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

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 regularized version, but semiclassical arguments suggest that the topological susceptibility ${\chi }_{\mathrm{t}}$ does not scale towards a finite continuum limit. Previous numerical studies confirmed that the quantity ${\chi }_{\mathrm{t}}{\xi }^2$ diverges at large correlation length $\xi$. 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 ${\chi }_{\mathrm{t}}$. However, even when the flow time is so long that the GF impact range—or smoothing radius—attains $\xi /2$, we still do not observe evidence of continuum scaling.

Figure 1

The correlation function $C(r)=⟨{\overrightarrow{s}}_{x_2}·{\overrightarrow{s}}_{x_2+r}⟩$, measured before and after the GF, at $L=120$ (as an example). At fixed separation $r$, $C(r)$ rises under GF, but the value of $\xi$—obtained from a fit to relation (2.2) in the interval $40\le r\le 80$—remains practically constant.

Figure 2

A comparison of the correlation length $\xi$, and the second moment correlation length ${\xi }_2$, under GF up to flow time $10t_0$. We see that they initially agree well, but as the GF proceeds, ${\xi }_2$ increases, whereas $\xi$ remains stable [this is visualized with horizontal lines at the values of $\xi (t=0)$]. We show three volumes as examples for this generic effect.

Figure 3

The quantity $⟨\epsilon ⟩=⟨S⟩/\beta$ measured in volumes $V=L\times L$, at $L/\xi \approx 6$. In each volume, and at each flow time $t/t_0=0,1\dots 10$, we used 50 000 configurations with topological charge $|Q|=1$. We also show the minima of $\epsilon$ within this set. Even after a long GF, up to flow time $10t_0$, these minima are still more than a factor of 5 above the dislocation value ${\epsilon }_{\text{disloc}}\simeq 6.69$. We infer that dislocations and their vicinities hardly contribute to the statistics.

Figure 4

GF time evolution of the expectation value $⟨Q^2⟩={\chi }_{\mathrm{t}}V$.

Figure 5

Illustration of the nonscaling of the term ${\chi }_{\mathrm{t}}{\xi }^2$. The scale used in this plot is the second moment correlation length ${\xi }_2$.

Figure 6

The constants $a_i(t)$ and $b_i(t)$ of the fits to the logarithmic function (4.1) and to the power law (4.2), respectively, along with the ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$. ratios. The fits were performed at any flow time $t/t_0=1\dots 10$, in the range $L=54\dots 404$ ($\mathrm{d}.\mathrm{o}.\mathrm{f}.=3$). In all cases the logarithmic fits have a better quality, in contrast to the case $t=0$. The constants $a_1$, $a_2$, $b_1$, and $b_2$ are positive and incompatible with 0 at any GF time up to $10t_0$. Hence our data are incompatible with continuum scaling.

Figure 7

A schematic illustration of the expectation value $\mathrm{i}⟨Q⟩$ as a function of the vacuum angle $\theta$, in the continuum limit. The peculiarity of the 2D O(3) model is that its slope—which is proportional to ${\chi }_{\mathrm{t}}$—seems to diverge at $\theta =0$. This picture corresponds to Refs. [41, 42]; our results suggest that its qualitative features persist under the GF.

Figure 8

The mean value $⟨Q^2⟩(t)$, obtained from 100 configurations at $L=404$, at flow times $t=0\dots 10t$. The results coincide for all three GF implementations under consideration.

Figure 9

The step size evolution of the Dormand-Prince algorithm with an initial time step $d{t}_0={10}^{-4}$ and with tolerance parameter $ϵ={10}^{-6}$ or ${10}^{-7}$.