Geometrical phase shift in Friedel oscillations

Abstract

This work addresses the problem of elastic scattering through a localized impurity in a one-dimensional crystal with sublattice freedom degrees. 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 the Bloch band structure topology in the absence of impurity. Therefore, 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.Electric screening in metals arises as a collective response of the conduction electrons to the Coulomb potential of a charged impurity. A long wavelength description of the problem captures the exponential screening of the impurity by the surrounding electrons, in agreement with the classical picture that depicts electrons as point charges. In quantum mechanics, however, particles are described by wavefunctions and may interfere. In the 1950s, J. Friedel actually understood that a charged impurity additionally yields a long-range interference pattern in the electronic density [1]. It consists of algebraically decaying 2k F -wavevector oscillations, and results from Fermi surface nesting associated to twice the Fermi momentum k F . Thus, these oscillations reported by Friedel for charges in metals rely on wave features, and they have subsequently been revisited in other contexts, such as magnetic interactions [2][3][4][5] and noninteracting electrons [6][7][8]. In particular, Friedel oscillations have been observed in nonrelativistic electron gases via scanning tunnelling microscopy (STM); an experimental technique that images the local density of states (LDOS), i.e., the electronic density with atomic-scale and energy resolutions [9][10][11]. These experiments have confirmed that backscattering was the most efficient process involved in the elastic scattering through short-range impurities. For noninteracting electrons in a one-dimensional crystal, backscattering is indeed responsible for 2k 0 -wavevector Friedel oscillations that behave asHere δρ denotes the correction to the LDOS induced by a localized impurity, m labels the distance to the impurity in units of the Bravais lattice vector, V(ω) is some real function, and wavevector 2k 0 refers to the elastic backscattering between the time-reversed states −k 0 and k 0 at energy ω ( = 1). Such backscattering wavevectors are illustrated in Fig. 1 by the double arrows. The reader may find the details of the derivation of Eq. (1) in Supplemental Material (SM) [12]. In this letter, we would like to highlight the effects of internal degrees of freedom on the long-range interference pattern induced b...

Wavefront dislocations reveal the topology of quasi-1D photonic insulators

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“…In contrast, the oscillations in ρ A imply the additional phase shift δ A ðkÞ ¼ 2Arg½hðkÞ (see ref. 34 ).…”

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