Topologically Protected Surface Altermagnetism on Antiferromagnets

TL;DR

The paper investigates how surface symmetry breaking in spin-degenerate antiferromagnets can realize altermagnetism (AM) at the crystal surface, and whether such surface AM can be topologically protected. It develops both a simple trivial model and a topological framework, showing that spin-split surface states can be robust when protected by bulk symmetries or by topological invariants such as Dirac nodal lines and Dirac points in AFMs. Two robust routes are demonstrated: (i) AM drumhead states arising from Dirac nodal line semimetals and (ii) AM Fermi arcs in Dirac semimetals, with CuMnAs serving as a concrete material realization. The work broadens the material landscape for AM and proposes surface state spin polarization as a robust, topology- and symmetry-driven phenomenon, with potential implications for spintronic applications in antiferromagnets such as CuMnAs.

Abstract

Altermagnetism (AM) and its associated spin-transport phenomena are typically linked to spin-split electronic band structures in bulk materials. However, the crystal surface has a reduced symmetry with respect to the bulk, which can induce AM at the surface of conventional antiferromagnets (AFMs) $\unicode{x2013}$ a local effect which cannot be detected using bulk properties. In this work we define the symmetry conditions necessary for surface AM and show how it can be topologically protected, rendering it a robust effect. We provide a minimal model for one trivial and two topological examples of surface AM. We show that the spin spectral density, accessible by spin- and angle-resolved photoemission spectroscopy, can exhibit a $d$-wave-like altermagnetic character at the surface, even when the full band structure is completely spin degenerate. Our topological model describes the Dirac semimetal CuMnAs, which provides an existing realization of our theory. Our results identify crystal surfaces as a platform to realize robust, topology- and symmetry-driven unconventional magnetism beyond the bulk classification of magnetic materials.

Figures (7)

• Figure 1: Altermagnetic surface states on spin-degenerate AFMs. (a) The antiferromagnetic bulk symmetry group can break into an altermagnetic surface symmetry group, leading to a spin-split surface Fermi surface (red,blue is the spin character) where the bulk is spin degenerate. (b) In AFM Dirac nodal line semimetals spin-degeneracy is lost at the surface resulting in topologically protected altermagnetic drumhead states. (c) In AFM Dirac semimetals altermagnetic Fermi arcs connect the bulk topological nodes. (d) To unearth the surface behavior described in (a-c) crystals are studied in a slab geometry, i.e., periodic boundary conditions along $x$ and $y$ direction and open boundary conditions in $z$ direction. Here, the crystal structure of Eq. (\ref{['eq:H_DLKK']}) is shown, where spin-up (spin-down) sites are red (blue). The bulk 4-site unit cell is highlighted in blue, labeled by sublattice, and the layer degree of freedom is in the $z$ direction.
• Figure 2: Topologically trivial altermagnetic surface state in a spin degenerate AFM, defined in \ref{['eq:H_DLKK']}. (a) The 3D bulk band structure is spin-degenerate. (b-e) Colorplots of the spin-resolved spectral function with the following color scheme: $\mathcal{A}_{{\uparrow}} - \mathcal{A}_{{\downarrow}}$ sets the coloring from spin-down polarized (blue) over spin-degenerate (black) to spin-up polarized (red); $\mathcal{A}_{{\uparrow}} + \mathcal{A}_{{\downarrow}}$ sets the opacity. (b,c) The bulk spin spectral density $\mathcal{A}_{s}(z=L_z/2)$ in the slab geometry ($L_z = 30$) along a high symmetry path (c) and at the Fermi energy ($E=0$, dashed line) (d). (d,e) The spin spectral density $\mathcal{A}_{s}(z=0)$ in the slab geometry at the surface along a high symmetry path (c) and at the Fermi energy (d). (f) The layer-resolved spin conductivity, expressed as the spin splitter angle $\alpha = 2\arctan((\sigma^{\uparrow}_{xx}-\sigma^{\downarrow}_{xx})/(\sigma^{\uparrow}_{xx}+\sigma^{\downarrow}_{xx}))$ shows that the the spin splitting is only sizable at the surface.
• Figure 3: Topologically protected altermagnetic surface states in an AFM, described by $H_\text{topo}$ following \ref{['eqn:h0', 'eq:H CuMnAs magnetic hoppings']}. (a) The 3D bulk band structure of the Dirac nodal line semimetal (black, $t'=0$) and the Dirac semimetal (yellow, $t'=0.2$) is spin degenerate. Dashed lines indicate overlap of both band structures. The touching points at $E=-0.1$ (shifted with a small chemical potential $\mu$) are the nodal line and the Dirac points. The spin-resolved spectral function in a slab geometry ($L_z = 31$) of the Dirac nodal line semimetal is shown in (b-e) and the Dirac semimetal in (f-i). The color scheme is identical to Fig. \ref{['fig:DLKK_model']}. Panel (b,f) and (d,h) show the bulk and surface spin spectral density of each model along a high symmetry path respectively. The bulk spin spectral density $\mathcal{A}_s(z=15)$ at the Fermi energy ($E=0$, dashed line) shows the Dirac nodal line (c) and the Dirac cones (g). The surface spin spectral density $\mathcal{A}_s(z=0)$ at the Fermi energy shows the corresponding surface states, where closed contours are crossings of the drumhead states and the Fermi energy (e) and open arcs are crossings of the Fermi arcs and the Fermi energy (i). (j,k) The layer-resolved spin-splitter angles $\alpha_x$ ($\alpha_y$), as solid (dashed) line, which can be observed when applying an electric field in $x$-direction ($y$-direction). Panels (j) and (k) correspond to the Dirac nodal line (black) and the Dirac semimetals (orange), respectively.
• Figure S1: Real space mean field simulations of the slab confirm that assuming constant mean fields is a valid assumption. Mean field result of Eq. (\ref{['A:eq:H_DLKK MF slab']}) for $t' = 0.3 t$, $U=6t$, $\delta=1$, around half-filling $n=0.54$. (a) The AFM magnetization $m_z$ tends to increase slightly towards the surface, whereas the density $n_z$ is almost constant. (b) Comparison of the spin splitter effects for layer-dependent mean field results (solid) and using the average values for the mean fields (dashed). (c-f) Comparison of the surface (c-d) and bulk (e-f) spin spectral density (opacity encodes absolute spectral density and the red to blue encodes the spin character) when using the layer-dependent mean field values (c,e) and their mean values (d,f).
• Figure S2: Tight-binding model for Wyckoff position 4a of Pmna. Filled (empty circles) are sites on even (odd) layers accounted by $\tau$-Pauli matrices, yellow (purple) circles are sites of the A (B) sublattices accounted by $\sigma$-Pauli matrices (a) In-plane and out-of-plane hoppings which preserve all three accidental translations in Eqs. (\ref{['A:eq:cumnas:translation1']})-(\ref{['A:eq:cumnas:translation3']}). (b) Further neighbor hoppings which keep only $\bm{t}_{a/2,b/2,c/2}$. (c) Further neighbor hoppings which keep only $\bm{t}_{a/2,c/2}$. (d) (110) hopping $t'$, which gaps out the nodal line into 4 Dirac cones. (e-h) Spin-dependent hoppings corresponding to the magnetic order with odd $M_c\{\tfrac{1}{2},0,\tfrac{1}{2}\}$ symmetry.