Theoretical approaches to one-dimensional and quasi-one-dimensional quantum rings with a few electrons are reviewed. Discrete Hubbard-type models and continuum models are shown to give similar results governed by the special features of the one-dimensionality. The energy spectrum of the many-body states can be described by a rotation-vibration spectrum of a 'Wigner molecule' of 'localized' electrons, combined with the spin-state determined from an effective antiferromagnetic Heisenberg Hamiltonian. The persistent current as a function of the magnetic flux through the ring shows periodic oscillations arising from the 'rigid rotation' of the electron ring. For polarized electrons the periodicity of the oscillations is always the flux quantum Φ0. For nonpolarized electrons the periodicity depends on the strength of the effective Heisenberg coupling and changes from Φ0 first to Φ0/2 and eventually to Φ0/N when the ring gets narrower.

The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e. a two dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be close to reaching the quantum Hall regime. This paper reviews the theoretical and experimental work done to date in exploring quantum Hall physics in cold bosonic gases. Future perspectives are discussed briefly, in particular the idea of exploiting some of these strongly correlated states in the context of topological quantum computing.

We investigate the operational characteristics of a nanorelay based on a conducting carbon nanotube placed on a terrace in a silicon substrate. The nanorelay is a three terminal device that acts as a switch in the GHz regime. Potential applications include logic devices, memory elements, pulse generators, and current or voltage amplifiers.Nanoelectromechanical systems (NEMS) are a rapidly growing research field with substantial potential for future applications. The basic operating principle underlying NEMS is the strong electromechanical coupling in nanometer-size electronic devices in which the Coulomb forces associated with device operation are comparable with the chemical forces that hold the devices together. Carbon nanotubes (CNT) 1 are ideal candidates for nanoelectromechanical devices due to their well-characterized chemical and physical structures, low masses, exceptional directional stiffness, and good reproducibility. Nanotubebased NEMS have internal operating frequencies in the gigahertz range, which makes them attractive for a number of applications. Recent progress in this direction includes fabrication of CNT nanotweezers, 2,3 CNT based random access memory, 4 and super-sensitive sensors. 5,6In this paper we consider another example of CNT based NEMS, a so-called nanorelay. This three-terminal device consists of a conducting CNT placed on a terraced Si substrate and connected to a fixed source electrode. A gate electrode is positioned underneath the CNT so that charge can be induced in the CNT by applying a gate voltage. The resulting capacitive force between the CNT and the gate bends the tube and brings the tube end into contact with a drain electrode on the lower terrace, thereby closing an electric circuit. We describe the system with a model based on classical elasticity theory 7 and the orthodox theory of Coulomb blockade,8,9 and study its IV-characteristics and switching dynamics. Theoretical studies of a related two-terminal structure have recently been reported. 10,11Model system. The geometry of the nanorelay is depicted in Fig. 1. We model the CNT as an elastic cantilever using continuum elasticity theory:7 Assuming that only the lowest vibrational eigenmode is excited, and that the bending profile upon applying an external force is the same as that of free oscillations, one can express the potential energy of the bent tube in terms of the deflection x of its tip as V = kx 2 /2. The effective spring constant k depends on the geometry of the tube and is approximately given by k ≈ 3EI/L 3 . Here E is Young's modulus, experimentally determined to be approximately 1.2 TPa, 12,13 L is the tube length and I = π(D 2 , where M is the total tube mass and Ω its lowest eigenfrequency.7 It is known experimentally that Q-factors of CNT cantilevers are of the order of 170-500.14 We model this by a phenomenological damping force −γ dẋ in the equations of motion.

The fractional quantum Hall effect, being one of the most studied phenomena in condensed matter physics during the past thirty years, has generated many groundbreaking new ideas and concepts. Very early on it was realized that the zoo of emerging states of matter would need to be understood in a systematic manner. The first attempts to do this, by Haldane and Halperin, set an agenda for further work which has continued to this day. Since that time the idea of hierarchies of quasiparticles condensing to form new states has been a pillar of our understanding of fractional quantum Hall physics. In the thirty years that have passed since then, a number of new directions of thought have advanced our understanding of fractional quantum Hall states, and have extended it in new and unexpected ways. Among these directions is the extensive use of topological quantum field theories and conformal field theories, the application of the ideas of composite bosons and fermions, and the study of nonabelian quantum Hall liquids. This article aims to present a comprehensive overview of this field, including the most recent developments.arXiv:1601.01697v2 [cond-mat.str-el]

We exploit the analogy with the Quantum Hall (QH) system to study weakly interacting bosons in a harmonic trap. For a $\delta$-function interaction potential the ``yrast'' states with $L\ge N(N-1)$ are degenerate, and we show how this can be understood in terms of Haldane exclusion statistics. We present spectra for 4 and 8 particles obtained by numerical and algebraic methods, and demonstrate how a more general hard-core potential lifts the degeneracies on the yrast line. The exact wavefunctions for N=4 are compared with trial states constructed from composite fermions (CF), and the possibility of using CF-states to study the low L region at high N is discussed.Comment: 6 pages, including 4 figures and 3 table

We show that the quantum Hall wave functions for the ground states in the Jain series nu=n/(2np+1) can be exactly expressed in terms of correlation functions of local vertex operators Vn corresponding to composite fermions in the nth composite-fermion (CF) Landau level. This allows for the powerful mathematics of conformal field theory to be applied to the successful CF phenomenology. Quasiparticle and quasihole states are expressed as correlators of anyonic operators with fractional (local) charge, allowing a simple algebraic understanding of their topological properties that are not manifest in the CF wave functions. Moreover, our construction shows how the states in the nu=n/(2np+1) Jain sequence may be interpreted as condensates of quasiparticles.

In the conformal field theory (CFT) approach to the quantum Hall effect, the multi-electron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires non-local operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts new ones. In particular we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state. Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a longtime controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states, and argue that our construction of localized quasielectron states sheds new light on their statistics. At the technical level we introduce a generalized normal ordering, that allows us to "fuse" an electron operator with the inverse of an hole operator, and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.

The yrast spectra (i.e. the lowest states for a given total angular momentum) of quantum dots in strong magnetic fields, are studied in terms of exact numerical diagonalization and analytic trial wave functions. We argue that certain features (cusps) in the many-body spectrum can be understood in terms of particle localization due to the strong field. A new class of trial wavefunctions supports the picture of the electrons being localized in Wigner molecule-like states consisting of consecutive rings of electrons, with low-lying excitations corresponding to rigid rotation of the outer ring of electrons. The geometry of the Wigner molecule is independent of interparticle interactions and the statistics of the particles.