Tunable two-dimensional Dirac-Weyl semimetal phase induced by altermagnetism

Abstract

We demonstrate a tunable Dirac-Weyl semimetal phase in two dimensions, realized by introducing in-plane d-wave altermagnetism into a Dirac system. This phase hosts both a central Dirac point and momentumseparated Weyl points connected by Fermi line edge states. The Weyl point positions--and thus the edge-state connectivity--can be continuously tuned by rotating the altermagnetic axis. In contrast, out-of-plane altermagnetism gaps part of the bulk spectrum while preserving a single Dirac point accompanied by chiral edge modes, as evidenced by quantized edge polarization. Our findings provide a tunable platform for manipulating Dirac-Weyl physics and topological edge transport in two dimensions.

Figures (4)

• Figure 1: (a) Schematic illustration of the $\sqrt{2} \times \sqrt{2}$ square lattice. The unit cell is consist of two sublattices in black and blue as indicated inside the dashed red rectangle. The dashed lines and black solid lines represent the nearest-neighbor and second-nearest-neighbor couplings, respectively. (b) Bulk band structure along high-symmetry lines in the absence of altermagnetic order, showing Dirac points at $X_1$, $X_2$, and $M$ points. (c) Bulk band structure under $x$-direction $d$-wave altermagnetism, where the Dirac points at $X_1$ and $X_2$ split into pairs of Weyl points. Bulk bands are plotted using blue solid and red dashed lines to highlight their degeneracy. (d) Schematic distribution of Dirac and Weyl points in the Brillouin zone. The parameters used are $t = -1$, $t_2 = 0$, $t_s= 0.5$, and $J = 0.3$.
• Figure 2: (a) Energy spectrum as a function of $k_y$ with open boundary conditions along the $x$ direction, showing Fermi line edge states connecting Weyl points. The two edge states highlighted by labels A (blue) and B (red), respectively. (b) Energy spectrum as a function of $k_x$ under open boundary conditions along $y$, where no Fermi line states are observed. (c), (d) Real-space probability distributions $|\phi|^2$ along $x$ for the edge states A and B marked in (a). The parameters used are $t = -1$, $t_2 = 0$, $t_s = 0.5$, and $J = 0.3$.
• Figure 3: Momentum-space distributions of Dirac and Weyl points in the Brillouin zone under different orientations of the $d$-wave altermagnetic order. (a) For an altermagnetic field aligned along the $y$ direction, pairs of Weyl points (red and blue dots) emerge near $X_1$ and $X_2$ and split along the $x$ direction, while the Dirac point (cyan) at $M$ remains intact. (b) Rotating the altermagnetic direction causes the Weyl points to shift accordingly, with their splitting direction always perpendicular to the altermagnetic orientation. The red arrow indicates the direction of the in-plane altermagnetism.
• Figure 4: (a) Bulk band structure along high-symmetry lines under out-of-plane $d$-wave altermagnetic order, showing that only the Dirac point at $M$ remains gapless. (b) Schematic distribution of the residual Dirac point in the Brillouin zone. (c) Energy spectrum as a function of $k_x$ under open boundary conditions along $y$, revealing a pair of chiral edge states (labeled A and B). (d1), (d2) Probability distributions $|\phi|^2$ along the $y$ direction for the edge states A and B, respectively, confirming their localization at opposite boundaries. The parameters used are $t = -1$, $t_2 = 0$, $t_s= 0.5$, and $J = 0.4$.