Effective pseudospin wave model for the coherent stripe phase in a bilayer system

Hartree–Fock theory predicts a stripe-like ground state for the two-dimensional electron gas in a bilayer quantum Hall system in a quantizing magnetic field at filling factor 4N+1 (with N>0). This stripe state contains quasi-1D linear coherent regions where electrons are delocalized across both wells and which support low-energy collective excitations in the form of phonons and pseudospin waves. We have recently computed the dispersion relation of these low-energy modes in the generalized random phase approximation. In this work, we propose an effective pseudospin model in which the stripe state is modeled as an array of coupled 1D anisotropic X–Y systems. The coupling constants and stiffness of our model are extracted from the density and pseudospin response functions computed in the GRPA.     

Static electric dipole polarizabilities of Na clusters

The static electric dipole polarizability of Na _N clusters with even N has been calculated in a collective, axially averaged and a three-dimensional, finite-field approach for , including the ionic structure of the clusters. The validity of a collective model for the static response of small systems is demonstrated. Our density functional calculations verify the trends and fine structure seen in a recent experiment. A pseudopotential that reproduces the experimental bulk bond length and atomic energy levels leads to a substantial increase in the calculated polarizabilities, in better agreement with experiment. We relate remaining differences in the magnitude of the theoretical and experimental polarizabilities to the finite temperature present in the experiments. Received 8 November 1999     

On the Ground State Energy of the¶Fractional Quantum Hall Effect

Let I(N,R) be the ground state energy of N electrons confined to a disc of radius R with a constant magnetic field B in the perpendicular direction. We show that, in the limit and , where ν is the Landau level filling factor, we have with . The factor is obtained through the solution of an extreme-value problem in measure theory. Received: 10 November 1998 / Accepted: 27 January 1999     

Constructive Approximations of Markov Operators

We construct piecewise linear Markov finite approximations of Markov operators defined on L ¹([0, 1] ^N ) and we study various properties, such as consistency, stability, and convergence, for the purpose of numerical analysis of Markov operators.     

The influence of Coulomb interaction on transport through mesoscopic two-barrier structures

We investigate the influence of the Coulomb interaction on the energy spectrum of a finite number of electrons in a geometrically confined quantum mechanical system. The spectrum is calculated numerically using the Slater determinants of the one-electron states as basis set. It is found to be dominated by the Coulomb repulsion when the system is large. Coulomb and exchange matrix elements for a given combination of four one-electron states are of the same order of magnitude. As a consequence, the energy difference between the ground states of the (N+1)- and theN-electron system is an order of magnitude smaller than each of the matrix elements, although being much larger than the separation of the one-electron energy levels. We discuss the importance of the interaction effects for the explanation of the recently observed resonant behavior of the electronic transport through quantum dots.     

Split-shell molecular orbital calculations on the diatomic alkali metal molecules

Absolute g-tensor calculations for planar hydrocarbon and for non-planar phenyl substituted hydrocarbon radicals are reported. The relevant interactions determining g are discussed. Calculations are performed on the basis of a second-order perturbation expansion. The electronic wavefunctions are obtained from a simplified version of Hoffmann's extended Hückel model (SEH), where all valence electrons are taken into account explicitly. For planar systems the observed linear dependence of g on the energy of the half filled π orbital is well reproduced. A qualitative analysis of this dependence, making restrictive assumptions about the σ electrons, was given earlier by Stone. The calculations for non-planar model systems reproduce the g-factor anomalies which are observed for highly twisted phenyl substituted hydrocarbon radicals. The results show the necessity of direct π-σ mixing and are consistent with recent investigations of the proton hyperfine couplings in such systems.     

Spectral Analysis for Systems of Atoms and Molecules Coupled to the Quantized Radiation Field

We consider systems of static nuclei and electrons – atoms and molecules – coupled to the quantized radiation field. The interactions between electrons and the soft modes of the quantized electromagnetic field are described by minimal coupling, p→p−e A (x), where A(x) is the electromagnetic vector potential with an ultraviolet cutoff. If the interactions between the electrons and the quantized radiation field are turned off, the atom or molecule is assumed to have at least one bound state. We prove that, for sufficiently small values of the fine structure constant α, the interacting system has a ground state corresponding to the bottom of its energy spectrum. For an atom, we prove that its excited states above the ground state turn into metastable states whose life-times we estimate. Furthermore the energy spectrum is absolutely continuous, except, perhaps, in a small interval above the ground state energy and around the threshold energies of the atom or molecule. Received: 3 September 1998 / Accepted: 17 March 1999     

N = 4 mechanics,WDVV equations and polytopes

N = 4 superconformal n-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial nonlinear differential equations generalizing the Witten—Dijkgraaf—Verlinde—Verlinde (WDVV) equation for F. The solutions are encoded by the finite Coxeter systems and certain deformations thereof, which can be encoded by particular polytopes. We provide A _n and B ₃ examples in some detail. Turning on the prepotential U in a given F background is very constrained for more than three particles and nonzero central charge. The standard ansatz for U is shown to fail for all finite Coxeter systems. Three-particle models are more flexible and based on the dihedral root systems.     

Two-Tape Finite Automata with Quantum and Classical States

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and two-way two-tape deterministic finite automata (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called two-way two-tape finite automata with quantum and classical states (2TQCFA). First, we give efficient 2TFA algorithms for identifying languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as L_square={aⁿbⁿ² | n ? N}L_{\mathit{square}}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}. Third, we show that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by (k+1)-tape deterministic finite automata ((k+1)TFA). Finally, we introduce k-tape automata with quantum and classical states (kTQCFA) and prove that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by kTQCFA.     

Solution and ellipticity properties of the self-duality equations of Corriganet al. in eight dimensions

We study the two sets of self-dual Yang-Mills equations in eight dimensions proposed in 1983 by E. Corriganet at. and show that one of these sets forms an elliptic system under the Coulomb gauge condition, and the other (overdetermined) set can have solutions that depend at most onN arbitrary constants, whereN is the dimension of the gauge group, hence the global solutions of both systems are finite dimensional. We describe a subvarietyP ₈ of the skew-symmetric 8 x 8 matrices by an eigenvalue criterion and we show that the solutions of the elliptic equations of Corriganet al. are among the maximal linear submanifolds ofP ₈. We propose an eighth-order action for which the elliptic set is a maximum.     

Pairing mechanism in the doped Hubbard antiferromagnet: the 4×4 model as a test case

We introduce a local formalism, in terms of eigenstates of number operators, having well defined point symmetry, to solve the Hubbard model at weak coupling on a N × N square lattice (for even N). The key concept is that of W = 0 states, that are the many-body eigenstates of the kinetic energy with vanishing Hubbard repulsion. At half filling, the wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. Further, we state a general theorem which allows to find all the W = 0 pairs (two-body W = 0 singlet states). We show that, in special cases, this assigns the ground state symmetries at least in the weak coupling regime. The N = 4 case is discussed in detail. To study the doped half filled system, we enhance the group theory analysis of the 4×4 Hubbard model introducing an Optimal Group which explains all the degeneracies in the one-body and many-body spectra. We use the Optimal Group to predict the possible ground state symmetries of the 4×4 doped antiferromagnet by means of our general theorem and the results are in agreement with exact diagonalization data. Then we create W = 0 electron pairs over the antiferromagnetic state. We show analitycally that the effective interaction between the electrons of the pairs is attractive and forms bound states. Computing the corresponding binding energy we are able to definitely predict the exact ground state symmetry. Received 24 October 2000     

Gravitational Collapse and Ergodicity in Confined Gravitational Systems

The ergodic properties of many-body systems with repulsive-core interactions are the basis of classical statistical mechanics and are well established. This is not the case for systems of purely-attractive or gravitational particles. Here we consider two examples, (i) a family of one-dimensional systems with attractive power-law interactions, , and (ii) a system of N gravitating particles confined to a finite compact domain. For (i) we deduce from the numerically-computed Lyapunov spectra that chaos, measured by the maximum Lyapunov exponent or by the Kolmogorov–Sinai entropy, increases linearly for positive and negative deviations of ν from the case of a non-chaotic harmonic chain (ν = 2). For there is numerical evidence for two additional hitherto unknown phase-space constraints. For the theoretical interpretation of model (ii) we assume ergodicity and show that for a small-enough system the reduction of the allowed phase space due to any other conserved quantity, in addition to the total energy, renders the system asymptotically stable. Without this additional dynamical constraint the particle collapse would continue forever. These predictions are supported by computer simulations. PACS numbers: 05.45.Pq, Numerical simulation of chaotic systems, 05.20.−y, Classical statistical mechanics, 36.40.Qv, Stability and fragmentation of clusters, 95.10.Fh, Chaotic dynamics.     

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G _D(r,r′, N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r′ . The expression for G(r,r′, s) is found to be the sum of the commonly accepted Dirichlet result G _D(r,r′, N) = G ₀(r,r′, N) - G ₀(r,r”, N) , where r” is the mirror image of r′ , and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.     

Thermodynamics of small systems embedded in a reservoir: a detailed analysis of finite size effects

We present a detailed study on the finite size scaling behaviour of thermodynamic properties for small systems of particles embedded in a reservoir. Previously, we derived that the leading finite size effects of thermodynamic properties for small systems scale with the inverse of the linear length of the small system, and we showed how this can be used to describe systems in the thermodynamic limit [Chem. Phys. Lett. 504, 199 (2011)]. This approach takes into account an effective surface energy, as a result of the non-periodic boundaries of the small embedded system. Deviations from the linear behaviour occur when the small system becomes very small, i.e. smaller than three times the particle diameter in each direction. At this scale, so-called nook- and corner effects will become important. In this work, we present a detailed analysis to explain this behaviour. In addition, we present a model for the finite size scaling when the size of the small system is of the same order of magnitude as the reservoir. The developed theory is validated using molecular simulations of systems containing Lennard-Jones and WCA particles, and leads to significant improvements over our previous approach. Our approach eventually leads to an efficient method to compute the thermodynamic factor of macroscopic systems from finite size scaling, which is for example required for converting Fick and Maxwell–Stefan transport diffusivities.     

Solvable and/or Integrable and/or Linearizable N-Body Problems in Ordinary (Three-Dimensional) Space. I

Abstract

Several N -body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (“acceleration equal force;” in most cases, the forces are velocity-dependent) and are amenable to exact treatment (“solvable” and/or “integrable” and/or “linearizable”). These equations of motion are always rotation-invariant, and sometimes translation-invariant as well. In many cases they are Hamiltonian, but the discussion of this aspect is postponed to a subsequent paper. We consider “few-body problems” (with, say, N =1,2,3,4,6,8,12,16,...) as well as “many-body problems” (N an arbitrary positive integer). The main focus of this paper is on various techniques to uncover such N -body problems. We do not discuss the detailed behavior of the solutions of all these problems, but we do identify several models whose motions are completely periodic or multiply periodic, and we exhibit in rather explicit form the solutions in some cases.     

How Does an Electron Tell the Time?

We examine a model of a digital clock to clarify the origin of the spacetime approach in special relativity. Specifically, we consider a two photon clock and assemble a statistical mechanics of such clocks to see how Minkowski space relates to local finite frequency clock behaviour. The result suggests that finite frequency clocks measure spacetime area and it is this feature that provides a simple mechanism behind Minkowski space on large scales. The same feature appears to implicate quantum mechanics on small scales.     

Quantum World with Quantum Time

Since the beginning of quantum mechanics there have been a lot of attempts to quantize time. In this paper we refer to the little known concept of quantum time proposed by E.Kapuscik [Hadronic J. 8 (1985) 75]. We analyze some properties of systems with quantum time. Moreover we comment and discuss the idea of quantum time.     

Rigorous Estimates of the Tails of the Probability Distribution Function for the Random Linear Shear Model

In previous work Majda and McLaughlin, and Majda computed explicit expressions for the 2Nth moments of a passive scalar advected by a linear shear flow in the form of an integral over R ^N . In this paper we first compute the asymptotics of these moments for large moment number. We are able to use this information about the large-N behavior of the moments, along with some basic facts about entire functions of finite order, to compute the asymptotics of the tails of the probability distribution function. We find that the probability distribution has Gaussian tails when the energy is concentrated in the largest scales. As the initial energy is moved to smaller and smaller scales we find that the tails of the distribution grow longer, and the distribution moves smoothly from Gaussian through exponential and stretched exponential. We also show that the derivatives of the scalar are increasingly intermittent, in agreement with experimental observations, and relate the exponents of the scalar derivative to the exponents of the scalar.