Fermi arcs for generic nodal points hosting monopoles or dipoles

TL;DR

The paper addresses the problem of unambiguously characterizing Fermi arcs on the surfaces of 3D topological semimetals hosting generic nodal points. It develops a generic, Hermiticity-preserving boundary-condition framework that yields analytic arc equations for arbitrary N-fold nodes, including multifold and anisotropic or nonlinear dispersions, and even zero-monopole Berry-dipoles. By applying the method to a range of models—from untitled/tilted Weyl and multifold nodes to Hopf-dipoles and quadruple-Weyl points—the authors show that the number and geometry of Fermi arcs reflect the bulk Chern numbers (Berry-monopole charges) and remain robust across diverse band structures. This work provides a unified toolkit for predicting and interpreting ARPES experiments and other probes of surface states in complex 3D semimetals, with potential implications for engineered photonic and phononic systems. The approach lays the groundwork for exploring dynamical responses and quantum oscillations of surface states under external fields.

Abstract

Fermi arcs represent the surface states at the boundary of a three-dimensional topological semimetal with the vacuum, illustrating the notion of bulk-boundary correspondence playing out in real materials. Their special character is tied up with the topological charges carried by the nodes of the semimetal in the momentum space, where two or more bands cross. In fact, they are constrained to begin and end on the perimeters of the projections of the Fermi surfaces of the bands tangentially, signalling their mixing with the bulk states. The number of Fermi arcs grazing onto the tangents of the outermost projection about a given node also reflects the magnitude of the charge at the node (equalling the Berry-curvature monopole), revealing the intrinsic topology of the underlying bandstructure, which can be visualised in experiments like ARPES. Here we take upon the task of unambiguously characterising the analytical structure of these states for generic nodal points, (1) whose degeneracy might be twofold or multifold; and (2) the associated bands might exhibit isotropic or anisotropic, linear- or nonlinear-in-momentum dispersion. Moreover, we also address the question of whether we should get any Fermi arcs at all for topological nodes carrying zero values of monopoles, but representing ideal dipoles.

Figures (6)

• Figure 1: (a) Dispersion ($E$) in isotropic twofold (viz. Weyl node) and fourfold (viz. Rarita-Schwinger-Weyl node) band-crossings against the $k_x k_y$-plane. (b) Schematics of BC-flux lines in the $k_x = 0$ planes for a monopole and an ideal dipole. Here, the dipole-axis is orientied along the $k_z$-axis and, hence, the BC goes to zero on the $k_z=0$ plane in the 3d BZ.
• Figure 2: A pair of conjugate Weyl nodes: The Fermi arcs represent modes with $E = \mu$, where $\mu =0.5$ and $k_w = 1$. Each arc corresponds to a distinct value of $b_1$, colour-coded as shown in the plot-legends. The dashed red curves represent the projections of the bulk FSs on the $k_y k_z$-plane.
• Figure 3: (a) A pair of triple-point nodes: The Fermi arcs represent modes with $E = \mu$, where $\mu = 0.5$ and $k_w =1$. The dashed red curves represent the projections of the bulk FSs (of the non-flat bands) on the $k_y k_z$-plane. The two Fermi arcs emanating tangentially from each FS-projection reflect the magnitude Chern number at each node equalling two. The cyan arc has been obtained by setting $b_1 = -1$. (b) Four Fermi arcs emerging from the FS-projects of the two conduction bands of an isotropic RSW node with $\mu=0.5$ and $b_1 = 0$, indicating the Chern number to be four.
• Figure 4: The Fermi arcs represent modes with $E = \mu$, where $\mu = 0.5$ and $k_w =1$. (a) A pair of double-Weyl nodes source two arcs from the perimeter of each FS- projection. Here, $t = \pi/6$ and $s=1$. (b) A pair of triple-Weyl nodes: source two arcs from the perimeter of each FS- projection. Here, $t = 0$ and $s=1$.
• Figure 5: The Fermi arcs represent modes with $E = \mu$, where $\mu =0.5$ and $k_w =1$. The dashed red closed contours represent the projections of the bulk FSs on the $k_y k_z$-plane. Each arc corresponds to a distinct value of $b_1$, colour-coded as shown in the plot-legends. A pair of nodes carrying ideal BC-dipoles using the model in (a) Eq. \ref{['eqbcd1']}, using $v_p= v_z=1$; (b) Eq. \ref{['eqbcd2']}. In each case, Fermi arcs graze through each FS-projection at two distinct points, meeting tangentially along a FS-projection.