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Ferroelectric Altermagnetic Chern Insulator in magnetic field: electrical control of the Chern number
Meysam Bagheri Tagani ID 1, 2, ∗ and Carmine Autieri ID 1, † 1 International Research Centre Magtop, Institute of Physics,
arXiv:2606.11862v1 [cond-mat.mes-hall] 10 Jun 2026
Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland 2 Department of Physics, University of Guilan, P. O. Box 41335-1914, Rasht, Iran
Department of Physics, University of Guilan, P. O. Box 41335-1914, Rasht, Iran (Dated: June 11, 2026) The quantum anomalous Hall effect in altermagnets is difficult to realize because spin-up and spin-down states remain degenerate at the Γ point in the nonrelativistic limit. We start from the Bernevig–Hughes–Zhang model to incorporate nontrivial band topology. We demonstrate that the combined effects of an external magnetic field, spin canting, and ferroelectric orbital hybridization lift the degeneracy at the Γ point, enabling electric-field control of the Chern number. A minimal two-dimensional d -wave altermagnetic model with band inversion then realizes a ferroelectrically tunable Chern insulator with spontaneous spin canting. The ferroelectric polarization controls the topological phase and the orbital angular momentum, enabling a rich phase diagram with C = ± 1 and C = ± 2 through a Berry-curvature reorganization linked to the spin canting response and ferroelectricity. Our results establish a symmetry-consistent route to electrically tunable Chern insulating phases in altermagnetic materials, opening opportunities for low-power topological and orbitronic devices.
I. INTRODUCTION The realization of dissipationless edge transport re- mains a central objective in condensed matter physics, which can be useful for metrology applications 1 . Since the theoretical proposal and experimental observation of the quantum anomalous Hall effect (QAHE), Chern in- sulating phases have been predominantly associated with ferromagnets 2 –7 . However, the QAHE in ferromagnets was observed only at very low temperatures so far 8 . The altermagnetism has emerged as a magnetic phase characterized by zero net magnetization in the non- relativistic limit and momentum-dependent spin split- ting protected by rotational symmetry. 9 –18 Spi-orbital ef- fects as relativistic spin-momentum locking 1 9 and orbital magnetization 2 0 has been shown to play an important role in altermagnets, where the relevance of the orbital angular momentum was supported by recent experimen- tal observations 2 1 . The altermagnetism has stimulated intense interest in their transport and topological re- sponses. In particular, altermagnets provide a promising route toward realizing Chern insulating phases without large macroscopic magnetization. Nevertheless, a fun- damental obstacle persists. In altermagnets, the spin-up and spin-down states remain degenerate at the Γ point in the nonrelativistic limit. This degeneracy obstructs the opening of a full topological gap required for the QAHE. Previous strategies to overcome this limitation relied on explicitly breaking rotational symmetry, thereby render- ing magnetic sublattices inequivalent and driving the sys- tem into a ferrimagnetic phase. The rotational symmetry was broken by uniaxial strain 2 2–24 , by ferrimagnetism on the surface 2 5,26 , by stacking fault 2 7 , or by an external magnetic field 2 8,29 . In other materials with zero magne- tization, the Chern insulating phase has been predicted in two-dimensional magnets with mirror symmetry 3 0 and in noncollinear magnets on a kagome lattice 3 1 . The ma- nipulation of the quantum anomalous Hall effect via fer- roelectricity has been widely investigated primarily in MnBi 2 T e 4 , where time-reversal symmetry can be broken in different ways 2 7,32–35 , whereas the corresponding phe- nomenon in altermagnets has received considerably less attention. Among other topological effects in altermagnets or in system based on altermagnets, we can mention the quan- tum spin Hall 3 6,37 , the quantization of the spin circu- lar photogalvanic effect 3 8 , the topological piezomagnetic effect 3 9 , the layer Hall effect 4 0 , axion insulator 4 1 , almost half-quantized planar Hall effects 4 2,43 , and higher-order topology with corner states 4 4,45 . In this work, we demonstrate that the interplay among the external magnetic field, ferroelectricity, and spin canting can lift the Γ-point degeneracy, yielding a rich phase diagram with Chern insulating phases. Specifi- cally, we construct and analyze a two-dimensional d -wave altermagnetic model coupled to a ferroelectric order pa- rameter. The resulting phase is a ferroelectric Chern in- sulator that exhibits a fully gapped bulk spectrum and quantized Hall conductivity in the absence of any exter- nal magnetic field. Crucially, the ferroelectric polariza- tion acts as an electrically tunable control parameter be- tween different topological phases C = ± 1 and C = ± 2; the system therefore realizes an electrically switchable topological phase based on altermagnets. II. MODEL AND SYMMETRY ANALYSIS To capture the essential interplay among altermag- netism, ferroelectricity, and weak ferromagnetism, we construct the minimal symmetry-allowed Hamiltonian for a two-orbital system on a square lattice. The model acts using the Pauli matrices in the tensor-product space of an orbital (or sublattice) pseudospin τ and the physi-
2 cal spin s , and is written as H ( k ) = H 0 ( k ) + H A M ( k ) + H C AN + H F E + H R ( k ) . (1) Each term is constrained by the crystalline symmetries of the square lattice and by the microscopic mechanisms relevant to ferroelectric altermagnets. The baseline Hamiltonian H 0 ( k ) = [ M 0 − 2 B (2 − cos k x − cos k y )] τ z ⊗ s 0 + A sin k x τ x ⊗ s 0 + A sin k y τ y ⊗ s 0 (2) is the standard two-band Dirac (BHZ-type) model 4 6,47 . It preserves time-reversal symmetry ( T ), inversion ( P ), and fourfold rotation ( C 4 ) , and provides the Dirac mass and orbital structure on which symmetry-breaking per- turbations act. At the Γ point, we have one site with +M 0 and another site with -M 0 , which, with opposite spins on the two sites, means that M 0 behaves as an ex- ternal magnetic field. Altermagnetism is introduced through the momentum- dependent spin splitting H A M ( k ) = ∆ d (cos k x − cos k y ) τ z ⊗ s z . (3) This term has a d x 2 −y 2 form factor that changes sign un- der C 4 rotation, while the spin structure changes sign un- der time reversal. As a result, the combined antiunitary operation C 4 T remains a symmetry of the altermagnetic sector, but it is broken by the external magnetic field M 0 . To introduce spin canting for the S x component, we include H C AN = m C AN τ z ⊗ s x . (4) This term breaks time-reversal symmetry and spin- rotation symmetry, while preserving lattice translational symmetry. In the orbital basis chosen here, it does not generate a net uniform magnetization due to the stag- gered τ z structure. However, it reduces the symmetry of the pure altermagnetic sector and, in general, does not preserve the C 4 T symmetry. Ferroelectricity is modeled by the Peierls dimerization H F E = P p olar τ x ⊗ s 0 . (5) The interplay between this Peierls dimerization and the magnetic field M 0 term breaks inversion symmetry ( P ) by mixing the two orbital sectors. Its behavior under C 4 depends on the orbital representation and is generally symmetry-breaking once ferroelectric order is present. Importantly, H F E enables coupling between the alter- magnetic spin texture and band-resolved spin polariza- tion through orbital hybridization. Finally, we include the Rashba spin–orbit coupling, H R ( k ) = λ R [sin k y τ 0 ⊗ s x − sin k x τ 0 ⊗ s y ] . (6) The Rashba term arises once inversion symmetry is bro- ken by H F E . These SOC terms do not modify the al- termagnetic origin of the spin splitting but contribute to spin texture reconstruction and band topology. In summary, Eq. (1) defines a minimal model in which the pure altermagnetic sector H 0 + H A M preserves the combined C 4 T symmetry, while ferroelectric and spin canting perturbations act as controlled symmetry- breaking fields that modify the spin–orbital entangle- ment. The full Hamiltonian, therefore, does not retain all symmetries of the altermagnetic limit, but continu- ously connects the altermagnetic phase to regimes with ferroelectric control and spin canting. III. RESULTS The spectrum of the minimal 4 × 4 Hamiltonian de- scribing a ferroelectric altermagnet can be obtained an- alytically in the absence of spin–orbit coupling. In this limit, the Hamiltonian contains four essential ingredients: the BHZ Dirac term, the d x 2 −y 2 altermagnetic spin split- ting, a weak-ferromagnetic canting field, and a ferroelec- tric orbital-mixing term. Introducing M k = M 0 − 2 B (2 − cos k x − cos k y ) , (7) f k = cos k x − cos k y , (8) and defining the orbital and spin vectors a ( k ) = A sin k x + P p olar , A sin k y , M k , (9) b ( k ) = m C AN , 0 , ∆ d f k , (10) the Hamiltonian can be written compactly as H ( k ) = τ · a ( k ) + τ z s · b ( k ) , (11) with τ = ( τ x , τ y , τ z ) and s = ( s x , s y , s z ). The ferro- electric polarization enters exclusively through the shift P p olar of the τ x component, reflecting inversion-breaking hybridization between the two orbitals. Analytic spectrum. — Squaring the Hamiltonian and us- ing the Pauli algebra yields H 2 ( k ) = | a ( k ) | 2 + | b ( k ) | 2 + 2 a z ( k ) s · b ( k ) , (12) where a z ( k ) = M k and | b ( k ) | = m 2 C AN + ∆ 2 d f 2 k . Since s · b has eigenvalues σ| b | with σ = ± 1, the four energy bands are E σ ,± ( k ) = ± A sin k x + P p olar 2 + A 2 sin 2 k y + M k + σ m 2 C AN + ∆ 2 d ( cos k x − cos k y ) 2 2 , σ = ± 1 . (13)
3 This expression makes explicit how the three symmetry-breaking ingredients reshape the Dirac spec- trum. The quantity in square brackets defines a spin- dependent effective mass, m σ ( k ) = M k + σ m 2 CAN + ∆ 2 d f 2 k , (14) while the ferroelectric polarization modifies the kinetic term through the shift A sin k x → A sin k x + P p olar . Im- portantly, P p olar does not directly enter the mass term, but instead displaces the Dirac cone and alters the gap structure indirectly. Altermagnetic limit. — In the purely altermagnetic case m C AN = P p olar = 0, the spin splitting is governed solely by the d x 2 −y 2 form factor f k , which vanishes at the Γ and M points but is maximal and of opposite sign at X and Y . Consequently, the spectrum remains spin-degenerate at Γ, E ↑ ,± ( Γ) = E ↓ ,± ( Γ) = ±|M 0 | , (15) while the two spin sectors acquire opposite mass shifts at X and Y . Each spin block, therefore, realizes a Qi–Wu– Zhang Hamiltonian with a spin-dependent Dirac mass, but the two blocks remain symmetry-related and con- tribute identical Chern numbers. As a result, the total Chern number is always an even integer, taking the val- ues C t ot = 0 or C t ot = ± 2 depending on the ordering of the TRIM masses, see section I of SI for more informa- tion. Role of spin canting. — A finite weak–ferromagnetic com- ponent m C AN qualitatively changes the topological struc- ture of the altermagnetic BHZ model. At the Γ point, where f Γ = 0 and M Γ = M 0 , the two hybridized Dirac cones acquire masses m ± ( Γ) = M 0 ± m C AN , E ± ( Γ) = ±|M 0 ± m C AN | . (16) Thus, the spin degeneracy of the altermagnetic limit is lifted uniformly, and the two cones undergo band inver- sion at different values of M 0 . Physically, weak ferromag- netism converts the purely momentum-dependent alter- magnetic splitting into a net topological imbalance be- tween the two hybridized cones. Each cone then carries an independent Chern number C ± = 0 , ± 1, so that the total Chern number C = C + + C − can take the values 0 , ± 1 , ± 2. This mechanism produces intermediate phases with |C| = 1 that have no analogue in either the pristine BHZ model or the altermagnetic case, see section I of SI for more information, and enables a sequence of topolog- ical transitions C = 0 → ± 1 → ± 2 (and the reverse) as M 0 is tuned. Role of ferroelectricity. — Ferroelectricity enters the Hamiltonian through the inversion-breaking orbital hy- bridization H F E = P p olar τ x , which shifts the Dirac dis- persion as A sin k x → A sin k x + P p olar . Although this term does not modify the Dirac mass directly, it re- shapes the spectrum in two essential ways. First, the (a) (b) (c) (d) FIG. 1. Band structures along high-symmetry lines for A = 1, B = 0 . 4, and M 0 = 0 . 05, with color indicating the spin projection ⟨s z ⟩ . (a) Pristine BHZ model, showing fully spin-degenerate bands. (b) Inclusion of altermagnetism (∆ d = 0 . 5), which induces momentum-dependent spin split- ting while preserving degeneracy at Γ. (c) Addition of spin canting ( m C AN = 0 . 1), lifting the Γ-point degeneracy and producing inequivalent Dirac masses. (d) Spin texture of the highest occupied band for the parameters in (c), showing a d -wave pattern in S z together with a uniform S x component. shift of the kinetic term generally enlarges the direct gap at Γ, stabilizing the insulating phase. Second, by mixing the two orbital sectors, ferroelectricity makes the weak-ferromagnetic canting m C AN τ z s x visible in the band structure, thereby enabling the magnetic and alter- magnetic contributions to combine into a tunable topo- logical mass. In the absence of ferroelectricity, the gap closes only at the TRIM, where the effective masses m ± ( k ) determine the Chern numbers of the two hybridized Dirac cones. A finite P p olar displaces the band extrema away from these high-symmetry points, but the topological transitions re- main governed by the zeros of m ± ( k ) as long as the gap closes near the TRIM. Thus, ferroelectricity does not shift the mass-inversion boundaries in the small- P p olar regime; instead, it modifies the dispersion anisotropically and redistributes Berry curvature in momentum space. The combined action of the three symmetry-breaking fields can therefore be summarized succinctly. The al- termagnetic term ∆ d ( cos k x − cos k y ) imposes a d -wave pattern of spin splitting and determines where band in- versions occur. Weak ferromagnetism introduces a uni- form component that separates the two hybridized Dirac masses, allowing them to invert at distinct values of M 0 and producing intermediate phases with |C| = 1. Ferro- electricity activates the orbital mixing required for this canting to manifest spectrally and provides an efficient electric knob to stabilize and tune the resulting Chern- insulating phases. Together, these ingredients yield a minimal and symmetry-consistent route to electrically controllable topological order emerging from a collinear altermagnet. Figure 1 illustrates how altermagnetism and spin cant- ing progressively reshape the Dirac spectrum and spin structure of the minimal model. In the pristine BHZ
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(a) (b) (c) (d) (e) FIG. 2. Topological phase diagram and Berry curvature of the altermagnetic Chern insulator. (a) Total Chern number C as a function of the band-inversion parameter M 0 and the altermagnetic gap ∆ d for m C AN = 0. Only even Chern phases ( C = 0 , ± 2) are realized due to spin degeneracy. (b) Phase diagram for finite spin canting ( m C AN = 0 . 1), where spin mix- ing splits the Dirac masses at Γ and stabilizes an intermediate |C| = 1 phase within |M 0 | < m C AN . (c)–(e) Berry curvature Ω z ( k ) for representative phases with (c) C = 0, (d) C = 1, and (e) C = 2. Weak ferromagnetism generates a single dom- inant Berry-curvature hotspot in the C = 1 phase, whereas the C = 2 phase exhibits two symmetry-related contributions. Parameters are the same as in Fig. 1. limit [Fig. 1(a)], the two spin sectors are exactly degen- erate throughout the Brillouin zone, including at the Γ point, where the gap is controlled by the Dirac mass M 0 . For A = 1, B = 0 . 4, and M 0 = 0 . 05, each spin block carries C B HZ = 1, yielding a total Chern number C = 2, consistent with the doubly degenerate chiral edge modes of the standard BHZ phase. Upon introducing altermagnetism [Fig. 1(b)], the d x 2 −y 2 form factor generates a pronounced momentum- dependent spin splitting, most visible along the X – Y di- rection, while vanishing at Γ. As a result, the spectrum remains spin-degenerate at Γ and the two spin blocks stay decoupled and topologically equivalent. For the pa- rameters shown, the total Chern number remains C = 2, in agreement with the analytic BHZ+AM phase diagram presented in Section I of SI. A qualitative change occurs when spin canting is in- cluded [Fig. 1(c)]. The canting term m C AN τ z s x lifts the degeneracy at Γ and splits the Dirac masses into m ± ( Γ) = M 0 ± m C AN , so that the two hybridized Dirac cones undergo band inversion at different values of M 0 . For m C AN = 0 . 1 and ∆ d = 0 . 5, only one cone remains inverted, and the total Chern number is reduced from C = 2 to C = 1, realizing an odd-Chern phase that has no analogue in either the pristine BHZ or purely alter- magnetic limits. The corresponding spin texture of the highest occu- pied band is shown in Fig. 1(d). The S z component ex- hibits the characteristic d -wave pattern inherited from altermagnetism, while a nearly uniform S x component emerges due to canting, showing a d -wave pattern in S z together with a uniform S x component. Figure 2 summarizes how altermagnetism and spin canting shape the topological phase diagram of the model. In the absence of spin canting, Fig. 2(a), the two spin sectors remain exactly degenerate, and only even Chern numbers C = 0 , ± 2 appear. The altermagnetic term redistributes the Dirac masses across the Brillouin zone but leaves the Γ-point degeneracy intact, in agree- ment with the analytical conditions derived in the Sup- plemental Information. A qualitatively new regime emerges once spin canting is introduced [Fig. 2(b)]. The canting term m C AN τ z s x lifts the residual degeneracy at Γ and splits the Dirac masses as m ± ( Γ) = M 0 ± m C AN . The two hybridized Dirac cones, therefore, invert at different values of M 0 , producing an intermediate window |M 0 | < m C AN in which only one cone is topological. This mechanism sta- bilizes an odd-Chern phase with |C| = 1, in direct corre- spondence with the analytical phase boundaries obtained from the TRIM mass conditions. The momentum-space origin of these phases is revealed by the Berry-curvature patterns in Figs. 2(c)–(e). In the trivial phase ( C = 0), positive and negative contribu- tions cancel by symmetry. In the C = 1 phase, spin canting breaks this balance and generates a single dom- inant hotspot near Γ, yielding a net Berry flux of one quantum. In contrast, the C = 2 phase exhibits two symmetry-related hotspots, each contributing approxi- mately one quantum of Berry curvature, consistent with two effectively decoupled Dirac cones. Together, these results demonstrate that altermag- netism controls the momentum structure of the Dirac masses, while spin canting lifts their degeneracy and se- lects a single topological channel, thereby enabling robust odd-Chern phases that are absent in both the pristine BHZ and purely altermagnetic limits. Figure 3 demonstrates how ferroelectricity provides an efficient and symmetry-consistent route to electrically reconstruct both the topology and geometric responses of an altermagnetic Chern insulator. In the inversion- symmetric limit ( P p olar = 0), all gap closings remain pinned to the time-reversal-invariant momenta, and the two hybridized Dirac cones invert only when their Γ-point masses m ± ( Γ) = M 0 ± m C AN change sign. As a result, the odd-Chern sector is restricted to the narrow window |M 0 | < m C AN , and the phase diagram is governed en- tirely by TRIM band inversions. Ferroelectricity qualitatively alters this mechanism. The inversion-odd hybridization H F E = P p olar τ x shifts the gap-closing momenta away from the TRIM, relocat- ing the Dirac criticality to finite wave vectors. At these displaced points the altermagnetic form factor f k be- comes finite, causing the two hybridized Dirac masses to renormalize differently. Their inversion boundaries, therefore, separate in parameter space, substantially en- larging the region in which only a single cone is inverted. As shown in Fig. 3(a), this momentum-space splitting stabilizes broad odd-Chern sectors that extend far be-
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(b) σ x y = 1 e 2 /h σ x y = 2 e 2 /h σ x y = 0 e 2 /h (a) (c) (d) FIG. 3. Ferroelectric control of topology and geometric responses in an altermagnetic Chern insulator. (a) Chern- number map as a function of the band-inversion parame- ter M 0 and ferroelectric polarization P p olar . The inversion- odd hybridization shifts the Dirac criticality to finite mo- mentum, separates the inversion boundaries of the two hy- bridized cones, and substantially widens the odd-Chern re- gion ( |C| = 1), in agreement with the analytic P 4 p olar scaling derived in the Supplemental Information. (b) Ribbon spectra for representative C = 0, C = 1, and C = 2 phases, showing the corresponding one or two chiral edge channels. (c) Orbital magnetization M z versus P p olar for several Rashba couplings λ R . In the inversion-symmetric limit ( P p olar = 0), an an- tiunitary symmetry enforces M z = 0, while ferroelectrically induced Rashba coupling generates a finite, even-in- P p olar or- bital response. (d) Anomalous Hall conductivity σ x y as a function of P p olar for different λ R , illustrating Rashba-driven shifts of the ferroelectric phase boundaries and the resulting reconstruction of quantized Hall plateaus. Together, the pan- els demonstrate that ferroelectricity provides an efficient elec- trical knob for tuning Dirac criticality, stabilizing odd-Chern phases, and activating geometric responses in altermagnetic Chern insulators. yond the weak-ferromagnetic scale. The analytic expan- sion derived in the Supplemental Information shows that the width of the |C| = 1 phase acquires a positive correc- tion proportional to P 4 p olar , demonstrating that ferroelec- tricity reshapes the topology not by reversing the Dirac mass, but by shifting the location of the topological crit- ical points in the momentum space. The bulk-boundary correspondence is confirmed by the ribbon spectra in Fig. 3(b). In the C = 1 phase, a single chiral edge channel traverses the gap, whereas the C = 2 regime hosts two co-propagating modes, reflecting the inversion of both hybridized Dirac cones. Although the ferroelectric shift displaces the Dirac points and partially overlaps the edge dispersions, the multiplicity of the chi- ral channels remains clearly resolved, providing a direct spectroscopic signature of the ferroelectrically stabilized odd-Chern phase. Ferroelectricity also activates geometric responses that are forbidden in the inversion-symmetric limit. At P p olar = 0, an antiunitary symmetry enforces an exact FIG. 4. Schematic illustration of electrically controlled or- bital and topological responses in a ferroelectric altermagnetic Chern insulator. In the nonpolar state ( P p olar = 0), alter- magnetic orbital currents cancel and the orbital magnetiza- tion vanishes. Applying a ferroelectric polarization activates Rashba spin–orbit coupling, reconstructs the Berry curvature near the hybridized Dirac crossings, and generates finite cir- culating orbital currents and electrically tunable orbital mag- netization together with chiral edge transport. cancellation of the orbital-magnetization kernel, yield- ing M z = 0 despite finite Berry curvature and nontriv- ial Chern bands. Once ferroelectricity induces a Rashba field, this cancellation is lifted and a finite orbital magne- tization emerges. Figure 3(c) shows that M z ( P p olar ) is an even function of polarization and grows rapidly with the Rashba strength, reflecting a redistribution of Berry cur- vature near avoided crossings of the altermagnetic bands. This mechanism is intrinsically geometric: it converts the altermagnetic spin texture into circulating orbital cur- rents through inversion-odd relativistic coupling, rather than through conventional spin polarization. Additional information about the effect of ferroelectricity on topol- ogy and orbital magnetization is presented in Sections III and IV of the Supplementary Information. The anomalous Hall conductivity in Fig. 3(d) exhibits a similar ferroelectric reconstruction. Even at λ R = 0, a quantized Hall response persists due to the broken time-reversal symmetry of the weak-ferromagnetic state. Increasing the Rashba coupling shifts the ferroelectric phase boundaries and reorganizes the Berry curvature weight, producing a strongly polarization-dependent pro- file of σ x y while preserving its quantized plateaus in the insulating regions. Together, these results establish fer- roelectric altermagnets as a platform in which electric polarization controls not only the topological phase di- agram but also the geometric responses associated with Berry curvature and orbital magnetism. To summarize the physical picture emerging from our results, Fig. 4 presents a schematic illustration of the fer- roelectric altermagnetic device proposed in this work. In the nonpolar state ( P p olar = 0), the system hosts alter- magnetic spin textures and topological edge transport,
6 but the residual symmetry of the Hamiltonian forces the momentum-space orbital currents to cancel, yielding zero orbital magnetization. Applying an external elec- tric field switches the ferroelectric polarization and si- multaneously activates inversion-odd Rashba spin–orbit coupling. This ferroelectrically induced Rashba interac- tion reconstructs the Berry curvature near the hybridized Dirac crossings, converts the altermagnetic spin texture into circulating orbital currents, and generates a finite, electrically tunable orbital magnetization. At the same time, the displacement of the Dirac criticality away from the time-reversal-invariant momenta reshapes the topo- logical phase boundaries and controls the multiplicity of chiral edge channels. This mechanism provides a direct route for electric- field manipulation of both orbital and topological re- sponses in altermagnetic Chern insulators. Unlike con- ventional quantum anomalous Hall platforms, where magnetism and orbital effects arise from uniform ferro- magnetism, the present system exploits the cooperative interplay among altermagnetism, spin canting, ferroelec- tricity, and inversion-odd spin–orbit coupling. Our re- sults, therefore, identify ferroelectric altermagnets as a promising materials platform for electrically switchable orbitronics, nonvolatile topological devices, and multi- functional quantum transport controlled by ferroelectric polarization. From an experimental perspective, the proposed mech- anism may be relevant to emerging altermagnetic materi- als integrated with ferroelectric environments. In partic- ular, altermagnetic thin films, magnetic quantum wells, and van der Waals heterostructures placed on ferroelec- tric substrates provide natural settings where inversion symmetry can be broken and Rashba spin–orbit coupling can be engineered. The additional spin canting required by our model may arise from external magnetic fields or intrinsic relativistic interactions, suggesting feasible routes toward realizing electrically tunable altermagnetic Chern phases. IV. CONCLUSIONS We demonstrate that the quantum anomalous Hall ef- fect can be realized in a two-dimensional d -wave alter- magnet under an applied magnetic field based on the BHZ model. By incorporating ferroelectricity and ex- ploiting its coupling to spin canting, the ferroelectric polarization provides a deterministic and reversible tun- ing parameter for topological phase transitions between states with Chern numbers C = ± 1 and C = ± 2. By switching the polarization, the total Chern number can be controlled, enabling electric-field tunability of chiral edge transport. A notable consequence of the ferroelectric altermag- netic Chern-insulator phase is the emergence of a siz- able and electrically tunable orbital magnetization. We showed that the orbital magnetization is highly sensi- tive to the topological phase transitions driven by spin canting and Rashba coupling, displaying characteristic changes associated with Berry-curvature redistribution in momentum space. Because both the Chern number and orbital magnetization can be controlled through fer- roelectric polarization, our results reveal a route toward electric-field manipulation of orbital magnetic properties in topological altermagnets. These findings identify or- bital magnetization not only as a diagnostic of the under- lying topological state but also as a functional degree of freedom for future orbitronic and topological-electronic applications. Our findings uncover a route to stabilizing Chern in- sulating phases in altermagnetic systems and establish a direct link between ferroelectric order, spin canting, and Berry-curvature engineering. This interplay provides a viable platform for electrically programmable topological phases and low-power quantum devices that operate without external magnetic fields. Moreover, given the rapidly expanding materials landscape of altermagnets and their potentially robust exchange energy scales, the QAHE in altermagnetic systems may be achievable at temperatures higher than those realized in conventional ferromagnetic Chern insulators. ACKNOWLEDGMENTS M.B.T. acknowledges the funding support by Nar- odowa Agencja Wymiany Akademickiej (NAWA) under the ULAM program with project number BPN/ULM/2025/1/00156/U/00001. M.B.T. acknowl- edges the funding support by Iran National Science Foundation (INSF) under project No.4043973. This research was supported by the Foundation for Polish Science project “MagTop” no. FENG.02.01-IP.05- 0028/23 co-financed by the European Union from the funds of Priority 2 of the European Funds for a Smart Economy Program 2021–2027 (FENG). We acknowledge the access to the computing facilities of the Poznan Supercomputing and Networking Center, Grant No. pl0807. ∗ [email protected] † [email protected] 1 N. J. Hu´ang, J. L. Boland, K. M. Fijalkowski, C. Gould, T. Hesjedal, O. Kazakova, S. Kumar, and H. Scherer, Applied Physics Letters 126 , 040501 (2025). 2 J. Wang, B. Lian, and S.-C. Zhang, Physica Scripta 2015 ,
7 014003 (2015). 3 C.-Z. Chang and M. Li, Journal of Physics: Condensed Matter 28 , 123002 (2016). 4 Y.-H. Wan, P.-Y. Liu, and Q.-F. Sun, Phys. Rev. Lett. 135 , 186302 (2025). 5 S. Majewski, M. Wierzbicki, and T. Dietl, “From many valleys to many topological phases - quantum anomalous hall effect in iv-vi semiconductor quantum wells,” (2026), arXiv:2601.16137 [cond-mat.mes-hall]. 6 M. M. Wysoki´nski and W. Brzezicki, Phys. Rev. B 108 , 035121 (2023). 7 A. Tripathi, M. Trif, and T. Posske, “Universal topolog- ical power transfer with arbitrarily large chern number in driven quantum spin chains,” (2026), arXiv:2604.24851 [cond-mat.str-el]. 8 J. Zhang, B. Zhao, T. Zhou, and Z. Yang, Chinese Physics B 25 , 117308 (2016). 9 L. ˇ Smejkal, R. Gonz´alez-Hern´andez, T. Jungwirth, and J. Sinova, Science Advances 6 , eaaz8809 (2020). 10 S. Hayami, Y. Yanagi, and H. Kusunose, journal of the physical society of japan 88 , 123702 (2019). 11 L. ˇ Smejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12 , 031042 (2022). 12 C. Autieri, R. M. Sattigeri, G. Cuono, and A. Fakhredine, Phys. Rev. B 111 , 054442 (2025). 13 C. Autieri, G. Cuono, D. Chakraborty, P. Gentile, and A. M. Black-Schaffer, Phys. Rev. B 112 , 014412 (2025). 14 A. Fakhredine, G. Cuono, J. Skolimowski, S. Picozzi, and C. Autieri, Mater. Horiz. , (2026). 15 J. W. Gonz´alez, T. Brumme, E. S. Morell, and A. M. Le´on, NPJ 2D Mater. Appl. (2025). 16 K. Tenzin, B. Kilic, R. M. Sattigeri, Z. He, C. C. Ye, M. Costa, M. B. Nardelli, C. Autieri, and J. Sl awi´nska, Npj Spintron. 3 , 46 (2025). 17 M. Yarmohammadi, M. Berritta, M. Bukov, L. ˇ Smejkal, J. Linder, and P. M. Oppeneer, Phys. Rev. B 113 , L060403 (2026). 18 A. Le´on, C. Autieri, T. Brumme, and J. W. Gonz´alez, npj Quantum Materials 10 (2025), 10.1038/s41535-025-00814- y. 19 C. Autieri and A. Fakhredine, The Journal of Physical Chemistry Letters 17 , 449 (2026), pMID: 41469325. 20 C. Chen Ye, K. Tenzin, J. Sl awi´nska, and C. Autieri, Phys. Rev. B 113 , 014413 (2026). 21 G. Ren, J. M. DeStefano, X.-W. Zhang, A. S. Thind, R. Giridharagopal, J. A. Castellanos-Reyes, P. M. Zeiger, N. Kamm, S. Xu, Z. Liu, Y. Xie, F. Krizek, J. Michalicka, R. Campion, P. Dai, P. Wadley, D. S. Ginger, T. Jung- wirth, R. F. Klie, J. Rusz, D. Xiao, J.-H. Chu, and J. C. Idrobo, “Atomic-scale observation of symmetry breaking in altermagnetic MnTe,” (2026), arXiv:2605.27543 [cond- mat.mtrl-sci]. 22 P.-J. Guo, Z.-X. Liu, and Z.-Y. Lu, npj Computational Materials 9 (2023), 10.1038/s41524-023-01025-4. 23 X. Chen, D. Wang, L. Li, and B. Sanyal, Applied Physics Letters 123 , 022402 (2023). 24 B. Wu, Y.-l. Song, W.-x. Ji, P.-j. Wang, S.-f. Zhang, and C.-w. Zhang, Phys. Rev. B 107 , 214419 (2023). 25 R. M. Sattigeri, G. Cuono, and C. Autieri, Nanoscale 15 , 16998 (2023). 26 X. Jiang, S. A. Akbar Ghorashi, D. Lu, and J. Cano, Nano Letters 26 , 1327 (2026). 27 J. Li, Q. Wu, and H. Weng, npj Computational Materials 11 , 53 (2025). 28 M. Bagheri Tagan, A. Fakhredine, and C. Autieri, The Journal of Physical Chemistry Letters 17 , 6132 (2026). 29 X. Chen, J. Zhang, B. Hao, J. Qian, Z. Zhu, I. Zu- tic, Z. Zhang, and T. Zhou, “Altermagnets enable gate- switchable helical and chiral topological transport with spin-valley-momentum-locked dual protection,” (2026), arXiv:2603.06487 [cond-mat.mes-hall]. 30 Y. Liu, J. Li, and Q. Liu, Nano Letters 23 , 8650 (2023). 31 W. Ahmed, S. Schaeffer, P. Lombardo, R. Hayn, and I. Makhfudz, “Large chern-number quantum anoma- lous hall effect from canted antiferromagnetic order in d -electron system on kagome lattice,” (2025), arXiv:2509.10976 [cond-mat.mes-hall]. 32 J. Li, X. Du, C. Lou, and J. Zhong, Phys. Rev. B 112 , 014418 (2025). 33 M. U. Muzaffar, K.-Z. Bai, W. Qin, G. Cao, B. Fu, P. Cui, S.-Q. Shen, and Z. Zhang, Nano Letters 25 , 7361 (2025). 34 Y. Liang, P. Zhao, F. Zheng, and T. Frauenheim, Phys. Rev. B 110 , 205421 (2024). 35 K.-Z. Bai, B. Fu, and S.-Q. Shen, “Ferroelectrically switch- able chirality in topological superconductivity,” (2026), arXiv:2505.01759 [cond-mat.supr-con]. 36 D. S. Antonenko, R. M. Fernandes, and J. W. F. Vender- bos, Phys. Rev. Lett. 134 , 096703 (2025). 37 R. Gonz´alez-Hern´andez and B. Uribe, Phys. Rev. B 112 , 184101 (2025). 38 H. Yoshida, J. Priessnitz, L. ˇ Smejkal, and S. Murakami, Phys. Rev. Lett. 136 , 096701 (2026). 39 H. Radhakrishnan, B. Bell, C. Ortix, and J. W. F. Vender- bos, “Topological piezomagnetic effect in two-dimensional dirac quadrupole altermagnets,” (2026), arXiv:2602.05894 [cond-mat.str-el]. 40 F. Qin and R. Chen, Phys. Rev. B 113 , 165418 (2026). 41 N. Pournaghavi, M. F. Islam, R. Islam, C. Autieri, T. Di- etl, and C. M. Canali, Phys. Rev. B 103 , 195308 (2021). 42 M. Ezawa, Phys. Rev. B 112 , 235307 (2025). 43 S. Korrapati, S. Nandy, and S. Tewari, Phys. Rev. B 112 , 195429 (2025). 44 X. Huo, X. Zhu, C.-A. Li, S. Feng, S.-B. Zhang, S. A. Yang, and H. Guo, Phys. Rev. B 113 , 024420 (2026). 45 Y.-K. Wang, S. Qian, A.-D. Fan, and S. Li, Phys. Rev. B 112 , 245148 (2025). 46 B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, science 314 , 1757 (2006). 47 D. Wang, A. K. Ghosh, Y. Tao, F. Ma, and C. Song, Advanced Science 13 , e22203 (2026).