Through the introduction of a class of appropriate translationally invariant trial wave functions, we show that the strong correlations in the lowest Landau level (LLL) reflect in finite systems the emergence of intrinsic point-group symmetries associated with rotations and vibrations of molecules formed through particle localization. This molecular description is universal, being valid for both bosons and fermions, for both the yrast and excited states of the LLL spectra, and for both low and high angular momenta. This physical picture is fundamentally different from the "quantum-fluid" one associated with Jastrow-type trial functions.
Through the introduction of a class of appropriate translationally invariant trial wave functions, we show that the strong correlations in the lowest Landau level (LLL) reflect in finite systems the emergence of intrinsic point-group symmetries associated with rotations and vibrations of molecules formed through particle localization. This molecular description is universal, being valid for both bosons and fermions, for both the yrast and excited states of the LLL spectra, and for both low and high angular momenta. This physical picture is fundamentally different from the "quantum-fluid" one associated with Jastrow-type trial functions.
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430(Dated: 19 February 2009)Through the introduction of a class of appropriate translationally invariant trial wave functions,we show that the strong correlations in the lowest Landau level (LLL) reflect in finite systems theemergence of intrinsic point-group symmetries associated with rotations and vibrations of moleculesformed through particle localization. This quantal molecular description is universal, being valid forboth bosons and fermions, for both the yrast and excited states of the LLL spectra, and for bothlow and high angular momenta. ”Quantum-fluid” physical pictures associated with Jastrow-typetrial functions are shown to be reducible to the molecular description introduced in this paper.
Through the introduction of a class of trial wave functions portraying combined rotations and vibrations of molecules formed through particle localization in concentric polygonal rings, a correlated basis is constructed that spans the translationally invariant part of the lowest-Landau-level (LLL) spectra. These trial functions, referred to as rovibrational molecular (RVM) functions, generalize our previous work that focused exclusively on electronic cusp states, describing them as pure vibrationless rotations. From a computational viewpoint, the RVM correlated basis enables controlled and systematic improvements of the original strongly correlated variational wave function. Conceptually, it provides the basis for the development of a quantal molecular description for the full LLL spectra. This quantal molecular description is universal, being valid for both bosons and fermions, for both the yrast and excited states of the LLL spectra, and for both low and high angular momenta. Furthermore, it follows that all other translationally invariant trial functions (e.g., the Jastrow-Laughlin, compact composite-fermion, or Moore-Read functions) are reducible to a description in terms of an excited rotating and vibrating quantal molecule.
Exploring strongly correlated spinful states of few fermionic ultracold atoms in a rapidly rotating trap, an example of which was recently realized for two fermionic 6Li atoms in an optical tweezer, we derive analytical (algebraic) total-spin-eigenstate wavefunctions through the development and employment of a theoretical platform that integrates exact numerical diagonalization (full configuration interaction, FCI) with symbolic language processing. For such rapid rotations, where the atoms occupy the lowest Landau level (LLL), the obtained algebraic expressions can address the full LLL spectrum in all its complexity, demonstrating that their spatial, spectral, and spin characteristics manifest formation of collectively rotating and vibrating Wigner molecules. The explicitly exhibited analytic wavefunctions (for two and three spinful 6Li atoms) reproduce precisely the corresponding numerical FCI results, and they are shown to reach beyond the limited range of applicability of previous Jastrow-type treatments. These results, and their extension to bosonic systems, provide the impetus and analysis tools for future experimental and theoretical simulations of larger mesoscopic systems
Investigations of emergent symmetry breaking phenomena occurring in small finite-size systems are reviewed, with a focus on the strongly correlated regime of electrons in two-dimensional semiconductor quantum dots and trapped ultracold bosonic atoms in harmonic traps. Throughout the review we emphasize universal aspects and similarities of symmetry breaking found in these systems, as well as in more traditional fields like nuclear physics and quantum chemistry, which are characterized by very different interparticle forces. A unified description of strongly correlated phenomena in finite systems of repelling particles (whether fermions or bosons) is presented through the development of a two-step method of symmetry breaking at the unrestricted Hartree-Fock level and of subsequent symmetry restoration via post Hartree-Fock projection techniques. Quantitative and qualitative aspects of the two-step method are treated and validated by exact diagonalization calculations.Strongly-correlated phenomena emerging from symmetry breaking include the following.Chemical bonding, dissociation and entanglement (at zero and finite magnetic fields) in quantum dot molecules and in pinned electron molecular dimers formed within a single anisotropic quantum dot, with potential technological applications to solid-state quantum-computing devices.Electron crystallization, with particle localization on the vertices of concentric polygonal rings, and formation of rotating electron molecules (REMs) in circular quantum dots. Such electron molecules exhibit ro-vibrational excitation spectra, in analogy with natural molecules.At high magnetic fields, the REMs are described by parameter-free analytic wave functions, which are an alternative to the Laughlin and composite-fermion approaches, offering a new point of view of the fractional quantum Hall regime in quantum dots (with possible implications for the thermodynamic limit).Crystalline phases of strongly repelling bosons. In rotating traps and in analogy with the REMs, such repelling bosons form rotating boson molecules (RBMs). For a small number of bosons, the RBMs are energetically favored compared with the Gross-Pitaevskii solutions describing vortex formation.We discuss the present status concerning experimental signatures of such strongly correlated states, in view of the promising outlook created by the latest experimental improvements that are achieving unprecedented control over the range and strength of interparticle interactions.
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.
A constructive theoretical platform for the description of quantum space-time crystals uncovers for N interacting and ring-confined rotating particles the existence of low-lying states with proper space-time crystal behavior. The construction of the corresponding many-body trial wave functions proceeds first via symmetry breaking at the mean-field level followed by symmetry restoration using projection techniques. The ensuing correlated many-body wave functions are stationary states and preserve the rotational symmetries, and at the same time they reflect the point-group symmetries of the mean-field crystals. This behavior results in the emergence of sequences of select magic angular momenta Lm. For angular-momenta away from the magic values, the trial functions vanish. Symmetry breaking beyond the mean-field level can be induced by superpositions of such good-Lm many-body stationary states. We show that superposing a pair of adjacent magic angular momenta states leads to formation of special broken-symmetry states exhibiting quantum space-time-crystal behavior. In particular, the corresponding particle densities rotate around the ring, showing undamped and nondispersed periodic crystalline evolution in both space and time. The experimental synthesis of such quantum space-time-crystal wave packets is predicted to be favored in the vicinity of ground-state energy crossings of the Aharonov-Bohm-type spectra accessed via an externally applied, natural or synthetic, magnetic field. These results are illustrated here for Coulomb-repelling fermionic ions and for a lump of contact-interaction attracting bosons.