Ivan Hip

Approximate Ginsparg-Wilson fermions: A First test

Abstract Wexa0construct 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 ofxa0 D . Wexa0solve these equations for a finite parametrization of D and find an approximate solution of the Ginsparg–Wilson equation. Wexa0analyze 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 ourxa0 D .

The scaling of exact and approximate Ginsparg–Wilson fermions

Abstract We construct a number of lattice fermions, which fulfill the Ginsparg–Wilson relation either exactly or approximately, and test them in the framework of the 2-flavor Schwinger model. We start from explicit approximations within a short range, and study this formulation, as well as its correction to an exact Ginsparg–Wilson fermion by the “overlap formula”. Then we suggest a new method to realize this correction perturbatively, without using the tedious square root operator. In this way we combine many favorable properties: good chiral behavior, small mass renormalization, excellent scaling and rotational invariance, as well as a relatively modest computational effort, which makes such formulations most attractive for QCD.

Eigenvalue spectrum of massless Dirac operators on the lattice

Abstract We present a detailed study of the interplay between chiral symmetry and spectral properties of the Dirac operator in lattice gauge theories. We consider, in the framework of the Schwinger model, the fixed point action and a fermion action recently proposed by Neuberger. Both actions show the remnant of chiral symmetry on the lattice as formulated in the Ginsparg-Wilson relation. We check this issue for practical implementations, also evaluating the fermion condensate in a finite volume by a subtraction procedure. Moreover, we investigate the distribution of the eigenvalues of a properly defined anti-hermitian lattice Dirac operator, studying the statistical properties at the low lying edge of the spectrum. The comparison with the predictions of chiral Random Matrix Theory enables us to obtain an estimate of the infinite volume fermion condensate.

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

Abstract 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 π -particle implying almost chirally symmetric fermions. The generalization of our method to 4-d is straightforward.

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 (124, β = 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 β 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.

Topological charge and the spectrum of the fermion matrix in lattice QED2

We investigate the interplay between topological charge and the spectrum of the fermion matrix in lattice QED2 using analytic methods and Monte Carlo simulations with dynamical fermions. A new theorem on the spectral decomposition of the fermion matrix establishes that its real eigenvalues (and corresponding eigenvectors) play a role similar to the zero eigenvalues (zero-modes) of the Dirac operator in continuous background fields. Using numerical techniques we concentrate on studying the real part of the spectrum. These results provide new insights into the behavior of physical quantities as a function of the topological charge. In particular we discuss the fermion determinant, the effective action and pseudoscalar densities.

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

Abstract 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 QED2 as an example.

Effects of Topology in the Dirac Spectrum of Staggered Fermions

Abstract We compare the lower edge spectral fluctuations of the staggered lattice Dirac operator for the Schwinger model with the predictions of chiral random matrix theory (chRMT). We verify their range of applicability, checking in particular the role of non-trivial topological sectors and the flavor symmetry of the staggered fermions for finite lattice spacing. Approaching the continuum limit we indeed find clear signals for topological modes in the eigenvalue spectrum. These findings indicate problems in the verification of the chRMT predictions.

Quantum fluctuations versus topology — a study in U(1)2 lattice gauge theory

Abstract Using the geometric definition of the topological charge we decompose the path integral of 2-dimensional U(1) lattice gauge theory into topological sectors. In a Monte Carlo simulation we compute the average value of the action as well as the distribution of its values for each sector separately. These numbers are compared with analytic lower bounds of the action which are relevant for classical configurations carrying topological charge. We find that quantum fluctuations entirely dominate the path integral. Our results for the probability distribution of the Monte Carlo generated configurations among the topological sectors can be understood by a semi-phenomenological argument.

Wilson, fixed point and Neuberger's lattice Dirac operator for the Schwinger model

Abstract We perform a comparison between different lattice regularizations of the Dirac operator for massless fermions in the framework of the single and two flavor Schwinger model. We consider a) the Wilson-Dirac operator at the critical value of the hopping parameter; b) Neubergers overlap operator; c) the fixed point operator. We test chiral properties of the spectrum, dispersion relations and rotational invariance of the mesonic bound state propagators.