Continuum limit of ${\mathrm{HP}}^1$-based topological charge density distribution

The bulk distribution of the topological charge density, constructed via the ${\mathrm{HP}}^1$ $\sigma$-model embedding method, is investigated. We argue that the specific pattern of leading power corrections to gluon condensate hints on a particular UV divergent structure of ${\mathrm{HP}}^1$ $\sigma$-model fields, which in turn implies the linear divergence of the corresponding topological density in the continuum limit. We show that under testable assumptions, the topological charge is to be distributed within three-dimensional sign-coherent domains and, conversely, the dimensionality of sign-coherent regions dictates the leading divergence of the topological density. Confronting the proposed scenario with lattice data we present evidence for indeed peculiar divergence of the embedded fields. Then the UV behavior of the topological density is studied directly and is found to agree with our proposition. Finally, we introduce a method to investigate the dimensionality of relevant topological fluctuations and show that indeed topological charge sign-coherent regions are likely to be three-dimensional.

Figure 1

Scaling of the correlators ${\mathcal{M}}_n$, Eqs. (28, 29, 30), with diminishing lattice spacing. Lines represent the best fits according to Eq. (31).

Figure 2

Volume density of points with $|q_x|>{\Lambda }_q$ as a function of lattice spacing at different ${\Lambda }_q$ scaling laws (38).

Figure 3

Positive core of topological density correlation function $⟨q_0{q}_x⟩$ at distances $|x|<R_0\approx 0.4\mathrm{fm}$.

Figure 4

Negative part of the correlation function $⟨q_0{q}_x⟩$ at distances $|x|\ge R_0=0.4\mathrm{fm}$ at various lattice spacings.

Figure 5

Correlation length $R_{qq}$, Eq. (40), as a function of lattice spacing. Line represents the best linear fit.

Figure 6

The mean squared topological density $⟨q^2⟩$ as a function of lattice spacing. Curves represent the best power law fits, Eq. (44).

Figure 7

Return probability of biased random walk $\Phi (t,0)$, Eqs. (45, 46), at different $\gamma$ values for $\beta =2.500$ data set. Solid lines are the best fits of $\gamma =\{0.25,0.27,0.30\}$ data points to Eq. (49), fitting range is $t<100$.

Figure 8

Quality of the power law fits of the return probability $\Phi (t,0)$ to Eq. (49) in the range $t<100$ at various $\gamma$ values. Measurements were performed on $\beta =2.500$ data set.

Figure 9

Return probability $\Phi (t,0)$, Eqs. (45, 46), calculated on $\beta =2.500$ data set with random permutation of topological density records on each configuration.

Figure 10

Diffusion dimensions $D({\gamma }^{*})$, Eqs. (45, 46, 49), at various lattice spacings. Line represents the fit to the constant behavior.

Figure 11

Fluctuations of the topological charges $Q_{\pm }$ at various lattice spacings. Circles: spacing (in)dependence of the ratio $\mathcal{A}(Q_{\pm })$, Eq. (59); squares: the magnitude of $Q_{\pm }$ relative fluctuations characterized by $\mathcal{B}(Q_{\pm })$, Eq. (59).

Figure 12

Cumulative distribution of $V_{\mathrm{MC}}(T)$ and $V_{\det }(T)$ (see text).