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 τ. 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.

For field theories with a topological charge Q, it is often of interest to measure the topological susceptibility χ t = ( Q 2 − Q 2 )/V . If we manage to perform a Monte Carlo simulation where Q changes frequently, χ t can be evaluated directly. However, for local update algorithms and fine lattices, the auto-correlation time with respect to Q tends to be extremely long, which invalidates the direct approach. Nevertheless, the measurement of χ t is still feasible, even when the entire Markov chain is topologically frozen. We test a method for this purpose, based on the correlation of the topological charge density, as suggested by Aoki, Fukaya, Hashimoto and Onogi. Our studies in non-linear σ-models and in 2d Abelian gauge theory yield accurate results for χ t , which confirm that the method is applicable. We also obtain promising results in 4d SU(2) Yang-Mills theory, which suggest the applicability of this method in QCD.

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 τ . 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.

Lattice QCD simulations tend to become stuck in a single topological sector at fine lattice spacing or when using chirally symmetric overlap quarks. In such cases physical observables differ from their full QCD counterparts by finite volume corrections. These systematic errors need to be understood on a quantitative level and possibly be removed. In this paper we extend an existing relation from the literature between two-point correlation functions at fixed and the corresponding hadron masses at unfixed topology by calculating all terms proportional to $1/V^2$ and $1/V^3$, where $V$ is the spacetime volume. Since parity is not a symmetry at fixed topology, parity mixing is comprehensively discussed. In the second part of this work we apply our equations to a simple model, quantum mechanics on a circle both for a free particle and for a square-well potential, where we demonstrate in detail, how to extract physically meaningful masses from computations or simulations at fixed topology.Comment: 38 pages, 7 figures, addition of a discussion on parity mixing and correction of typo

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