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arXiv:1607.02672 · Physical review. B./Physical review. B

Few-body collective excitations beyond Kohn's theorem in quantum Hall systems

Rachel Wooten, Bin Yan, Chris H. Greene

Published Jan 27, 2017• 3 citations
arXivHTMLPDF

Abstract

A relative coordinate breathing mode in the quantum Hall system is predicted to exist with different behavior under either Coulomb or dipole-dipole interactions. While Kohn's theorem [W. Kohn, Phys. Rev. 123, 1242 (1961)] predicts that any relative coordinate interaction will fail to alter the center-of-mass energy spectrum, it can affect excitations in the relative coordinates. One such collective excitation, which we call the hyperradial breathing mode, emerges naturally from a few-body, hyperspherical representation of the problem and depends on the interparticle interactions, the ground state wave function, and the number of particles participating in the excitation. Possible observations of this excitation will be discussed in the context of both conventional quantum Hall experiments and cold, rotating atomic simulations.

DOI
10.1103/physrevb.95.035150

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A relative coordinate breathing mode in the quantum Hall system is predicted to exist with different behavior under either Coulomb or dipole-dipole interactions. While Kohn's theorem [W. Kohn, Phys. Rev. 123, 1242 (1961)] predicts that any relative coordinate interaction will fail to alter the center-of-mass energy spectrum, it can affect excitations in the relative coordinates. One such collective excitation, which we call the hyperradial breathing mode, emerges naturally from a few-body, hyperspherical representation of the problem and depends on the interparticle interactions, the ground state wave function, and the number of particles participating in the excitation. Possible observations of this excitation will be discussed in the context of both conventional quantum Hall experiments and cold, rotating atomic simulations.
We show that rotating two-dimensional Fermi gases possess a nonrelativistic scale and conformal invariance at weak but nonzero interactions, where the scale invariance of universal short-range interactions is not yet broken by quantum effects. We demonstrate the symmetry in the excitation spectrum of few-fermion ensembles in a harmonic trap obtained by exact diagonalization. The excitation spectrum is shown to split in a set of primary states and derived excited states that consist of breathing modes as well as two different center-of-mass excitations, which describe cyclotron and guiding-center excitations of the total particle cloud. Furthermore, the conformal symmetry is manifest in the many-body wave function, where it dictates the form of the hyperradial component, which we demonstrate using Monte Carlo sampling of few-body wave functions. Published by the American Physical Society 2024
We investigate the quantum breathing mode (monopole oscillation) of trapped fermionic particles with Coulomb and dipole interaction in one and two dimensions. This collective oscillation has been shown to reveal detailed information on the many-particle state of interacting trapped systems and is thus a sensitive diagnostics for a variety of finite systems, including cold atomic and molecular gases in traps and optical lattices, electrons in metal clusters and in quantum confined semiconductor structures or nanoplasmas. An improved sum rule formalism allows us to accurately determine the breathing frequencies from the ground state of the system, avoiding complicated time-dependent simulations. In combination with the Hartree–Fock and the Thomas–Fermi approximations this enables us to extend the calculations to large particle numbers N on the order of several million. Tracing the breathing frequency to large N as a function of the coupling parameter of the system reveals a surprising difference of the asymptotic behavior of one-dimensional and two-dimensional harmonically trapped Coulomb systems.
Excitation of an atomic-gas Bose-Einstein condensate (BEC) in the zeroth-order ground-state channel is studied with the hyperspherical adiabatic method of Bohn et al. [Bohn et al., Phys. Rev. A 58, 584 (1998)] suitably generalized to accommodate the anisotropic trapping potential. The method exploits the system's size as an adiabatic parameter so that the explicit size dependence is immediately conducive to the virial theorem. The oscillation frequencies associated with the monopole (breathing) and quadrupole modes thus emerge naturally and converge to the well-known Thomas-Fermi limits. Analysis of the single-particle density and the projected excitation wave function shows that the excitation in the single hyperspherical ground-state channel merely represents a progressive increase in occupancy of the first excited single-particle state. The work paves the way for applying the adiabatic picture to other BEC phenomena.
Collective excitations of many-body electron systems can carry internal structure, tied to the quantum geometry of the Hilbert space in which they are embedded. This has been shown explicitly for particle-hole-like excitations, which carry a ``quantum geometric dipole'' (QGD) that is essentially an electric dipole moment associated with the state. We demonstrate in this work that this property can be formulated in a generic way, which does not require wave functions expressed in terms of single particle-hole states. Our formulation exploits the density matrix associated with a branch of excitations that evolves continuously with its momentum K\mathbf{K}K, from which one may extract single-particle states allowing a construction of the QGD. We demonstrate the formulation using the single-mode approximation for excited states of two quantum Hall systems: the first for an integrally filled Landau level, and the second for a fractional quantum Hall state at filling factor $\ensuremath{\nu}=1/m$, with mmm an odd integer. In both cases we obtain the same result for the QGD, which can be attributed to the translational invariance assumed of the system. Our study demonstrates that the QGD is an intrinsic property of collective modes which is valid beyond approximations one might make for their wave functions.
We deal with the effects of an anharmonic trap on an interacting two-boson system in one dimension. Our primary focus is on the role of the induced coupling between the center of mass and the relative motion as both anharmonicity and the (repulsive) interaction strength are varied. The ground state reveals a strong localization in the relative coordinate, counteracting the tendency to fragment for stronger repulsion. To explore the quantum dynamics, we study the system's response upon (i) exciting the harmonic ground state by continuously switching on an additional anharmonicity and (ii) displacing the center of mass, this way triggering collective oscillations. The interplay between collective and internal dynamics materializes in the collapse of oscillations, which are explained in terms of few-mode models.
Abstract Particles spatially confined in trapping potentials have attracted increasing interest over the recent decade. Of particular importance are systems of charged particles, such as non‐neutral plasmas, nanoplasmas, electrons in metal clusters, electrons in quantum‐confined semiconductor structures (“artificial atoms”), electrons on the surface of liquid helium, ions in traps or highly charged particles (grains) in dusty plasmas. A second example of recent interest are systems with other kinds of pair interactions, including dipole interaction, which is important for excitons in quantum wells or ultracold Fermi and Bose gases in traps and optical lattices. Trapped systems are fundamentally different from macroscopic systems since they are dominated by strong spatial inhomogeneity and finite size effects (the properties depend on the exact particle number). Furthermore, by changing the strength of the confinement potential, the many‐particle state of the system can be externally controlled—from weak coupling (gas‐like) to strong coupling (crystal‐like). While trapped classical particles are meanwhile well understood and accessible to first‐principle computer simulations, their quantum counterparts still pose big challenges, both for experiment and theory. Therefore, collective properties that can be easily measured or computed and allow to diagnose the many‐particle state of the system are of prime importance. It has been found that the quantum breathing mode (monopole oscillation) is one of the most important such properties. In recent years a number of theoretical studies has demonstrated that the quantum breathing mode is ideally suited to measure the coupling strength (the degree of nonideality) of a trapped system, its kinetic and interaction energy and other key observables. This give rise to a novel kind of “spectroscopy” of trapped systems. In this review these developments are summarized. The quantum breathing mode is studied for trapped fermions and bosons with Coulomb and dipole interaction, respectively. A systematic description of collective oscillations and especially the breathing mode is provided. Making use of time‐dependent perturbation theory, it is shown how the corresponding breathing frequencies are connected to the properties of the initial equilibrium system. This gives rise to the application of the quantum mechanical sum rules. It is demonstrated how an improved version of the conventional sum rule formulas is suitable for an accurate description of the breathing mode in small systems. Finally, the dependence of the breathing mode on the particle number N is analyzed and the limit of large N is studied for one‐dimensional and two‐dimensional systems. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)