Berry curvature associated to Fermi arcs in continuum and lattice Weyl systems

Recently it has been discovered that in Weyl semimetals the surface state Berry curvature can diverge in certain regions of momentum. This occurs in a continuum description of tilted Weyl cones, which for a slab geometry results in the Berry curvature dipole associated to the surface Fermi arcs growing linearly with slab thickness. Here we investigate analytically incarnations of lattice Weyl semimetals and demonstrate this diverging surface Berry curvature by solving for their surface states and connect these to their continuum descriptions. We show how the shape of the Fermi arc and the Berry curvature hot-line is determined and confirm the $1/k^2$ divergence of the Berry curvature at the end of the Fermi arc as well as the finite-size effects for the Berry curvature and its dipole, using finite-slab calculations and surface Green's function methods. We further establish that apart from affecting the second-order, nonlinear Hall effect, the divergent Berry curvature has a strong impact on other transport phenomena as the Magnus-Hall effect and the nonlinear chiral anomaly.

Figure 1

Berry curvature, Fermi arc (red line), and hot-line (black line) of the tilted Weyl cone pair Hamiltonian $H_{\text{L}}\left(\mathit{k}\right)=H_{\text{cone}}+H_{\text{tilt}}$ of Eq. (25) on a cubic lattice. The parameters are $V=\mathbb{1}, k_0=\pi /2, \chi =\mathrm{\Delta }=1$ and the tilt is $\mathit{u}=(0.4,0,0.5)$. The gray area denotes delocalized bulk states.

Figure 2

Fermi arcs of the lattice model with four Weyl nodes on surface (001) for different values of $\gamma$ and tilt $u_x$. The projections of the bulk Weyl cones are marked in black. A red (yellow) arc is located on the top (bottom) surface while for orange both arcs are on top of each other.

Figure 3

(a) Fermi arc with Berry curvature of the lattice model with a single tilted Weyl pair on the (001) surface. The tilt is $u_x=1/2$ and $u_z=1/4$, with $V=\mathbb{1}$ and slab size $L=4$. The black dots mark the projections of the Weyl nodes. (b) Surface Berry curvature dipole $\mathit{D}$ in dependence of slab size $L$. The blue line corresponds to $D_x/L$ while the dashed line shows the behavior for large $L$.

Figure 4

Surface Berry curvature (SBC) ${\mathrm{\Omega }}_z^s$ and SBC dipole density ${\mathit{d}}^S$ of the tilted Weyl cone pair surface (001) associated with a surface layer of thickness (a)–(c) $N=1$ and (d)–(f) $N=4$ of a semi-infinite slab with $L\to \infty$. The Fermi arcs are marked in black. The SBC dipole $D_y^S$ vanishes due to the dashed green mirror line $M_x$. (g) SBC of the Fermi arc for different $N$ using Green's functions in dependence of the distance $|\mathbf{k}-{\mathbf{k}}_0|$ from the Weyl node at ${\mathbf{k}}_0=(\pi /2,0)$. The dashed line is the result for the surface state of a finite slab with thickness $L=500$.