Correlated fermions on a lattice in high dimensions

The limit of infinite dimension of the Hubbard model was recently introduced by Metzner and Vollhardt as a new type of model with interesting implications. In the present paper the same limit is applied to a family of lattice fermion models with generalized kinetic and potential energies. A simple method is introduced for investigating the important issue of handling momentum conservation at the interaction vertices. It is shown that the irrelevance of momentum conservation found by Metzner and Vollhardt applies more generally than their Gaussian density of states. Interactions between particles on different sites are shown to simplify to their Hartree substitute. This leaves on-site interactions as the only dynamical interactions in the limit of infinite dimension.     

Variational evaluation of correlation functions for lattice electrons in high dimensions

We present a general formalism for the diagrammatic calculation of correlation functions for Hubbard-type models in terms of projected wave functions. It is shown that in the limit of high spatial dimensionsd only diagrams with bubble-structure remain. This causes correlation functions to have an overall RPA-type form ind. Exact evaluations are performed for the Gutzwiller wave function. Nearest neighbor correlations are shown to be proportional to their value in the non-interacting case, i.e. are renormalized. However, their absolute value is only of order 1/d. Hence this wave function does not describe spin correlations adequately in high dimensions. The asymptotic behavior of the spin-correlation function is extracted and is found to have a scaling form similar tod=1. Assuming this form to hold in all dimensions we show that the Brinkman-Rice transition only occurs ind=. Finite orders of perturbation theory in 1/d around this singular point are not sufficient to remove the transition.     

Finite-element lattice fermions in perturbation theory

A lattice Thirring model is defined using a finite-element differencing scheme. The momentum-space propagator is calculated to second order in the coupling, and it is shown that the form of the ultraviolet divergence as the lattice spacing is taken to zero is identical to that found in continuum perturbation theory. This provides further evidence for the absence of fermion species-doubling present in most other lattice formulations.     

Exact results for strongly correlated fermions in 2 + 1 dimensions

We derive exact results for a model of strongly interacting spinless fermions hopping on a two-dimensional lattice. By exploiting supersymmetry, we find the number and type of ground states exactly. Exploring various lattices and limits, we show how the ground states can be frustrated, quantum critical, or combine frustration with a Wigner crystal. We show that on generic lattices the model is in an exotic "superfrustrated" state characterized by an extensive ground-state entropy.     

Scaling behavior of disordered lattice fermions in two dimensions

We propose a lattice model for Dirac fermions which allows us to break the degeneracy of the node structure. In the presence of a random gap we analyze the scaling behavior of the localization length as a function of the system width within a numerical transfer-matrix approach. Depending on the strength of the randomness, there are different scaling regimes for weak, intermediate and strong disorder. These regimes are separated by transitions that are characterized by one-parameter scaling.     

Variational study of vacuum structure with fermions ind+1 dimensional Hamiltonian lattice gauge theory

The physical vacuum state and general expression for the Hamiltonian ofd+1 dimensional lattice gauge theory are given by incorporating the exact ground state of pure gauge theory and the variational fermion vacuum state. The applications toSU(2) andSU(3) gauge theories in 2+1 and 3+1 dimensions are demonstrated and the fermion condensates \(\left\langle {\bar \psi \psi } \right\rangle \) as functions of 1/g ² are calculated.     

Finite lattice fermions and gauge fields in 1+1 dimensions

The finite hamiltonian method (with Lanczos diagonalisation) is used in a numerical comparison of different formulations of the free and interacting Dirac field in 1 + 1 dimensions. We present new results for the Schwinger model and pay special attention to the choice of gauge and of boundary conditions.     

Variational cluster approach to correlated electron systems in low dimensions

A self-energy-functional approach is applied to construct cluster approximations for correlated lattice models. It turns out that the cluster-perturbation theory [Phys. Rev. Lett. 84, 522 (2000)]] and the cellular dynamical mean-field theory [Phys. Rev. Lett. 87, 186401 (2001)]] are limiting cases of a more general cluster method. The results for the one-dimensional Hubbard model are discussed with regard to boundary conditions, bath degrees of freedom, and cluster size.     

On the bosonization of interacting fermions in high dimensions

It is shown that the recently proposed bosonization procedure of interacting fermions in dimensions higher than one is equivalent to the RPA in the long wavelength limit.     

Ising lattice gauge theory in three dimensions

By raising the transfer matrix to a finite power the partition function for a finite lattice Z(2) gauge model is obtained exactly. The zeros of the resultant polynomial are found and some plaquette-plaquette expectation values are extracted. An exponential fit for the inverse correlation length matches onto both strong- and weak-coupling results but breaks down close to the second-order phase transition point.Similar calculations for the three-dimensional Ising model are also discussed.     

Interacting fermions in two dimensions: beyond the perturbation theory

We consider a system of 2D fermions with a short-range interaction. A straightforward perturbation theory is shown to be ill defined even for an infinitesimally weak interaction, as the perturbative series for the self-energy diverges near the mass shell. We show that the divergences result from the interaction of fermions with the zero-sound collective mode. By resumming the most divergent diagrams, we obtain a closed form of the self-energy near the mass shell. The spectral function exhibits a threshold feature at the onset of the emission of the zero-sound waves. We also show that the interaction with the zero sound does not affect a nonanalytic, T2 part of the specific heat.     

On confinement of fermions in strongly coupled lattice gauge theory

A lattice theory of Fermi fields of massm coupled to gauge fields in the region wherem and the gauge field coupling constantg are large is studied. It is shown that the energy of some states composed of a fermion and a distant antifermion with a string in between grows at least linearly with the distance if 1<g ⁶<m<g ^logg .On leave from Department of Mathematical Methods of Physcis, Warsaw University, Hoa 74, PL 00-682 Warsaw, Poland     

Progress and bottlenecks in simulating lattice gauge theory with fermions

We present a progress report in lattice gauge theory computer simulations which includes the effects of light, dynamical fermions. Microcanonical and hybrid microcanonical-Langevin alogrithms are presented and discussed. A method for “accelerating” stochastic differential equations and defeating critical slowing down is reviewed. Physics applications such as the thermodynamics of quantum chromodynamics, hierarchal energy scales in unified gauge theories, and the phase diagram of theories with many fermion species are discussed. Prospects for future research are assessed.     

Generalized bogoliubov transformations in lattice gauge theory with staggered fermions

A transformation is investigated that mixes quarks with composites of N−1 antiquarks in a gauge-invariant way for QCD with gauge group SU (N). An infinite family of identities among fermionic Green functions is derived in the form of a generating functional.     

Dynamical fermions in lattice gauge theories: Metropolis and Langevin techniques in two dimensions

A variety of techniques for the inclusion of dynamical fermions in lattice gauge theory is examined. Three pseudo-fermionic techniques that have the characteristics desirable for an unquenched simulation of four-dimensional QCD are studied in detail. Langevin and Metropolis pseudo-fermionic techniques are implemented for a 64×64 lattice on the Distributed Array Processor and their relative merits examined both for free fermions and the lattice Schwinger model.