Contents - Max-Planck-Institut für Physik komplexer Systeme

power law for the Anderson overlap,
|∆| 2 M −δ2 /π 2
eff , (1)
where Meff is the number of participating electrons and
δ is the phase shift induced by the perturbation, at the
Fermi energy, still holds with Meff now being the number
of electrons on the Fermi shell. Since this number
is of the order √ N, N being the total number of dot
levels, the effective AOC response is characterized by
|∆| 2 N −0.5 δ2 /π 2
, i.e. by an additional factor of 1/2 in
the exponent. This behaviour is denoted by the solid
line in the topmost curve of Fig. 2.
|∆| 2
1
0.5
0.05
10 100 1000 10000
N
Figure 2: Modified Anderson power law in parabolic quantum
dots.Vc/ωc increases from top to bottom and takes the value 0.1, 1,
10, 100 and 10 5 . The solid and dashed curves represent analyitcal
forms of the Anderson overlap. See text for details. Note that the
spatial dependence of the wave functions is omitted here for clarity.
When (ii) very strong perturbations strengths (Vc/ω0
large) are considered, the whole system participates,
and we find the usual Anderson power law, Eq. (1), for
the AOC response, see the lowermost curve in Fig. 2.
The transition regime of (iii) intermediate perturbation
strengths is the most interesting, as here the size of the
quantum dot (i.e., the number of electrons on it) also
plays a role, simply because the inter- and intra-shell
energy scales depend on N. This is illustrated by the
central curves of Fig. 2 that clearly show a transition
in the power law. For small N, the shell seperation remains
relatively large, and one is back to case (i). The
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[5] S. Bandopadhyay and M. Hentschel, Phys. Rev. B 83 (2011) 035303.
[6] G. Röder, PhD thesis, TU Dresden, Germnany, 2011.
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opposite holds for (very) large N, where the small shell
spacing results in a response of the whole, or at least of
a large part of the system as in (ii), of the large N-part
(dashed) of the last but one curve. For the transition
regime of intermediate N we have analytically computed
the Anderson overlap assuming that the shells
right below and above the Fermi shell participate [5].
Taking these three shells into account, we were able to
analytically describe the transition regime, see Fig. 2,
that is characterized by an exponential correction term
in the Anderson power law that enhances the effect of
AOC compared to the case-(i) behaviour. It is denoted
by the dashed lines in the 2nd, 3rd and 4th curve from
to as well as the solid line in the last but one curve
and cleanly illustrates the deviation from the Anderson
power law.
Eventually, we briefly mention the importance of shell
effects as the number of electrons on the parabolic dot is
varied. Most remarkably, AOC is enhanced especially
for small perturbations, where the Anderson overlap
drops to a value 1/degeneracy, rather than being close
to one, whenever a new shell is opened. This is understood
by a phase-space argument: The sudden perturbation
“shakes up” the electrons on the dot, and
a single electron on an otherwise empty shell can rechoose
its level, resulting in a correction factor 1/degeneracy
[5]. We have studied this behaviour also in
detail for circular quantum dots [4] where the characteristic
two-fold level degeneracies leaves its traces also
in the photoabsorption cross section [6]. We point out
that the effects persist if the degeneracy is slightly perturbed.
Another direction of our research on the group’s manybody
topic is graphene. Here, the special properties of
the density of states – no states at the Dirac point and
the possibility of having edge states – provide new features
not available or known in bulk metals that turn
into specific properties of the many-body responses.
For example, an additional singularity develops in the
photoabsorption cross section at the Dirac point.
In summary, we have found that the variety of
mesoscpopic systems considerably enriches the field of
many-body physics and contributes to its deeper understanding.
The availability of high quality samples
raises the hope for experimental confirmation of our
predicitions in the near future.
2.20. Many-body effects in mesoscopic systems: Beyond the Anderson power law 81