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2.20 Many-body effects in mesoscopic systems: Beyond the Anderson power law MARTINA HENTSCHEL, SWARNALI BANDOPADHYAY Introduction. Many-body phenomena such as the Kondo effect, Fermi-edge singularities, or Anderson orthogonality catastrophe (AOC), have attracted a lot of interest in condensed matter physics for many decades. Here we will focus on AOC, an universal many-body response of a system to a sudden perturbation, for example the appearance of a localized hole potential after the excitation of a core electron into the conduction band. Many-body effects have always been an inspiration to deepen and widen our physical understanding. We shall see here that this remains true when shrinking the system sizes to the mesoscopic scale (determined by the phase coherence length of the system and typically in the micrometer range), i.e. when we consider quantum dots, metallic nanoparticles or graphene rather than bulk metals. Instead of energy bands and Bloch waves, the electrons occupy now discrete energy levels. Their wave functions depend, among others, also on the system geometry (we confine ourselves to the ballistic case in the following). Moreover, the treasure box of mesoscopic systems allows to customize systems with properties that are not known from the macroscopic, metallic case: the possibility to have degenerate levels such as in circular or parabolic quantum dots, for example. Will these new, mesoscopic features – discrete energy levels, finite number of particles (as compared to 10 23 ), role of system geometry, degeneracies – leave their traces in many-body signatures? The answer is a clear Yes. We have seen the importance of mesoscopic properties in a number of cases, for example in Fermi-edge singularities determining the photoabsorption response of quantum dots [1, 6] and graphene, the mesoscopic Kondo box [2], in AOC of chaotic [3], integrable [4], and parabolic [5] quantum dots. We have found a broad distribution of photoabsorption rates at threshold, as well as of Kondo temperatures and Anderson overlaps, that directly reflect the importance of the well-known mesoscopic fluctuations. Besides this sample-specific properties, we also made predictions about the behaviour of the averages, e.g., of the photoabsorption cross section. These data can be compared to the metallic case, and we found considerably deviating behaviour between a metallic sample and a quantum dot. Surprisingly, in addition to the above-mentioned mesoscopic features, the mere existence of a system boundary turns out to be mainly responsible for a peaked photoabsorption at the Fermi threshold [6], that has to be contrasted to a rounded Fermi edge singularity in the metallic case (the socalled K-edge is considered in both cases). Anderson orthogonality catastrophe in parabolic quantum dots. In this report we will focus on AOC in parabolic quantum dots (PQDs). The shell structure characteristic for such typically few electron quantum dots was confirmed in experiments [7]. As in the harmonic oscillator, the lowest shell contains just one level, the next shell hosts two levels and so on. This introduces two energy scales, see Fig. 1: The intershell spacing ω0 and the intra-shell spacing ωc that is (very close to) zero in the (quasi-) degenerate case (that we consider for practical reasons) and increases as an external magnetic field lifts the degeneracies. The strength Vc of a sudden and localized perturbation that induces the AOC response has to be compared to these two energy scales. We shall see that this, together with the intrinic size dependence of the harmonic oscillator energy scales, is the origin of the very different bahaviour of PQDs and general ballistic quantum dots (e.g., chaotic dots), respectively; in the latter the energy levels are characterized by only one scale, namely the mean level spacing d. N M ωc i 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1100 11 01 00 110 1 01 00 11 00 11 00 11 0 00 11 01 00 11 01 00 11 01 01 01 01 00 11 00 110 1 00 110 1 1 ε ωo 00 11 00 11 00 110 1 00 1101 00 11 00 11 00 11 0 00 1100 11 00 1100 11 01 1 0 01 1 01 00 11 01 00 11 00 11 00 11 01 01 00 11 00 11 Figure 1: Energy levels in a parabolic quantum dot, before (left) and after (right) a sudden perturbation leading to AOC is applied. EF denotes the Fermi energy, M the number of electrons on the PQD, and N the total number of levels. This results in a rich behaviour of AOC in PQDs and we have to distinguish three regimes (we assume half filling as usual): For (i) very small perturbation strengths (Vc/ωc small), the physics is governed by the Fermi shell, i.e., the shell containing the Fermi energy level - and interestingly, the other shells do not play a role. Consequently, the 80 Selection of Research Results 01 01 01 λ E F
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 [1] M. Hentschel, D. Ullmo, and H. Baranger, Phys. Rev. Lett. 93 (2004) 176807. [2] R. Bedrich, S. Burdin, and M. Hentschel, Phys. Rev. B 81 (2010) 174406. [3] M. Hentschel, D. Ullmo, and H. Baranger, Phys. Rev. B 72 (2005) 035310. [4] G. Röder and M. Hentschel, Phys. Rev. B 82 (2010) 125312. [5] S. Bandopadhyay and M. Hentschel, Phys. Rev. B 83 (2011) 035303. [6] G. Röder, PhD thesis, TU Dresden, Germnany, 2011. [7] L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog. Phys. 64 (2001) 701. 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
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Chapter1 ScientificWorkanditsOrgani
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2009-2010 •In2009Dr.K.Hornbergera
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possibilityforyoungscientiststotake
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manipulationofchargequbits.TakingCo
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Contents - Max-Planck-Institut für Physik komplexer Systeme

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