Unconventional Altermagnetism in Quasicrystals: A Hyperspatial Projective Construction

Abstract

Altermagnetism, a novel magnetic phase characterized by symmetry-protected, momentum-dependent spin splitting and collinear compensated magnetic moments, has thus far been explored primarily in periodic crystals. In this Letter, we extend the concept of altermagnetism to quasicrystals -- aperiodic systems with long-range order and noncrystallographic rotational symmetries. Using a hyperspatial projection framework, we construct decorated Ammann-Beenker and Penrose quasicrystalline lattices with inequivalent sublattices and investigate a Hubbard model with anisotropic hopping. We demonstrate that interaction-induced Néel order on such lattices gives rise to alternating spin-polarized spectral functions that reflect the underlying quasicrystalline symmetry, revealing the emergence of unconventional $g$-wave (octagonal) and $h$-wave (decagonal) altermagnetism. Our symmetry analysis and low-energy effective theory further reveal unconventional altermagnetic spin splitting, which is compatible with quasicrystalline rotational symmetry. Our work shows that quasicrystals provide a fertile ground for realizing unconventional altermagnetic phases beyond crystallographic constraints, offering a platform for novel magnetisms and transport phenomena unique to quasiperiodic systems.

Figures (23)

• Figure 1: Schematic illustration of the hyperspatial projective construction of the quasicrystal lattice for altermagnetism. (a) Construction of 1D bipartite Fibonacci quasicrystal from a 2D square lattice with AB sublattices. (b) Construction of a 2D Ammann-Beenker quasicrystal from the hypercubic lattice with two sublattices in 4D. The projective selection window in perpendicular space ($V_\perp$) is an origin-centered regular octagon with a side length of 1 and 2 for hypercubic sites and decorations, respectively. The 2D Ammann-Beenker quasicrystal exhibits two sublattices in physical space ($V_\parallel$). The inset shows the AB sublattices and different hoppings.
• Figure 2: (a) Order parameter $\delta m$ in the plane of [$U,\bar{t}_2=(t_{2r}+t_{2b})/2$] for Eq. \ref{['H_hubbard']} from the self-consistent Hartree-Fock calculations on the Ammann-Beenker quasicrystal lattice with $N=329$ sites under the open boundary condition. The parameters used are $t_1=1$ and $\delta_2=0.2$. (b) Real-space magnetization distribution at $U=3$ and $\bar{t}_2=0.3$. (c) Spin-difference spectral function $\mathcal{A}_\uparrow-\mathcal{A}_\downarrow$ calculated from $H_\mathrm{MF}$ in Eq. \ref{['H_MF']}, where parameters are $|\bm{N}|=3, t_1=1, \bar{t}_2=0, \delta_2=2$, $\iota=0.01$, and $\omega$ is set at the lowest eigenenergy of the spectrum for better visualization. (d) Spin-polarized eigenvalues around $\Gamma$ of the $g$-wave altermagnetic term $H_\mathrm{AM}^{(g-wave)}$ in Eq. \ref{['H_g-wave']}.
• Figure 3: (a) Penrose tiling lattice with two sublattices (A and B), where the red and blue arrows represent nearest intra-sublattice hopping vectors with positive imaginary part within each sublattice, respectively. (b) Rhombic icosahedron as the selection window in 3D $V_\perp$. (c) Spin-difference spectral function $\mathcal{A}_\uparrow-\mathcal{A}_\downarrow$, where parameters are $|\bm{N}|=3, t_1=1, t_2=0, \delta_2=10$, $\iota=0.01$, and $\omega$ is set at the lowest eigenenergy of the spectrum for better visualization. (d) Spin-polarized eigenvalues around $\Gamma$ of the unconventional $h$-wave term $H_\mathrm{AM}^{(h-wave)}$ in Eq. \ref{['H_h-wave']}.
• Figure 4: (a) (b) Initial ABT in (a) the physical and (b) perpendicular space. (c) (d) The 1st order hopping (green lines) and the first class of the 2nd order hopping (red lines) at the center of ABT in (c) the physical and (d) perpendicular space. The octagon represents the selection window. (e) All possible 1st hopping vectors $\mathcal{R}_\mathrm{NNN}$ for vertices in the physical space. (f) All possible 2nd hopping vectors $\mathcal{R}_\mathrm{NN}=\mathcal{R}_\mathrm{NN}^1\cup\mathcal{R}_\mathrm{NN}^2$ for vertices in physical space, where orange vectors $\mathcal{R}_\mathrm{NN}^1$ with polar angles $\theta=\pi/8\pm n\pi /2$, while blue vectors $\mathcal{R}_\mathrm{NN}^2$ with $\theta=-\pi/8\pm n\pi /2$ ($n\in\mathbb{Z}$).
• Figure 5: (a) (b) Initial Penrose tiling in (a) the physical space and (b) the perpendicular space. (c) All possible 1st-order hopping vectors for vertices in the physical space. (d) All possible 2nd-order hopping vectors for the vertices in the physical space.