Pumping in quantum dots and non-Abelian matrix Berry phases

Abstract

We have investigated pumping in quantum dots from the perspective of non-Abelian (matrix) Berry phases by solving the time dependent Schr{\"o}dinger equation exactly for adiabatic changes. Our results demonstrate that a pumped charge is related to the presence of a finite matrix Berry phase. When consecutive adiabatic cycles are performed the pumped charge of each cycle is different from the previous ones

Nonadiabatic charge pumping by oscillating potentials in one dimension: Results for infinite system and finite ring

“…Note that the corresponding eigenvalue remains e −iE1T to this order; this is a consequence of the choice made in Eq. (17). We can now calculate the current averaged over one time period for one of these eigenstates,…”

“…We then find that c 1 (T ) = c 1 (0) = c 10 and c 2 (T ) = c 2 (0) = c 20 due to Eq. (17). Further, there is now no relation between c 10 and c 20 ; we can choose c 10 and c 20 in an arbitrary way to obtain two orthonormal states v ± which are eigenstates of U (T ) with the same eigenvalue e −iE1T .…”

“…We now turn on the time-dependent perturbation V which will be assumed to satisfy Eq. (17). There are two different possibilities which we will discuss separately: (i) Non-resonant case where E j − E k is not an integer multiple of ω, i.e., e −i(Ej−E k )T = 1, for any pair of states j, k, and (ii) Resonant case where E j − E k is an integer multiple of ω, i.e., e −i(Ej −E k )T = 1, but E j = E k , for some pair of states j, k. (iii) Resonant case where E j = E k for some pair of states j, k.…”

“…For the case of non-interacting electrons, theoretical studies of this phenomenon have used adiabatic scattering theory [9][10][11][12][13][14][15][16][17], Floquet scattering theory [21][22][23], the nonequilibrium Green function formalism [24][25][26][27], and the equation of motion approach [36,37]. The case of interacting electrons has been studied using a renormalization group method for weak interactions [50], and the method of bosonization for arbitrary interactions [51][52][53][54][55][56][57][58][59][60][61].…”

“…The idea that oscillating potentials applied to certain points in a one-dimensional system can pump a net charge between two reservoirs at the same chemical potential has been studied extensively for many years, both theoretically and experimentally [43][44][45][46][47][48][49]. For the case of non-interacting electrons, theoretical studies of this phenomenon have used adiabatic scattering theory [9][10][11][12][13][14][15][16][17], Floquet scattering theory [21][22][23], the nonequilibrium Green function formalism [24][25][26][27], and the equation of motion approach [36,37]. The case of interacting electrons has been studied using a renormalization group method for weak interactions [50], and the method of bosonization for arbitrary interactions [51][52][53][54][55][56][57][58][59][60][61].…”