Measuring the topological susceptibility in a fixed sector

Abstract

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.

Topology in the 2d Heisenberg Model under Gradient Flow

“…The explicit formulae are given in Refs. [7][8][9]. Parity symmetry implies Q = 0, hence the topological susceptibility takes the form Figure 2 shows an example for a histogram of the topological charge distribution, which tends to be approximately Gaussian.…”

Topological Susceptibility under Gradient Flow

“…[13] and tested in Refs. [9,12,14,15]. Here we consider the slab method as another way to evaluate χ t from data obtained at fixed Q (actually data from ±Q can be combined).…”

Frontiers of finite temperature lattice QCD

“…Moreover, an algorithmic difficulty, the topological freezing, preventing the crossing between topological sectors, worsens the problem. Suggestions include artificial decreasing of the weight of the trivial sector [90] and the use of the topological charge density correlator [92]. In Refs.…”