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Consequences of different discretizations of the two-dimensional Dirac operator on low energy properties (e.g., the number of nodes) and their relations to gauge properties are discussed. Breaking of the gauge invariance was suggested in a recent work by M. Bocquet, D. Serban, and M.R. Zirnbauer [cond-mat/9910480] in order to destroy an intermediate metallic phase of lattice Dirac fermions with random mass. It is explained that such a procedure is inconsistent with the underlying lattice physics. Previous results point out that the logarithmic growth of the slope of the average density of states with the system size, obtained in the field-theoretical calculation of M. Bocquet et al., could be a precursor for the appearence of an intermediate metallic phase., Comment: 10 pages, 1 figure

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., Comment: 4 pages, 4 figures

A grand canonical ensemble of interacting bosons is considered. The zero temperature phase diagram is evaluated from the mean-field approximation of the functional integral. Three phases are found: a superfluid, a normal fluid and a Mott insulator., Comment: Latex, 5 pages, 1 figure

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., Comment: 9 pages, 2 figures included, Plain TeX and psbox

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., Comment: extended introduction and additional chapter on 'species multiplication' including two figures

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., Comment: 12 pages, Plain-Tex, 5 figures

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., Comment: 10 pages, RevTeX, no figures

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., Comment: 8 pages, plain-TeX, figures available upon request

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., Comment: 9 pages, Plain-TeX

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., Comment: 6 pages,plainTEX

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 $\epsilon$ for OH/FL, the probability for the formation of these objects is studied for $\epsilon\to0$. In the pure limit this probability vanishes with $\sqrt{\epsilon}$, 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., Comment: 10 pages, RevTex 3.0, 1 figure (postscript)

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 $\nu_e=1/2$ in contrast to the pure system where $\nu_e=1$. The calculation is based on a previously developed effective field theory for the energy density fluctuations., Comment: 8 pages, RevTeX 3.0, TKM-preprint

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}