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Christof Gattringer

Publications and source records attributed to Christof Gattringer.

At least 163 records · Page 9Linked to original sources
arXiv2001

Properties of near-zero modes and chiral symmetry breaking

We study localization and chirality properties of eigenvectors of the lattice Dirac operator. In particular we focus on the dependence of our observables on the size of the corresponding eigenvalue, which allows us to study the transition of a near-zero mode into a bulk mode. We analyze ensembles of quenched SU(3) configurations using a Dirac operator which is a systematic expansion in path length of a solution of the Ginsparg-Wilson equation. Our results support the interpretation of the excitations relevant for chiral symmetry breaking as interacting instantons and anti-instantons.

hep-lat↗
arXiv2001

Chirally improved Dirac operators: Studying the sensitivity to topological excitations for zero and finite temperature

We discuss the construction and properties of an approximate solution of the Ginsparg-Wilson equation, the so-called chirally improved lattice Dirac operator. In particular we study the behavior of its eigenmodes in smooth instanton backgrounds as well as for thermalized gauge configurations on both sides of the QCD phase transition. We compare with results from other Dirac operators including the overlap operator. The results support the picture of chiral symmetry breaking being closely related to instantons.

hep-lat↗
arXiv2000

Approximate Ginsparg-Wilson fermions: A first test

We construct a 4-d lattice Dirac operator D using a systematical expansion in terms of simple operators on the lattice. The Ginsparg-Wilson equation turns into a system of coupled equations for the expansion coefficients of D. We solve these equations for a finite parametrization of D and find an approximate solution of the Ginsparg-Wilson equation. We analyze the spectral properties of our D for various ensembles of quenched SU(3) configurations. Improving the gauge field action considerably improves the spectral properties of our D.

hep-lat↗
arXiv2000

New approximate solutions of the Ginsparg-Wilson equation - tests in 2-d

A new method for finding approximate solutions of the Ginsparg-Wilson equation is tested in 2-d. The Dirac operator is first constructed and then used in a dynamical simulation of the 2-flavor Schwinger model. We find a very small mass of the pi-particle implying almost chirally symmetric fermions. The generalization of our method to 4-d is straightforward.

hep-lat↗
arXiv2000

A new approach to Ginsparg-Wilson fermions

We expand the most general lattice Dirac operator D in a basis of simple operators. The Ginsparg-Wilson equation turns into a system of coupled quadratic equations for the expansion coefficients. Our expansion of D allows for a natural cutoff and the remaining quadratic equations can be solved numerically. The procedure allows to find Dirac operators which obey the Ginpsparg-Wilson equation with arbitrary precision.

hep-lat↗
arXiv1999

The lattice Schwinger model as a discrete sum of filled Wilson loops

Using techniques from hopping expansion we identically map the lattice Schwinger model with Wilson fermions to a model of oriented loops on the lattice. This is done by first computing the explicit form of the fermion determinant in the external field. Subsequent integration of the gauge fields renders a sum over all loop configurations with simple Gaussian weights depending on the number of plaquettes enclosed by the loops. In our new representation vacuum expectation values of local fermionic operators (scalars, vectors) can be computed by simply counting the loop flow through the sites (links) supporting the scalars (vectors). The strong coupling limit, possible applications of our methods to 4-D models and the introduction of a chemical potential are discussed.

hep-lat↗
arXiv1999

A formula for the hopping expansion of 8-vertex models coupled to an external field

We study a generalized 8-vertex model where the vertices are coupled to a locally varying field. We rewrite the partition function as an integral over Grassmann variables. In this form it is possible to explicitly evaluate all terms of the hopping expansion. Applications of the resulting formula, in particular its relation to 2-D lattice field theories with fermions are discussed.

cond-mat.stat-mech↗
arXiv1998

Clover improvement, spectrum and Atiyah-Singer index theorem for the Dirac operator on the lattice

We study the role of the O(a)-improving clover term for the spectrum of the lattice Dirac operator using cooled and thermalized SU(2) gauge field configurations. For cooled configurations we observe improvement of the spectral properties when adding the clover term. For the thermalized case (12^4, beta = 2.4) without clover term we find a rather bad separation of physical and doubler branches making a probabilistic interpretation of the Atiyah-Singer index theorem on the lattice questionable for this beta and lattice size. Adding the clover term leads to the creation of additional real eigenvalues which come in pairs of opposite chirality thus further worsening the situation for the index theorem.

hep-lat↗
arXiv1998

Loop representation for 2-D Wilson lattice fermions in a scalar background field

We show that the fermion determinant for 2-D Wilson lattice fermions coupled to an external scalar field is equivalent to self avoiding loops interacting with the external field. In an application of the resulting formula we integrate the scalar field with a Gaussian action to generate the N-component Gross-Neveu model. The loop representation for this model is discussed.

hep-lat↗
arXiv1998

Hopping expansion as a tool for handling dual variables in lattice models

The hopping expansion of 8-vertex models in their Grassmann representation is studied. We use the functional similarity of the Ising model in this expansion with the hopping expansion of 2-D Wilson fermions to show that the lattice fermions are equivalent to the Self-avoiding Loop Model at bending rigidity 1/sqrt{2}.

hep-lat↗
arXiv1998

Analyzing the spectrum of general, non-hermitian Dirac operators

We discuss the computational problems when analyzing general, non-hermitian matrices and in particular the un-modified Wilson lattice Dirac operator. We report on our experiences with the Implicitly Restarted Arnoldi Method. The eigenstates of the Wilson-Dirac operator which have real eigenvalues and correspond to zero modes in the continuum are analyzed by correlating the size of the eigenvalues with the chirality of the eigenstates.

hep-lat↗
arXiv1998

On the spectrum of the Wilson-Dirac lattice operator in topologically non-trivial background configurations

We study characteristic features of the eigenvalues of the Wilson-Dirac operator in topologically non-trivial gauge field configurations by examining complete spectra of the fermion matrix. In particular we discuss the role of eigenvectors with real eigenvalues as the lattice equivalents of the continuum zero-modes. We demonstrate, that those properties of the spectrum which correspond to non-trivial topology are stable under adding fluctuations to the gauge fields. The behavior of the spectrum in a fully quantized theory is discussed using QED_2 as an example.

hep-lat↗
arXiv1996

Lattice Regularization of the Chiral Schwinger Model

We analyze the chiral Schwinger model on an infinite lattice using the continuum definition of the fermion determinant and a linear interpolation of the lattice gauge fields. For non-compact and Wilson formulation of the gauge field action it is proven that the effective lattice model is Osterwalder-Schrader positive, which is a sufficient condition for the reconstruction of a physical Hilbert space from the model defined on a Euclidean lattice. For the non-compact model we furthermore establish the existence of critical points where the corresponding continuum theory can be reconstructed. We show that the continuum limit for the two-point functions of field strength and chiral densities can be controlled analytically. The article ends with some remarks on fermionic observables.

hep-lat↗
arXiv1996

Lattice Fermions Coupled to Interpolated Gauge Fields - Results in 2 Dimensions

We discuss a new approach for putting gauge theories on the lattice. The gauge fields are defined on the lattice only, but are interpolated to the interior of the lattice cells, where they couple to continuum fermions. The purpose of this approach is to avoid doublers and keep the chiral symmetry of the action intact. In two dimensional models (Schwinger model and chiral Schwinger model) many results for this hybrid approach can be obtained analytically. For the vectorlike Schwinger model we concentrate on proving the existence of a critical point where the continuum limit can be constructed, on analyzing the chiral properties and on proving the Osterwalder-Schrader positivity which allows the reconstruction of the physical Hilbert space. We conclude with outlining how the results can be generalized to the chiral Schwinger model.

hep-lat↗
arXiv1996

Discussing the U(1)-Problem of QED_2 without Instantons

We construct QED_2 with mass and flavor and an extra Thirring term. The vacuum expectation values are carefully decomposed into clustering states using the U(1)-axial symmetry of the considered operators and a limiting procedure. The properties of the emerging expectation functional are compared to the proposed theta-vacuum of QCD. The massive theory is bosonized to a generalized Sine-Gordon model (GSG). The structure of the vacuum of QED_2 manifests itself in symmetry properties of the GSG. We study the U(1)-problem and derive a Witten-Veneziano-type formula for the masses of the pseudoscalars determined from a semiclassical approximation.

hep-th↗
arXiv1995

Lattice Schwinger Model with interpolated Gauge Fields

We analyze the Schwinger model on an infinite lattice using the continuum definition of the fermion determinant and a linear interpolation of the lattice gauge fields. The possible class of interpolations for the gauge fields, compatible with gauge invariance is discussed. The effective action for the lattice gauge field is computed for the Wilson formulation as well as for non-compact lattice gauge fields. For the non-compact formulation we prove that the model has a critical point with diverging correlation length at zero gauge coupling e. We compute the chiral condensate for e > 0 and compare the result to the N-flavor continuum Schwinger model. This indicates that there is only one flavor of fermions with the same chiral properties as in the continuum model, already before the continuum limit is performed. We discuss how operators have to be renormalized in the continuum limit to obtain the continuum Schwinger model.

hep-lat↗
arXiv1995

On the U(1)-Problem of QED_2

QED_2 with mass and flavor has in common many features with QCD, and thus is an interesting toy model relevant for four dimensional physics. The model is constructed using Euclidean path integrals and mass perturbation series. The vacuum functional is carefully decomposed into clustering states being the analogue of the theta-vacuum of QCD. Finally the clustering theory can be mapped onto a generalized Sine-Gordon model. Having at hand this bosonized version, several lessons on the theta-vacuum, the U(1)-problem and Witten-Veneziano-type formulas will be drawn. This sheds light on the corresponding structures of QCD.

hep-th↗