in the\n1 2\nstate. Red and blue represent opposite spins. The inset on the right highlights the spin positions,\n+\n𝑷\ncl1early illustrating the spin and stacking order. The out-of-plane polarization direction is indicated\nby a black arrow, while the in-plane polarization direction aligns along the x-axis and is parallel\nto the mirror plane (cyan plane). (b) On the left side of the panel, the real-space sliding along path\n1 for the two magnetic orderings are shown, with the corresponding symmetry operations\ndisplayed on the right. (c) Symmetry relations for the three sliding paths. The black spheres\nrepresent the Mn sites, showing that the three sliding paths are related to each other through the\nC symmetry and are independent of the magnetic order. Therefore, the ferroelectric structures\n3z\nobtained by sliding along paths 2 and 3 can be combined with the structure from path 1 using the\nC symmetry.\n3z\nReviewer #1: 10) Authors should add xyz coordinates in figure 5 to indicate coordinates system.\nReply: We thank the reviewer for the suggestion. We have added the xyz coordinates in Figure 4\nto clearly indicate the coordinate system.\nFig. 4. Spin-resolved charge conductivity under different magnetic orderings and polarization\nconfigurations. The green dashed line indicates the Fermi level, with the inset showing the\ntransport response at -0.1 eV. For the states, opposite spin channels exhibit conductivities of\n±\nopposite signs, with the direction deter𝑷m1ined by the type I or type II spin–ferroelectric coupling.\nIn contrast, the transition from the to states display an anisotropic modification that\n+ −\ndisrupts this spin-dependent signature\n𝑷\na1nd ge\n𝑷\nn2erate an asymmetric transverse spin current.\nReviewer #1: 11) Similar comments for Figure 5. Authors should try to correlate equations and\ncorresponding figure 4 and figure 5 to clearly show connections between different sections of the\nmanuscript.\nReply: We appreciate the reviewer’s insightful comment. The spin-charge conductivity based on\nthe Boltzmann transport theory is directly related to the band structure shown in Figure 3.\nTherefore, the symmetry-related currents in Figure 4 are consistent with the spin splitting in the\nfirst Brillouin zone, as illustrated in the inset of Figure 3. The discussion in the fourth paragraph\nof Section 2 already includes an analysis of how symmetry affects the band mapping in the\nBrillouin zone. To address the reviewer's suggestion, we have now added the correlation between\nthe analysis in Figure 4 and the corresponding equations. (Results, Section 2, Last Paragraph, Lines\n1-3 and 6-13)\nWithin the Boltzmann transport framework, the band structure—by governing the nonequilibrium\ndistribution function—directly translates into electrical signals, providing a direct electrical\nresponse measurement to probe the aforementioned band mapping relationship.\nDuring ferroelectric switching along path 1, the spin Hall current follows the symmetry-locked\ncharacteristics of Type I or Type II coupling, either remaining unchanged in the L magnetic\n1\nordering according to Eq. (11), or reversing its overall sign in the L magnetic ordering according\n2\nto Eq. (13). In contrast, along path 2, the involvement of an additional symmetry\nmodifies the anisotropic characteristics of the spin-band, thereby giving rise t\n𝐶\no3 a𝑧 T\n∉\nyp e\n𝑮\nIII strongly\ndirection-asymmetric transverse current response in the L and L magnetic orderings, as\n1 2\ndescribed by Eq. (12) and (14).\nReviewer #1: 12) Can authors add comments about all the presented theory for its applicability or\nlimitations for prototypical multiferroic (possible altermagnetic) 3D ferroelectric such as BiFeO3\nor any other relevant standard system in ferroelectric world in last portion of the manuscript.\nReply: We thank the reviewer for this insightful suggestion. As requested, we have added a\ndiscussion in the last portion of the manuscript regarding the applicability of the proposed theory\nto prototypical 3D multiferroic systems. Using BiFeO as an example, we demonstrate that the\n3\naltermagnetic–ferroelectric coupling in 3D systems also adheres to the symmetry framework\npresented in our work. This further strengthens our argument. (Results, Section 3)\n3. Proof of Universality Using BiFeO as a 3D Exemplar\nwe extend our analysis to a classic 3D multiferroic material, G-type antiferromagnetic BiFeO\n3\nmultiferroics with R3c space group, the altermagnetism was confirmed recently. From a symmetry\nperspective, the corresponding crystallographic Laue group of this system is 3m, and its complete\nset of symmetry operations is G = {E, I, C , , S , , , , ,\n3[111] 3[111]\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅]\n, , }. The halving subgroup H = {E, I, C , , S , }\n3[111] 3[111]\n2 2\n𝑀pr [ e 1̅ s 1 e 0 r ] ves𝑀 e [ a 0 c 1 h 1̅] spi𝑀n [ s 1 u 0 b 1̅] lattice. Therefore, the system is described by the a𝐶l3t [ e1r1m1] agnetic gro𝑆u3p [1 1R1]\nAM\n= [E||{E, I, C , , S , }] + [C ||{ , , , , ,\n3[111] 3[111] 2\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅] 𝑀 [1̅10] 𝑀[011̅]\n}], representing a g-wave altermagnetic system, as shown in Fig. 5a-b.\n𝑀[10\nI\n1̅\nn\n]\nBiFeO , the switching of ferroelectric polarization can occur through two possible\n3\nmicroscopic paths. The path 1 is mainly realized by the displacement of Bi3+ ions along the [111]\ndirection, with the overall effect equivalent to an O = I inversion operation, i.e., [E||I]P = P’\nFE\n(Fig. 5b). The path 2 involves the displacement of Bi3+ ions accompanied by the rotation of the\noxygen octahedra. In this case, inversion symmetry cannot map the lattice structure while\npreserving the system’s magnetic ordering. Instead, path 2 must map the system in a O =\nFE\nmanner, i.e., [E|| ]P = P’ (Fig. 5b). According to Eq. 9, the path 1, where O\nFE\n, do\n𝐶\ne2s[ 1̅n1o0]t\nalter the spin spl\n𝐶\nit2ti[n1̅1g0 d]uring polarization reversal, and is classified as Type I (de\n∈\nc\nH\noupled). The\npath 2, where O , maps the opposite spin channels and is classified as Type II (pseudo-\nFE\ntime-reversal coupling). The band structure calculations shown in Fig. 5 clearly confirm these two\n∈ AH\ndistinct responses.\nThis analysis not only provides an intuitive symmetry explanation for the recently proposed\nidea that spin splitting in BiFeO requires oxygen octahedral rotation for electric control, but also\n3\nclarifies the universal applicability of this symmetry mechanism in various multiferroic materials.\nBy identifying the symmetry of the ferroelectric switching operation, one can predict and classify\nits magnetoelectric coupling functionality, independent of the specific system under study.\nFig. 5. The crystal structure of BiFeO and its corresponding Brillouin zone of the rhombohedral\n3\nlattice. Red and blue represent Fe ions with opposite spins, with k (k’) located at the midpoint\nbetween Γ and P (P ’) in the Brillouin zone. (b) Spin-splitting band structure with alternating\n2 2\nmagnetism. (c) and (d) represent two ferroelectric switching paths, where the spin band structure\nremains unchanged under O = I, but reverses under O = .\nFE FE\n𝐶2[1̅10]\nReviewer #2 (Remarks to the Author):\nSun and co-workers present a symmetry analysis enabling the prediction of couplings between\nspins and ferroelectric displacements in altermagnets. They identify three different types of\ncoupling: a first mechanism in which spins and ferroelectric displacements are decoupled, and two\nadditional paths in which polarization reversal can reverse the spin texture. Using these couplings,\nthey further demonstrate that, in one of these mechanisms, the spin texture can generate a finite\ncharge current. This could be highly desirable for spintronic applications.\nOverall, I find the paper interesting and relevant for the spintronics community. Nevertheless, I\nalso find it rather technical in places and perhaps too specialized for a broad audience. In addition,\nI miss a clear recipe for identifying compounds that satisfy the optimal criteria — the reader is left\nsomewhat “starved” at the end. Therefore, I feel the manuscript falls a bit short for publication in\nNature Communications. I would instead recommend submission to a more specialized journal\nsuch as Physical Review Letters or a similar outlet. The paper also appears to be a modified\nmanuscript transferred from another journal, with “Section IV (Methods)” unexpectedly placed in\nthe middle of the text.\nReply: We sincerely thank the reviewer for the thoughtful and constructive feedback, and we\nappreciate the positive assessment of our work's relevance to the spintronics community.\nWe fully acknowledge the concern regarding the technical nature of the original manuscript. In\nresponse, we have undertaken a comprehensive revision aimed at making the content more\naccessible to a broad readership while strengthening its practical utility. Importantly, our symmetry\nframework does provide a clear, experimentally actionable “recipe”: based on different spin-\nferroelectric coupling types, it directly predicts distinct electrical transport signatures, such as the\nspin Hall response. This bridges theoretical symmetry analysis with tangible experimental\nvalidation.\nTo address the reviewer’s suggestions, we have implemented the following key improvements in\nthe revised manuscript:\n1. Enhanced clarity and accessibility:\n(1) The Introduction has been rewritten to provide a more intuitive physical overview,\n(2) Technical derivations and the detailed tight-binding model have been moved to\nSupplementary Information, allowing the main text to focus on conceptual understanding\nand physical implications.\n(3) Figures have been enriched with clearer labels, schematics, and coordinate annotations to\nguide readers through the symmetry mappings and switching pathways.\n2. Expanded discussion on material applicability:\n(1) We have added a dedicated section on BiFeO as a 3D exemplar, demonstrating how our\n3\nsymmetry classification applies directly to a well-known multiferroic system. This\nillustrates the framework's generality beyond 2D materials and provides a concrete\nexample of how to identify coupling types in real compounds.\n(2) The discussion of sliding ferroelectric mechanisms, symmetry mapping, and switching\npathways has been strengthened to offer clearer guidance on how different switching\noperations relate to observable spin–texture reconstructions.\n3. Structural improvements:\n(1) The “Methods” section has been relocated to its proper position after the Conclusion.\n(2) The flow of the manuscript has been reorganized to better connect symmetry analysis,\ncomputational validation, and transport predictions.\nWe believe these revisions have significantly increased the manuscript’s appeal and readability for\na wide scientific audience, while preserving the rigorous theoretical foundation that specialists will\nvalue. The work now not only presents a fundamental symmetry-based classification but also\nprovides experimentally verifiable predictions—making it both conceptually novel and practically\nuseful for guiding the design of next-generation spin–ferroelectric devices.\nWe are confident that the revised manuscript meets the high standards of Nature\nCommunications in terms of broad interest, clarity, and potential impact across the condensed\nmatter physics and materials science communities.\nReviewer #2: Beyond these comments, I also find the duplication between the tight-binding and\ndensity-functional-theory approaches somewhat puzzling. Wouldn’t it be simpler to present a\nsingle test case based solely on DFT?\nReply: We appreciate the reviewer’s comment. In response, we have moved the tight-binding\nmodel section to the Supplementary Information to avoid redundancy and ensure a more\nstreamlined presentation. This allows the focus to remain on the density-functional theory\napproach, as suggested. (Results, Section 1, Penultimate Paragraph)\nWe further obtained numerical results from the tight-binding model constructed under the 2/m\nLaue group (Fig. S1), which are in perfect agreement with the aforementioned group-theoretical\nclassification. This further validates the impact of different ferroelectric switching symmetries on\nthe three characteristic types of spin bands in alternating magnetic materials.\nReviewer #2: I list below a few additional remarks that the authors may find relevant:\n(I) Please clarify the antiferromagnetic orders used in the calculations.\nReply: We thank the reviewer for this comment. We have clearly presented the distinction between\nthe two different antiferromagnetic orders used in the calculations to ensure clarity. (Results,\nSection 2, Paragraph 3, Lines 9-10; Supplementary Information, Part 5)\nThe detailed structure mapping process including magnetic ordering is presented in Fig. S5.\nFig. S5. (a) Top and side views of the L and L antiferromagnetic ordering configurations in the\n1 2\nstate. Red and blue represent opposite spins. The inset on the right highlights the spin positions,\n+\n𝑷\ncl1early illustrating the spin and stacking order. The out-of-plane polarization direction is indicated\nby a black arrow, while the in-plane polarization direction aligns along the x-axis and is parallel\nto the mirror plane (cyan plane).\nReviewer #2: (II) The units used for the spontaneous polarization are unusual and likely incorrect\n(µC·cm-2 are standard).\nReply: We thank the reviewer for the comment. Generally, µC·cm-2 is used to describe the\npolarization of 3D materials, representing the electric dipole moment (C·m) per unit volume (m3).\nHowever, for the 2D materials discussed in the manuscript, it is more common to use pC/m, which\nrepresents the electric dipole moment (C·m) per unit area (m2). This distinction reflects the\nstandard conventions for describing polarization in 3D and 2D systems, respectively. For example,\nthe same units are used in the description of Figure 2 in the article PHYSICAL REVIEW\nLETTERS 133, 126801 (2024), which also investigates sliding ferroelectrics.\nReviewer #2: (III) Energy barriers are discussed in meV per formula unit, whereas Fig. 3(c) uses\na different normalization.\nReply: We thank the reviewer for pointing this out. We have now standardized the units and\npresented all energy barriers consistently in meV per formula unit (meV/f.u.) throughout the\nmanuscript, including Figure 2c.\nReviewer #2: (IV) Sketches for the different pathways are highly recommended, as these are not\neasy to infer from Fig. 3(b) alone.\nReply: We have included detailed sketches of the polarization reversal pathways in the\nSupplementary Information to clearly illustrate the specific processes involved in the polarization\nswitching. (Results, Section 2, Paragraph 2, Lines 18-20; Results, Section 2, Paragraph 3, Lines 9-\n10; Supplementary Information, Part 4 and 5)\nThe theory of bilayer stacking ferroelectricity (BSF) provides a method for selectively choosing\nsliding paths using an external electric field,33 allowing for controllable polarization switching, as\nshown in Fig. S4.\nThe detailed structure mapping process including magnetic ordering is presented in Fig. S5.\nPart 4. Controllable switching of the six states achieved by the external electric field.\nSince each ferroelectric ground state corresponds to three distinct translational directions, it\nis not possible to achieve stable switching between all six states using only an out-of-plane electric\nfield. We demonstrate the controllable polarization switching method introduced by the bilayer\nstacking ferroelectricity (BSF) theory through an external electric field using the state. In the\n+\ncase of path 1, where only out-of-plane polarization switching is involved, th\n𝑷\ne1 transition is\nachieved by applying an out-of-plane electric field while keeping the in-plane electric field fixed\nalong the direction of in-plane polarization, as shown in Fig. S4a. For paths 2 and 3, which involve\nthe transition between in-plane and out-of-plane polarization, the coupling of these two\npolarization components is exploited. In this approach, an in-plane electric field is applied\nperpendicular to the switching path, aligning with the direction of in-plane polarization change\n(the [ ] direction for path 2 or the [2 0] direction for path 3). This approach reduces the\nintera\n2\nct 1̅io\n0\nn energy in the form of −E·P, thus\n1\nproviding the optimal switching pathway, as shown in\nFig. S4b-c.\nFig. S4. Switching paths selected by different external electric fields, enabling the switching of six\npolarization states. The simplified schematic of the bilayer stacking uses black spheres to represent\nMn atoms, highlighting the interlayer configuration. Panels (a), (b), and (c) represent paths 1, 2,\nand 3, respectively. The green arrows represent the direction of the applied electric field, where\nfor path 1, a combination of in-plane and out-of-plane fields is required, while only an in-plane\nelectric field is needed for paths 2 and 3. The red arrows indicate the direction of ionic\ndisplacement required to complete the polarization switching. The cyan plane illustrates the\ncrystal’s mirror symmetry, which changes depending on the sliding path.\nPart 5. Spin Structure Mapping for Polarization Switching.\nFigure S5a shows the magnetic ordering configurations of L and L , both of which exhibit\n1 2\nthe same intralayer Néel-type magnetic ordering, but with different interlayer magnetic couplings.\nFor clarity in illustrating the magnetic ordering and related symmetry operations, we simplify\nthese configurations into stacking models that only include the magnetic sites. Based on the sliding\npaths shown in Fig. S4, the sliding direction of path 1 is aligned with the mirror plane, causing the\ntop layer to slide relative to the bottom layer in-plane, thus achieving electric-field controlled\npolarization switching. For the L magnetic ordering, by applying the [E||M ] mapping to the\n1 z\n+\nstate, we obtain a structure with the same position and magnetic ordering as the sta\n𝑷\nte1.\n−\nHowever, for the L\n2\nmagnetic ordering, the [E||M\nz\n] mapping only transforms the lattice\n𝑷\nst1ructure;\nan additional M operation is required to restore the original magnetic ordering, specifically an\ny\n[E||M M ] operation, as shown in Fig. S5b.\nz y\nNext, we consider paths 2 and 3, as shown in Fig. S5c, another state is generated from the\nsame polarization state by following three different sliding paths. These states have the same out-\nof-plane polarization but a 3/π angle between their in-plane polarizations. The stacking structure\nreveals that the three states can be mapped onto each other via the C symmetry. Therefore, the\n3z\npolarization switching achieved through these paths can be viewed as the result of applying an\nadditional [E||C ] operation to the structure from path 1, yielding [E||M C ] for the L magnetic\n3z z 3z 1\nordering and [E||M M C ] for the L magnetic ordering.\nz y 3z 2\nFig. S5. (a) Top and side views of the L and L antiferromagnetic ordering configurations in the\n1 2\nstate. Red and blue represent opposite spins. The inset on the right highlights the spin positions,\n+\n𝑷\ncl1early illustrating the spin and stacking order. The out-of-plane polarization direction is indicated\nby a black arrow, while the in-plane polarization direction aligns along the x-axis and is parallel\nto the mirror plane (cyan plane). (b) On the left side of the panel, the real-space sliding along path\n1 for the two magnetic orderings are shown, with the corresponding symmetry operations\ndisplayed on the right. (c) Symmetry relations for the three sliding paths. The black spheres\nrepresent the Mn sites, showing that the three sliding paths are related to each other through the\nC symmetry and are independent of the magnetic order. Therefore, the ferroelectric structures\n3z\nobtained by sliding along paths 2 and 3 can be combined with the structure from path 1 using the\nC symmetry.\n3z\nReviewer #3:\nIn the paper: A unified Symmetry Framework for Spin – Ferroelectric Coupling in Altermagnetic\nMultiferroics, Wei Sun and coauthors discuss a general framework of alter magnetic splitting in\nmultiferroic materials using spin space group symmetries. While the work is timely and interesting,\nI have a hard time following some of the points that the authors make, which means that for now,\nI don’t feel like this framework is a general as the authors claim. If the following points are clarified,\nthe paper can be reconsidered for publication:\nReviewer #3: (1) The relationship between the Laue group and the broken inversion symmetry\nneeds to be clarified. Of course one can use the formulation of the Laue group for the symmetry\nof the bands, the persistence of inversion symmetry in the Laue group makes the problem difficult\nto describe. A more clear introduction how to deal with the broken inversion symmetry and the\ncomparison to the crystallographic point groups is necessary.\nReply: We sincerely thank the reviewer for raising this important point regarding the relationship\nbetween the Laue group and broken inversion symmetry. We agree that a clear explanation is\nessential. Our response below aims to clarify the conceptual separation between real space crystal\nsymmetry and momentum space band symmetry, which lies at the heart of our analysis.\n1. Laue group describes momentum space band symmetry, not the real space crystal structure. The\nLaue group (the crystallographic point group augmented by inversion) is the appropriate\nsymmetry group for describing electronic bands in momentum space. A key feature of non-\nrelativistic collinear magnets—including altermagnets—is that their band structure preserves\ninversion symmetry in k space even when the real space crystal lacks inversion symmetry. This\nis a direct consequence of the symmetry, as stated in the manuscript: “\n[𝐶2̅ ||𝑇] 𝐸(𝑠,𝐤) =\n. This indicates that the band structure always preserves inversion symmetry I in\nmomentum space, even if the crystal lacks real-space inversion symmetry.”\n[𝐸||𝐼]𝐸(𝑠,𝐤)\n2. Real space ferroelectricity is described by the crystallographic point group. The broken inversion\nsymmetry associated with ferroelectric order is a real space property, described by the\ncrystallographic point group of the atomic arrangement. This real space symmetry breaking does\nnot conflict with the preservation of inversion in the Laue group, because the two operate in\ndifferent “spaces”: real space (atomic positions) vs. momentum space (electronic bands).\n3. The ferroelectric switching operation O is a mapping between two polarization states, not a\nFE\nsymmetry of either state. The operator O describes how the system transforms from one\nFE\nferroelectric state (P) to its opposite (P’). It is not required to be a symmetry element of the\ncrystal’s Laue group or point group. Considering the goal of our study, the effect of ferroelectric\nreversal in real space on the spin splitting in momentum space, we need to match the O with\nFE\nthe Laue group that describes momentum space in order to distinguish between the three\ndifferent symmetry couplings. Therefore, O can belong to the Laue group (O G), or it may\nFE FE\nlie outside of it (O G). This distinction is precisely what gives rise to our three coupling\nFE ∈\ntypes. Our analysis starts from the Laue group G of the bands, identifies the halving subgroup\n∉\nH and its complement AH, and then classifies O according to whether it belongs to H, AH, or\nFE\nneither.\nTo make this multi layered relationship more intuitive, we have added a schematic (Fig. R1) in the\nrevised manuscript, which illustrates how real space symmetry breaking (ferroelectricity),\nmomentum space band symmetry (Laue group), and the switching operator O interconnect. The\nFE\ndiagram helps clarify that O acts in real space but its effect on spin textures is classified via the\nFE\nmomentum space Laue group.\nIn summary, the “persistence of inversion in the Laue group” does not prevent us from describing\nbroken inversion in real space; rather, it provides the correct symmetry framework for the bands.\nThe switching operation O is treated as an external mapping whose relation to the Laue group\nFE\ndetermines the resulting spin texture reconstruction. We have refined the relevant passages in the\nmanuscript (Results, Section 1, Paragraph 3) to present this logic more explicitly.\nFig. R1. The relationship between Laue group, point group, and ferroelectric switching O .\nThe\nFE\nschematic illustrates how altermagnetic multiferroics (non-relativistic collinear magnets) exhibit\ninversion symmetry breaking in real space and the presence of inversion symmetry in momentum\nspace. Additionally, while the ferroelectric switching operation O occurs in real space, the\nFE\ncoupling effects we focus on take place in momentum space. These phenomena occur in different\ndimensions and are not contradictory.\nWe now consider the ferroelectric switching operation O acting in real space, represented\nFE\nas a symmetry operation = [E||O ], which reverses the ferroelectric polarization P while\nFE\npreserving the system’s ma\n𝑈\ng netic configuration, i.e., . Note that the operation\nis independent of its own symmetry; rather, it describe\n[\ns\n𝐸\na\n|| 𝑂\ntr𝐹a𝐸n\n]\ns\n𝑷\nfo\n=\nrm\n𝑷\nat\n’\nion that maps one system t\n𝑈\no\nanother with different polarization. During this process, the spins remain unchanged in real space,\nbut exhibit distinct responses in momentum space. By matching the real-space ferroelectric\nswitching operation with the Laue group in momentum space, three distinct types of effects on the\nspin-resolved band structure E(s, k) of an altermagnet are observed, as illustrated in Fig. 1.\nReviewer #3: (2) Finally, I am not convinced that the authors deal with the “polarisation switching”\ncorrectly. The authors define a polarisation switching operation OFE. The they discuss if the\nswitching operation is part of the the halving group H that maps the same spin onto each other or\nthe complementary group AH that maps sub lattices of different spins, or of neither. If a\nferroelectric, the switching operations always include inversion. While inversion is always part of\nthe Laue group, the authors claim further that beyond that, inversion is always part of the halving\ngroup H, meaning that altermagnetism and polarisation is decoupled. I do not think there is a good\nreason for this claim (see e.g ref 28, where it is demonstrated how flipping the polarisation flips\nthe altermagnetic splitting. \nReply: We appreciate the reviewer’s thoughtful comment. We acknowledge that some ferroelectric\nswitching operations, such as in materials like BaTiO and PbTiO , are indeed mediated by the\n3 3\ninversion operation I. However, this does not mean that ferroelectric switching is always\nexclusively represented by inversion. From the perspective of symmetry, the inversion operation\nbelongs to the overall symmetry of the crystal, while ferroelectric switching is a local process\nrelated to ionic displacements associated with polarization. Therefore, inversion is not necessarily\ninvolved in the process. For example, in the case of sliding ferroelectrics, the switching operation\nis mediated by the M symmetry rather than inversion. As shown in Fig. R2, we provide examples\nz\nof the mapping of the sliding ferroelectric structure under M and I, as well as the real ferroelectric\nz\nswitch structure.\nFig. R2. Sliding ferroelectric structure through (a) real ionic displacements, and the structural\nmapping under (b) I and (c) M symmetry operations. The red arrows indicate the direction of ionic\nz\ndisplacements. This clearly demonstrates that the sliding ferroelectric switching is accomplished\nthrough M , rather than I.\nz\nSpecifically, within the framework of the spin space groups discussed in the manuscript, the\nferroelectric switching operation not only requires the lattice sites to map to corresponding\npositions with reversed polarization but also requires the corresponding magnetic sites to be\nrestored. This leads to different mapping operations for the two magnetic orders (L and L )\n1 2\ndiscussed in the manuscript. For example, for the L magnetic order, the mapping is carried out by\n1\n[E||M ], whereas for the L magnetic order, it requires [E||M M ]. Figure S5 further elaborates on\nz 2 z y\nhow these mappings are performed. (Results, Section 2, Paragraph 3, Lines 9-10; Supplementary\nInformation, Part 5)\nThe detailed structure mapping process including magnetic ordering is presented in Fig. S5.\nPart 5. Spin Structure Mapping for Polarization Switching.\nFigure S5a shows the magnetic ordering configurations of L and L , both of which exhibit\n1 2\nthe same intralayer Néel-type magnetic ordering, but with different interlayer magnetic couplings.\nFor clarity in illustrating the magnetic ordering and related symmetry operations, we simplify\nthese configurations into stacking models that only include the magnetic sites. Based on the sliding\npaths shown in Fig. S4, the sliding direction of path 1 is aligned with the mirror plane, causing the\ntop layer to slide relative to the bottom layer in-plane, thus achieving electric-field controlled\npolarization switching. For the L magnetic ordering, by applying the [E||M ] mapping to the\n1 z\n+\nstate, we obtain a structure with the same position and magnetic ordering as the sta\n𝑷\nte1.\n−\nHowever, for the L\n2\nmagnetic ordering, the [E||M\nz\n] mapping only transforms the lattice\n𝑷\nst1ructure;\nan additional M operation is required to restore the original magnetic ordering, specifically an\ny\n[E||M M ] operation, as shown in Fig. S5b.\nz y\nNext, we consider paths 2 and 3, as shown in Fig. S5c, another state is generated from the\nsame polarization state by following three different sliding paths. These states have the same out-\nof-plane polarization but a 3/π angle between their in-plane polarizations. The stacking structure\nreveals that the three states can be mapped onto each other via the C symmetry. Therefore, the\n3z\npolarization switching achieved through these paths can be viewed as the result of applying an\nadditional [E||C ] operation to the structure from path 1, yielding [E||M C ] for the L magnetic\n3z z 3z 1\nordering and [E||M M C ] for the L magnetic ordering.\nz y 3z 2\nFig. S5. (a) Top and side views of the L and L antiferromagnetic ordering configurations in the\n1 2\nstate. Red and blue represent opposite spins. The inset on the right highlights the spin positions,\n+\n𝑷\ncl1early illustrating the spin and stacking order. The out-of-plane polarization direction is indicated\nby a black arrow, while the in-plane polarization direction aligns along the x-axis and is parallel\nto the mirror plane (cyan plane). (b) On the left side of the panel, the real-space sliding along path\n1 for the two magnetic orderings are shown, with the corresponding symmetry operations\ndisplayed on the right. (c) Symmetry relations for the three sliding paths. The black spheres\nrepresent the Mn sites, showing that the three sliding paths are related to each other through the\nC symmetry and are independent of the magnetic order. Therefore, the ferroelectric structures\n3z\nobtained by sliding along paths 2 and 3 can be combined with the structure from path 1 using the\nC symmetry.\n3z\nRegarding the claim in reference 28, where it is shown that ferroelectric switching inverts spin\nsplitting, this actually corresponds to the Type-II coupling mentioned in our manuscript. This\ncoupling requires the ferroelectric switching operation to belong to the AH group rather than\ninversion. Furthermore, in another paper by the same authors of reference 28, PHYSICAL\nREVIEW X 12, 031042 (2022), they describe the nontrivial spin Laue group in altermagnetic\nmaterials, where inversion symmetry is explicitly part of the halving group H. This is because in\nthe definition of altermagnetism, sublattices with opposite spins can only be related by rotations\nor mirror symmetries in the AH group. This means that inversion is not included in the AH group,\nbut rather in the H group. We have cited and explained this point in the results discussion section\nof the manuscript:\nIn altermagnets, however, specific spatial operations exchange opposite-spin sublattices,\ncorresponding to a nontrivial spin Laue group7:\nHere, is a h\n𝑅\na𝐴lv𝑀in\n=\ng s\n[\n𝑢\n𝐸\nb\n|\ng\n|𝑯\nro\n]\nu\n+\n[𝐶\nf 2th\n||\ne\n𝐴\nc\n]\nr\n[\ny\n𝐸\ns\n|\nt\n|\nH\nll\n]\nog\n=\n[\n𝐸\nh\n|\ni\n|c\n𝑯\nL\n]\na\n+\n[ 𝐶\ng2r\n||\no\n|c\n𝑮\np\n−\n, re\n𝑯\np\n]\nresenting operati(o\n2\nn)s\nthat preserve the same spin sublattice. The coset G − H = AH is generated by an element A, which\nH ⊂ G\nis a real-space rotation or mirroring (excluding pure inversion or translation; hence, the inversion\noperation in G is contained within H).\nAdditionally, a similar situation in BiFeO is described in PHYSICAL REVIEW B 112, 104411\n3\n(2025), where ferroelectric switching must occur alongside octahedral rotations to achieve a switch\nin spin splitting. This is a clear example of the point we make in our manuscript, as a polarization\noperation accompanied only by inversion and non-octahedral rotations does not change spin\npolarization. In contrast, a polarization switch with octahedral rotations cannot map the lattice\nstructure of the opposite polarization state under inversion while preserving the system’s magnetic\nordering. Instead, it corresponding to O = , induces the necessary spin polarization\nFE\nswitching. We have added this discussion in the\n𝐶\nre2v[1̅i1se0]d manuscript. (Results, Section 3)\n3. Proof of Universality Using BiFeO as a 3D Exemplar\nwe extend our analysis to a classic 3D multiferroic material, G-type antiferromagnetic BiFeO\n3\nmultiferroics with R3c space group, the altermagnetism was confirmed recently.32 From a\nsymmetry perspective, the corresponding crystallographic Laue group of this system is 3m, and its\ncomplete set of symmetry operations is G = {E, I, C , , S , , , ,\n3[111] 3[111]\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅]\n, , , }. The halving subgroup H = {E, I, C , , S , }\n3[111] 3[111]\n2 2\n𝐶pr 2 e [1 s 0 e 1̅ r ] ves𝑀 e [ a 1̅ c 10 h ] sp𝑀in [ 0 s 1 u 1̅ b ] la𝑀tti [ c 1 e 0 . 1̅ T ] herefore, the system is described by the alt𝐶er3m [1a11g ] netic grou𝑆p3 [ R111]\nAM\n= [E||{E, I, C , , S , }] + [C ||{ , , , , ,\n3[111] 3[111] 2\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅] 𝑀 [1̅10] 𝑀[011̅]\n}], representing a g-wave altermagnetic system, as shown in Fig. 5a-b.\n𝑀[10\nI\n1̅\nn\n]\nBiFeO , the switching of ferroelectric polarization can occur through two possible\n3\nmicroscopic paths. The path 1 is mainly realized by the displacement of Bi3+ ions along the [111]\ndirection, with the overall effect equivalent to an O = I inversion operation, i.e., [E||I]P = P’\nFE\n(Fig. 5b). The path 2 involves the displacement of Bi3+ ions accompanied by the rotation of the\noxygen octahedra. In this case, inversion symmetry cannot map the lattice structure while\npreserving the system’s magnetic ordering. Instead, path 2 must map the system in a O =\nFE\nmanner, i.e., [E|| ]P = P’ (Fig. 5b). According to Eq. 9, the path 1, where O\nFE\n, do\n𝐶\ne2s[ 1̅n1o0]t\nalter the spin spl\n𝐶\nit2ti[n1̅1g0 d]uring polarization reversal, and is classified as Type I (de\n∈\nc\nH\noupled). The\npath 2, where O , maps the opposite spin channels and is classified as Type II (pseudo-\nFE\ntime-reversal coupling). The band structure calculations shown in Fig. 5 clearly confirm these two\n∈ AH\ndistinct responses.\nThis analysis not only provides an intuitive symmetry explanation for the recently proposed\nidea that spin splitting in BiFeO requires oxygen octahedral rotation for electric control, but also\n3\nclarifies the universal applicability of this symmetry mechanism in various multiferroic materials.\nBy identifying the symmetry of the ferroelectric switching operation, one can predict and classify\nits magnetoelectric coupling functionality, independent of the specific system under study.\nFig. 5. The crystal structure of BiFeO and its corresponding Brillouin zone of the rhombohedral\n3\nlattice. Red and blue represent Fe ions with opposite spins, with k (k’) located at the midpoint\nbetween Γ and P (P ’) in the Brillouin zone. (b) Spin-splitting band structure with alternating\n2 2\nmagnetism. (c) and (d) represent two ferroelectric switching paths, where the spin band structure\nremains unchanged under O = I, but reverses under O = .\nFE FE\n𝐶2[1̅10]\nReviewer #3: Further “polarisation changing” operations could be a rotation into a different\npolarisation domain. However, unlike inversion, any other “polarisation switching operation”\ncannot be part of the Laue groups (neither H or AH), since these effectively switch the system into\na new domain state, which in fact means that you apply the symmetry elements that were lost in\nthe ferroelectric transition, thus these operations are not anymore part of the crystals symmetry\ngroup. (Inversion here is a special element, because it is recovered in k-space in the absence of\nspin orbit. Thus with this logic, type II (the “switchable” alter magnetic ferroelectric wouldn’t exist,\nwhich we know already not to be true from ref 28. The authors need to clarify all these points in\norder to provide a “general theory”.\nReply: We thank the reviewer for their insightful comments. We would like to reiterate that, as\nwith sliding ferroelectrics, polarization switching achieved through partial ionic displacements\ndoes not necessarily correspond to the overall spatial inversion operation of the crystal. In other\nwords, non-spatial inversion operations, such as rotations and mirror, can also serve as mapping\noperations for ferroelectric polarization switching, and are unrelated to polarization domains.\nAdditionally, the theory presented in our manuscript does not contradict reference 28, which does\nnot suggest that ferroelectric switching is mediated by spatial inversion. Specifically, within the\nframework described in our manuscript, the operation not only requires mapping the lattice sites\nto positions with opposite ferroelectric polarization but also needs to preserve the spin structure of\nthe system. This point has been further clarified in the manuscript. (Results, Section 1, Paragraph\n3, Lines 1-5)\nWe now consider the ferroelectric switching operation O acting in real space, represented as a\nFE\nsymmetry operation = [E||O ], which reverses the ferroelectric polarization P while\nFE\npreserving the system’s\n𝑈\nmagnetic configuration, i.e., . Note that the operation\nis independent of its own symmetry; rather, it describe\n[𝐸\ns a\n|| 𝑂\ntr𝐹a𝐸n\n]\ns\n𝑷\nfo\n=\nrm\n𝑷\nat\n’\nion that maps one system t\n𝑈\no\nanother with different polarization.\nReviewer #3: Further, there is some smaller points:\n- “Furthermore, collinear magnets possess a fundamental symmetry , which implies [..].\nThis is only true in absence of spin orbit” [𝐶2̅ ||𝑇]\nReply: We thank the reviewer for pointing out this important detail. Since altermagnetism is\ndescribed within the framework of non-relativistic spin space groups, the manuscript is analyzed\nwithin a non-relativistic framework. We have added the relevant description to clarify this point.\n(Results, Section 1, Paragraph 1, Lines 2; Results, Section 1, Paragraph 1, Lines 7; Results, Section\n1, Paragraph 2, Lines 1)\nFor any non-relativistic spin space operation …\nIn non-relativistic altermagnets, however, specific spatial operations exchange opposite-spin\nsublattices, …\nFurthermore, non-relativistic collinear magnets possess a fundamental symmetry , …\n[𝐶2̅ ||𝑇]\nReviewer #3: - In order to help us understand the relationship between the Laue group and the\nCrystallographic point group they should put both in their example of MnPS3, and they should\nin addition tell us also the paraelectric parent phase, so we can compare.\nReply: We thank the reviewer for the suggestion. We have provided the crystallographic point\ngroup and Laue group for MnPS3, along with its paraelectric phase, to facilitate comparison.\n(Results, Section 2, Paragraph 2, Lines 3-4 and 9-14; Supplementary Information, Part 3)\nwe construct the bilayer by applying the mapping {M |τ } together with an out-of-plane translation\nz o\nτ , resulting in a crystallographic point group of m and a crystallographic Laue group of 2/m, with the full\nz\nset of symmetry operations G = {E, C , I, M }. The halving subgroup H = {E, I} preserves each\n2y y\nspin sublattice.\nDFT total-energy scans identify a global maximum for the paraelectric phase at τ (0,0), along\no\nwith six degenerate global minima for the ferroelectric phase at τ {(va,0), (0,vb), (va,vb) | v =\no =\n±1/3}, as shown in Fig. 2b, leading to uncompensated interfacial charge transfer and a combined\n∈\npolarization of 0.92 μC/m (in-plane) and 0.26 μC/m (out-of-plane), as demonstrated by the charge\ndifference between the paraelectric and ferroelectric phases in Fig. S3.\nPart 3. Differential charge distribution of MnPS bilayer.\n3\ninversion symmetry due to the stacking, resulting in a combination of in-plane and out-of-plane\npolarization. The polarization direction is denoted as , where the superscript ± denotes the\n±\nsign of the out-of-plane component, the subscript n =\n𝑷\n1𝑛, 2, 3 labels the in-plane polarization\ndirections [100], [010], [110]. For comparison, we also constructed the paraelectric phase P at\n0\nτ = (0,0), where the charge redistribution maintains mirror symmetry in the out-of-plane direction,\no\nwhile the in-plane charge transfer follows the C rotational symmetry, thereby prohibiting any\n3z\nform of polarization. The out-of-plane ferroelectricity of MnPS is demonstrated using the\n3\n±\nstates at τ {(a/3, b/3), (-a/3, -b/3)}, where charge accumulates at the interface S atoms a𝑷n3d\no\ndepletes around the P atoms, generating out-of-plane polarization that can be reversed via sliding,\n∈\nas shown in Fig. S3a. Additionally, the three in-plane polarization directions are demonstrated by\nstates at τ {(-a/3, -b/3), (a/3, 0), (0, b/3)}, where these directions are controlled by the\no\n+\n𝑷\nsli𝑛ding paths. T\n∈\nhe different paths determine the final orientation of the mirror symmetry. Due to\nthe presence of mirror symmetry, any polarization component perpendicular to the mirror plane is\nforbidden, meaning the polarization is controlled by the mirror plane direction and can only occur\nalong its orientation, as shown in Fig. S3b.\nFig. S3. Differential charge distribution of MnPS bilayer. (a) The side view clearly shows electron\n3\naccumulation at the interfacial S atoms and depletion around the P atoms, establishing a vertical\nout-of-plane dipole moment. (b) In the in-plane direction, mirror symmetry M enforces a symmetric\ndistribution on both sides, thereby aligning the polarization direction with the mirror plane. The\nblack solid line represents the unit cell lattice, and the blue dashed line highlights the crystal's\nmirror symmetry.\nReviewer #3: In addition to this 2D material the authors should also provide an example of a bulk\nmultiferroic with alter magnetic phase such as BFO or one of the systems in ref 28, so we can\nverify their model, and provide the proper analysis in terms of the Laue groups.\nReply: We thank the reviewer for this suggestion. We have added BiFeO as an example of a 3D\n3\nmaterial to verify the applicability of the proposed theory and provide a proper analysis in terms\nof the Laue groups. (Results, Section 3)\n3. Proof of Universality Using BiFeO as a 3D Exemplar\nwe extend our analysis to a classic 3D multiferroic material, G-type antiferromagnetic BiFeO\n3\nmultiferroics with R3c space group, the altermagnetism was confirmed recently.32 From a\nsymmetry perspective, the corresponding crystallographic Laue group of this system is 3m, and its\ncomplete set of symmetry operations is G = {E, I, C , , S , , , ,\n3[111] 3[111]\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅]\n, , , }. The halving subgroup H = {E, I, C , , S , }\n3[111] 3[111]\n2 2\n𝐶pr 2 e [1 s 0 e 1̅ r ] ves𝑀 e [ a 1̅ c 10 h ] sp𝑀in [ 0 s 1 u 1̅ b ] la𝑀tti [ c 1 e 0 . 1̅ T ] herefore, the system is described by the alt𝐶er3m [1a11g ] netic grou𝑆p3 [ R111]\nAM\n= [E||{E, I, C , , S , }] + [C ||{ , , , , ,\n3[111] 3[111] 2\n2 2\n𝐶3[111] 𝑆3[111] 𝐶2[1̅10] 𝐶2[011̅] 𝐶2[101̅] 𝑀 [1̅10] 𝑀[011̅]\n}], representing a g-wave altermagnetic system, as shown in Fig. 5a-b.\n𝑀[10\nI\n1̅\nn\n]\nBiFeO , the switching of ferroelectric polarization can occur through two possible\n3\nmicroscopic paths. The path 1 is mainly realized by the displacement of Bi3+ ions along the [111]\ndirection, with the overall effect equivalent to an O = I inversion operation, i.e., [E||I]P = P’\nFE\n(Fig. 5b). The path 2 involves the displacement of Bi3+ ions accompanied by the rotation of the\noxygen octahedra. In this case, inversion symmetry cannot map the lattice structure while\npreserving the system’s magnetic ordering. Instead, path 2 must map the system in a O =\nFE\nmanner, i.e., [E|| ]P = P’ (Fig. 5b). According to Eq. 9, the path 1, where O\nFE\n, do\n𝐶\ne2s[ 1̅n1o0]t\n𝐶2[1̅10]\n∈ H\nalter the spin splitting during polarization reversal, and is classified as Type I (decoupled). The\npath 2, where O , maps the opposite spin channels and is classified as Type II (pseudo-\nFE\ntime-reversal coupling). The band structure calculations shown in Fig. 5 clearly confirm these two\n∈ AH\ndistinct responses.\nThis analysis not only provides an intuitive symmetry explanation for the recently proposed\nidea that spin splitting in BiFeO requires oxygen octahedral rotation for electric control, but also\n3\nclarifies the universal applicability of this symmetry mechanism in various multiferroic materials.\nBy identifying the symmetry of the ferroelectric switching operation, one can predict and classify\nits magnetoelectric coupling functionality, independent of the specific system under study.\nFig. 5. The crystal structure of BiFeO and its corresponding Brillouin zone of the rhombohedral\n3\nlattice. Red and blue represent Fe ions with opposite spins, with k (k’) located at the midpoint\nbetween Γ and P (P ’) in the Brillouin zone. (b) Spin-splitting band structure with alternating\n2 2\nmagnetism. (c) and (d) represent two ferroelectric switching paths, where the spin band structure\nremains unchanged under O = I, but reverses under O = .\nFE FE\n𝐶2[1̅10]\nWe sincerely thank the reviewers for their valuable feedback and positive evaluation of our work,\nwhich have played a crucial role in enhancing the quality of the manuscript. Below, we provide a\ndetailed point-by-point response to the reviewers’ comments.\nReviewer #1 (Remarks to the Author): Authors have responded in sufficient details and included\nmost of the suggested edits.\nReply: We sincerely thank Reviewer #1 for their positive feedback and for recognizing the detailed\nresponses and revisions we have made. We are glad that the revisions have addressed the majority\nof the suggested edits.\nReviewer #2 (Remarks to the Author): The authors have significantly improved the manuscript\nand have adequately addressed my previous concerns. Overall, the paper represents a substantial\nadvance for the spintronics community and is likely to be of interest to a broad audience. The\nimprovements go beyond my initial expectations, in particular through the inclusion of an\nadditional test-case material, the widely studied multiferroic BiFeO3. This addition helps to\nbroaden the scope of the manuscript and to connect the proposed framework with well-known\noxide systems. Regarding BiFeO3, it would be important to clarify the physical mechanism\nresponsible for the modification of the spin texture. In bulk BiFeO3, the crystal structure arises\nfrom the condensation of two primary order parameters: a polar distortion along the pseudocubic\n111 direction and an antiphase oxygen octahedral rotation described by the R4+ irreducible\nrepresentation (a⁻a⁻a⁻ in Glazer notation). From a symmetry and energetic standpoint, it is the\n⟨ ⟩\noctahedral rotation that governs the magnetic texture, whereas the ferroelectric polarization itself\nis not the relevant order parameter controlling this effect. In this context, the discussion of BiFeO3\nwould benefit from being framed more explicitly in terms of the role of the octahedral rotation in\ndetermining the spin texture, rather than suggesting a direct causal link with polarization reversal.\nReference 32 is fully consistent with this interpretation, as it highlights the impact of the R4+\nrotational mode on the magnetic structure. Clarifying this point would improve the accuracy of the\ndiscussion and help avoid potential confusion for the reader. While it is true that Nature 516, 371\n(2014) proposed that polarization reversal may, under specific conditions, be accompanied by a\nreversal of the octahedral rotation—potentially enabling electrical control of spin textures—this\nscenario remains subtle and should be discussed with appropriate caution. Aside from this\nclarification, the model proposed by the authors appears sound, and the manuscript is now much\nclearer and stronger than in its initial version. Should the authors revise the discussion on BiFeO3\naccordingly, I would be happy to recommend the manuscript for publication.\nReply: We sincerely thank Reviewer #2 for their positive evaluation of the revisions and for their\ninsightful feedback. We are pleased that the manuscript has improved and now meets the\nreviewer’s expectations.\nRegarding the clarification of the physical mechanism responsible for the modification of the spin\ntexture in BiFeO , we deeply appreciate the reviewer’s detailed comments. We now explicitly\n3\nacknowledge that the octahedral rotation, rather than the polarization reversal, governs the\nmagnetic texture in bulk BiFeO . To address this, we have revised the manuscript to emphasize\n3\nthe role of the rotational mode in determining the spin texture, in alignment with Reference\n+\n32. Furthermore\n𝑅\n, 4we have reworded the discussion to avoid suggesting a direct causal link between\npolarization reversal and spin texture modification, while framing the potential connection with\noctahedral rotation more cautiously. (Results, Section 3)\nWe believe these revisions enhance the clarity and accuracy of the manuscript, and we are grateful\nfor the reviewer’s constructive input.\nIn bulk BiFeO , the R3c crystal structure arises from by the condensation of two primary\n3\norder parameters: a polar distortion along the [111\n𝑃𝑚\nd 3̅i\n𝑚\nrection and a rotation of the oxygen\noctahedra along the [111 direction, described by the irreducible representation. For\n]\n+\npolarization switching ach\n]\nieved by independent displacemen\n𝑅\nt4 of Bi3+ ions (referred to as path 1),\nthe overall effect is equivalent to an inversion operation O = I, i.e., [E||I]P = P’\nFE\n(Fig. 5b). However, in certain conditions, polarization switching may be accompanied by the rotation of the\noxygen octahedra (referred to as path 2).34 In this case, inversion symmetry cannot map the lattice\nstructure while preserving the system’s magnetic ordering; instead, path 2 affects the system in an\nO = manner, i.e., [E|| ]P = P’ (Fig. 5b).\nFE\n𝐶2[1̅10] 𝐶2[1̅10]\nThis analysis not only emphasizes that the spin splitting control in BiFeO is directly related to\n3\noctahedral rotation rather than polarization switching, providing an intuitive symmetry\nperspective for electric control of spin splitting in BiFeO , …\nfor the reviewer’s constructive input."}, {"round": 2, "reviewer_comments": {"reviewer_1": "Authors have responded in sufficient details and included most of the suggested edits.", "reviewer_2": "The authors have significantly improved the manuscript and have adequately addressed my previous concerns. Overall, the paper represents a substantial advance for the spintronics community and is likely to be of interest to a broad audience. The improvements go beyond my initial expectations, in particular through the inclusion of an additional test-case material, the widely studied multiferroic BiFeO3. This addition helps to broaden the scope of the manuscript and to connect the proposed framework with well-known oxide systems.\nRegarding BiFeO3, it would be important to clarify the physical mechanism responsible for the modification of the spin texture. In bulk BiFeO3, the crystal structure arises from the condensation of two primary order parameters: a polar distortion along the pseudocubic ⟨111⟩ direction and an antiphase oxygen octahedral rotation described by the R4+ irreducible representation (a⁻a⁻a⁻ in Glazer notation). From a symmetry and energetic standpoint, it is the octahedral rotation that governs the magnetic texture, whereas the ferroelectric polarization itself is not the relevant order parameter controlling this effect.\nIn this context, the discussion of BiFeO3 would benefit from being framed more explicitly in terms of the role of the octahedral rotation in determining the spin texture, rather than suggesting a direct causal link with polarization reversal. Reference 32 is fully consistent with this interpretation, as it highlights the impact of the R4+ rotational mode on the magnetic structure.\nClarifying this point would improve the accuracy of the discussion and help avoid potential confusion for the reader. While it is true that Nature 516, 371 (2014) proposed that polarization reversal may, under specific conditions, be accompanied by a reversal of the octahedral rotation—potentially enabling electrical control of spin textures—this scenario remains subtle and should be discussed with appropriate caution.\nAside from this clarification, the model proposed by the authors appears sound, and the manuscript is now much clearer and stronger than in its initial version. Should the authors revise the discussion on BiFeO3 accordingly, I would be happy to recommend the manuscript for publication."}, "author_response": ""}, {"round": 3, "reviewer_comments": {"reviewer_2": "Authors have appropriately modified the manuscript. I recommand publication of the study in Nature Communications."}, "author_response": ""}]

Decision letter