Class Of Two-dimensional Systems Research Articles

Nonclassical scattering dynamics in the quantum Hall regime.

We consider a large class of two-dimensional systems of electrons in a static disorder potential and subject to an in-plane electric field and to a strong perpendicular magnetic field. The time evolution of the single-particle states is investigated and found to be crucial for quantum Hall behavior. The macroscopic Hall current is carried by the nonadiabatic states. Quantum Hall behavior results when the Fermi energy lies in a range of adiabatic levels. Linear-response theory is inadequate to describe the quantum-mechanical single-particle scattering process in the quantum Hall regime. The general results are illustrated by an explicit weak-disorder model, where the scattering process and the nature of dc-insulating and -conducting states can be understood in detail. We stress analogies with one-dimensional conductance in the presence of disorder.

On the Lyapunov Exponent of a Harmonic Oscillator Driven by a Finite-State Markov Process

This paper considers the asymptotic behavior of the random harmonic oscillator driven by a Markov process with a finite-state space. For a certain choice of variables, the motion of the system follows circles in the plane, with jumps from one circle to another at random times. The Lyapunov exponent is expressed in terms of the equilibrium measure of a related Markov chain. The latter satisfies an integralequation whose form permits an easy derivation of estimates in various limiting cases. This is demonstrated by computing the first-order approximation in a limit of weak noise. It is shown how some of the results can be extended to a certain class of two-dimensional systems.

Scattering mechanism of integer quantum Hall conductance in a model system

We consider a large class of two-dimensional systems of charged particles in crossed electric and strong magnetic fields in the presence of a static substrate potentialV(x,y), which contains disorder. The time dependence of the single-particle Schroedinger functions are investigated. It is found on general grounds, that adiabatic states are dc-insulating and, that the macroscopic Hall current is carried by the non-adiabatic states. Hall conductance plateaus correspond to spectral domains, which contain only adiabatic states. The plateaus disappear, when these states become non-adiabatic at sufficiently high electric fields, i.e., at sufficiently large Hall currents. An explicit model system is investigated with a substrate potential composed of disorder and of a sequence of smooth barriers and wells. Here the non-adiabatic states can be calculated in a weak-disorder approximation, whenever the electric field (and hence the Hall current) is sufficiently low. In this case the quantum mechanical scattering process leading, to the quantisation of the Hall conductance can be understood in detail. Non-classical particle propagation plays a crucial role in this process. Our analysis opens new perspectives for the theory of the integer quantum Hall effect.

Sedimentation and dispersion of Brownian particles in spatially periodic potential fields

A generalized Taylor dispersion analysis is presented of the combined diffusion and sedimentation of a Brownian particle subjected to a spatially periodic potential plus a uniform force. At long times this transport process is macroscopically characterized by a uniform mean drift velocity vector of the Brownian particle, and by a dispersion dyadic that quantifies stochastic ‘‘spreading’’ about the (traveling) mean position. A general explicit integral formula is derived for the dispersion coefficient in the one-dimensional case, which supplements known results for the mean velocity. Illustrative calculations show that, for sufficiently large values of the drift force, the dispersivity exceeds the molecular diffusivity, which characterizes dispersion in the absence of any potential barriers. These calculations also lead to the surprising conclusion that an increase in the height of the potential barriers can sometimes lead to an increase in the dispersivity. A general asymptotic analysis is developed for the limiting case where the drift force tends to infinity. Although in one dimension the dominance of the drift force over the potential leads to a spatially uniform, steady-state, periodic distribution, the same behavior does not generally obtain in higher-dimensional cases. Asymptotic expressions are derived for the mean velocity and dispersivity characterizing one-dimensional systems and a class of two-dimensional systems. The analysis is subsequently applied to the related problem of sedimentation and dispersion of a particle within a medium for which its molecular diffusivity is a periodic function of position (in the absence of any potential). Although a periodic potential and a periodic diffusivity lead to equivalent mathematical problems in the absence of a sedimentation force, the presence of drift leads to important qualitative differences.

Monte Carlo study of 4He in two dimensions.

The Green's function Monte Carlo and variational methods have been used to calculate the properties of the ground state of two-dimensional liquid and solid $^{4}\mathrm{He}$ described by the HFDHE2 potential. The equation of state, melting freezing transition, radial distribution functions in the liquid and solid, and the momentum distribution in the liquid are all presented. Comparisons are made with three-dimensional $^{4}\mathrm{He}$ and two-dimensional classical systems.

New universality classes in one-dimensional antiferromagnets

New universality classes in two-dimensional critical phenomena have recently been discovered using methods of quantum field theory and conformal invariance. These can equally well describe classical two-dimensional systems at their critical temperature or one-dimensional gapless quantum systems at (or near) zero temperature. (The two types of systems are related by the transfer matrix or Feynman path integral.) It is argued that isotropic one-dimensional antiferromagnets are realizations of these new universality classes. Exact predictions for the low temperature specific heat, susceptibility, correlation length, and other critical properties are made. These agree exactly with exactly solveable models and well with experimental results on CuCl2⋅2(NC5H5).

New approach to observer design for two-dimensional systems

A new method of observer design for shift-invariant two-dimensional digital systems is presented. The method is applicable to a large class of two-dimensional systems and the condition of applicability is very simple to test.

Further examples of integrable systems in two dimensions

The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q 1 +iq 2 andq=q 1 −iq 2. As a result a class of Toda-type potentials admitting second order invariants is explored.

On some variational approximations in two-dimensional classical lattice systems

A new derivation is presented of some variational approximations for classical lattice systems that belong to the class of cluster-variation methods, among them the well-known Bethe-Peierls and Kramers-Wannier approximations. The limiting behavior of a hierarchical sequence of cluster-variation approximations, the so-calledC hierarchy, is discussed. It is shown that this hierarchy provides a monotonically decreasing sequence of upper boundsfn on the free energy per lattice sitef and thatfn → f asn → ∞. Our results are based on extension theorems for states given on subsets of the lattice, which might be of some independent interest, and on an application of transfer matrix concepts to the variational characterization of translation-invariant equilibrium states.

Construction of the second constant of motion for two-dimensional classical systems

A general method for the construction of the second constant of motion of third and fourth orders is given for two-dimensional systems in terms of z=q1+iq2, and z̄=q1−iq2. Correspondingly, the third- and fourth-order potential equations are obtained whose solutions directly provide the integrable systems. Using the Holt ansatz, the potential equation corresponding to the third-order invariants has been reduced to a pair of potential equations whose solutions yield a large class of integrable systems.

Microscopic Structure of Two-Dimensional Electron Glass

Structural properties of a classical two-dimensional system of electrons on a corrugated surface are investigated by the molecular dynamics method. Rapid quenching of the electron liquid leads to glass formation. Direct evidence for two-level states is obtained in the electron glass and the transitions between them are found to arise from displacements of a small number of spatially localized electrons.

Critical behavior of (1+1)-dimensional transverse Ising models for spinS≥12

The phase transitions in the ground state of (1+1)-dimensional Ising models in a transverse field are investigated for spin $S=\frac{1}{2}, 1, \frac{3}{2}, \mathrm{and} 2$. The critical fields and "thermal" and correlation exponents are calculated as a function of $S$ by the finite-size scaling method. The universality of the transition with respect to $S$ is fully confirmed. Furthermore, we elaborate on a recent conjecture by Luck and others relating the correlation exponent $\ensuremath{\eta}$ to the amplitude of a correlation length for a two-dimensional (2D) classical isotropic system. The extension of this relation to quantum chains as extremely anisotropic 2D systems has been considered. We find that while no simple prescription can be given to calculate various correlation exponents from the amplitudes, the ratios of the amplitudes of different quantities still have universal character.

Dynamical invariants for two-dimensional time-dependent classical systems

General equations are formulated to determine all potentials for two-dimensional systems of the type L= (1)/(2) ( p21 +p22) −V(q1,q2,t), which admits invariants of the form I=a0+aiξi + (1)/(2) aijξiξj, i, j=1,2, where ξ1 =ż=q̇1+iq̇2, ξ2=z̄=q̇1−iq̇2, a0, ai, aij are arbitrary functions of t, z=q1+iq2, and z̄=q1−iq2. Simplifying restrictions reduce the general equation to a tractable form. The resulting equations are solved for a special class of time-separable potentials and derive (i) the van der Waals-type long-range potential, V(r,t)=β(t)(b/r4+d) and (ii) the quark-confining logarithmic potential, V(r,t)=β(t)λ(ln r+b1/r4+d1). Invariants I for the resulting dynamical systems are found. Some observations on the present method in the context of Katzin and Levine and of Lewis and Leach analyses have also been made.

Two-dimensional time-dependent classical systems with confining potentials

The construction of invariants for two-dimensional, time-dependent classical systems has been carried out with special reference to linearly and harmonically confining potentials.

On potential and field fluctuations in two-dimensional classical charged systems

We supplement a previous paper on three-dimensional systems by studying the electric potential and field fluctuations in two-dimensional Coulomb systems. The novelty in two dimensions is that the fluctuations of the potential at a point are infinite in the thermodynamic limit. However, the potential difference between two points has finite fluctuations, which resemble the ones which occur in the three-dimensional case. The field fluctuations are also rather similar in both cases. The correlations do not have a fast decay. Explicit results are obtained for a solvable model; the fluctuations of the potential are Gaussian with an infinite variance.

The mermin-wagner phenomenon and cluster properties of one- and two-dimensional systems

We give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one- and two-dimensional quantum and classical lattice and continuum systems. For a class of models verifying infrared bounds our conditions are necessary and sufficient. Using the same techniques we obtain “a priori” bounds on clustering for systems with continuous symmetry, improving results of Jasnow and Fisher.

Commensurate-incommensurate transitions in two dimensions-a low-temperature expansion

The commensurate-incommensurate transition of a two-dimensional classical sine-Gordon system is studied by the equivalent one-dimensional quantum system. The latter is expanded around its classical limit, corresponding to a low-temperature expansion of the two-dimensional system. It is found that both mass and wavefunction renormalisation are required. The critical exponent for incommensurability is beta =0, but if the momentum cut-off is kept finite one obtains beta =1. The classical limit is a singular point and the results are reliable when they are not too close to the transition.

Efficient Monte Carlo Procedure for Systems with Fermions

A new Monte Carlo approach applicable to systems with fermion and boson degrees of freedom is proposed. The main advantage of the method is its speed: For systems in one space and one time dimension (1 + 1) the calculation reduces to doing Monte Carlo computations on a two-dimensional classical system with local interactions. Here the potential of the method is illustrated by application to various (1 + 1)-dimensional systems.

The two-dimensional Lennard-Jones system: Sublimation, vaporization, and melting

The solid–vapor equilibrium line and the liquid–vapor equilibrium line near the melting region for classical two-dimensional systems of particles interacting by Lennard-Jones (12,6) pair potentials are determined from Monte Carlo simulations with 224 particles. Structural and thermodynamic properties are evaluated for homogeneous single phases on both sides of the melting transition. The Monte Carlo results show that a cell model approximation to the solid phase has good accuracy for most of the anharmonic 2D solid conditions of this work. Free energies for solid states are constructed starting from the low temperature quasiharmonic solid; for the 2D liquid states the starting point is the virial expansion for the dilute gas. From these free energies the triple-point melting of the periodically extended 224-particle system is found to be first order and the latent heat of melting is determined. Graphic displays are obtained showing the occurrence of dislocation pairs, voids, particle exchange, and mixed phases. The implementation of the simulation is discussed, including problems of superheating and the achievement of equilibrium.

Dynamical fluctuation spectra of two-dimensional classical electron liquids in magnetic fields

Dynamical fluctuation spectra of a two-dimensional classical system of electrons in a magnetic field are obtained by numerical experiments in the domain of a strongly coupled liquid. Excitations in these systems have dispersion relations similar to those in the Wigner lattice of the same number density.