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arXiv:cond-mat/0309606 · Springer eBooks

Explicit and Hidden Symmetries in Quantum Dots and Quantum Ladders

K. Kikoin, Y. Avishai, Mikhail N. Kiselev

Published Jan 1, 2004• 4 citations
arXivHTMLPDF
DOI
10.1007/978-1-4020-2093-3_17

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Strong repelling interactions between a few fermions or bosons confined in two-dimensional circular traps lead to particle localization and formation of quantum Wigner molecules (QWMs) possessing definite point-group space symmetries. These point-group symmetries are "hidden" (or emergent), namely they cannot be traced in the circular single-particle densities (SPDs) associated with the exact many-body wave functions, but they are manifested as characteristic signatures in the ro-vibrational spectra. An example, among many, are the few-body QWM states under a high magnetic field or at fast rotation, which are precursor states for the fractional quantum Hall effect. The hidden geometric symmetries can be directly revealed by using spin-resolved conditional probability distributions, which are extracted from configuration-interaction (CI), exact-diagonalization wave functions. The hidden symmetries can also be revealed in the CI SPDs by reducing the symmetry of the trap (from circular to elliptic to quasi-linear). In addition the hidden symmetries are directly connected to the explicitly broken-symmetry (BS) solutions of mean-field approaches, such as unrestricted Hartree-Fock (UHF). A companion step of restoration of the broken symmetries via projection operators applied on the BS-UHF solutions produces wave functions directly comparable to the CI ones, and sheds further light into the role played by the emergence of hidden symmetries in the exact many-body wave functions. Illustrative examples of the importance of hidden symmetries in the many-body problem of few electrons in semiconductor quantum dots and of few ultracold atoms in optical traps (where unprecedented control of the interparticle interaction has been experimentally achieved recently) will be presented.
We study a number of strongly correlated many-body quantum systems. Most of the thesis is devoted to effects of pairing between fermions. In particular, we develop an explicit description of a time-dependent response of fermionic condensates to external perturbations. The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model. We obtain a general explicit solution for the dynamics of the BCS model. We also solve a closely related dynamical problem – the central spin model, which describes a localized spin coupled to a ‘spin bath’. This model has a range of interesting applications, in particular to the issue of electron spin coherence in semiconductor quantum dots. We evaluate finite size corrections for the BCS model, which is relevant for small systems such as metallic grains. In the reminder of the thesis, we study the connection between symmetry and degeneracy in quantum systems. We propose a method for extracting hidden symmetry from the energy spectrum and clarify the origin of crossings and degeneracies in the 1d Hubbard model.