Dynamics of a quantum particle in a random potential

Abstract

The motion of a quantum particle in a random potential with short-range fluctuations is analyzed using a generalized mode-coupling approximation. The original theory of GUtze was extended by introducing a wave-vector dependent current relaxation kernel. Here we apply the theory to the special case of exciton dynamics in a binary molecular crystal. We find a localization-delocalization transition in three dimensions with a critical exponent of 1 for the inverse localization length. In the intermediate time regime, our theory predicts a region of anomalous diffusion. This can be interpreted as a precursor to localization. At the transition point it leads to the known asymptotic low-frequency behaviour D"(o) 0: o "~. Ever since Anderson [l]introduced the concept of localization of electrons by disorder, physicists have tried to describe the implications of this phenomenon on the behaviour of measurable quantities of metallic conductors such as conductivity, chargecarrier diffusion coefficient, etc. In this case two problems are intimately entangled: the dynamics of suppressed diffusion (localization) in a random potential, on the one hand, and the statistics of a degenerate fermion gas in this potential (particle statistics), on the other. Allowing to disentangle these two problems in a simple way is one of the appealing aspects of mode-coupling theory, which was introduced in the present context by Gotze [2] in his pioneering work on electron localization.Since earlier attempts failed to render a correct description of the localization transition within mode-coupling theory, we will in this note report on an improved version (details will be published in a forthcoming paper). We calculate the probability density P(r, t ) for a single particle to move by r in time t. In P(r, t ) the random potential has already been averaged over. It exhibits translational symmetry, thus depending on r = r2 -rl , only. This probability density ("incoherent" density relaxation function) will in the following play a fundamental role in understanding the dynamical properties of a system of non-interacting quantum particles (including the ac-conductivity of a dilute gas of such particles). In contrast to earlier self-consistent theories of the Anderson transition [2, 3, 61, which calculate the "coherent" density-relaxation function @ (9, t ) and thus require an a-priori knowledge of the isothermal susceptibility @(q, t = 01, we benefit here from the known initial value P(q, t = 0), Eq. (8) below. This simplification stems from the fact that P ( q , t ) is a one-particle quantity.

On the excited state wave functions of Dirac fermions in the random gauge potential

“…Localization of a particle by a random potential has been extensively investigated for the past several decades [1][2][3][4][5][6][7][8][9][10][11][12]. It is known that, for the strong disorder, singleparticle wave functions are confined and have exponentially decaying tails beyond the scale of the localization length ξ [4,13,14].…”