Weak antilocalization of ultrarelativistic fermions

The Letter discusses the Berry phase influence on the weak localization phenomenon at the adiabatic backscattering of ultrarelativistic particles in a random medium. We demonstrate that bosons that pass along a certain closed path in opposite directions come back always in phase (an example: the backscattering enhancement of electromagnetic waves), whereas fermions come exactly in antiphase. This produces a “complete weak antilocalization” of ultrarelativistic fermions: the backscattering field intensity vanishes.  2005 Elsevier B.V. All rights reserved. PACS: 42.25.Dd; 72.15.Rn; 03.65.Vf; 11.80.-m The phenomenon of backscattering enhancement has been much studied both for electromagnetic waves (photons), which are scattered in randomly inhomogeneous media (for review see [1]), and for electrons in solids (for review see [2]). In the latter case, the effect is commonly referred to as weak localization of quantum particles. The essence of the phenomenon consists in the following. In the course of multiple scattering of a wave in a random medium at an arbitrary (different from π ) finite angle, the partial waves traveling along E-mail address: [email protected] (K.Yu. Bliokh). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.06.062 various trajectories come to an observation point with random phases. As a result, an incoherent summation takes place, and the total intensity of scattered field is determined as a sum of the partial intensities. On the contrary, in the case of the backscattering, i.e. scattering at π angle, all the trajectories are loop-shaped, and any chosen trajectory can be associated with the same trajectory passed in opposite direction (Fig. 1). As a result, all the trajectories are grouped in pairs, and the phase increments coincide for both directions of the loop passage. Hence a coherent summation of fields occurs, and one should sum the field amplitudes from the paired trajectories. It is easy to see that in this case the backscattering intensity is exactly twice as large as the intensity of the field scattered in other direc128 K.Yu. Bliokh / Physics Letters A 344 (2005) 127–130 Fig. 1. Paired loop trajectories at the backscattering. tions [1] (the scattering indicatrix is considered to be isotropic) (see Fig. 3 below). Incorporation of various interactions can essentially change the features of the backscattering enhancement. If the phase difference of the waves traveling along a loop-trajectories in opposite directions changes, the backscattering enhancement can be reduced, vanish or even be replaced by attenuation [2,3]. In the latter case, the weak antilocalization of particles is said to occur. For electrons, two basic mechanisms for the interaction’s influence on the weak localization are known. The first one is associated with the introduction of an external magnetic field. If a magnetic flux Φ penetrates a loop-trajectory, the electrons travelling this loop in opposite directions acquire the Dirac phase difference 2eΦ/h̄c (Aharonov–Bohm effect). Due to a random character of the trajectories, the flux Φ , as well as the phase difference, turn out to be random values also. As a consequence of this, the backscattering peak reduces as the magnetic field increases, and it disappears when the magnetic field is strong enough: the backscattered fields become incoherent. In solids, the weakening of the electron backscattering leads to the growth of the conductivity as the magnetic field increases, or in other words, to the negative magnetoresistance (magnetoconductance) effect [2,3]. Another mechanism of the change in the phase difference on the paired trajectories is the appearance of the Berry geometric phase. Berry phase for electrons arises owing to the presence of a spin and is connected to the Zeeman and spin–orbit interactions. Therefore, the weak electron antilocalization is frequently associated with the spin–orbit interaction. In contrast to Dirac phase, Berry phase does not depend explicitly on the loop shape and is determined only by the geometry of the particle motion in the momentum space. Due to this, the Berry phase can be a regular quantity even for random loop-trajectories (see below). Then the backscattered fields are summed coherently as before, but the Berry phase’s difference may cause both the enhancement and weakening of the backscattered field, i.e., it may cause the weak antilocalization [2,3]. The weak antilocalization is not observed in the electromagnetic wave scattering [4]. Meanwhile, electromagnetic waves consist of photons, i.e. relativistic particles with spin 1. They also possess the Berry geometric phase [5]; moreover, as it was recently shown, the Berry phase can be presented as a consequence of the spin–orbit interaction of photons [6–10]. Why does the presence of the Berry phase not result in the weak antilocalization in photons? Is the weak antilocalization effect possible for other ultrarelativistic particles? First, let us recall that in the case of adiabatic evolution (if the helicity is conserved), the Berry phase for photons and other ultrarelativistic particles (i.e. a particle with the energy much larger than the rest energy) of arbitrary spin s is determined by a simple geometric law [5,11]. Let us consider a closed trajectory of a particle motion in the momentum p-space. The presence of spin freedom degrees in the relativistic wave equations entails that the particle moves as if an effective ‘magnetic monopole’ of charge σ (where σ =−s,−s + 1, . . . , s − 1, s is the helicity of the particle and σ = 0 for photons) is placed at the origin of p-space [5,8,10,11]. As a result, the Berry phase θB is an analogue of the Dirac phase generated by this monopole; it is numerically equal to the flux passing through the contour of the particle trajectory, or