Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

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 β = 2 the intermediate ensembles exhibit a mixing of orthogonal polynomials and skeworthogonal 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 the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson-Dirac operator in the ǫ-regime of lattice quantum chromodynamics.

Dirac spectrum of the Wilson-Dirac operator for QCD with two colors

“…Since the analytical derivation of ρ real for β = 2 was a tour de force [16,20], it is not surprising that we did not succeed to find an analytical result for the density of the real eigenvalues of the non-Hermitian Wilson Dirac operator D W for two color QCD with β = 1. Nonetheless we can extract some information via the inequality (25) and thus via the distributions ρ 5 (m, λ = 0) and ρ χ (λ).…”

“…Such a closed expression was indeed derived for QCD with three colors in [16,20] but seems to be much harder to obtain for the two color case.…”

“…Using mean field theory it was shown that the gap of the Hermitian Wilson Dirac operator closes when entering the Aoki phase. The microscopic Dirac spectrum was evaluated in great detail by means of a supersymmetric extension of the chiral Lagrangian and exploiting its equivalence with chiral random matrix theory [15,16]. One of the main results is that the Gaussian broadening of the topological zero modes scales as √ V a = a for small values of the lattice spacing a which was first observed by lattice simulations [17].…”

Phase diagram of dynamical twisted-mass Wilson fermions at finite isospin chemical potential

“…[44][45][46][47]. The third low energy constant associated with the a 2 terms, usually denoted by W 7 tr 2 (U − U † ) vanishes for the two flavor SU(2) partition function, see Ref.…”

New ways to determine low-energy constants with Wilson fermions

“…In this paper we wish to advocate a new strategy towards determining the leading physical and Wilson lowenergy constants which is based on the new analytical results for spectral correlators of the Wilson Dirac operator [8][9][10][11][12][13][14][15]. The idea is very simple, and relies on using the wealth of detailed information contained in the spectral density and spectral correlation functions of the Wilson Dirac operator D W and its Hermitian counterpart D 5 .…”