Topological charge using cooling and the gradient flow

TL;DR

This paper generalizes the equivalence of cooling and gradient flow methods for computing topological charge to include actions with rectangular terms, demonstrating their consistency across different gauge actions and lattice spacings.

Contribution

It extends the theoretical relation between cooling and gradient flow to more general gauge actions and validates this relation through numerical simulations.

Findings

Perturbative rescaling relates cooling steps to gradient flow time.

Cooling and gradient flow produce equivalent topological charge results.

Results are consistent across different gauge actions and lattice spacings.

Abstract

The equivalence of cooling to the gradient flow when the cooling step $n_c$ and the continuous flow step of gradient flow $\tau$ are matched is generalized to gauge actions that include rectangular terms. By expanding the link variables up to subleading terms in perturbation theory, we relate $n_c$ and $\tau$ and show that the results for the topological charge become equivalent when rescaling $\tau \simeq n_c/({3-15 c_1})$ where $c_1$ is the Symanzik coefficient multiplying the rectangular term. We, subsequently, apply cooling and the gradient flow using the Wilson, the Symanzik tree-level improved and the Iwasaki gauge actions to configurations produced with $N_f=2+1+1$ twisted mass fermions. We compute the topological charge, its distribution and the correlators between cooling and gradient flow at three values of the lattice spacing demonstrating that the perturbative rescaling $\tau \simeq n_c/({3-15 c_1})$ leads to equivalent results.