Altermagnetism and strain induced altermagnetic transition in Cairo pentagonal monolayer

Abstract

Altermagnetism, a recently discovered class of magnetic order characterized by vanishing net magnetization and spin-splitting band structures, has garnered significant research attention. In this work, we introduce a novel two-dimensional system that exhibits g-wave altermagnetism and undergoes a strain-induced transition from g-wave to d-wave altermagnetism. This system can be realized in an unconventional monolayer Cairo pentagonal lattice, for which we present a realistic tight-binding model that incorporates both magnetic and non-magnetic sites. Furthermore, we demonstrate that non-trivial band topology can emerge in this system by breaking the symmetry that protects the spin-polarized nodal points. Finally, ab initio calculations on several candidate materials, such as FeS₂ and Nb₂FeB₂, which exhibit symmetry consistent with the proposed tight-binding Hamiltonian, are also presented. These findings open new avenues for exploring spintronic devices based on altermagnetic systems.

Introduction

Recently, altermagnetism has attracted growing research interest due to its unconventional behavior, which is distinct from traditional collinear ferromagnetism and antiferromagnetism^{1,2,3,4,5,6,7,8,9}. In altermagnet, magnetic moments form an antiferromagnetic-like order with zero net magnetization, while it exhibits energy splitting between states with opposite spins, similar to ferromagnets. A variety of materials have been proposed or confirmed to exhibit altermagnetism through first-principles calculations and experiments, including RuO₂^10,11,12, MnTe^13,14,15,16, FeSb₂¹⁷, and CrSb^18,19. The unique electronic structure in altermagnets leads to numerous interesting effects and potential applications, such as spin-splitting torque phenomena^20,21, unconventional superconductivity^{22,23,24,25,26,27,28,29}, and distinct variants of Hall effect^{30,31,32,33,34,35,36,37,38}. In addition, altermagnets hold great promise for spintronic applications due to their large spin-splitting and robustness against magnetic field perturbations^2,39,40.

One of the most fundamental questions in the study of altermagnetic materials is how to understand the origin of this emergent behavior, which is essential for predicting new physical properties and exploring potential applications. At the microscopic level, several mechanisms for the emergence of altermagnetism have been proposed, including the interplay between magnetic and nonmagnetic atoms and the anisotropic ordering of local orbitals^24,41. Thus, identifying realistic tight-binding models from a microscopic perspective is crucial for fully understanding this unconventional phenomenon. While several studies have explored altermagnetism using effective models, only a few have investigated the microscopic origin of altermagnetic properties^{24,31,41,42,43}.

In our work, we investigate the microscopic theory for the origin of altermagnetism on a novel two-dimensional pentagonal structure⁴⁴, known as the Cairo Pentagon. We uncover the mystery of altermagnetism on the Cairo pentagonal lattice at a microscopic level by constructing a simple but realistic tight-binding model containing both magnetic and non-magnetic atoms, and demonstrate that the interplay between them plays a key role in the origin of altermagnetism. Another remarkable feature of the pentagonal lattice is its sensitivity to strain, which induces strong in-plane anisotropy^45,46,47. This anisotropy alters the spin-lattice symmetry, thereby impacting the structure of spin-splitting. Calculated band structure using the tight-binding model shows that this strong dependence on strain will lead to a transition between g-wave and d-wave altermagnetism. In addition, we find that this pentagonal altermagnet hosts symmetry protected polarized nodal points^37,38,48,49, and we give examples demonstrating how breaking this symmetry could gap out these nodal points and lead to non-trivial topological bands. Finally, we examine two candidate materials FeS₂ and Nb₂FeB₂ through ab initio calculations, the results successfully reproduce the altermagnetism and its transition under strain as we expect.

Our results represent the first demonstration of strain tuning to achieve different altermagnetic orders in realistic systems, along with a microscopic understanding of the underlying mechanisms. This paves a new avenue for the design and application of strain-tuned spintronic devices based on altermagnetism.

Results

Cairo pentagonal lattice structure and model Hamiltonian

We consider a Cairo pentagonal lattice with space group P4/mbm, shown in Fig. 1a. Due to the presence of non-magnetic sites, the opposite spin sublattices can not be mapped to each other by the combination of time-reversal with translation or inversion, which makes it an ideal platform to investigate the relationship between atomic interplay and altermagnetism.

Fig. 1: Cairo pentagonal lattice structure and its electronic energy bands.

a A schematics of the pentagonal lattice with space group P4/mbm and the tight-binding model parameters. The magnetic sites with spin-up, spin-down, and non-magnetic (nm) sites are represented by red, blue, and gray colors. t, t_m, t_nm,1 and t_nm,2 represent the hopping strengths between the nearest-neighbor magnetic and non-magnetic sites, the nearest-neighbor magnetic sites, the first nearest-neighbor non-magnetic sites, and the second nearest-neighbor non-magnetic sites, respectively. The reflection planes corresponding to the mirror operators ${{\mathcal{M}}}_{x}$, ${{\mathcal{M}}}_{y}$, ${{\mathcal{M}}}_{xy}$ and ${{\mathcal{M}}}_{\bar{x}y}$ are represented by dashed lines. b A schematics of the same pentagonal lattice under the diagonal strain along xy or $\bar{x}y$. c, d Visualization of the higher energy bands from the pair of spin-split bands on this lattice, without and with strain, in the first Brillouin zone. The bands with spin up and down are represented by red and blue, respectively. The yellow lines represent the spin-degenerate nodal lines that cross the Γ point. A number of high-symmetry momentum points are highlighted.

The lattice is centrosymmetric due to the four-fold rotation symmetry C_4z. In one primitive unit cell, there are two magnetic atoms with opposite collinear spins and four non-magnetic atoms. The presence of non-magnetic sites breaks symmetry $\{{C}_{2\perp }| | {\mathcal{P}}{\boldsymbol{\tau }}\}$ while preserving symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{x}(\frac{1}{2},\frac{1}{2})\}$, $\{{C}_{2\perp }| | {{\mathcal{M}}}_{y}(\frac{1}{2},\frac{1}{2})\}$, $\{{C}_{2\perp }| | {{\mathcal{M}}}_{xy}(\frac{1}{2},\frac{1}{2})\}$ and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{\bar{x}y}(\frac{1}{2},\frac{1}{2})\}$ relating two magnetic sublattices, where C_2⊥ is the 180° rotation operator around an axis perpendicular to the spins, ${\mathcal{P}}{\boldsymbol{\tau }}$ is an operator combined by the inversion ${\mathcal{P}}$ with any translation τ, ${{\mathcal{M}}}_{x}$, ${{\mathcal{M}}}_{y}$, ${{\mathcal{M}}}_{xy}$ and ${{\mathcal{M}}}_{\bar{x}y}$ are the mirror operators about the x axis, y axis, and the diagonals xy and $\bar{x}y$, and $\left(\frac{1}{2},\frac{1}{2}\right)$ is a translation of $\frac{1}{2}{{\bf{a}}}_{1}+\frac{1}{2}{{\bf{a}}}_{2}$, where a₁ and a₂ are primitive vectors. Due to symmetry considerations, the pair of electronic states with opposite spins split at general k points but remain degenerate along k_y = 0, k_x = 0, k_x = − k_y and k_x = k_y in the first Brillouin zone, as shown in Fig. 1c. This unconventional spin-splitting ensures that the magnetic pentagonal crystal is classified as an altermagnet.

To model the electronic structure in this pentagonal lattice, we consider a tight-binding Hamiltonian including both magnetic and non-magnetic sites in Eq. (1):

$$\begin{array}{lll}H\,=-\mathop{\sum}\limits _{\langle i,j\rangle ,\sigma }{t}_{ij}{c}_{i\sigma }^{\dagger }{c}_{j\sigma }-J\mathop{\sum}\limits _{i\in \,\text{m}\,,\sigma ,{\sigma }^{{\prime} }}{{\bf{S}}}_{i}\cdot {c}_{i\sigma }^{\dagger }{{\boldsymbol{\sigma }}}_{\sigma {\sigma }^{{\prime} }}{c}_{i{\sigma }^{{\prime} }}\\\qquad \,+\,({\epsilon }_{m}-\mu )\mathop{\sum}\limits _{i\in m,\sigma }{c}_{i\sigma }^{\dagger }{c}_{i\sigma }+({\epsilon }_{nm}-\mu )\mathop{\sum}\limits _{i\in nm,\sigma }{c}_{i\sigma }^{\dagger }{c}_{i\sigma },\end{array}$$

(1)

where ${c}_{i\sigma }^{(\dagger )}$ is the annihilation (creation) operator of an electron at site i with spin σ = ↑, ↓, and the hopping strength between electrons at sites i and j is described by t_ij. In our model, we take into account the hopping t between the nearest-neighbor magnetic site and non-magnetic site, the hopping t_m between the nearest-neighbor magnetic sites, and the hopping t_nm,1 (t_nm,2) between the first (second) nearest-neighbor non-magnetic sites. J is the coupling between the electronic spins and localized magnetic moments S_i, and σ is the Pauli matrix. In this work, we set ${{\bf{S}}}_{i}=(0,0,S{e}^{i{\bf{q}}{{\bf{r}}}_{i}})$ with magnetic wave vector q = (2π/a, 0) and S = 1. The on-site energy of magnetic and non-magnetic sites and their chemical potential are denoted by ϵ_m, ϵ_nm and μ.

We obtain the electronic band structure by diagonalizing the Hamiltonian matrix at each momentum point (see details in Supplementary Note 1). Without spin-orbit coupling, there are no interactions between electrons with opposite spins, and thus spin σ is a good quantum number. Finally, the Hamiltonian in Eq. (1) becomes

$$H=\sum _{n,{\bf{k}},\sigma }{E}_{n,\sigma }({\bf{k}}){f}_{n,{\bf{k}},\sigma }^{\dagger }{f}_{n,{\bf{k}},\sigma },$$

(2)

where ${f}_{n,{\bf{k}},\sigma }\,(\,{f}_{n,{\bf{k}},\sigma }^{\dagger })$ is a fermionic annihilation (creation) operator and E_n,σ(k) represents the corresponding energy dispersion of the nth band with spin σ (n = 1, …, 6, with energy increasing from low to high). For simplicity, we set t as the energy unit in the following discussion.

Altermagnetic band splittings of lattice with C _4z symmetry

The band dispersion of the above pentagonal lattice Hamiltonian exhibits g-wave altermagnetism, as shown in Fig. 2. Figure 2a shows six pairs of bands with inverse spin-splitting along Γ-C and Γ-D. Figure 2b, c plot the energy spin-splitting ΔE_n(k) = E_n,↑(k) − E_n,↓(k) for the 4th and 5th pair of bands. In the first Brillouin zone, there are four spin-degenerate nodal lines, k_y; = 0, k_x = 0, k_x = k_y and k_x = − k_y, crossing the Γ point, as expected from the symmetry. Additionally, the spin-splitting ΔE_n(k) for isolated pairs of bands near the Γ point is directly proportional to ${k}_{x}{k}_{y}({k}_{x}^{2}-{k}_{y}^{2})$ (see details in Supplementary Note 2). All these results illustrate that the tight-binding model successfully realizes g-wave altermagnetism.

Fig. 2: Band dispersion of pentagonal lattice Hamiltonian with C_4z symmetry.

a The electronic band structure along the chosen k path in the first Brillouin zone. The parameters used are t_m = 0.2, t_nm,1 = 0.9, t_nm,2 = 0.6, J = 1, ϵ_m = ϵ_nm = 0 and μ = 0. The spin up and spin down bands are shown in red and blue, respectively. b, c False color plots of the spin-splitting energy ΔE_n(k) = E_n,↑(k) − E_n,↓(k) for the 4th and 5th pair of bands in the first Brillouin zone. Four dashed lines represent the spin-degenerate nodal lines that cross the Γ point.

To understand the origin of g-wave altermagnetism in this pentagonal lattice, we investigate the dependence of spin-splitting on parameters of the tight-binding Hamiltonian in Eq. (1). To quantify the spin-splitting, we introduce two measures: the maximal spin-splitting of the n-th pair of bands ∣ΔE_n∣_max, and the mean spin-splitting over the entire Brillouin zone ∣ΔE∣_mean, which are expressed as

$$\begin{array}{l}| \Delta {E}_{n}{| }_{max}=Max[| \Delta {E}_{n}({\bf{k}})| ],\\ | \Delta E{| }_{mean}=\sqrt{\frac{1}{6{(2\pi )}^{2}}{\int}_{1BZ}{d}^{2}k\sum _{n}| \Delta {E}_{n}({\bf{k}}){| }^{2}}.\end{array}$$

(3)

Figure 3 summarizes how these two quantities vary as functions of J, t_m, t_nm,1 and t_nm,2. The spin-splitting exhibits a complex dependence on the parameters of the tight-binding model, and the different pairs of electronic bands show significant distinctions. It is evident that the hopping t between magnetic and non-magnetic sites, and the coupling J between the electronic spins and localized magnetic moments are necessary for the system to exhibit altermagnetism. If t = 0, the magnetic sites form a normal Néel antiferromagnet; while if J = 0, there is no distinction between the sub-Hamiltonians for spin up and down.

Fig. 3: Dependence of spin-splitting energy on tight-binding model parameters.

Plots of the maximal spin-splitting for the n-th pair of bands, ∣ΔE_n∣_max, and the mean spin-splitting over the entire Brillouin zone, ∣ΔE∣_mean, as functions of a J, b t_m, c t_nm,1, and d t_nm,2, while other parameters are fixed as t_m = 0.2, t_nm,1 = 0.9, t_nm,2 = 0.6, J = 1, ϵ_m = ϵ_nm = 0 and μ = 0. Different colors represent the band pairs, ordered from lower to higher energy, while gray represents the mean spin-splitting ∣ΔE∣_mean.

In addition to these factors, all spin-splittings ∣ΔE_n∣_max and ∣ΔE∣_mean vanish when either t_nm,1 or t_nm,2 becomes zero. In these cases, the Hamiltonian in Eq. (1) always exhibits band degeneracy E_n,σ(k_x, k_y) = E_n,σ(k_x, − k_y) = E_n,σ( − k_x, k_y), as well as symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{x}(\frac{1}{2},\frac{1}{2})\}$ and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{y}(\frac{1}{2},\frac{1}{2})\}$ (see details in Supplementary Note 3), which lead to band degeneracy E_n,↑(k) = E_n,↓(k) at all k points. The above reasoning indicates that the hopping terms between non-magnetic sites are also crucial for the emergence of altermagnetic spin-splitting in the pentagonal lattice.

Altermagnetic band splittings of lattice under x y and ${\bar{x}}y$ strains

Next, we study the impact of strain along the diagonal directions xy and $\bar{x}y$ on the altermagnetism of the Cairo pentagonal lattice while keeping the Néel antiferromagnetic spin arrangement, as shown in Fig. 1b. After applying the strain, a significant in-plane anisotropy emerges due to the lattice’s sensitivity to it. This anisotropy breaks the lattice symmetry C_4z, and the symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{x}(\frac{1}{2},\frac{1}{2})\}$ and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{y}(\frac{1}{2},\frac{1}{2})\}$ relating two magnetic sublattices, while preserving two other symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{xy}(\frac{1}{2},\frac{1}{2})\}$ and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{\bar{x}y}(\frac{1}{2},\frac{1}{2})\}$. The change in symmetry results in a transformation in the structure of unconventional spin-splitting, as shown in Fig. 1d.

To incorporate the effect of strain into the tight-binding Hamiltonian, we introduce anisotropic hopping between sites along the xy and $\bar{x}y$ directions. For simplicity, we keep the positions of atoms unchanged. Consequently, the hopping term in Eq. (1) is modified as

$${H}_{S}^{hp}=-(1+\delta )\sum _{{\langle i,j\rangle }_{xy}}{t}_{ij}{c}_{i\sigma }^{\dagger }{c}_{j\sigma }-(1-\delta )\sum _{{\langle i,j\rangle }_{\bar{x}y}}{t}_{ij}{c}_{i\sigma }^{\dagger }{c}_{j\sigma },$$

(4)

where the anisotropy δ reflects the strength of strain.

The band dispersion relations are shown in Fig. 4a. Unlike the previous case, six pairs of bands now exhibit inverse spin-splitting along X-Γ-C and Y-Γ-D. The energy spin-splittings ΔE_n(k) shown in Fig. 4b, c demonstrate that there are only two spin-degenerate nodal lines, kx = ± ky, crossing the Γ point, while the two former nodal lines are shifted away. The change in nodal lines is consistent with the symmetry breaking induced by diagonal strain, while the two preserved symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{xy}(\frac{1}{2},\frac{1}{2})\}$ and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{\bar{x}y}(\frac{1}{2},\frac{1}{2})\}$ enforce the existence of the remaining two nodal lines. In Fig. 4d and e, we zoom in on the above plots near the Γ point, where the spin-splitting ΔE_n(k) is directly proportional to ${k}_{x}^{2}-{k}_{y}^{2}$ (see details in Supplementary Note 2) and thus exhibits d-wave symmetry. Consequently, as a result of the applied strain, the system transitions from a g-wave to a d-wave altermagnetism in the neighborhood of the Γ point.

Fig. 4: Band dispersion of pentagonal lattice Hamiltonian under xy and ${\bar{x}}y$ strains.

a The electronic band structure along the chosen k path in the first Brillouin zone. The parameters used are δ = 0.1, t_m = 0.2, t_nm,1 = 0.9, t_nm,2 = 0.6, J = 1, ϵ_m = ϵ_nm = 0 and μ = 0. The bands of spin up and down are labeled by red and blue. b, c False color plots of the spin-splitting energy ΔE_n(k) = E_n,↑(k) − E_n,↓(k) for the 4th and 5th pair of bands in the first Brillouin zone. d, e Plots of ΔE_n(k) near the Γ points, within ∣k_x∣ ≤ 10⁻²π/a and ∣k_y∣ ≤ 10⁻²π/a.

Additionally, we examine the dependence of spin-splitting on the strain strength. Figure 5a illustrates the maximal spin-splitting ∣ΔE_n∣_max and the mean spin-splitting ∣ΔE∣_mean as functions of anisotropy δ. The sensitivity of the band splitting to strain varies among different pairs of bands, while the variation in the mean spin-splitting ∣ΔE∣_mean is relatively small. Within the range of ∣δ∣≤0.5, the maximum splitting can reach slightly higher than the energy level of t. For comparison, Fig. 5b depicts the case where the next-nearest-neighbor hopping between the magnetic and nonmagnetic sites t_nm,2 is zero. As discussed earlier, the unstrained system exhibits normal spin-degeneracy in this case. Upon applying diagonal strain, the system transitions from a normal antiferromagnetic phase to a d-wave altermagnetic phase, with the maximal spin-splitting showing an almost linear dependence on δ for small values of δ.

Fig. 5: Dependence of spin-splitting on strain strength.

Plots of the maximal spin-splitting for the n-th pair of bands, ∣ΔE_n∣_max, and the mean spin-splitting over the entire Brillouin zone, ∣ΔE∣_mean, as functions of anisotropy δ under a t_nm,2 = 0.6 and b t_nm,2 = 0, with other parameters being t_m = 0.2, t_nm,1 = 0.9, _nm,2 J = 1, ϵ_m = ϵ_nm = 0 and μ = 0. Different colors represent the band pairs, ordered from lower to higher energy, while gray represents the mean spin-splitting ∣ΔE∣_mean.

Polarized nodal points and band topology

In addition to the nodal lines formed by spin-degeneracy, several spin-polarized nodal points are present in this altermagnetic pentagonal lattice. As shown in Fig. 6a, eight spin-polarized nodal points emerge at the band crossings of the 4th and 5th bands with the same spin. These nodal points are protected by the symmetry $\{{\mathcal{T}}{C}_{2\perp }| | {\mathcal{T}}{\mathcal{P}}\}$, with each nodal point carrying a π Berry phase, where ${\mathcal{T}}$ represents the time-reversal operator. Thus, nontrivial band topology can be achieved by breaking this symmetry. Here, we explore two mechanisms that gap out the spin-polarized nodal points using toy models: breaking the lattice inversion symmetry $\{E| | {\mathcal{P}}\}$ through the anisotropic hopping term between magnetic atoms, represented by

$${H}_{1}=\sum _{i\in (m\uparrow ),\sigma }\sum _{{\boldsymbol{\xi }}}\left({t}_{1}{c}_{i,\sigma }^{\dagger }{c}_{i+{\boldsymbol{\xi }},\sigma }-{t}_{1}{c}_{i,\sigma }^{\dagger }{c}_{i-{\boldsymbol{\xi }},\sigma }\right)+h.c.,$$

(5)

and breaking the lattice time-reversal symmetry $\{{\mathcal{T}}{C}_{2\perp }| | {\mathcal{T}}\}$ through the complex hopping term between magnetic atoms, represented by

$${H}_{2}=-\sum _{i\in (m\uparrow ),\sigma }{(-1)}^{\sigma }\sum _{{\boldsymbol{\xi }}}i\left({t}_{2}{c}_{i,\sigma }^{\dagger }{c}_{i+{\boldsymbol{\xi }},\sigma }+{t}_{2}{c}_{i,\sigma }^{\dagger }{c}_{i-{\boldsymbol{\xi }},\sigma }\right)+h.c.,$$

(6)

where ${\boldsymbol{\xi }}=\frac{1}{2}a\hat{x}\pm \frac{1}{2}a\hat{y}$.

Fig. 6: Band structure and Berry curvature after gapping out the polarized nodal points.

a Eight spin-polarized nodal points formed by band crossing between the 4th and 5th pairs of bands. b The electronic band structure for the 4th and 5th pairs of bands under inversion symmetry $\{E| | {\mathcal{P}}\}$ breaking with t₁ = 0.2, and c, d the corresponding Berry curvature Ω(k) of the 5th pair of spin-up and spin-down bands. e The electronic band structure for the 4th and 5th pairs of bands under time-reversal symmetry $\{{\mathcal{T}}{C}_{2\perp }| | {\mathcal{T}}\}$ breaking with t₂ = 0.2, and f, g the corresponding Berry curvature Ω(k) of the 5th pair of spin-up and spin-down bands. The other parameters in the calculations are t_m = 0.2, t_nm,1 = 0.9, t_nm,2 = 0.6, J = 1, ϵ_m = ϵ_nm = 0 and μ = 0.

The impact of breaking the symmetry $\{E| | {\mathcal{P}}\}$ using Eq. (5) is illustrated in Fig. 6b. All spin-polarized nodal points are gapped out, while the spin-degeneracy nodal lines are preserved. We plot the Berry curvature Ω(k) of the 5th pairs of bands with opposite spins after inversion symmetry breaking in Fig. 6c, d, where the peaks of Ω(k) appear at the former nodal points (see details in Supplementary Note 4). In this case, the Berry curvature satisfies Ω(k) = − Ω( − k) for all k points, resulting in Chern number ${{\mathcal{C}}}_{\sigma }=0$. In contrast, the band structure after $\{{\mathcal{T}}{C}_{2\perp }| | {\mathcal{T}}\}$ symmetry breaking using Eq. (6) and the Berry curvature of the 5th pairs of bands are shown in Fig. 6e, f and g. In each band, this symmetry-breaking term in Eq. (6) leads to an equivalent contribution from four nodal points. This mechanism mirrors the behavior seen in Haldane’s model, making the 5th pair of bands topologically non-trivial with a Chern number ${{\mathcal{C}}}_{\sigma }=\mp 2$.

Candidate materials

A two-dimensional form of pyrite, FeS₂, has been predicted to exhibit a Cairo pentagonal structure with antiferromagnetic order⁵⁰. It belongs to space group P4/mbm (No.127) and point group D_4h. Fig. 7a shows the crystal structure, where each Fe atom is surrounded by four S atoms. To examine the magnetic ground state, we perform ab initio calculations to calculate the energy of different collinear magnetic states on this planar structure under geometry optimizations (see details in Supplementary Note 5). The result indicates that the most stable configuration exhibits the Néel antiferromagnetic spin arrangement on Fe atoms while S atoms are non-magnetic, as shown in Fig. 7a.

Fig. 7: Crystal structure and electronic bands of FeS₂.

a Crystal structure of FeS₂. The red and blue colors of Fe atoms indicate the directions of their localized magnetic moments, while the orange arrows represent the direction of diagonal strain. b The electronic band structure and the spin-splitting of the 30th pair of bands of FeS₂ without strain. c, d Spin density isosurfaces for two pairs of bands without strain, illustrating narrow and large spin-splitting, respectively (the inset zooms in on energy by a factor of 5). e The electronic band structure of FeS₂ with diagonal strain, where the angle γ between lattice vectors a and b is tuned to γ = 88°. f, g False color plot of the spin-splitting of the 30th pair of bands of FeS₂ without and with diagonal strain (indicated by the blue arrow in b and e, see details in Supplementary Note 6). The coordinate (k₁, k₂) corresponds to reciprocal lattice vector b_r = k₁b₁ + k₂b₂.

Thus, this planar FeS₂ structure corresponds to the pentagonal lattice system discussed above. In particular, it exhibits the symmetries $\{{C}_{2\perp }| | {{\mathcal{M}}}_{x}(\frac{1}{2},\frac{1}{2})\}$, $\{{C}_{2\perp }| | {{\mathcal{M}}}_{y}(\frac{1}{2},\frac{1}{2})\}$, $\{{C}_{2\perp }| | {{\mathcal{M}}}_{xy}(\frac{1}{2},\frac{1}{2})\}$, and $\{{C}_{2\perp }| | {{\mathcal{M}}}_{\bar{x}y}(\frac{1}{2},\frac{1}{2})\}$, which relate two magnetic sublattices. The electronic band structure and the spin-splitting of one pair of the bands obtained from ab initio calculations are shown in Fig. 7b, f, displaying features consistent with the tight-binding model result in Fig. 2, that inverse spin-splitting occurs along the Γ-C and Γ-D paths, while spin-degeneracy is observed along the Γ-X, Γ-Y, and Γ-M paths. These behaviors characterize planar pentagonal FeS₂ as a g-wave altermagnet as expected. Additionally, we present the spin density isosurfaces for bands with different levels of spin-splitting in Fig. 7c, d, where a larger spin-splitting coincides with a more anisotropic spin density.

Next, we study the effect of strains along the diagonal directions on planar FeS₂, as indicated by the arrows in Fig. 7a. We introduce the strain in ab initio calculations by slightly reducing the angle γ between lattice vectors a and b to less than 90°. For a small change Δγ = 2°, our ab initio calculation shows that the Néel antiferromagnetic spin arrangement is still the most stable configuration (see details in Supplementary Note 5). Fig. 7e and g present the electronic band structure and the spin-splitting for γ = 88. 0°, where inverse spin-splitting appears along the Γ-X-C-Γ and Γ-Y-D-Γ paths, while the bands remain degenerate along the Γ-M path, displaying the same features as in the model result shown in Fig. 4. This behavior indicates a transition from g-wave to d-wave altermagnetism, as predicted by the tight-binding model analysis.

In addition to the two-dimensional cases, we highlight Nb₂FeB₂, a three-dimensional material predicted to exhibit g-wave altermagnetism^51,52. Figure 8a shows the crystal structure of Nb₂FeB₂, which consists of alternate layers of Fe-B and Nb atoms along c axis. Each Fe-B layer exhibits the same Cairo pentagonal structure discussed earlier, and the magnetic Fe atoms carry an in-plane Néel antiferromagnetic spin arrangement, which is also examined through ab initio calculation (see details in Supplementary Note 5). Thus, the previous analysis of symmetry-induced altermagnetism also applies to Nb₂FeB₂. Its electronic band structures and spin-splitting without and with the diagonal strain are shown in Fig. 8c–f, where the angle γ between lattice vectors a and b is 90° and 88° respectively. These results demonstrate that Nb₂FeB₂ is a g-wave altermagnet and it undergoes a transition from g-wave to d-wave altermagnetism under diagonal strain, as expected. Furthermore, in both altermagnetic materials, the spin-splitting increases significantly after applying strain, consistent with the model analysis.

Fig. 8: Crystal structure and electronic bands of Nb₂FeB₂.

a Crystal structure of Nb₂FeB₂ shown from two perspectives. The purple arrows indicate the directions of localized magnetic moments on Fe atoms. b The first Brillouin zone of Nb₂FeB₂ and high symmetry moment points of our interest. c, d The electronic band structures of Nb₂FeB₂ under two conditions: without diagonal strain (where the angle γ between lattice vectors a and b is 90°) and with diagonal strain (where γ = 88°). e, f False color plot of the spin-splitting of the 45th pair of bands of Nb₂FeB₂ without and with diagonal strain (indicated by the blue arrow in (c, d), see details in Supplementary Note 6). The coordinate (k₁, k₂) corresponds to reciprocal lattice vector b_r = k₁b₁ + k₂b₂.

Discussion

Our study uncovers the first microscopic mechanism underlying g-wave altermagnetism in two dimension, realized in Cairo pentagonal lattice. Based on the symmetry analysis, we propose a simple but realistic tight-binding model containing both magnetic and non-magnetic sites that successfully realizes the altermagnetic band spin-splitting with g-wave symmetry. The lattice-spin symmetry ensured by the presence of non-magnetic atoms guarantees the existence of altermagnetism in this antiferromagnetically ordered pentagonal system. Our analysis of the dependence of spin-splitting on parameters in the tight-binding model provides one of the mechanisms, that both the hopping between magnetic site and non-magnetic site, and the hopping between non-magnetic sites are indispensable for the altermagnetic behavior. We note that there are other possible origins of altermagnetism, such as the anisotropic ordering of local orbitals, which are beyond our discussion in this work.

Another important finding from our model is that applying strain to the crystal can induce a transformation in the type of altermagnetism. In general, strain reduces the original lattice-spin symmetry and introduces anisotropic interactions, which makes the spin-splitting behavior more complex, even changes magnetism of the ground state. In this work, we focus on the cases where the strain is relatively small so that it does not affect the Néel antiferromagnetic spin arrangement in the ground state, which is confirmed by our ab initio calculations. Under this premise, near the Γ point we observe that the strain applied along the diagonal direction of this pentagonal structure causes a transition from g-wave to d-wave altermagnetism. Additionally, we find that this pentagonal altermagnet contains spin-polarized nodal points protected by $\{{\mathcal{T}}{C}_{2\perp }| | {\mathcal{T}}{\mathcal{P}}\}$ symmetry. Breaking this symmetry could lead to non-trivial topological bands and the quantum Hall effect for electrons.

Finally, we identify several candidate materials through ab initio calculations, including monolayer FeS₂ and bulk Nb₂FeB₂, which show promise for exploring altermagnetism in Cairo pentagonal crystal structures. In particular, we examine the effect of diagonal strain on these crystals, observing that altermagnetism transforms from g-wave to d-wave as expected. These findings suggest new paradigm of tuning the type of altermagnetism by strain, opening new possibilities for enabling strain-tuned spintronic devices and the interplay with other degrees of freedom.

Methods

We use Vienna Ab initio Simulation Package (VASP)⁵³ for all density functional theory calculations. Projector-augmented-wave (PAW) potential was utilized for ion-electron interaction, and the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional⁵⁴ was applied to describe electron exchange-correlation interaction. The energy cutoff was set to be 600 eV. More details are in Supplementary Note 5.