Electric Hidden Unconventional Collinear Magnetism in 2D $\mathcal{PT}$-Symmetric Antiferromagnets

Introduction.—The intimate connection between magnetic order and crystalline symmetry is a central theme in condensed matter physics for understanding and designing novel magnetic states [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The long-established paradigm holds that a system’s magnetic symmetry classification and electronic spin structure are constrained by the symmetry operations connecting its spin sublattices in the nonrelativistic collinear limit, an approach that has successfully categorized current crystal-symmetry-compensated magnets [13, 14, 15, 16, 17, 18, 19]. Yet, what if this seemingly complete classification is merely part of a larger hidden landscape? What if symmetries beyond the primitive cell, concealed beneath the protection of these dominant operations, hold the key to emergent phases previously invisible to the conventional framework?

Among various magnetic systems [5, 11, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], conventional $\mathcal{PT}$-symmetric antiferromagnets precisely embody this tension [5, 11, 33, 34, 35, 36, 37, 38]. These materials possess the combined symmetry of spatial inversion ($P$) and time reversal ($T$) in relativistic case, which protects their degenerate band structures and ensures zero net magnetization [11, 36, 37, 38]. Yet, it is this very protection that also masks the influence of other crystallographic operations on the electronic structure, leading to a common perception that conventional $\mathcal{PT}$-symmetric antiferromagnets form a single rigid class with very limited tunability [33, 34, 35]. In three dimensions, $\mathcal{PT}$-symmetric antiferromagnets have been extensively studied, but the effect of an out-of-plane electric field ($\mathcal{E}_z$) on breaking $P$ is often limited by sample thickness and electrostatic screening. In two dimensions, however, the situation changes dramatically and field control is emerging to serve as an efficient method for manipulating spin transport [55, 56, 57, 58, 59]. The reduced dimensionality not only enriches the possible magnetic ground states but also allows $\mathcal{E}_z$ to efficiently penetrate the material and selectively break a subset of crystallographic operations while preserving others, thereby offering an unprecedented route to unveil hidden magnetic phases that are strictly forbidden in the pristine $\mathcal{PT}$-symmetric state [60, 61, 62, 63, 64, 65]. Despite its great potential, a complete symmetry-based framework that systematically classifies how $\mathcal{E}_z$ transforms conventional $\mathcal{PT}$-symmetric antiferromagnets into unconventional collinear magnetic phases is still lacking. Previous classifications either rely on spin space groups and are limited to three dimensions, or do not consider the selective symmetry breaking by the electric field, leaving the $\mathcal{E}_z$-driven magnetic phase transitions and their full classification in two-dimensional (2D) $\mathcal{PT}$-symmetric antiferromagnets largely unexplored.

To fill this gap, we establish a complete symmetry classification of conventional 2D $\mathcal{PT}$-symmetric antiferromagnets within the collinear spin layer groups (cSLGs) framework [66], and show that an $\mathcal{E}_z$ can drive them into three distinct classes of hidden unconventional collinear magnetism. According to the symmetry operations that survive the $\mathcal{E}_z$-induced selective symmetry breaking, the 88 cSLGs describing 2D $\mathcal{PT}$-symmetric antiferromagnetism fall into three categories: Class-I (electric type-IV magnetism, 23 cSLGs), Class-II (electric altermagnetism, 34 cSLGs), and Class-III (electric fully compensated ferrimagnetism, 31 cSLGs). To illustrate these categories, we select the most common or feasibly exfoliated 2D candidates, including Sr$_2$CaCr$_2$P$_2$O$_9$ and Sr$_2$CaCr$_2$As$_2$O$_9$ (Class-I), BaMnGe and MnPS$_3$ (Class-II), and Sr$_3$Fe$_2$O$_5$Cl$_2$ and bilayer MnBi$_2$Te$_4$ (Class-III). We demonstrate that $\mathcal{E}_z$ efficiently lifts Kramers degeneracy and generates qualitatively different spin patterns, which manifest as distinct spin-polarized responses: Class-I systems can exhibit the electric Hall effect, Class-II systems can exhibit spin-dependent transport, and Class-III systems can exhibit the electric reversible magneto-optical Kerr effect (MOKE). These responses depend on the specific symmetry and electronic structure and are not universal to all systems within each class. Our work establishes a unified framework bridging conventional antiferromagnets and emerging magnetic phases, opening a viable route for low-power multifunctional spintronics based on 2D magnets.

Theoretical analysis.—The cSLGs theory provides the natural language for classifying 2D collinear magnets [66]. Compared with spin space groups [17, 18, 19, 67], cSLGs more accurately capture the non-relativistic symmetries of 2D systems. Within this framework, conventional 2D $\mathcal{PT}$-symmetric antiferromagnetism is defined as collinear magnetism carrying $\{C_{2}\Vert\mathcal{P}\Vert\tau\}$ symmetry, which, in the relativistic limit, reduces to the combined $\mathcal{PT}$ (representing the combined operation of spatial inversion $\mathcal{P}$ and time reversal $\mathcal{T}$) symmetry and enforces Kramers spin degeneracy. Our previous cSLG classification of 2D magnets has already identified type-IV magnetism as a phase absent in three dimensions [41, 42]. Here, we generalize this approach to all 88 cSLGs compatible with conventional $\mathcal{PT}$-symmetric antiferromagnetism, which can be expressed as $\mathcal{L}_{\mathcal{P}}=\{\mathcal{E}\Vert\mathcal{L}\}\cup\{C_{2}\Vert\mathcal{P}\mathcal{L}\}$, where $\mathcal{L}$ is an index-two normal subgroup of $\mathcal{L}_{\mathcal{P}}$ consisting of spin-invariant operations.

Previous studies have frequently grouped conventional $\mathcal{PT}$-symmetric antiferromagnetism as a single class, as the $\{C_{2}\Vert\mathcal{P}\}$-enforced spin degeneracy hides the influence of other symmetries. In 2D systems, however, an $\mathcal{E}_z$ breaks $\mathcal{P}$. This selective symmetry breaking divides all symmetry operations into two sets: $\mathcal{O}_{I}=\{\mathcal{E}, C_{nz}, M_{\parallel}\}$, which remain intact under $\mathcal{E}_z$, and $\mathcal{O}_{II}=\{\mathcal{P}, S_{nz}, M_z, C_{2}\}$, which are broken [68]. Here, $\mathcal{E}$ represents identity element, $C_{nz}$ is an $n$-fold in-plane rotation, and $M_{\parallel}$ denotes any mirror perpendicular to the plane; $S_{nz}=C_{nz}M_z$ denotes an improper rotation. Consequently, the symmetry operations in $\mathcal{P}L$ (after field application) that survive the field dictate the band structure. This yields three distinct scenarios (Figure 1): (i) if $\exists S_1 \in\mathcal{P}L$, $S_1=C_{2z}$, the spin degeneracy cannot be lifted by $\mathcal{E}_z$. In this case, the $\mathcal{E}_z$-applied system can then be regarded as an electric type-IV magnetism (Class-I); (ii) if $\exists S_2 \in\mathcal{P}L$, $S_2 \in O_{\mathrm{I}}\setminus\{C_{2z}\}$, then after $\mathcal{E}_z$ is applied the system still possesses the $\{C_{2\parallel}|S_2\}$ symmetry, which can be regarded as electric altermagnetism (Class-II); (iii) if $\forall S_3 \in\mathcal{P}L$, $S_3 \in O_{\mathrm{II}}$, then every operation that connects opposite spin sublattices will be destroyed by $\mathcal{E}_z$. The system is regarded as electric ferrimagnetism (Class-III). If the system remains semiconducting under $\mathcal{E}_z$, it becomes an electrically induced fully compensated ferrimagnet [69].

Applying these criteria to all 88 cSLGs compatible with 2D conventional $\mathcal{PT}$-symmetric antiferromagnetism yields a complete classification: 23 belong to Class-I, 34 to Class-II, and 31 to Class-III (see Supplemental Material (SM) [70]). Two representative examples illustrate the physical consequences. Bilayer MnBi$_2$Te$_4$, as a widely studied intrinsic magnetic topological insulator, is described by cSLG L.72.69.1.1, which falls into Class-III; an $\mathcal{E}_z$ thus induces ferromagnetic spin splitting in its originally degenerate bands. Monolayer MnPS$_3$, a layered antiferromagnetic semiconductor, belongs to cSLG L.71.67.1.1 (Class-II), and hence an applied electric field produces altermagnetic spin splitting. These examples demonstrate that the classification is not merely formal, and each class directly maps onto experimentally accessible electric-field responses in known 2D materials.

The above classification is not limited to specific compounds but applies universally to all 88 cSLGs describing 2D conventional $\mathcal{PT}$-symmetric antiferromagnetism. To explicitly demonstrate the distinct electronic responses of each class, we perform first-principles calculations on three representative materials, one for each class: monolayer Sr$_2$CaCr$_2$P$_2$O$_9$ (Class-I), BaMnGe (Class-II) and Sr$_3$Fe$_2$O$_5$Cl$_2$ (Class-III). Our classification thus unveils a purely electrical paradigm for 2D $\mathcal{PT}$-symmetric antiferromagnets: a simple static means $\mathcal{E}_z$ enables selective and reversible switching between three distinct unconventional magnetic phases, paving the way for multifunctional all-electrical spintronics.

Class-I: electric type-IV magnetism.—Monolayer Sr$_2$CaCr$_2$X$_2$O$_9$ (X=P, As) family serves as a potential representative, belonging to the electric type-IV cSLG L.41.31.1.1. Taking Sr$_2$CaCr$_2$P$_2$O$_9$ as an example, its magnetic atoms occupy the 4k Wyckoff position, and the two spin sublattices are related by $\{C_{2\parallel}|\mathcal{P}\}$, $\{C_2||C_{2z}\}$, $\{C_2||C_{2\parallel}||\mathcal{T}\}$, and $\{C_2||M_{\parallel}||\mathcal{T}\}$ [Figure 2(a) and Figure 2(b)]. In the absence of $\mathcal{E}_z$, the spin-up and spin-down bands remain degenerate along the $\Gamma$-$X$-$M$-$Y$-$\Gamma$ path [Figure 2(c)]. This degeneracy follows from the symmetry constraint $E(\mathbf{s},\mathbf{k})=E(\mathbf{s},-\mathbf{k})$ enforced by $\{\widetilde{C}_{2\parallel}||\mathcal{T}\}$ for all collinear magnets [27, 71, 72]; together with the symmetry relations above, it yields:

Upon applying $\mathcal{E}_z$, the band dispersions are significantly modified, while the spin-up and spin-down bands remain degenerate because $\{C_2||\mathcal{P}\}$ is broken whereas $\{C_2||C_{2z}\}$ is preserved [Figure 2(d)]. In the nonrelativistic limit, this $\mathcal{E}_z$-unbreakable spin degeneracy identifies the system as Class-I.

When relativistic effects are taken into account, type-IV magnets often give rise to fascinating physical properties [41, 42]. In particular, after considering spin-orbit coupling (SOC), many physical phenomena may emerge in Class-I. For example, in monolayer Sr$_2$CaCr$_2$P$_2$O$_9$ with the magnetic moment oriented along the [10] direction, the recently defined electric-field-induced Hall effect [68] can also appear. Moreover, the Hall conductivity $\sigma_{xy}$ exhibits a sign reversal when the electric field is reversed. Without $\mathcal{E}_z$, monolayer Sr$_2$CaCr$_2$P$_2$O$_9$ will maintain Kramers degeneracy due to its $\mathcal{PT}$ symmetry, and the Hall conductivity $\sigma_{xy}$ is also constrained to be 0 (see details in SM[70]). When $\mathcal{E}_z=0.2$ eV/Å is further applied, the band exhibits a subtle splitting [see Figure 2(e) and the enlarged view in the green box]. Consequently, the sum of the Berry curvatures contributed by the band-inversion points in the conduction band and the valence band is no longer zero, thereby giving rise to a significant Hall conductivity [Figure 2(f)]. When the electric-field direction is reversed ($\mathcal{E}_z=-0.2$ eV/Å), the spin component $S_y$ near the gaps also changes sign and the corresponding

Hall conductivity reverses sign accordingly (see details in SM[70]). From a symmetry perspective, a Class-I under an applied $\mathcal{E}_z$ essentially belongs to the recently discovered type-IV magnetism [41, 42]. Therefore, it may host many of the physical properties recently uncovered in this class of magnetism and holds great promise for applications.

Class-II: electric altermagnetism.—Monolayer BaMnGe belongs to cSLG L.64.59.1.1, where the two spin sublattices are connected by the symmetry operations $\{C_2\Vert P\}$ and $\{C_2\Vert S_2\}$ [Figure 3(a) and Figure 3(b)]. Here, $S_2=\{M_{\parallel}\vert\tau,C_{4z}\tau,M_z\tau,C_{2\parallel}\tau\}$, with $\tau$ denoting a distinct sliding operation. In the absence of $\mathcal{E}_z$, $\{C_2\Vert P\}$ protects the degeneracy between the spin channels (see details in SM[70]).

Upon applying an $\mathcal{E}_z$, the symmetries $\{C_2\Vert P\}$, $\{C_2\Vert C_{2\parallel}\}$, and $\{C_2\Vert M_z\}$ are broken, whereas $\{C_2\Vert M_{\parallel}\}$ and $\{C_2\Vert C_{4z}\}$ are retained. The remaining symmetries impose directional constraints on the splitting, leading to an alternating, momentum-dependent spin splitting throughout the Brillouin zone [Figure 3(c) and Figure 3(d)]. Such a symmetry-driven evolution is characteristic of the electric altermagnetic response discussed within the cSLG framework.

Without $\mathcal{E}_z$, the spin-up and spin-down contributions to both $\sigma_{xx}$ and $\sigma_{yy}$ are essentially identical, consistent with the spin-degenerate band structure (see details in SM[70]). We define the spin polarization of conductivities as:

Accordingly, the corresponding spin polarizations $SP_{xx}$ and $SP_{yy}$ remain zero. A strikingly different behavior emerges under $\mathcal{E}_z$ [Figure 3(e) and Figure 3(f)]: the spin-up and spin-down longitudinal conductivities become clearly separated in both the $x$ and $y$ directions, accompanied by finite $SP_{xx}$ and $SP_{yy}$. This demonstrates that the electric field converts the initially spin-degenerate charge transport into spin-selective transport. Importantly, $SP_{xx}$ and $SP_{yy}$ exhibit distinct energy dependencies and opposite tendencies, indicating that the spin polarization is strongly direction-dependent. Therefore, the system supports electrically tunable spin-polarized currents along orthogonal directions.

Class-III: electric fully compensated ferrimagnetism.—In the monolayer $\mathrm{Sr}_{3}\mathrm{Fe}_{2}\mathrm{O}_{5}\mathrm{Cl}_{2}$ shown in Figure 4(a) and Figure 4(b), the two spin sublattices are correlated by the symmetry operations $\{C_{2}\Vert S_{3}\}$, where $S_{3}=\{P,S_{4z},M_{z},C_{2\parallel}\}$. In the absence of $E_{z}$, the symmetry $\{C_{2}\Vert P\}$ protects the degeneracy between the two opposite-spin channels (see details in SM[70]). Upon applying $E_{z}$, all $\{C_{2}\Vert S_{3}\}$ symmetries are broken and the spin degeneracy is lifted, as shown in Figure 4(c) and Figure 4(d). The band exhibits ferromagnetic splitting while the overall magnetic moment of the system remains zero, a phenomenon known as fully compensated ferrimagnetism [69]. The $E_{z}$-induced ferromagnetic band structure and ferromagnetic properties characterize monolayer $\mathrm{Sr}_{3}\mathrm{Fe}_{2}\mathrm{O}_{5}\mathrm{Cl}_{2}$ as Class-III.

As an effective probe of magnetic moments, MOKE can also reveal the properties of Class-III [73, 74, 75, 76]. When SOC is included, the $\mathcal{P}\mathcal{T}$ symmetry of $\mathrm{Sr}_{3}\mathrm{Fe}_{2}\mathrm{O}_{5}\mathrm{Cl}_{2}$ simultaneously protects the band spin degeneracy and forbids all Hall photoconductivity components; consequently, the pristine system exhibits no MOKE (see details in SM[70]). Since monolayer $\mathrm{Sr}_{3}\mathrm{Fe}_{2}\mathrm{O}_{5}\mathrm{Cl}_{2}$ is a soft magnet (see details in SM[70]), its magnetization direction can be easily manipulated. When its magnetic moment is oriented along [1] and an $E_{z}$ is applied, the $\mathcal{P}\mathcal{T}$ symmetry is broken, pronounced spin splitting appears [Fig. 4(c)], and MOKE emerges. The calculated results for $E_{z}=0.2\ \mathrm{eV}/\mathring{\mathrm{A}}$ are shown in Figure 4(e); when the electric field is reversed to $E_{z}=-0.2\ \mathrm{eV}/\mathring{\mathrm{A}}$, the band exhibits opposite chiral splitting and MOKE undergoes a sign reversal as well [Figs. 4(d) and (f)].

Conclusion

We have unveiled a hidden landscape of unconventional collinear magnetism in 2D $\mathcal{P}\mathcal{T}$-symmetric antiferromagnets, where the combined $\mathcal{P}\mathcal{T}$ symmetry has long obscured the role of other crystallographic symmetries in shaping the electronic structure. By constructing the full classification of 88 cSLGs, we demonstrate that an $E_{z}$ serves as a universal knob to release these hidden constraints, yielding three exhaustive classes of emergent magnetic phases. Notably, the distinct spin-splitting patterns and spin-dependent transport responses, including electric Hall effect, spin-dependent transport, and electric reversible MOKE, are not accidental but are dictated by the underlying symmetry of each class. Our work thus establishes a symmetry-based taxonomy for electric hidden unconventional magnetism, transforming what was previously a set of disparate phenomena into a coherent, complete, and predictive framework. The identified candidate materials, all amenable to exfoliation, provide an experimentally accessible testbed for realizing and controlling these phases; moreover, our classification scheme can be readily extended to other layered magnets with similar hidden-symmetry structures.

References