Geometrical phase shift in Friedel oscillations

This work addresses the problem of elastic scattering on a localized impurity in a one-dimensional crystal with sublattice degrees of freedom. The impurity yields long-range interferences in the local density of states known as Friedel oscillations. Here we show that the internal degrees of freedom of Bloch waves are responsible for a geometrical phase shift in Friedel oscillations. The Fourier transform of the energy-resolved interference pattern reveals a topological property of this phase shift, which is intrinsically related to a wave-function topological property (Zak phase) in the absence of impurity. As a result, Friedel oscillations in the local density of states can be regarded as a probe of wave topological properties in a broad class of classical and quantum systems, such as acoustic and photonic crystals, ultracold atomic gases in optical lattices, and electronic compounds.

Figure 1

Shown on top are the monatomic (left) and diatomic (right) pattern crystals with a localized impurity $V$. On the bottom are Bloch spectra of the monatomic (left) and diatomic (right) pattern crystals in the first Brillouin zone. The horizontal double arrows depict elastic scattering processes between time-reversed states $-k_0$ and $k_0$.

Figure 2

Diagrammatic representation of the $T$ matrix within a perturbation theory in the impurity potential $V$. The oriented lines between two elastic scattering processes denote the bare Green's functions in momentum space.

Figure 3

Energy-resolved interference pattern induced in the LDOS of sublattices $A$ (top) and $B$ (bottom) by a localized impurity. The latter, which is simulated by a potential of $V=1$, belongs to sublattice $A$ and unit cell $m=0$. Its position is marked by the vertical white dashed line. Energy is given in units of hopping $t_1$.

Figure 4

Modulus (top) and argument (bottom) of the Fourier transform of the energy-resolved interference pattern on sublattice $B$ for $\alpha <1$ (left) and $\alpha >1$ (right). The white dashed lines mark the wave vectors $q=\pm 2k_0$ at energy $\omega$.

Figure 5

Energy-resolved interference patterns in the LDOS on sublattice $A$ (top) and sublattice $B$ (bottom) obtained numerically (left) and analytically (right). Only the scattering occurring in the conduction band is shown, i.e., $|1-\alpha |\le \omega \le 1+\alpha$, where $\alpha =0.9$. The vertical white dashed line marks the site of the impurity that belongs to sublattice $A$ and unit cell $m=0$. The impurity potential is simulated by $V=1$. Energy is given in units of the hopping amplitude $t_1$.

Figure 6

Illustration of the two possible definitions of the unit cell. The site of the impurity (gray square) unambiguously belongs to the site $A$ (blue disk) of the unit cell $m=0$. Then there are two nearest-neighbor sites $B$ (red disks) and we have to pick one of them to obtain the diatomic pattern of the unit cell, which leads to the two configurations depicted in the figure.