Quantum Wigner molecules in semiconductor quantum dots and cold-atom optical traps and their mathematical symmetries

TL;DR

This paper explores the hidden geometric symmetries in quantum Wigner molecules formed by few particles in traps, revealing their signatures in spectra and wave functions, and connecting them to broken-symmetry mean-field solutions.

Contribution

It introduces methods to detect and analyze hidden symmetries in many-body wave functions of quantum dots and cold-atom traps, linking them to broken-symmetry mean-field approaches.

Findings

Hidden symmetries manifest in ro-vibrational spectra and conditional probability distributions.

Restoring broken symmetries via projection clarifies the role of hidden symmetries.

Examples include quantum dots and ultracold atomic systems with controllable interactions.

Abstract

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.