Anomalous shift in scattering from topological nodal-ring semimetals

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.

💡 Research Summary

In this work the authors theoretically investigate the anomalous spatial shift experienced by an electron beam upon reflection from a mirror‑symmetry‑protected nodal‑ring semimetal. Starting from a generic two‑band Hamiltonian that respects Mz symmetry, they construct a versatile model (Eq. 5) in which the parameters A, B, C, D and an integer exponent n control the existence, geometry and topological charge χh of the nodal ring. When B·D < 0 a ring of radius R = −D/B appears in the kz = 0 plane; the sign of C determines whether the ring coexists with Weyl points (C > 0) or stands alone (C < 0). By evaluating the Berry curvature on a mirror‑symmetric hemisphere the authors define an integer topological charge χh = (1/2π)∮SN Ω·dσ, which can be non‑zero even though the total Chern number on a full sphere vanishes because of mirror symmetry. Wilson‑loop calculations confirm that χh equals −n for the “vortex‑ring” case (n ≥ 1) and vanishes for conventional rings (n = 0).

The scattering problem is set up at a flat x = 0 interface separating an incident medium (x < 0) from the target nodal‑ring material (x > 0). An incident wave packet centered at momentum kc is written as a superposition of Bloch states weighted by a Gaussian envelope. Upon hitting the interface each partial wave acquires a reflection amplitude r(k) with phase ϕ(k). The anomalous shift ℓ, a vector lying in the y–z interface plane, is given by ℓ = Ar − Ai − ∂ϕ/∂k∥, where Ai and Ar are the Berry connections of the incident and reflected Bloch states, respectively. By defining a shift‑vector field κ(k∥) ≡ ∇k∥ × ℓ(k∥) the authors study its circulation over closed contours in the conserved‑parallel‑momentum plane.

A key result is that the circulation over a semicircle (i.e., the boundary of the mirror‑symmetric hemisphere) obeys a simple quantization rule κs = −2π χh. This is the direct analogue of the previously known relation κ = 2π N for Weyl points, but the factor of ½ reflects that the nodal ring’s topological charge is defined on a hemisphere rather than a full sphere. Consequently, measuring the semicircular circulation of the shift provides an immediate experimental read‑out of the integer charge χh of the vortex ring.

Numerical simulations are performed for the four phases identified in the C–D parameter space: (i) nodal ring only, (ii) nodal ring coexisting with Weyl points, (iii) fully gapped insulator, and (iv) Weyl points only. The Berry curvature distribution shows that both the ring and Weyl points act as sources (or sinks) of Ω, and the anomalous shift magnitude is strongly enhanced near these singularities. In the ring‑only phase the shift vectors form a vortex‑like pattern whose radius scales with the physical ring radius R; the direction of the vortex reverses when χh changes sign. When Weyl points appear, the shift field exhibits additional singular features, and the semicircular circulation jumps by ±2π n as a Weyl point of charge ±n crosses the integration contour, thereby preserving the total χh.

The authors also discuss topological phase transitions driven by varying C or D. For example, fixing C < 0 and sweeping D from negative to positive shrinks the ring to a point; because χh is non‑zero the system cannot become fully gapped, and a pair of Weyl points emerges once D > 0, keeping χh unchanged. Conversely, changing C from negative to positive while keeping D < 0 creates Weyl points at infinity that then move toward the Brillouin‑zone center, causing a discrete change in χh when a Weyl point crosses the hemisphere boundary. These transitions are directly reflected in abrupt changes of κs, offering a clear experimental signature of topological phase changes.

From an experimental standpoint, the anomalous shift can be accessed using electron‑beam techniques such as angle‑resolved photoemission spectroscopy with spatial resolution, low‑energy electron microscopy, or electron diffraction setups where the beam’s lateral displacement upon reflection is measured. By mapping ℓ(k∥) across the interface momentum plane, one can reconstruct the ring’s radius, the exponent n (which determines χh), and monitor the evolution of χh across phase transitions. This provides a non‑invasive probe of topological band structures that complements conventional spectroscopic methods, especially for features that are difficult to resolve in energy‑momentum space alone.

In summary, the paper establishes that (1) anomalous spatial shifts are dramatically amplified by the presence of mirror‑protected nodal rings, (2) the shift‑vector field’s semicircular circulation encodes the integer topological charge χh via the universal relation κs = −2π χh, and (3) the shift profile serves as a powerful diagnostic for both the geometry of the nodal ring and its topological phase transitions. These insights open a new avenue for probing and manipulating topological semimetals through electron‑beam scattering phenomena.