Finite-size clusters with periodic boundary conditions resemble isolated clusters for a small number of sites, and infinite lattices for a large number of sites. The transition from a self-contained system to an infinite lattice passes through an intermediate region with increased (hidden) symmetry. In this high-symmetry region irreducible representations of the space group may stick together to form higher-dimensional representations of the complete symmetry group. This transition is examined for a class of simple-, body-centered-, and face-centered-cubic lattice clusters and the two-dimensional square lattice cluster. The implications of an enlarged symmetry group are also studied for a model of strongly correlated electrons interacting on eight-site clusters.Finite-size clusters with periodic boundary conditions resemble isolated clusters for a small number of sites, and infinite lattices for a large number of sites. The transition from a self-contained system to an infinite lattice passes through an intermediate region with increased (hidden) symmetry. In this high-symmetry region irreducible representations of the space group may stick together to form higher-dimensional representations of the complete symmetry group. This transition is examined for a class of simple-, body-centered-, and face-centered-cubic lattice clusters and the two-dimensional square lattice cluster. The implications of an enlarged symmetry group are also studied for a model of strongly correlated electrons interacting on eight-site clusters.

Physical review. B, Condensed matterAug 15, 19917 citations

Symmetry of Many-Electron Systems

Elsevier eBooksJan 1, 197532 citations

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

Constantine Yannouleas, Uzi Landman

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.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.

arXiv:1604.01749Apr 6, 20160 citations

Renormalization Group and Interacting Electron Systems

Zhizhong Zhang

Jan 1, 19930 citations

Infinite Symmetry of the Spin Systems with Inverse Square Interactions

Kazuhiro Hikami, Miki Wadati

One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the $U(\nu )\cong SU(\nu )\otimes U(1)$ current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the $U(\nu )\cong SU(\nu )\otimes U(1)$ current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.

Journal of the Physical Society of JapanDec 15, 199365 citations

Electron Correlations in Strongly Interacting Systems

J. J. Quinn, George Simion

WORLD SCIENTIFIC eBooksMay 1, 20101 citation

One-dimensional quantum particle system with S U (ν) spins interacting through inverse square interactions is studied. We reveal algebraic structures of the system: hidden symmetry is the

U(\nu )\cong SU(\nu )\otimes U(1)

current algebra. This is consistent with the fact that the ground state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, which is the w 1+∞ algebra. With this W algebra we can clarify simultaneously the structures of the Calogero type (1/ x 2 -interactions) and Sutherland type (1/sin 2 x -interactions). The Yangian symmetry is briefly discussed.