Spectral flow, condensate and topology in lattice QCD

Abstract

We study the spectral flow of the Wilson-Dirac operator H(m) with and without an additional Sheikholeslami-Wohlert (SW) term on a variety of SU(3) lattice gauge field ensembles in the range 0 ≤ m ≤ 2. We have used ensembles generated from the Wilson gauge action, an improved gauge action, and several two-flavor dynamical quark ensembles. Two regions in m provide a generic characterization of the spectrum. In region I defined by m ≤ m 1 , the spectrum has a gap. In region II defined by m 1 ≤ m ≤ 2, the gap is closed. The level crossings in H(m) that occur in region II correspond to localized eigenmodes and the localization size decreases monotonically with the crossing point down to a size of about one lattice spacing. These small modes are unphysical, and we find the topological susceptibility is relatively stable in the part of region II where the small modes cross. We argue that the lack of a gap in region II is expected to persist in the infinite volume limit at any gauge coupling. The presence of a gap is important for the implementation of domain wall fermions.

Spectrum of the Hermitian Wilson Dirac operator

“…The numbers for the topological susceptibility obtained in this manner are consistent with the results obtained by field theoretic methods [8]. The modes that cross zero to the left of the peak in ρ(0; m) correspond to small modes [6,3] and this is consistent with Fig. 4.…”

“…This implies that the topological charge of a lattice gauge field configuration defined as the net level crossings in H L (m) in the range [m, 0] will depend on m. The topology of a single lattice gauge field configuration is not interesting in a field theoretic sense. One has to obtain an ensemble average of the topological susceptibility and study its dependence on m. This has been studied on a variety of ensembles [3,6] and the results show that the topological susceptibility is essentially independent of m in the region to the left of the peak in ρ(0; m). This indicates that the levels that cross zero to the left of the peak are not physically relevant consistent with the result that ρ(0; m) goes to zero in the continuum limit.…”

“…This has direct implications for how the spectral density at zero behaves on the lattice. A careful study of the spectral density at zero has been performed on a variety of SU(3) pure gauge ensembles [6]. This is done by computing the low lying eigenvalues of H L (m) using the Ritz functional [7].…”

Study of chiral symmetry in quenched QCD using the overlap Dirac operator

“…The massless Overlap-Dirac operator has exact zero eigenvalues, due to topology, in topologically non-trivial gauge fields, and these zero eigenvalues are always paired with eigenvalues exactly equal to unity [1,6]. All exact zero eigenvalues are chiral, and their partners at unity are also chiral but with opposite chirality.…”

New ways to determine low-energy constants with Wilson fermions

“…The few cases were the two determinations differed were usually associated with multiple crossings, with the last one at a fairly large λ W real . We now remind the reader of a few basic facts about the spectrum of the Hermitian Wilson Dirac operator [17]. In the continuum, the spectrum of the Dirac operator D is chiral: for every nonzero eigenvalue iλ there is a matching eigenvalue −iλ.…”