Topology, Random Matrix Theory and the spectrum of the Wilson Dirac operator

Source: arXiv


We study the spectrum of the hermitian Wilson Dirac operator in the
epsilon-regime of QCD in the quenched approximation and compare it to
predictions from Wilson Random Matrix Theory. Using the distributions of single
eigenvalues in the microscopic limit and for specific topological charge
sectors, we examine the possibility of extracting estimates of the low energy
constants which parametrise the lattice artefacts in Wilson chiral perturbation
theory. The topological charge of the field configurations is obtained from a
field theoretical definition as well as from the flow of eigenvalues of the
hermitian Wilson Dirac operator, and we determine the extent to which the two
are correlated.

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    ABSTRACT: Recently, random matrix theory predictions for the distribution of low-lying Dirac operator eigenvalues have been extended to include lattice effects for both staggered and Wilson fermions. We computed low-lying eigenvalues for the Hermitian Wilson-Dirac operator and for improved staggered fermions on several quenched ensembles with size $\approx 1.5$ fm. Comparisons to the expectations from RMT with lattice effects included are made. Wilson RMT describes our Wilson data nicely. For improved staggered fermions we find strong indications that taste breaking effects on the low-lying spectrum disappear in the continuum limit, as expected from staggered RMT.
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    ABSTRACT: The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble as well as the Gaussian unitary ensemble are associated to the Dyson index $\beta=2$ the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian as well as the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel-Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With help of these results we derive the unquenched $k$-point correlation function of the Hermitian and then non-Hermitian Wilson random matrix in terms of two flavour partition functions only. This representation is due to a Pfaffian factorization drastically simplifying the expressions for numerical applications. It also serves as a good starting point for studying the Wilson-Dirac operator in the $\epsilon$-regime of lattice quantum chromodynamics.
    Journal of Physics A Mathematical and Theoretical 02/2012; 45(20). DOI:10.1088/1751-8113/45/20/205203 · 1.58 Impact Factor
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    ABSTRACT: We summarize recent analytical results obtained for lattice artifacts of the non-Hermitian Wilson Dirac operator. Hereby we discuss the effect of all three low energy constants. In particular we study the limit of small lattice spacing and also consider the regime of large lattice spacing which is closely related to the mean field limit. Thereby we extract simple relations between measurable quantities like the average number of additional real modes and the low energy constants. These relations may improve the fitting for the low energy constants.
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