Topological Superconductivity in Altermagnetic Heterostructures on a Honeycomb Lattice

TL;DR

This work investigates topological superconductivity in altermagnet–superconductor heterostructures on a honeycomb lattice. It develops a two-dimensional tight-binding Bogoliubov–de Gennes model with $d$-wave altermagnetism, Rashba spin–orbit coupling, and proximity-induced $s$-wave pairing to map a rich topological phase diagram. The authors demonstrate both first-order topology with chiral edge modes (nonzero $C$) and higher-order topology featuring Majorana corner modes, with disorder robustness varying between edge and corner states. They also analyze how lattice geometry and microscopic details influence observable signatures of corner modes, highlighting implications for experimental realization and potential quantum computation applications. Overall, the honeycomb lattice enables a versatile platform where tunable parameters such as the chemical potential control access to distinct topological boundary states.

Abstract

Altermagnet-superconductor heterostructures have been shown, in principle, to provide a route towards realising topological superconductivity, and therefore host topologically protected boundary states. In this work we demonstrate that the topological states observed are dependent on the structure of the underlying lattice. By deriving and analysing a model on a honeycomb lattice, we demonstrate that the topological phase diagram has a rich structure containing both chiral edge modes and Majorana corner modes, the latter of which are an indication of higher-order topology. We analyse the effect of disorder on these states and find that whilst the edge modes are robust to a disordered system, any potential observation of the corner modes may be sensitive to the microscopic details. In particular, we show that vacancies can lead to other low energy bound states that may be difficult to distinguish from the corner modes.

Figures (11)

• Figure 1: The altermagnet-superconducting heterostructure is shown in (a). The altermagnet is $d$-wave so that the exchange terms follow the symmetries depicted in (b). Furthermore, this work considers a honeycomb lattice, see (c), with sublattices $A$ (red) and $B$ (blue). The size of the lattice is $N_x \times N_y$, with $N_x=7$ being the number of sites in a row (rows separated by dashed lines) and $N_y=6$ is the number of rows. The vectors, ${\bm R_j}$, define the nearest neighbours. This system hosts various topological regimes, including chiral edge modes which can be seen using periodic boundary conditions (i.e. on a cylinder as in (d)) or in real space, as in (e). Zero-energy modes localised to the corners of the sample can also be observed as depicted in (f).
• Figure 2: The Chern number as a function of the magnitude of the coupling vector $J$ and chemical potential $\mu$ for the $d_{x^2-y^2}$ symmetry ($\alpha_1=1$, $\alpha_2=0$ in the Hamiltonian). The phase diagram, plotted for $500 \times 500$ uniformly spaced data points, contains a trivial region with no edge modes (region I) as well as regions with edge modes and corner modes. The regions containing topological boundary states can be divided into two classes; first regions with $C\neq 0$ host $|C|$ chiral edge modes representing a first-order TSC, and second, regions with $C=0$. This can be due to either counter-propagating edge modes (region II) or a higher-order TSC (HOTSC) which hosts corner modes (region III). The additional labels correspond to other figures that show the topological states. The parameters used are $M=0$, $\lambda=\Delta=0.3$. The Néel vector is assumed to point in the $z$-direction.
• Figure 3: The energy spectrum of modes on a cylinder geometry. The cylinder is periodic in the $y$-direction and has $100$ sites in the $x$ direction. The average position of the modes along the cylinder, $\langle x \rangle$ is denoted by the colour. (a) corresponds to a regime where $C=1$ where there is a single chiral edge mode per edge. In (b), where $C=-2$ there are 2 edge modes per edge with opposite chirality to (a), and (c) corresponds to region II in Fig. \ref{['fig:HexagonalChern']} where $C=0$. In all plots, $\mu=-0.4$. The values of $J$ used are: (a) $J=0.5$, and (b) $J=0.98$, and (c) $J = 0.7$.
• Figure 4: The eigenstates with the lowest absolute energy plotted as a sum of Gaussian functions centred on each site weighted by the real-space on-site probability $|\psi(x)|^2 = |\langle x | \psi\rangle|^2$. The size of the lattice is $51 \times 50$ sites. The parameters shown are: (a) $J=0.5$$\mu=-0.4$, resulting in Chern number $|C|=1$ and therefore a gapless chiral edge mode; (b) $J=0.8$, $\mu=-0.1$, which corresponds to region III. Here $|C|=0$ and there are zero-energy (see inset) modes localised to the corners of the sample.
• Figure 5: (a): The phase diagram for symmetric hopping, that is $t_1=t_2=t_3=1$, with the sublattice asymmetry given by $M=0.1$. The orbital symmetry used is $d_{x^2-y^2}$. The lattice can still host zero-energy corner modes, for example at $\mu=-0.15, J=0.85$. (b): The phase diagram for the $d_{xy}$-orbital ($\alpha_1=0, \alpha_2=1$ in the Hamiltonian) is shown for $M=0$ and asymmetric hopping. The diamond at the centre of the plot contains corner modes for $\mu \neq 0$. The top, right, and bottom $C=0$ regions contain gapped-out edge modes.