On topological frustration and graphene magnonics

Abstract

The graph-theoretic topological frustration is a peculiar situation on a finite piece of the honeycomb lattice that prevents a full pairwise coupling of the lattice sites via nearest neighbor links, even when the total number of sites is an even number. This type of frustration is inherent for organic molecules that are classified as concealed non-Kekulean hydrocarbons, representing peculiar diradicals. Here we show that this topological frustration persists in 2D systems based on honeycomb lattice. Such systems exhibit fully flat electronic energy bands located at the Fermi level. Therefore, 2D ultimately flat bands can be systematically and predictably constructed for graphene monolayer nanomeshes. These systems are prone to antiferromagnetic ordering and hybrid spin-wave excitations mixing weak ferromagnetic and strong antiferromagnetic features, which could pave the way towards low-power, compact, and ultrafast organic spintronics with near room-temperature operation.

Figures (4)

• Figure 1: Topological frustration on torus. (a) A torus embedment of bipartite hexagonal graph based on Mothra Saroka2025, featuring equal number of vertices in subsets $A$ (red) and $B$ (blue): $N_A = N_B = 77$. (b) Mothra-based graph from panel (a) unrolled into a plane. Edges connecting bottom to top and left to the right are not shown for clarity of the picture. Vertices in $A$ (red) and $B$ (blue) subsets are numbered, sequentially. The AI-generated background artwork explains a formal 'Thor' name chosen for this graph. (c) The maximal matching (yellow) returned by Hopcorft-Karp-Karzanof algorithm. The two non-covered vertices represent graph deficit $\eta=2$. (d) A segment of an amchair carbon nanotube $(6,6)$, decorated with nanocarving equivalent to the one in panel (b). A 2D side projection is shown below. Black arrow: view direction. (e) A torus based on long $(6,6)$ nanotube with an embedment of a single unit (black arrow) from panel (b). (f) A piece of 2D topologically frustrated graphene nanomesh, which unit cell is based on Thor-graph in panel (b).
• Figure 2: Obstruction and extension rules. (a) The Tutte-Berge decomposition of Mothra graph, see Ref. Saroka2025, showing obstruction set $S$, together with even subgraph $C$ and odd subgraphs $D$. Edges incident on $S$ are depicted with reduced opacity. (b) The Thor graph obtained by extension from the Mothra graph in panel (a) by adding an even number of vertices and edges between them, which are both highlighted as semitransparent. Black arrows: further possible extension directions.
• Figure 3: Flat bands and magnetism. (a) The TB energy bands along a 1D $k$-path through the high symmetry points. (b) The 2D plot of the four TB bands closest to the Fermi level. Light green: Brillouin zone. Black arrows: the $k$-path through the high symmetry points. (c) The TB+U band structure and the spin density for the initial FM spin configuration. Spin-up and -down species are presented by red and blue, accordingly. Dashed orange line marks the Fermi level adjusted to zero. (d) Same as panel (c) but for the initial AFM spin configuration. $\odot_a$ and $\otimes_b$ mark positions of the spin-up and down gravity centers, respectively. (e) The magnetization of lattice sites highlighted in the inset versus $U$. Dashed gray vertical line denotes the $U$-value used in panels (c) and (d). (f) The magnon dispersion from Eq. (\ref{['eq:MagnonDispersion']}). $\tilde{\epsilon}=0$ plane of the 3D plot hosts a 2D contour plot of the dispersion with the same color scheme. Light green: Brillouin zone. White: iso-energy curves.
• Figure : Antiferromagnetic magnon dispersion for topologically frustrated graphene nanomesh.