Anomalous shift in scattering from topological nodal-ring semimetals

TL;DR

The paper demonstrates that mirror-protected nodal rings in 3D topological semimetals imprint a distinct anomalous scattering shift on reflected electron beams. By constructing a two-band model with a tunable topological charge $χ_h$, it shows that the shift vector field is amplified near the ring and that a refined semicircular circulation $κ_s$ encodes $χ_h$ via $κ_s=-2π χ_h$. The work further shows that $κ_s$ tracks topological phase transitions between nodal rings, Weyl points, and gapped phases, offering a robust experimental signature for probing ring geometry and topology. Overall, anomalous scattering shifts emerge as a powerful diagnostic tool for topological band structures beyond Weyl points.

Abstract

An electron beam may experience an anomalous spatial shift during an interface scattering process. Here, we investigate this phenomenon for reflection from mirror-symmetry-protected nodal-ring semimetals, which are characterized by an integer topological charge $χ_h$. We show that the shift is generally enhanced by the presence of nodal rings, and the ring's geometry can be inferred from the profile of shift vectors in the interface momentum plane. Importantly, the anomalous shift encodes the topological information of the ring, where the circulation of the shift vector field $κ_s$ over a semicircle is governed by the topological charge, with a simple relationship: $κ_s=-2πχ_h$. Furthermore, we demonstrate that the shift and its circulation reflect distinct features of topological phase transitions of the charged rings. This study uncovers a novel physical signature of topological nodal rings and positions anomalous scattering shifts as a powerful tool for probing topological band structures.

Figures (9)

• Figure 1: (a) Illustration of a mirror-protected nodal ring, denoted by the red circle in $k_z=0$ plane. The ring is inside the disk $\mathcal{D}$ and covered by hemisphere $S_N$. (b) and (c) show the Wilson loop on a circle of latitude versus the spherical angle $\gamma$ (as shown in (1)) for model Eq. (\ref{['HH']}) with $n=1$ and $n=2$, respectively. Here, we set $A=1$, $B=1$, $C=-0.2$ and $D=-2$. (d) Phase diagram of model (\ref{['HH']}) in the $C$-$D$ parameter plane. Here, we take $B>0$.
• Figure 2: Distribution of the Berry curvature field $\boldsymbol{\Omega}(\boldsymbol{k})$ for the four phases in the phase diagram Fig. \ref{['nodalstructure']}(d). The plot is for the $k_x=0$ plane. The color map indicates the magnitude of Berry curvature. (a) The phase with nodal ring plus Weyl points, for $C>0$ and $D<0$ . (b) The gapped phase, for $C>0$ and $D>0$. (c) The phase with nodal ring, for $C<0$ and $D<0$. (d) The phase with Weyl points, for $C<0$ and $D>0$. In the calculation, we set $n=1$, $A=1$, $B=0.5$, $|C|=0.5$ and $|D|=0.5$.
• Figure 3: (a) Schematic figure showing the setup. An interface at $x=0$ is formed between an incident medium ($x<0$) and a target medium ($x>0$). (b) An electron beam $\Psi^i$ from the incident medium is scattered at the interface. The reflected beam is described by $\Psi^r$. The reflected beam may experience a spatial shift $\bm \ell$ from the incident point in the interface plane.
• Figure 4: (a) Schematic plot of the band structures of the incident and the target media. The target medium here has a mirror-protected nodal ring near the Fermi level. (b) Illustration of the CAS over a closed loop $C$ in the interface momentum plane.
• Figure 5: (a) Illustration of CAS $\kappa_s$ over a semicircle $C_+$ with symmetric ends $A$ and $\bar{A}$. The red line makes the range of the nodal ring. $\Gamma$ is another path with arbitrary shape but with the two ends $A$ and $\bar{A}$ fixed. (b) Numerical results for the shift vector field $\boldsymbol{\ell}$ in the interface momentum plane for a ring with a trivial charge. The magnitude and the direction of the shift vectors are indicated by the color map and the arrows, respectively. Here, we take $n=0$, $E_F=0.2$, $U=8$, $V=16$, $A=1$, $B=1$, $C=-1$ and $D=-1$. $\xi$ is a unit of length, defined as $\xi=A/B$.