Introduction

Almost two decades since the emergence of graphene1, two-dimensional layered materials (2DLM) have become a key area of research in materials science. These materials are characterized by their layers, each formed of a covalently bonded lattice free of dangling bonds. These layers are held together by van der Waals forces, allowing them to be easily separated and combined in various fashions. This flexibility in layering different atomic materials, such as metals, semiconductors, insulators, or multiferroics, creates versatile van der Waals heterostructures2,3,4,5.

Magnetoelectricity, regarded as an arcane phenomenon not long ago, has become a piece of technological promise6,7,8. The ability to control magnetization via electric fields that it conveys has great potential for memory and spintronics devices9,10,11,12. The pioneering materials that demonstrated a robust magnetoelectric effect were topological insulators13,14. This work investigates magnetoelectric multiferroic materials that exhibit ferroelectricity and ferromagnetism, leading to pronounced magnetoelectric effects15,16. Their significance is underscored by developing a room-temperature MRAM prototype based on multiferroic states17. More recently, control over the electric dipolar moment has been shown to induce magnon-mediated spin-transfer torques in reconfigurable logic memory18.

Along with the discovery of magnetic order in low-dimensional van der Waals (vdW) systems19,20, efforts to electrically steer their magnetic attributes have surged. However, understanding the subtle mechanisms involved requires further exploration. Focusing on vdW materials with intrinsic type II multiferroicity could facilitate the electrical manipulation of 2D magnetism. In these systems, the magnetic and ferroelectric order produces a powerful magnetoelectric response21,22,23,24. In particular, 2D vdW systems exhibiting ferroelectricity are of considerable interest25.

The quest for magnetoelectric effects is often based on symmetry principles10,12,26. Examples are abundant, such as topological insulators27 and the revelation of the multipolar magnetoelectric order in certain compounds28. This study introduces a simplified model of some topological multiferroic 2D systems, demonstrating how the synergy between the system’s magnetic and electric attributes plays out at the quantum level. Such insights aim to guide the discovery of materials with superior multiferroic properties. Over the past few decades, the field of topological band theory has expanded significantly, encompassing various systems such as topological insulators, topological crystalline insulators, and Weyl semi-metals. These materials exhibit topological boundary states linked to their non-trivial bulk topology, often attributed to Berry curvature. The two-dimensional (2D) extension of the SSH model has recently garnered significant attention29,30,31,32,33,34,35, revealing properties such as fractional wave polarization and high-order topological states. Unlike other topological systems, the 2D SSH model, due to the simultaneous presence of time-reversal and inversion symmetries, does not rely on Berry curvature for its topological characteristics. This finding improves our understanding of non-trivial topology and paves the way for upcoming research. However, most studies on the 2D SSH model have concentrated on its most symmetric configuration. Considering the versatility of the 1D SSH chain in forming various 2D configurations, there is the potential to discover intriguing topological properties in other 2D arrangements.

The Su-Schrieffer-Heeger (SSH) model is a fundamental topological model in its one-dimensional form36,37,38,39. It and its extended versions have been thoroughly researched over many years and realized in various physical platforms, such as ultra-cold atoms, quantum circuits, and mechanical granular chains. Its variant, the Rice-Mele model40,41, has become the standard model for ferroelectricity42,43,44,45,46,47 A basic one-dimensional model that incorporates type II topological multiferroic behavior is the antiferromagnetic version of the Rice-Mele Hamiltonian48:

$$\begin{array}{l}{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{SRM}}}}}\,=\,-\mathop{\sum}\limits_{i\sigma }\left({{{{\rm{t}}}}}_{i}\,{{{{\bf{c}}}}}_{i,\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{i+1,\sigma }+{{{{\rm{t}}}}}_{i}^{* }\,{{{{\bf{c}}}}}_{i+1,\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{i,\sigma }\right)\\ \,\qquad\qquad+\,\Delta \mathop{\sum}\limits_{i\sigma {\sigma }^{{\prime} }}{(-1)}^{i}{{{{\bf{c}}}}}_{i,\sigma }^{{\dagger} }{{{{\boldsymbol{\tau }}}}}_{\sigma {\sigma }^{{\prime} }}^{{{{\rm{z}}}}}{{{{\bf{c}}}}}_{{{{\rm{i}}}},{\sigma }^{{\prime} }}+{{{\rm{V}}}}\mathop{\sum}\limits_{i\sigma }{(-1)}^{{{{\rm{i}}}}}{{{{\bf{c}}}}}_{{{{\rm{i}}}},\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{{{{\rm{i}}}},\sigma }\end{array}$$
(1)

In Eq. (1), τz is the pseudo spin Pauli matrix on the axis z and the operator of creation (annihilation) of an electron at the site i with the spin component σ is denoted by \({{{{\bf{c}}}}}_{i,\sigma }^{{\dagger} }\) (\({{{{\bf{c}}}}}_{i,\sigma }\)). The physics of this model was thoroughly examined in ref. 48. In Eq. (1), \({{{{\rm{t}}}}}_{i}=\tilde{t}+{(-1)}^{i}\tilde{\delta t}\) and Δ is an antiferromagnetic spin splitting. In the absence of V, Eq. (1) describes a \({{{\mathcal{PT}}}}\)-invariant system similar to the inversion-invariant topological insulator described in ref. 49. Inclusion of V breaks the \({{{\mathcal{PT}}}}\) invariance leading to multiferroic behavior.

Results

Model

In ref. 32, a rather profound extension of the physics of the SSH model36 was developed. This research proposed a dimerized lattice model in two dimensions that displays a remarkable property: the emergence of a complex, non-trivial topological phase in a system with zero Berry curvature. Interestingly, this property is related to the Berry connection.

They analyzed a Hamiltonian defined as \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{SSH}}}}}\) (see “Methods” section for details). On a square grid, the electron operators at position (i, j) are denoted by \({{{{\bf{c}}}}}_{i,j}^{{\dagger} }\) and \({{{{\bf{c}}}}}_{i,j}\), which represent the creation and destruction operators of a spinless electron at that location. This model satisfies the time-reversal (\({{{\mathcal{T}}}}\)) and inversion (\({{{\mathcal{P}}}}\)) symmetries.

The proposed model highlights the Berry connection as the main contributor to the non-trivial topological phase. A fascinating relationship between the Berry connection and momentum space emerges when the former is integrated across the latter, resulting in the 2D Zak phase. This phase gives rise to fractional wave polarization in all directions, which has observable effects. The polarization produces edge states that exhibit the same energy levels but have opposite parities, directly demonstrating wave polarizations in the 2D lattice model.

Just as in the 1D system, we now regard a spinful system. We include the s-d exchange interaction, where each lattice site interacts with a local moment system that has a staggered arrangement of \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}\). This arrangement of the magnetic moments is depicted with a cross or a circle in Fig. 1a. In addition, a staggered local potential \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{ST}}}}}\) is included; see Fig. 1b. \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{ST}}}}}\) is a staggered potential that oscillates with the wave vector q = (π, π). The key factor is that the system is still diagonal in spin space and can be regarded as cohabiting two replicas, for the up and down spin, of the same two-dimensional spin-dependent Rice-Mele model, only differing by the sign of Δ. Despite the apparent similarities between these contributions, the one-dimensional Rice-Mele model in Eq. (1), there is a difference between them. Both \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}\) and \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{ST}}}}}\) preserve the inversion symmetry.

Fig. 1: Bulk structure of the square spin lattice with local potential configurations and Brillouine zone.
figure 1

a The 2D Rice-Mele model is implemented on a square lattice. This model consists of intracell and intercell hopping, represented by w and u, respectively. The spins at each site exhibit antiferromagnetic ordering, denoted by z. b A model with a staggered potential Vs (resembling a checkboard pattern) is considered. This potential possesses a C2 symmetry. c A model with a staggered potential \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{y}\) along the y-axis is investigated. This potential exhibits a C2v symmetry. d The first Brillouin zone of the 2D square lattice is depicted. The side length of the actual square lattice is simplified to a = 1. The points of maximum symmetry, namely Γ, X, Y, and M, are emphasized.

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To break \({{{\mathcal{P}}}}\), we need to add a parity breaking term in the form of a potential that oscillates with the wave vector q = (0, π): \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{{{{\rm{y}}}}}\), see Fig. 1c or q = (π, 0): \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{{{{\rm{x}}}}}\). Our final model is composed by the addition of the terms above: \({{{\boldsymbol{{{{\mathcal{H}}}}}}}}={{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{SSH}}}}}+{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{ST}}}}}+{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}+{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{{{{\rm{x}}}}}+{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{{{{\rm{y}}}}}.\)

Symmetry analysis and topological characterization

The contribution of each term of the Hamiltonian can be seen reflected in Fig. 2 which illustrates the band structure of the Rice-Mele model under varying conditions, highlighting the impact of different parameters on the system and showing how the system loses its degeneracy for the up and down spins, giving finally a magnetic order to the system when all the terms of the Hamiltonian are present. In Fig. 2a the band structure illustrates a degeneration for spin up and down, which is associated with symmetry PT. In Fig. 2b, we add a local perturbation through alternated potential along the y-axis given by \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}\). The band structure still degenerates for a spin, and the system preserves the PT symmetry.

Fig. 2: Band structures for the cases of preserved and broken \({{{\mathcal{P,\; T}}}}\) and \({{{\mathcal{PT}}}}\) symmetry.
figure 2

This panel showcases the Rice-Mele model’s band structure with u = 4.0 and w = 2.0 under different conditions, emphasizing the influence of various potentials on the system. Panel (a) reflects the SSH model’s conditions without external potentials, highlighting significant spin interactions with s-d coupling at Δ = 2.0, illustrating spin interactions in a potential-free setting. Panel (b) introduces a staggered potential \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}=2.5\), breaking the inversion symmetry along the y-axis and hinting at electronic properties. Panel (c) examines the effects of a staggered checkerboard potential Vs = 1.0 on the band’s structure, suggesting possible localized states or electronic modulations. Panel (d) explores a combination of staggered potentials Vs = 1.0 and \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}=2.5\), offering insights into the effects of multiple potentials on electronic and spin properties. Collectively, these panels provide a thorough understanding of the Rice-Mele model’s response to different potentials, underlining its utility in studying complex spin interactions and symmetry-breaking in condensed matter physics.

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The PT symmetry of the system ensures the absence of multiferroic features, in a clear resemblance to the spinless Rice-Mele 2D model. Figure 2c introduces a staggered potential Vs = 1.0, like a chessboard that breaks time-reversal symmetry (T). Finally, Fig. 2d delves into the combined effects of staggered potentials Vs and \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}\). The PT symmetry is completely broken, and the system becomes a topological multiferroic material.

By writing the Hamiltonian in momentum space, we reduce it to block-diagonal form. 2D-SSH, the staggered potential, and the parity-breaking contributions are encoded in the Hamiltonian matrix50:

$$\begin{array}{ll}{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{0}({{{\bf{k}}}})\,=\,{\mathbb{1}}\,\otimes {{{{\boldsymbol{\tau }}}}}_{x}(w+u\cos {{{{\rm{k}}}}}_{{{{\rm{x}}}}})+{\mathbb{1}}\,\otimes {{{{\boldsymbol{\tau }}}}}_{{{{\rm{y}}}}}\,{{{\rm{u}}}}\sin {{{{\rm{k}}}}}_{{{{\rm{x}}}}}\\ \qquad\qquad\quad+\,{{{{\boldsymbol{\tau }}}}}_{x}\otimes {\mathbb{1}}(w+u\cos {{{{\rm{k}}}}}_{{{{\rm{y}}}}})+{{{{\boldsymbol{\tau }}}}}_{{{{\rm{x}}}}}\otimes {\mathbb{1}}\,{{{\rm{u}}}}\sin {{{{\rm{k}}}}}_{{{{\rm{y}}}}}\\ \qquad\qquad\quad+\,{{{{\rm{V}}}}}_{{{{\rm{s}}}}}\,{{{{\boldsymbol{\tau }}}}}_{{{{\rm{z}}}}}\otimes {{{{\boldsymbol{\tau }}}}}_{{{{\rm{z}}}}}+{{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{x}}}}}\,{\mathbb{1}}\otimes {{{{\boldsymbol{\tau }}}}}_{{{{\rm{z}}}}}+{{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}\,{{{{\boldsymbol{\tau }}}}}_{{{{\rm{z}}}}}\otimes {\mathbb{1}}\end{array}$$
(2)

This interaction acts on each spin component, represented in each diagonal block of the Hamiltonian, being invariant in the dimension of the spins, expressed as \({\bar{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}}_{0}({{{\bf{k}}}})={\mathbb{1}}\otimes {{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{0}({{{\bf{k}}}})\), and the s-d interaction his expressed in this dimension, as \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}=\Delta \,{{{{\boldsymbol{\tau }}}}}_{z}\otimes ({{{{\boldsymbol{\tau }}}}}_{z}\otimes {{{{\boldsymbol{\tau }}}}}_{z})\). The complete Hamiltonian matrix becomes \({{{\boldsymbol{{{{\mathcal{H}}}}}}}}({{{\bf{k}}}})={\bar{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}}_{0}({{{\bf{k}}}})+{{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}\).

We now determine the system’s multiferroic properties described by \({{{\boldsymbol{{{{\mathcal{H}}}}}}}}({{{\bf{k}}}})\) and its topological aspects inherited from the 2D-SSH model.

Topological multiferroic phase

As long as Vd = 0, the two terms act as complementary staggered models segregated by spin species. They break the \({{{\mathcal{T}}}}\) symmetry but maintain the combined \({{{\mathcal{PT}}}}\) symmetry. This opens the door to a quantum effect known as the piezospintronic effect51,52. According to the tight-binding model described by Eq. (1), when hopping amplitudes depend on a spatial parameter, the system can produce pure spin currents when subjected to mechanical stress53. The two staggered components with opposite charge dipolar moments (one for each spin species) contribute their dipolar moment to a non-zero spin dipolar moment. The inversion symmetry imposes a significant constraint on the value of the wave polarization Pσ Eq. (3). Due to symmetry, \({{{{\rm{P}}}}}_{\sigma }^{{{{\rm{x}}}}}={{{{\rm{P}}}}}_{\sigma }^{{{{\rm{y}}}}}\). Through a band inversion, based on the above equation, one derives Pσ = (1/2, 1/2) for the two topologically non-trivial band gaps. On the one hand, the charge dipolar moment density \({{{{\bf{P}}}}}_{e}=({{{{\bf{P}}}}}_{\uparrow }+{{{{\bf{P}}}}}_{\downarrow })=0\) mod 1. In the same way, the spin dipolar moment density \({{{{\bf{P}}}}}_{S}=({{{{\bf{P}}}}}_{\uparrow }-{{{{\bf{P}}}}}_{\downarrow })=0\) mod 1.

$${{{{\bf{P}}}}}_{\sigma }=\frac{1}{{(2\pi )}^{2}}\int\int{{{{\rm{dk}}}}}_{{{{\rm{x}}}}}{{{{\rm{dk}}}}}_{{{{\rm{y}}}}}{{{\rm{Tr}}}}\left[{{{{\bf{A}}}}}_{\sigma }({{{{\rm{k}}}}}_{{{{\rm{x}}}}},{{{{\rm{k}}}}}_{{{{\rm{y}}}}})\right]{{{\rm{mod}}}}\,1$$
(3)

Where Aσ = 〈ψ∣i πσ∇k∣ψ〉 represents the Berry connection and the integration spans the first Brillouin zone (BZ) and πσ is a projector in the spin band σ.

We differentiate two topological phases how trivial and non-trivial, in ssh 2d like model this is defined by the hopping relations ∣w∣ < ∣u∣ (trivial phase) and ∣w∣ > ∣u∣ (non-trivial phase), the of origin of this is based on the band inversion, and the adiabatically disconnection for ∣w∣ = ∣u∣, this relation is shown in the Supplementary Fig. 1. The inversion symmetry imposes a significant constraint on the value of the wave polarization Pσ Eq. (3).

An intricate topological multiferroic phase in the material raises the possibility of electrically manipulating spin-conducting edge states. This feature may pave the way for creating innovative spintronic devices that offer more flexibility in information processing and spin transport by being able to be adjusted and controlled by external electric fields.

The findings from research on the multiferroic topological phase in the 2D Rice-Mele model pave the way for developing and using sophisticated spin-dependent transport devices. Researchers can investigate creative device architectures that take advantage of the interplay between spin, topology, and multiferroic behavior for improved performance and functionality by harnessing the material’s electrical and spin properties.

Symmetry-breaking effects on electronic and spin properties

Breaking a symmetry in quantum physics often leads to wonderful things. In this regard, the term proportional to Vd is fundamental, as it obliterates any remaining \({{{\mathcal{PT}}}}\)-symmetry. This, in turn, results in the emergence of a net electric dipole moment density and spin density (magnetization) that would not have been present in the symmetric regime. Ultimately, this leads to a topologically induced multiferroic, as illustrated in Fig. 3, which shows that the dipolar moment density and the spin dipolar moment density do not vanish.

Fig. 3: Non-quantized electric dipolar moment and spin dipolar moment.
figure 3

This panel illustrates the behavior of the electric charge dipolar moment density Pe (a, b) and spin dipolar moment density Ps (c, d) in the x, y components, for hopping u = 4.0, w = 2.0 with an s-d exchange interaction Δ = 2.0 and staggered potential Vs = 1.0, as a function of a staggered potential \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}\). Plots of Pe (e, f) and Ps (g, h) in the components x, y as a function of the hoppings u, w for different values of \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}\).

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Although the additional potential is spin-independent, a spin imbalance is created in the system because the potential favors electrons in the lower-energy region with a preferred spin orientation. As a result, the system develops an overall ferromagnetic moment. On the other hand, the correction ±Vd to the local energy at the sites left the Rice-Mele contribution unbalanced for the electric charge-dipolar moment of each spin species. Therefore, there is a non-zero net electric charge dipolar moment density Pe = (P↑ + P↓) ≠ 0.

The model discussed is an example of a multiferroic system that does not rely on spin-orbit coupling but rather on the topological nature of the polarization to grant permanent ferroelectric moment and a non-trivial magnetic response. Such materials are consistent with previous predictions48.

Likewise, a fundamental characteristic of topological insulators (TI) is the emergence of edge states in non-trivial topological phases, also observed in our system13,53. We draw the reader’s attention towards Fig. 4. This depiction reveals the intrinsic topological attributes of the band, which is especially remarkable in situations characterized by zero Chern numbers. Central to these scenarios is the manifestation of edge and corner states akin to those seen in the renowned 2D Su-Schrieffer-Heeger (SSH) model. What sets these observations apart is their spin-dependent nature, a feature that stems from the multiferroic character of the examined system. Multiferroic materials, known for their intertwined electric and magnetic orders, are fertile grounds for emergent physical phenomena. In this context, the bands sensitive to spin orientation are particularly intriguing, with the potential to lay the foundation for sophisticated spin devices such as spin filters and spin transistors. These devices leverage spin-selective band properties, offering significant prospects. Harnessing such properties could significantly enhance the precision and efficiency of electron spin control in various technological implementations.

Fig. 4: Band structure and edge states for semi-infinite nanoribbon and finite square spin lattice.
figure 4

The nanoribbon used for the following calculations is of finite size equal to 20 sites along the x-axis, and infinite along the y-axis, with hoppings u = 4.0 and w = 2.0, incorporates an s–d coupling Δ = 2.0 and a staggered potential Vs = 1.0. In (a) and (b), the staggered potential Vd is configured along the y-axis \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}=2.5\) and the x-axis \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{x}}}}}=2.5\), respectively. The corresponding band structures and the density of states for cases (a) and (b) are shown in (c) and (d), respectively. The band structure shows the spin-up in red and the spin-down in blue. The edge states for spin up and down for cases (a) and (b) are visually represented in (e) and (f), respectively, where the open and closed circles indicate the respective states in the band structures. g Eigenmodes of a sample for the finite system along the x and y axis for corner state calculations, where the size is 14 × 14 sites, for the up (red) and down (blue) components of the spin, for the parameters of the system u = 4.0 and w = 2.0, the s–d coupling Δ = 2.0, the staggered potential Vs = 1.0 and the staggered potential along the x-axis \({{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{x}}}}}=1.5\) (other configurations are shown in Supplementary 2, 3 and 4). The corner states are enclosed by a solid and segmented circle on the energy spectrum near the gap. In our system, the corner state is separated by the spin component and energy value (h), allowing the four corners to be exited separately.

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Discussion

Drawing parallels with the one-dimensional Rice-Mele Hamiltonian, our model acts as a two-dimensional counterpart to the spin-dependent antiferromagnetic Rice-Mele model in one dimension48. It showcases topological boundary states, inherently linked to its non-trivial bulk topology, a feature traditionally ascribed to Berry curvature. However, in our model, the Berry connection plays a crucial role in the genesis of these complex topological phases, despite the absence of Berry curvature. This model provides fertile ground for exploring the impacts of spatially varying potentials on materials’ electronic and spin properties.

The robust edge states arising from the topological multiferroic phase hold promise for a range of spintronics applications. These states, characterized by their resilience to external perturbations, can be leveraged in the design of spin filters, spin transistors, and other spin-dependent transport devices. By incorporating these edge states into device architectures, researchers can improve spin transport efficiency and reliability.

The topologically protected corner states54,55,56,57 that we reported are spin-polarized. This is a distinctive feature of the multiferroic nature of the system we propose. These states could be applied in the context of data storage, manipulation, and retrieval. The latter illustrates the great landscape of potentiality holding within our model.

Although it may seem an intricate endeavor, our ideas are not beyond the scope of the experimental reach. We may mention the experimental results in ref. 35. They studied the atomic and electronic structure of a rectangular Si lattice on Ag(001) using angle-resolved photoemission spectroscopy and theoretical calculations. They demonstrated that the Si lattice hosted gapped Dirac cones at the Brillouin zone corners. Their tight-binding analysis revealed that a 2D SSH model with anisotropic polarizations could describe the Dirac bands. Their results established an ideal platform for exploring the rich physical properties of the 2D SSH model.

Our ideas also have potential applications in related contexts, such as photonics, acoustics, and magnonics. In the former, the topology implementation opened up functionalities in photonic systems, such as topologically protected boundary modes. There are already implementations of 2D-SSH in photonic systems55. There, polarization-dependent topological properties in a 2D Su-Schrieffer-Heeger lattice are theoretically presented using a metallic nanoparticle array and considering the polarization degree of freedom. They demonstrated that when the eigenmodes were polarized parallel to the plane of the 2D lattice, it supported longitudinal edge modes isolated from the bulk states and transverse edge modes overlapping with the bulk states. The in-plane polarized modes also supported a second-order topological phase under an open boundary condition by breaking the four-fold rotational symmetry. This work offered polarization-based multifunctionality in compact photonic systems with topological features. In acoustics, we can mention33. In that work, they experimentally reported the acoustic realization of the two-dimensional Su-Schrieffer-Heeger model in a simple network of air channels. They analytically studied the steady-state dynamics of the system using a set of discrete equations for the acoustic pressure, leading to the two-dimensional Su-Schrieffer-Heeger Hamiltonian matrix without the use of the tight-binding approximation. By building an acoustic network operating in the audible regime, they experimentally demonstrated the existence of a topological band gap. Moreover, they observed the associated edge waves within this band gap, even though the system was open to free space. Their results provided a flexible platform for studying the topological properties of sound waves.

The implications of our findings are significant, especially in the realm of low-power information processing and the development of advanced electronic devices. In the field of spintronics, where the manipulation of spin properties is essential for device functionality, these states present promising avenues for innovation. Our research contributes to the theoretical understanding of topological phases and opens possibilities for practical applications in next-generation electronic and spintronic devices.

In conclusion, we have revealed topological multiferroicity by introducing the Rice-Mele 2D spin-dependent antiferromagnetic model. With a meticulous examination of bulk band polarizations, spin splitting, and spin-polarized edge and corner states, we emphasize the pivotal role of symmetry in driving the multiferroic state. Our analysis extends to potential systems, providing insight into spintronics and related physics domains. With a versatile focus, our model offers great implications for electronics, photonics, magnonics, and acoustics.

Methods

Explicit form of the model

In a square grid, the electron operators located at position (i, j) are symbolized as \({{{{\bf{c}}}}}_{i,j}^{{\dagger} }\) and \({{{{\bf{c}}}}}_{i,j}\), representing the creation and annihilation operators of a spinless electron at that specific point on the grid. The lattice features hopping amplitudes denoted as tx and ty along the x and y axes in a uniform pattern. The strengths of electron-lattice interactions are expressed as δtx and δty. The movement of electrons in the x or y direction depends on the parity of the site (i, j), leading to two types of electron transitions: intracell and intercell. This distinction results in four distinct atoms within a primary cell, distinguished by thick and thin links representing intracell and intercell transitions (refer to Fig. 1a. \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{SSH}}}}}={\sum }_{i,j}({{{{\rm{t}}}}}_{{{{\rm{x}}}}}+\delta {{{{\rm{t}}}}}_{{{{\rm{x}}}}}{(-1)}^{{{{\rm{i}}}}}){{{{\bf{c}}}}}_{{{{\rm{i}}}}+1,{{{\rm{j}}}}}^{{\dagger} }{{{{\bf{c}}}}}_{{{{\rm{i}}}},{{{\rm{j}}}}}+{\sum }_{{{{\rm{i}}}},{{{\rm{j}}}}}({{{{\rm{t}}}}}_{{{{\rm{y}}}}}+\delta {{{{\rm{t}}}}}_{{{{\rm{y}}}}}{(-1)}^{{{{\rm{j}}}}}){{{{\bf{c}}}}}_{{{{\rm{i}}}},{{{\rm{j}}}}+1}^{{\dagger} }{{{{\bf{c}}}}}_{{{{\rm{i}}}},{{{\rm{j}}}}}+\,{{\mbox{h.c.}}}\,\). For simplicity, we make the assumptions that tx = ty, δtx = δty, w = t − δt, and u = t + δt. This model adheres to time-reversal (\({{{\mathcal{T}}}}\)) and inversion (\({{{\mathcal{P}}}}\)) symmetries.

The interaction between electrons and local moments, with the strength Δ, is described by the following Hamiltonian: \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{sd}}}}}=\Delta {\sum }_{ij\sigma {\sigma }^{{\prime} }}{(-1)}^{i+j}{{{{\bf{c}}}}}_{ij,\sigma }^{{\dagger} }{{{{\boldsymbol{\tau }}}}}_{\sigma {\sigma }^{{\prime} }}^{z}{{{{\bf{c}}}}}_{ij,{\sigma }^{{\prime} }}\) breaking time-reversal (\({{{\mathcal{T}}}}\)) and preserving inversion (\({{{\mathcal{P}}}}\)) symmetries.

We include a scalar potential in the form of checkerboard \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{ST}}}}}={{{{\rm{V}}}}}_{{{{\rm{s}}}}}{\sum }_{ij\sigma }{(-1)}^{i+j}{{{{\bf{c}}}}}_{ij,\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{ij,\sigma }\) that preserves the time-reversal and inversion symmetries.

Finally, parity breaking terms are added for each direction y: \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{y}={{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{y}}}}}{\sum }_{ij\sigma }{(-1)}^{j}{{{{\bf{c}}}}}_{ij,\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{ij,\sigma }\), and x: \({{{{\boldsymbol{{{{\mathcal{H}}}}}}}}}_{{{{\rm{PB}}}}}^{x}={{{{\rm{V}}}}}_{{{{\rm{d}}}}}^{{{{\rm{x}}}}}{\sum }_{ij\sigma }{(-1)}^{i}{{{{\bf{c}}}}}_{ij,\sigma }^{{\dagger} }{{{{\bf{c}}}}}_{ij,\sigma }\).

Topology and polarization

A topological phase transition associated with the 2D Zak phase occurs when ∣w∣ = ∣u∣ as shown by the analysis in refs. 32,58 (see Supplementary Fig. 1 for the application of this analysis to the model at hand). This transition is quantified by the 2D extended Zak phase, which also represents the wave polarization, and is given by Eq. (3)44.