Abstract
In this paper, we show a comparison of different definitions of the topological charge on the lattice. We concentrate on one small-volume ensemble with 2 flavours of dynamical, maximally twisted mass fermions and use three more ensembles to analyze the approach to the continuum limit. We investigate several fermionic and gluonic definitions. The former include the index of the overlap Dirac operator, the spectral flow of the Wilson–Dirac operator and the spectral projectors. For the latter, we take into account different discretizations of the topological charge operator and various smoothing schemes to filter out ultraviolet fluctuations: the gradient flow, stout smearing, APE smearing, HYP smearing and cooling. We show that it is possible to perturbatively match different smoothing schemes and provide a well-defined smoothing scale. We relate the smoothing parameters for cooling, stout and APE smearing to the gradient flow time tau . In the case of hypercubic smearing the matching is performed numerically. We investigate which conditions have to be met to obtain a valid definition of the topological charge and susceptibility and we argue that all valid definitions are highly correlated and allow good control over topology on the lattice.
Highlights
E.g. the fluctuations of the topological charge are related to the mass of the flavour-singlet pseudoscalar η meson [1,2]
In this paper, we show a comparison of different definitions of the topological charge on the lattice
We investigate which conditions have to be met to obtain a valid definition of the topological charge and susceptibility and we argue that all valid definitions are highly correlated and allow good control over topology on the lattice
Summary
Many definitions of the topological charge of a lattice gauge field were proposed [7,8]. On the other hand, have shown numerically that the field theoretic topological susceptibility extracted with several smoothing techniques such as cooling, APE and HYP smearing give the same continuum limit. By expanding the gauge links perturbatively in the lattice spacing a, at subleading order, the two methods become equivalent if one sets τ = nc/(3 − 15b1) where b1 is the Symanzik coefficient multiplying the rectangular term of the smoothing action. It is, interesting to extend the study of Ref.
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