Altermagnetic Multiferroics: Symmetry & Control

• Altermagnetic multiferroics are defined by momentum-dependent spin splitting without net magnetization, enabling precise control of spin polarization.
• They combine various ferroic orders such as ferroelectric, antiferroelectric, ferroelastic, and antiferroaxial, which modulate electronic responses like Hall and Kerr effects.
• Diverse material platforms from molecular ferroelectrics to ultrathin oxides pave the way for low-power, reconfigurable spin transport and multifunctional device applications.

Searching arXiv for papers on altermagnetic multiferroics and closely related symmetry/mechanism papers. Searching arXiv for "altermagnetic multiferroics", "antiferroelectric altermagnets", and related 2D/perovskite work. Altermagnetic multiferroics are multiferroic systems in which a ferroic order parameter is coupled to altermagnetism, a symmetry-distinct magnetic phase with zero net magnetization but finite nonrelativistic momentum-space spin splitting. In these materials, the defining altermagnetic quantity is the band-resolved spin polarization $S_n(k)=\langle u_{n,k}|\sigma|u_{n,k}\rangle$, while the global magnetization still cancels through $\sum_{n,k} f(E_{n,k}) S_n(k)=0$. The field has expanded rapidly from symmetry proposals to first-principles materials realizations across molecular ferroelectrics, perovskites, Ruddlesden–Popper halides, van der Waals magnets, ultrathin oxides, and ferroelastic or antiferroaxial systems, with the central objective of controlling spin splitting, Hall responses, Kerr signals, and related observables by electric field, strain, interlayer sliding, or structural rotation rather than by net magnetization (Zhu et al., 9 Jul 2025, Sun et al., 10 Jul 2025, Šmejkal, 2024).

1. Definition and scope

Altermagnetism differs from both ferromagnetism and conventional collinear antiferromagnetism. Ferromagnets combine finite magnetization with generally uniform exchange-driven spin splitting, whereas conventional collinear antiferromagnets usually retain symmetries that enforce $E_\uparrow(k)=E_\downarrow(k)$ in the nonrelativistic limit. Altermagnets instead admit momentum-dependent spin splitting without net $M$, because opposite-spin sublattices are related by crystal rotations or mirrors rather than by the symmetry combinations that restore same-$k$ spin degeneracy (Zhu et al., 9 Jul 2025, Zhang et al., 18 Mar 2025).

Within multiferroics, this altermagnetic order can coexist with several distinct ferroic partners. Recent literature includes molecular ferroelectric altermagnets controlled by noncollinear molecular polarization, antiferroelectric altermagnets in which AFE-to-FE switching turns spin splitting on and off, type-II multiferroics in which Néel order directly generates polarization, type-III multiferroics in which ferroelectric and altermagnetic orders are symmetry-locked, ferroelastic altermagnets whose spin-splitting texture rotates under stress, and antiferroaxial altermagnets in which counter-rotating structural units induce and reverse the altermagnetic multipole (Zhu et al., 9 Jul 2025, Duan et al., 2024, Guo et al., 4 May 2025, Ding et al., 16 Oct 2025, Liu et al., 11 Feb 2026).

A recurrent theme is that the relevant magnetoelectric control acts on momentum-space spin structure rather than on a macroscopic magnetization. This is why the literature emphasizes electrically switchable $S(k)$, reversal of Kerr or Hall signals, and deterministic inversion of nonrelativistic spin splitting while preserving compensated magnetic order (Sun et al., 10 Jul 2025, Sun et al., 2024).

2. Symmetry architecture and microscopic couplings

The central symmetry distinction is whether opposite-spin sublattices are related by translation or inversion, which enforce same-$k$ spin degeneracy, or by a rotation or mirror that maps $k$ to a different symmetry-related momentum. In molecular ferroelectric altermagnets, collinear molecular polarization states preserve $[C_2|t]$ or $[C_2|I]$ and therefore remain spin-degenerate, while noncollinear molecular polarization breaks $\sum_{n,k} f(E_{n,k}) S_n(k)=0$0 and $\sum_{n,k} f(E_{n,k}) S_n(k)=0$1 but preserves a rotation-based relation such as $\sum_{n,k} f(E_{n,k}) S_n(k)=0$2, producing finite $\sum_{n,k} f(E_{n,k}) S_n(k)=0$3 with zero net magnetization. Reversing one molecular dipole in the noncollinear pattern interchanges the inequivalent hopping channels and reverses the sign of the spin polarization (Zhu et al., 9 Jul 2025).

This symmetry logic appears in several equivalent formulations. In antiferroelectric altermagnets, the exchange operation $\sum_{n,k} f(E_{n,k}) S_n(k)=0$4 satisfies $\sum_{n,k} f(E_{n,k}) S_n(k)=0$5; if $\sum_{n,k} f(E_{n,k}) S_n(k)=0$6, spin degeneracy is enforced, whereas if $\sum_{n,k} f(E_{n,k}) S_n(k)=0$7, nonrelativistic splitting is allowed. In the AFE state, a screw or roto-translation such as $\sum_{n,k} f(E_{n,k}) S_n(k)=0$8 connects opposite spins and permits altermagnetism; switching to the FE state restores pure translational connectivity and removes the splitting (Duan et al., 2024). In symmetry-locked type-III multiferroics, ferroelectric reversal acts as a pseudo-time-reversal operation on the altermagnetic spectrum: $\sum_{n,k} f(E_{n,k}) S_n(k)=0$9, so flipping polarization flips the momentum-space spin texture (Sun et al., 10 Jul 2025). In bilayer MnPSe$E_\uparrow(k)=E_\downarrow(k)$0, this appears explicitly as $E_\uparrow(k)=E_\downarrow(k)$1, meaning that ferroelectric switching alone fully inverts the altermagnetic spin polarization, equivalent to a $E_\uparrow(k)=E_\downarrow(k)$2 spin reversal in its action on the bands (Sun et al., 2024).

A second major route is spin-driven multiferroicity. In two-dimensional altermagnetic type-II multiferroics, the local dipole on magnetic sublattice $E_\uparrow(k)=E_\downarrow(k)$3 is written as $E_\uparrow(k)=E_\downarrow(k)$4, giving a macroscopic polarization $E_\uparrow(k)=E_\downarrow(k)$5 with $E_\uparrow(k)=E_\downarrow(k)$6. Because inversion does not connect the magnetic sublattices in an altermagnet, the cancellation that forbids polarization in conventional PT-symmetric antiferromagnets is removed (Guo et al., 4 May 2025). In antiferroaxial altermagnetism, Landau theory yields a trilinear invariant among Néel order $E_\uparrow(k)=E_\downarrow(k)$7, antiferroaxial order $E_\uparrow(k)=E_\downarrow(k)$8, and an altermagnetic multipole $E_\uparrow(k)=E_\downarrow(k)$9, with saddle-point relation $M$0. Reversing $M$1 therefore reverses $M$2, the sign of the spin splitting, and time-reversal-odd responses such as anomalous Hall conductivity (Liu et al., 11 Feb 2026).

Dimensionality can impose further symmetry restrictions. In layered perovskites reduced to the two-dimensional limit, only C-type antiferromagnetic order remains altermagnetic unless $M$3 is deliberately broken; A- and G-type orders recover same-$M$4 degeneracy because the surviving mirror symmetries reconnect opposite spins at identical momentum (Cui et al., 9 Jan 2026).

3. Materials platforms and realizations

One large family is molecular ferroelectric altermagnets. A symmetry-led design combined with tight-binding and first-principles calculations identified hybrid organic–inorganic perovskites and metal–organic frameworks in which molecular polarization controls altermagnetism. In monolayer [MA]$M$5MnCl$M$6, the noncollinear molecular-polarization phase is altermagnetic with spin splitting of order $M$7, while [PMA]$M$8MnCl$M$9 raises this to $k$0. In metal–organic frameworks, [DMA]Cu(HCOO)$k$1 reaches $k$2. The same symmetry recipe was extended to inorganic candidates such as BaFe$k$3Se$k$4 and Pb$k$5MnWO$k$6 (Zhu et al., 9 Jul 2025).

A second group centers on polar and antiferroelectric van der Waals systems. Bilayer FeCuP$k$7S$k$8 exhibits ferroelectricity-driven altermagnetism controlled by spin space group operations involving a nonsymmorphic screw axis or a twofold rotation, and interlayer sliding changes the spin space group, reverses the sign of $k$9, and switches the anomalous Hall response (Zhao et al., 1 Nov 2025). VOX$S(k)$0 monolayers are two-dimensional ferroelectric altermagnets; in VOI$S(k)$1, the Berry-phase polarization is $S(k)$2, the ferroelectric switching barrier is $S(k)$3, and the near-edge spin splitting reaches $S(k)$4 in the valence band and $S(k)$5 in the conduction band (Yang, 17 Mar 2025). Strained monolayer VCl$S(k)$6 realizes an orbital-order-driven ferroelectric altermagnet on the honeycomb lattice with a nematic $S(k)$7-wave spin splitting up to about $S(k)$8 along $S(k)$9–M, tied to an electronic polarization $k$0 (Camerano et al., 25 Mar 2025).

Two-dimensional multiferroic altermagnets now also include triferroic and ferroelastic cases. Pentagonal monolayer FeO$k$1 combines in-plane ferroelectricity, ferroelasticity, and altermagnetism in a single layer; the FE phase has $k$2, band gap $k$3, ferroelastic strain $k$4, and $k$5, while the competing AFE phase remains altermagnetic with $k$6 and ferroelastic strain $k$7 (Guo et al., 23 Jul 2025). Puckered pentagonal CoSe$k$8 realizes a ferroelastic altermagnet in which uniaxial stress induces a ferroelastic phase transition and a $k$9 rotation of a $k$0-wave spin-splitting pattern; the nonrelativistic splitting at the valence-band maximum is about $k$1 (Ding et al., 16 Oct 2025).

Oxide and halide platforms add both bulk and ultrathin realizations. BaCuF$k$2 and Ca$k$3Mn$k$4O$k$5 were identified as altermagnetic multiferroics in which polyhedral rotations mediate an altermagnetoelectric effect; BaCuF$k$6 shows nonrelativistic splitting in the $k$7–$k$8 range and has $k$9, while Ca$[C_2|t]$0Mn$[C_2|t]$1O$[C_2|t]$2 exhibits splitting above $[C_2|t]$3 and polarization $[C_2|t]$4 (Šmejkal, 2024). Ultrathin BiFeO$[C_2|t]$5 offers an oxide realization in the four-unit-cell limit, where a monoclinic $[C_2|t]$6 phase supports room-temperature multiferroicity, d-wave altermagnetic time-reversal symmetry breaking, and nonrelativistic spin splitting up to about $[C_2|t]$7 near the valence-band maximum (Fratian et al., 15 Jan 2026).

Additional chemically distinct families extend the field. In $[C_2|t]$8 Ruddlesden–Popper halides, K$[C_2|t]$9Cr$[C_2|I]$0F$[C_2|I]$1 hosts a ferrielectric altermagnetic phase stabilized by Jahn–Teller distortion plus octahedral rotations, with a low barrier of about $[C_2|I]$2 between ferrielectric and ferroelectric structures; the ferrielectric phase is altermagnetic, the ferroelectric phase is a conventional AFM, and strain or pressure produces sizable changes in weak ferromagnetism (Zhou et al., 19 Aug 2025). In Cr-doped wurtzite MnX $[C_2|I]$3, an A-type AFM phase becomes a g-wave altermagnet with large nonrelativistic splitting near the Fermi level and deterministic reversal of $[C_2|I]$4 under polarization switching, while the pristine compounds remain stripe-type, spin-degenerate antiferromagnets (Mavani et al., 30 Dec 2025).

4. Switching pathways and experimental observables

Electric-field switching is the most frequently emphasized control channel. In molecular ferroelectric altermagnets, twisting molecular polarization between PP/AP and NP/NP$[C_2|I]$5 toggles altermagnetism on and off and reverses the sign of the spin polarization without changing the antiferromagnetic order itself (Zhu et al., 9 Jul 2025). In antiferroelectric altermagnets such as CuWP$[C_2|I]$6S$[C_2|I]$7, an electric field converts the AFE altermagnetic state into an FE spin-degenerate AFM, so the AFE–FE transition functions as a symmetry switch for $[C_2|I]$8 (Duan et al., 2024). In bilayer MnPSe$[C_2|I]$9, FE sliding reversal between AB and BA stackings has a calculated barrier of $\sum_{n,k} f(E_{n,k}) S_n(k)=0$00 and fully inverts the altermagnetic spin polarization (Sun et al., 2024). In wurtzite Mn$\sum_{n,k} f(E_{n,k}) S_n(k)=0$01Cr$\sum_{n,k} f(E_{n,k}) S_n(k)=0$02Se, polarization reversal flips the sign of the g-wave spin splitting without reorienting the Néel vector (Mavani et al., 30 Dec 2025).

Mechanical control appears in several distinct forms. Interlayer sliding in FeCuP$\sum_{n,k} f(E_{n,k}) S_n(k)=0$03S$\sum_{n,k} f(E_{n,k}) S_n(k)=0$04 changes the spin space group, toggles altermagnetism on and off, and reverses both $\sum_{n,k} f(E_{n,k}) S_n(k)=0$05 and $\sum_{n,k} f(E_{n,k}) S_n(k)=0$06 at specific energies (Zhao et al., 1 Nov 2025). Uniaxial strain in CoSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$07 drives ferroelastic switching with a $\sum_{n,k} f(E_{n,k}) S_n(k)=0$08 rotation of the spin-splitting lobes, while cooperative versus noncooperative rotation of lattice and Néel vector preserves or reverses the Kerr sign (Ding et al., 16 Oct 2025). In FeO$\sum_{n,k} f(E_{n,k}) S_n(k)=0$09, in-plane uniaxial strain induces ferroelastic switching that rotates the FE polarization vector by $\sum_{n,k} f(E_{n,k}) S_n(k)=0$10 and simultaneously reverses the AM state, producing a six-state manifold selected by electric field and/or strain (Guo et al., 23 Jul 2025). In K$\sum_{n,k} f(E_{n,k}) S_n(k)=0$11Cr$\sum_{n,k} f(E_{n,k}) S_n(k)=0$12F$\sum_{n,k} f(E_{n,k}) S_n(k)=0$13, biaxial strain and hydrostatic pressure modulate the difference in weak ferromagnetism between altermagnetic and non-altermagnetic phases through symmetry-allowed piezomagnetism (Zhou et al., 19 Aug 2025).

Experimental identification has converged on a recurring set of probes. Spin-resolved ARPES and conventional ARPES are proposed throughout the literature as the most direct probes of nonrelativistic, momentum-dependent spin splitting, especially in systems with weak SOC such as organic ferroelectrics or VOX$\sum_{n,k} f(E_{n,k}) S_n(k)=0$14 monolayers (Zhu et al., 9 Jul 2025, Yang, 17 Mar 2025). Magneto-optical Kerr spectroscopy is a particularly important readout when FE switching reverses altermagnetic order; Kerr sign reversal is predicted for molecular NP/NP$\sum_{n,k} f(E_{n,k}) S_n(k)=0$15 states, for type-III bilayer MnPSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$16, and for ferroelastic CoSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$17 variants (Zhu et al., 9 Jul 2025, Sun et al., 2024, Ding et al., 16 Oct 2025). In ultrathin BiFeO$\sum_{n,k} f(E_{n,k}) S_n(k)=0$18, XMCD appears only in the $\sum_{n,k} f(E_{n,k}) S_n(k)=0$19 monoclinic collinear phase and, together with XMLD, resolves the domain structure associated with d-wave altermagnetism (Fratian et al., 15 Jan 2026).

Transport and optical nonlinearities offer complementary signatures. FeCuP$\sum_{n,k} f(E_{n,k}) S_n(k)=0$20S$\sum_{n,k} f(E_{n,k}) S_n(k)=0$21 shows AHE peaks below $\sum_{n,k} f(E_{n,k}) S_n(k)=0$22 in the altermagnetic AFE monolayer and a dominant shift-current component $\sum_{n,k} f(E_{n,k}) S_n(k)=0$23 at $\sum_{n,k} f(E_{n,k}) S_n(k)=0$24 (Zhao et al., 1 Nov 2025). VOI$\sum_{n,k} f(E_{n,k}) S_n(k)=0$25 displays a giant spin shift current $\sum_{n,k} f(E_{n,k}) S_n(k)=0$26 at $\sum_{n,k} f(E_{n,k}) S_n(k)=0$27, with sign reversal under ferroic switching, and a magnetoelectric coefficient $\sum_{n,k} f(E_{n,k}) S_n(k)=0$28 (Yang, 17 Mar 2025). In hidden-splitting Q-vector antiferromagnets such as MnS$\sum_{n,k} f(E_{n,k}) S_n(k)=0$29, macroscopic symmetry breaking without global spin splitting still produces a large Berry-curvature dipole and natural optical activity, clarifying that multiferroic-like responses can arise in a nearby but distinct regime (Matsuda et al., 2024).

5. Transport functionality and device architectures

The device literature focuses on electrically reconfigurable transport without stray fields. A proposed above-room-temperature CrSb/In$\sum_{n,k} f(E_{n,k}) S_n(k)=0$30Se$\sum_{n,k} f(E_{n,k}) S_n(k)=0$31/Fe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$32GaTe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$33 tunnel junction combines an altermagnetic electrode, a ferroelectric barrier, and a ferromagnetic electrode. First-principles NEGF calculations report TMR up to $\sum_{n,k} f(E_{n,k}) S_n(k)=0$34, TER of $\sum_{n,k} f(E_{n,k}) S_n(k)=0$35, and near-perfect spin filtering efficiency; in the specific In$\sum_{n,k} f(E_{n,k}) S_n(k)=0$36Se$\sum_{n,k} f(E_{n,k}) S_n(k)=0$37-barrier junction, the FE state strongly tunes TMR, while magnetic alignment tunes TER, establishing a dual-mode control architecture (Zhang et al., 18 Mar 2025). This proposal is device-oriented rather than a single-phase altermagnetic multiferroic, but it exemplifies how altermagnetic multiferroic concepts translate into switchable spin transport.

Ferroelectric and antiferroelectric van der Waals systems point toward lower-dimensional alternatives. In FeCuP$\sum_{n,k} f(E_{n,k}) S_n(k)=0$38S$\sum_{n,k} f(E_{n,k}) S_n(k)=0$39, FE/AFE switching and sliding directly reconfigure the AHE sign, suggesting electrically programmable Hall elements and spin filters without net magnetization (Zhao et al., 1 Nov 2025). In VOX$\sum_{n,k} f(E_{n,k}) S_n(k)=0$40, the coexistence of FE order, nonrelativistic altermagnetic splitting, large $\sum_{n,k} f(E_{n,k}) S_n(k)=0$41, and switchable charge and spin shift currents supports proposals for spin-photovoltaic devices, nonlinear optical logic, and multistate memory based on the four degenerate ferro-altermagnetic states $\sum_{n,k} f(E_{n,k}) S_n(k)=0$42 (Yang, 17 Mar 2025). Type-III bilayer MnPSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$43 adds an optical readout route, because the Kerr signal changes sign when FE switching flips the altermagnetic spin texture (Sun et al., 2024).

Bulk and ultrathin oxides offer a different device logic centered on robust ferroic order at reduced thickness. Ultrathin BiFeO$\sum_{n,k} f(E_{n,k}) S_n(k)=0$44 maintains room-temperature multiferroicity down to four unit cells with no dead layer, while the engineered monoclinic phase exhibits d-wave altermagnetic signatures and topological multiferroic textures, suggesting scalable oxide electronics with symmetry-selective optical readout of AFM domains (Fratian et al., 15 Jan 2026). Perovskite symmetry analysis further indicates that in the two-dimensional limit only C-type AFM naturally remains altermagnetic unless $\sum_{n,k} f(E_{n,k}) S_n(k)=0$45 is broken, so superlattices, substrate engineering, or shear strain become explicit design tools for device-compatible 2D altermagnetic multiferroics (Cui et al., 9 Jan 2026).

These proposals share a common systems-level advantage: the active order parameter is a switchable symmetry pattern rather than a uniform magnetization. This suggests low-power operation, reduced dipolar cross-talk, and compatibility with domain engineering, although many papers leave bias dependence, fatigue, interface termination, and long-cycle endurance to future work (Zhang et al., 18 Mar 2025, Zhu et al., 9 Jul 2025).

6. Conceptual boundaries, unresolved issues, and outlook

A recurrent misconception is that any polar or antiferroelectric antiferromagnet with unusual responses should be classified as an altermagnetic multiferroic. The recent literature is more restrictive. The defining feature is nonrelativistic momentum-dependent spin splitting, or an explicitly identified symmetry-locked hidden counterpart, at zero net magnetization. This is why conventional PT-symmetric AFMs remain outside the category even when they are multiferroic, and why Q-vector antiferromagnets with hidden altermagnetic split are treated as adjacent rather than identical phenomena (Matsuda et al., 2024).

Another misconception is that the ferroic partner must always be ferroelectricity. The field now includes antiferroelectric, ferrielectric, ferroelastic, and antiferroaxial mechanisms, all of which act as control knobs for the altermagnetic order. This broader usage is explicit in antiferroelectric altermagnets, ferroelastic CoSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$46, triferroic FeO$\sum_{n,k} f(E_{n,k}) S_n(k)=0$47, and antiferroaxial altermagnetism (Duan et al., 2024, Ding et al., 16 Oct 2025, Guo et al., 23 Jul 2025, Liu et al., 11 Feb 2026).

The main open questions are materials- and geometry-specific. Several proposals do not report coercive fields, switching times, or fatigue behavior, especially for molecular ferroelectrics and sliding ferroelectrics (Zhu et al., 9 Jul 2025, Zhao et al., 1 Nov 2025). Some chemically appealing systems have clear limitations: K$\sum_{n,k} f(E_{n,k}) S_n(k)=0$48Cr$\sum_{n,k} f(E_{n,k}) S_n(k)=0$49F$\sum_{n,k} f(E_{n,k}) S_n(k)=0$50 has a very small ferrielectric polarization of about $\sum_{n,k} f(E_{n,k}) S_n(k)=0$51 and a predicted $\sum_{n,k} f(E_{n,k}) S_n(k)=0$52, while CoSe$\sum_{n,k} f(E_{n,k}) S_n(k)=0$53 has $\sum_{n,k} f(E_{n,k}) S_n(k)=0$54 (Zhou et al., 19 Aug 2025, Ding et al., 16 Oct 2025). In transport proposals, interface termination, barrier thickness dependence, and finite-bias performance are often unspecified (Zhang et al., 18 Mar 2025). In ultrathin oxides and perovskites, the allowed altermagnetic order can be strongly constrained by residual mirror symmetries, so symmetry engineering becomes as important as chemical selection (Cui et al., 9 Jan 2026, Fratian et al., 15 Jan 2026).

A plausible implication is that the field will increasingly separate into two complementary directions. One is the search for experimentally simple, above-room-temperature platforms with large nonrelativistic splitting and robust switching, such as molecular ferroelectrics, wurtzite chalcogenides, or tunnel-junction heterostructures (Zhu et al., 9 Jul 2025, Mavani et al., 30 Dec 2025, Zhang et al., 18 Mar 2025). The other is the pursuit of symmetry-rich model systems—ultrathin BiFeO$\sum_{n,k} f(E_{n,k}) S_n(k)=0$55, antiferroaxial perovskites, or orbital-order-driven monolayers—that expose the full taxonomy of $\sum_{n,k} f(E_{n,k}) S_n(k)=0$56-, $\sum_{n,k} f(E_{n,k}) S_n(k)=0$57-, and $\sum_{n,k} f(E_{n,k}) S_n(k)=0$58-wave altermagnetism and its coupling to ferroelectric, ferroelastic, or axial order (Fratian et al., 15 Jan 2026, Liu et al., 11 Feb 2026, Camerano et al., 25 Mar 2025).

Across these directions, the defining contribution of altermagnetic multiferroics is not merely the coexistence of polarization and magnetic order, but the replacement of conventional weak magnetoelectricity by symmetry-mediated control of momentum-space spin structure. That replacement underlies the most distinctive results in the literature: FE reversal acting as pseudo-time reversal, AFE–FE transitions toggling $\sum_{n,k} f(E_{n,k}) S_n(k)=0$59 on and off, ferroelastic switching rotating spin-splitting lobes by $\sum_{n,k} f(E_{n,k}) S_n(k)=0$60, and structural axial order reversing anomalous Hall conductivity at zero net magnetization (Sun et al., 2024, Duan et al., 2024, Ding et al., 16 Oct 2025, Liu et al., 11 Feb 2026).