Topological susceptibility of the 2D O(3) model under gradient flow

Abstract

The 2d O(3) model is widely used as a toy model for ferromagnetism and for Quantum Chromodynamics. With the latter it shares -among other basic aspects -the property that the continuum functional integral splits into topological sectors. Topology can also be defined in its lattice regularised version, but semi-classical arguments suggest that the topological susceptibility χ t does not scale towards a finite continuum limit. Previous numerical studies confirmed that the quantity χ t ξ 2 diverges at large correlation length ξ. Here we investigate the question whether or not this divergence persists when the configurations are smoothened by the Gradient Flow (GF). The GF destroys part of the topological windings; on fine lattices this strongly reduces χ t . However, even when the flow time is so long that the GF impact range -or smoothing radius -attains ξ/2, we still do not observe evidence of continuum scaling.

Meron- and Semi-Vortex-Clusters as Physical Carriers of Topological Charge and Vorticity

“…[14] employed a (truncated) classically perfect lattice action, which has very small lattice artifacts, and found that dislocations are not present, while the logarithmic divergence persists. Recently this property was confirmed by applying the gradient flow to the standard action [15]. Remarkably, the same feature was also observed by simulating the topological action [4], for which the semiclassical argument would have suggested a strong power-law divergence.…”

Topology in the 2d Heisenberg Model under Gradient Flow

“…3 In contrast, at this point we conclude that the topology of the 2d O(3) model seems to be ill-defined in the continuum limit, even after the application of the Gradient Flow. However, this study is going to be extended to even larger L and ξ, in order to further check this conclusion [15] Table A1. A summary of our numerical results in the nine volumes V = L × L that we investigated.…”

“…In particular, at t = 0 a power-law cannot be ruled out by the present data, although a logarithmic divergence is expected. We hope for the extension of this study to even larger volumes [15] to be helpful also in this regard.…”

“…If the reference value is taken too large, it is not even attained for all parameter sets in our study (if one increases L and β more and more, this maximum decreases monotonously). The above reference value of 0.08 captures lattice sizes up to L = 606 [15], where β is always tuned such that relation (3.2) holds. Figure 3 shows the evolution of the term Figure 3.…”

Is N = 2 Large?

“…The most remarkable difference of the CP 1 model from other (N > 2) CP N−1 models is that the semi-classical calculation of the former leads to a UV divergence in the topological susceptibility. This is supported by lattice numerical calculations, unless a suitable counter term is added [50,[57][58][59][60][61][62][63][64]. 7…”