Quantized Circulation of Anomalous Shift in Interface Reflection

A particle beam may undergo an anomalous spatial shift when it is reflected at an interface. The shift forms a vector field defined in the two-dimensional interface momentum space. We show that, although the shift vector at individual momentum is typically sensitive to the system details, its integral along a close loop, i.e., its circulation, could yield a robust quantized number under certain conditions of interest. Particularly, this is the case when the beam is incident from a trivial medium, then the quantized circulation of anomalous shift (CAS) directly manifests the topological character of the other medium. We demonstrate that the topological charge of a Weyl medium as well as the unconventional pair potentials of a superconductor can be captured and distinguished by CAS. Our work unveils a hidden quantized feature in a ubiquitous physical process, which may also offer a new approach for probing topological media.

Figure 1

(a) Schematic figure showing the anomalous spatial shift $\mathit{\ell }$ for a beam reflected at an interface. (b) $\mathit{\ell }$ forms a vector field in the domain $S$ (where reflection occurs) in the interface momentum ${\mathit{k}}_{\parallel }$ space. The CAS ${\kappa }_C$ is defined for any closed loop $C\in S$.

Figure 2

(a) Schematic figure showing the interface formed between a trivial medium and a Weyl medium. (b) Numerical results for the $\mathit{\ell }$ field, for which the magnitude and the direction are indicated by the color and the streamline, respectively. (c) Plot of $\arg (r)$. The white circles in (b) and (c) indicate the region with radius $k_W$. Here, we set $m=0.04m_e$, $\mathrm{\Delta }=0.2\mathrm{eV}$, $v=1.5\times {10}^6\mathrm{m}/\mathrm{s}$, $U=0.6\mathrm{eV}$, $E=1.0\mathrm{eV}$, and $\delta ={10}^{-4}\mathrm{eV}{\mathrm{nm}}^2$.

Figure 3

Results for the target medium with (a)–(b) double-Weyl point, and (c)–(d) triple-Weyl point, corresponding to $n=2$ and $n=3$ in model (6), respectively. (a),(c) exhibit the $\mathit{\ell }$ field, and (b),(d) show the phase of $r$. The white circles indicate the region with radius $k_W$. Here, we take $w=1{\mathrm{nm}}^2\mathrm{eV}$, $U=0.8\mathrm{eV}$ in (a)–(b); and $w=1{\mathrm{nm}}^3\mathrm{eV}$, $U=0.92\mathrm{eV}$ in (c)–(d). Other parameters are the same as that in Fig. 2.

Figure 4

Calculated $\mathit{\ell }$ field in Andreev reflection for (a) $s$-wave, (b) chiral $p$-wave, and (d) $d_{x^2-y^2}$-wave pair potentials. (c) The phase of $r_A$ for the chiral $p$-wave case. Here, we take $ϵ=0.02\mathrm{eV}$, $m=0.04m_e$, ${\mathrm{\Delta }}_0=0.04\mathrm{eV}$, $E_F=0.1\mathrm{eV}$, and $U=0.3\mathrm{eV}$.