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K. Ziegler

Publications and source records attributed to K. Ziegler.

107 records · Page 6Linked to original sources
arXiv1997

Characterization of the Local Density of States Fluctuations near the Integer Quantum Hall Transition in a Quantum Dot Array

We present a calculation for the second moment of the local density of states in a model of a two-dimensional quantum dot array near the quantum Hall transition. The quantum dot array model is a realistic adaptation of the lattice model for the quantum Hall transition in the two-dimensional electron gas in an external magnetic field proposed by Ludwig, Fisher, Shankar and Grinstein. We make use of a Dirac fermion representation for the Green functions in the presence of fluctuations for the quantum dot energy levels. A saddle-point approximation yields non-perturbative results for the first and second moments of the local density of states, showing interesting fluctuation behaviour near the quantum Hall transition. To our knowledge we discuss here one of the first analytic characterizations of chaotic behaviour for a two-dimensional mesoscopic structure. The connection with possible experimental investigations of the local density of states in the quantum dot array structures (by means of NMR Knight-shift or single-electron-tunneling techniques) and our work is also established.

cond-mat.mes-hall↗
arXiv1997

Level statistics and localization in a 2D quantum percolation problem

A two dimensional model for quantum percolation with variable tunneling range is studied. For this purpose the Lifshitz model is considered where the disorder enters the Hamiltonian via the nondiagonal elements. We employ a numerical method to analyze the level statistics of this model. It turns out that the level repulsion is strongest around the percolation threshold. As we go away from the maximum level repulsion a crossover from a GOE type behavior to a Poisson like distribution is indicated. The localization properties are calculated by using the sensitivity to boundary conditions and we find a strong crossover from localized to delocalized states.

cond-mat↗
arXiv1997

Is the Peak Value of $σ_{xx}$ at the Quantum Hall Transition Universal?

The question of the universality of the longitudinal peak conductivity at the integer quantum Hall transition is considered. For this purpose, a system of 2D Dirac fermions with random mass characterised by variance $g$ is proposed as a model which undergoes a quantum Hall transition. Whilst for some specific models the longitudinal peak conductivity $σ_{xx}$ was found to be universal (in agreement with the conjecture of Lee et al. as well as with some numerical work), we find that $σ_{xx}$ is reduced by a factor $(1+g/2π)^{-1}$, at least for small $g$. This provides some theoretical evidence for the non-universality of $σ_{xx}$, as observed in a number of experiments.

cond-mat.mes-hall↗
arXiv1997

Reply to the Comment by A.A. Nersesyan and A.M. Tsvelik on the "Non--Zero Fermi Level Density of States for a Disordered d-Wave Superconductor in Two Dimensions" (PRL 77, 3013 (1996)) by K. Ziegler, M.H. Hettler and P.J. Hirschfeld

In a recent Comment (cond-mat/9701197) on our Letter (PRL 77, 3013 (1996), cond-mat/9604176) Nersesyan and Tsvelik questioned the relevance of our exact calculation of the density of states (DOS) of a 2D d-wave superconductor using a Lorentzian disorder distribution, claiming ``that models with Lorentzian and Gaussian disorder belong to different universality classes'', on grounds of a comparison with a straightforward perturbation expansion in the disorder potential. We showed in a very recent paper (cond-mat/9703047) that a rigorous nonzero lower bound for DOS can be established without relying on the tails of the disorder distribution, i.e. both the Gaussian and Lorentzian distributions belong to a large class of distributions which give a nonzero lower bound for the DOS. This proves that the result of our Letter is generic.

cond-mat.supr-con↗
arXiv1997

Lower Bound for the Fermi Level Density of States of a Disordered D-Wave Superconductor in Two Dimensions

We consider a disordered d--wave superconductor in two dimensions. Recently, we have shown in an exact calculation that for a lattice model with a Lorentzian distributed random chemical potential the quasiparticle density of states at the Fermi level is nonzero. As the exact result holds only for the special choice of the Lorentzian, we employ different methods to show that for a large class of distributions, including the Gaussian distribution, one can establish a nonzero lower bound for the Fermi level density of states. The fact that the tails of the distributions are unimportant in deriving the lower bound shows that the exact result obtained before is generic.

cond-mat.supr-con↗
arXiv1996

Scaling Behavior and Universality near the Quantum Hall Transition

A two-dimensional lattice system of non-interacting electrons in a homogeneous magnetic field with half a flux quantum per plaquette and a random potential is considered. For the large scale behavior a supersymmetric theory with collective fields is constructed and studied within saddle point approximation and fluctuations. The model is characterized by a broken supersymmetry indicating that only the fermion collective field becomes delocalized whereas the boson field is exponentially localized. Power counting for the fluctuation terms suggests that the interactions between delocalized fluctuations are irrelevant. Several quasi--scaling regimes, separated by large cross--over lengths, are found with effective exponents $ν$ for the localization length $ξ_l$. In the asymptotic regime there is $ν=1/2$ in agreement with an earlier calculation of Affleck and one by Ludwig et al. for finite density of states. The effective exponent, relevant for physical system, is $ν=1$ where the coefficient of $ξ_l$ is growing with randomness. This is in agreement with recent high precision measurements on Si MOSFET and AlGaAs/GaAs samples.

cond-mat.mes-hall↗
arXiv1996

Quantum Hall Transition in an Array of Quantum Dots

A two-dimensional array of quantum dots in a magnetic field is considered. The electrons in the quantum dots are described as unitary random matrix ensembles. The strength of the magnetic field is such that there is half a flux quantum per plaquette. This model exhibits the Integer Quantum Hall Effect. For $N$ electronic states per quantum dot the limit $N\to\infty$ can be solved by a saddle point integration of a supersymmetric field theory. The effect of level statistics on the density of states and the Hall conductivity is compared with the effect of temperature fluctuations.

cond-mat↗
arXiv1996

Dirac Fermions with Disorder in Two Dimensions: Exact Result

Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function which has only poles on the lower half-plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods.

cond-mat↗
arXiv1995

Phase Transitions of Fermions Coupled to a Gauge Field: a Quantum Monte Carlo Approach

A grand canonical system of non-interacting fermions on a square lattice is considered at zero temperature. Three different phases exist: an empty lattice, a completely filled lattice and a liquid phase which interpolates between the other two phases. The Fermi statistics can be changed into a Bose statistics by coupling a statistical gauge field to the fermions. Using a quantum Monte Carlo method we investigate the effect of the gauge field on the critical properties of the lattice fermions. It turns out that there is no significant change of the phase diagram or the density of particles due to the gauge field even at the critical points. This result supports a recent conjecture by Huang and Wu that certain properties of a three-dimensional flux line system (which is equivalent to two-dimensional hard-core bosons) can be explained with non-interacting fermion models.

cond-mat↗
arXiv1995

Two-Dimensional Electrons in a Strong Magnetic Field with Disorder: Divergence of the Localization Length

Electrons on a square lattice with half a flux quantum per plaquette are considered. An effective description for the current loops is given by a two-dimensional Dirac theory with random mass. It is shown that the conductivity and the localization length can be calculated from a product of Dirac Green's functions with the {\it same} frequency. This implies that the delocalization of electrons in a magnetic field is due to a critical point in a phase with a spontaneously broken discrete symmetry. The estimation of the localization length is performed for a generalized model with $N$ fermion levels using a $1/N$--expansion and the Schwarz inequality. An argument for the existence of two Hall transition points is given in terms of percolation theory.

cond-mat↗
arXiv1995

An Exactly Soluble Model of Directed Polymers with Multiple Phase Transitions

Polymer chains with hard-core interaction on a two-dimensional lattice are modeled by directed random walks. Two models, one with intersecting walks (IW) and another with non-intersecting walks (NIW) are presented, solved and compared. The exact solution of the two models, based on a representation using Grassmann variables, leads, surprisingly, to the same analytic expression for the polymer density and identical phase diagrams. There are three different phases as a function of hopping probability and single site monomer occupancy, with a transition from the dense polymer system to a polymer liquid (A) and a transition from the liquid to an empty lattice (B). Within the liquid phase there exists a self-dual line with peculiar properties. The derivative of polymer density with respect to the single site monomer occupancy diverges at transitions A and B, but is smooth across and along the self-dual line. The density-density correlation function along the direction $x$, perpendicular to the axis of directedness has a power law decay 1/$x^2$ in the entire liquid phase, in both models. The difference between the two models shows up only in the behavior of the correlation function along the self-dual line: it decays exponentially in the IW model and as 1/$x^4$ in the NIW model.

cond-mat↗
arXiv1994

Integer Quantum Hall Effect for Lattice Fermions

A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and $N$ levels per site is considered. This is a model which exhibits the Integer Quantum Hall Effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large $N$ limit can be solved: two Hall transitions appear and there is an interpolating behavior between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.

cond-mat↗
arXiv1994

Competition of Phonons and Magnetic Interaction in One-Dimensional Fermion Systems

We consider a system of spin-dependent fermions on a one-dimensional lattice which is coupled to phonons. The phonons create either a Peierls instability by breaking the translational invariance or create long range correlations. On the other hand, there is a spin-dependent Hubbard-like interaction which competes with the effective fermion-fermion interaction induced by the phonons. We investigate the competition of both interactions using a renormalization group calculation for weak interaction. The renormalization group transformation creates a third interaction. It turns out that non-interacting fermions represent a semi-stable fixed point of the fermion system.

cond-mat↗
arXiv1994

Incommensurate Phase on a Disordered Surface: Instability Against the Formation of Overhangs and Finite Loops

The stability of the quenched incommensurate phase in two dimensions against the creation of overhangs and finite loops (OH/FL) in the replica space is investigated for a model of domain walls with $N$ colors. Introducing a chemical potential $ε$ for OH/FL, the probability for the formation of these objects is studied for $ε\to0$. In the pure limit this probability vanishes with $\sqrtε$, whereas the fluctuations of this probability are long-range correlated in the quenched system. This indicates an instability related to the symmetry in replica space. It is accompanied by the creation of a massless boson. The latter leads to a power law decay with exponent $\propto 1/N$ for the product of the correlation functions along the domain walls.

cond-mat↗
arXiv1993

The Energy-Energy Correlation Function of the Random Bond Ising Model in Two Dimensions

The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is $ν_e=1/2$ in contrast to the pure system where $ν_e=1$. The calculation is based on a previously developed effective field theory for the energy density fluctuations.

cond-mat↗
arXiv1993

Stability of Insulating Phases in the Hubbard Model: a Cluster Expansion

The stability of the insulating regime of the Hubbard model on a $d$-dimensional lattice, which is characterized by an exponential decay of the Green's functions, is investigated in terms of a cluster expansion. This expansion for the Green's function is organized in terms of connected clustered transfer matrices. An upper bound for the expansion terms is derived for the hopping rate ${\bar t}$ depending on the coupling constant $U$ as ${\bar t}<U/4d$. This implies an upper bound for the decay length of the Green's function.

cond-mat↗