Lattice gauge fields topology uncovered by quaternionic $\sigma$-model embedding

We investigate SU(2) gauge fields topology using new approach, which exploits the well known connection between SU(2) gauge theory and quaternionic projective $\sigma$-models and allows to formulate the topological charge density entirely in terms of $\sigma$-model fields. The method is studied in details and for thermalized vacuum configurations is shown to be compatible with overlap-based definition. We confirm that the topological charge is distributed in localized four-dimensional regions which, however, are not compatible with instantons. Topological density bulk distribution is investigated at different lattice spacings and is shown to possess some universal properties.

Figure 1

Typical distribution of $\delta F_x$, Eq. (20), on thermalized ${12}^4$, $\beta =2.35$ lattices for various $\sigma$-model ranks.

Figure 2

Distribution of $\delta F_x$, Eq. (20), for ${\mathrm{HP}}^1$ $\sigma$-model on three highly cooled ${12}^4$ configurations containing, respectively, three instantons, one anti-instanton and no instantons at all.

Figure 3

$\delta F_x$ distribution on ${12}^4$ lattice for random input gauge fields.

Figure 4

Distribution of the minimal values of the functional (20) (with mean value subtracted) calculated on ${12}^4$ $\beta =2.35$ thermalized lattices and averaged over 20 configurations, see text.

Figure 5

Distribution of the topological charge $Q_{\mathrm{float}}$, Eq. (22), for single 5-dimensional simplex with random ${\mathrm{HP}}^1$ $\sigma$-model fields at its vertices.

Figure 6

Distribution of $Q_{\mathrm{float}}-Q$, Eqs. (22, 23) calculated on 55 configurations thermalized at $\beta =2.40$ on ${16}^4$ lattice.

Figure 7

Time slice of the action and the topological charge densities (21) on cooled ${12}^4$ configuration containing three instantons.

Figure 8

Time slice of the action and the topological charge densities (21) on cooled ${12}^4$ configuration containing one anti-instanton.

Figure 9

Correlation function (26) of the overlap- and ${\mathrm{HP}}^1$ $\sigma$-model based definitions of the topological charge density, see text.

Figure 10

The correlation function (27) in the embedded ${\mathrm{HP}}^1$ $\sigma$-model. Lines correspond to the best fitting curves (28).

Figure 11

Topological charge distribution calculated on $\beta =2.40$ configurations.

Figure 12

Visualization of the topological charge density distribution in a particular time slice of ${16}^4$, $\beta =2.40$ thermalized configuration.

Figure 13

Dependence of the lumps density $D_{\mathcal{L}}$ upon the cutoff ${\Lambda }_q$ for various values of $V_{\min }$, Eq. (30).

Figure 14

$D_{\mathcal{L}}({\Lambda }_q)$ dependence in the range ${10}^{-5}<{\Lambda }_q·a^4<1.5·{10}^{-3}$ for various $V_{\min }$, Eq. (30).

Figure 15

Distribution of lumps volumes, Eq. (33), for ${\Lambda }_q·a^4=1.5·{10}^{-3}$ calculated on ${16}^4$, $\beta =2.40$ lattices and normalized by the number of configurations. Solid line is the fit to Eq. (34).

Figure 16

Lumps charge, Eq. (33), distribution for ${\Lambda }_q·a^4=1.5·{10}^{-3}$ calculated on ${16}^4$, $\beta =2.40$ lattices and normalized by the number of configurations. Solid line is the fit to Eq. (34).

Figure 17

Lumps volume distribution for various ${\Lambda }_q$ given in lattice units. Solid line is the fit (35) with ${\alpha }_V=3/2$.

Figure 18

Distribution of lumps charges for different ${\Lambda }_q$ given in lattice units. Solid line is the fit (35) with ${\alpha }_Q=3/2$.

Figure 19

Lumps limited squared topological charge $⟨Q^2{⟩}_{\mathrm{lump}}$ versus cutoff ${\Lambda }_q$. Lines represent $⟨Q^2⟩$, Eq. (30), and the corresponding error bounds.

Figure 20

Lumps volume distribution for fixed cutoff ${\Lambda }_q·a^4=2·{10}^{-5}$.

Figure 21

Volumes of percolating lumps at various ${\Lambda }_q$. Note that percolating lumps disappear at ${\Lambda }_q·a^4\approx 7·{10}^{-4}$.

Figure 22

Distribution of the small volume lumps at various ${\Lambda }_q$. Note that it still follows the power law (34).

Figure 23

The power exponent ${\alpha }_V$, Eq. (34), as function of ${\Lambda }_q$ in the range (40). Line represents the conjectured universal value (36).

Figure 24

The topological charge density correlation function measured on ${16}^4$, $\beta =2.40$ lattices for ${\Lambda }_q·a^4=0$ and $1.5·{10}^{-3}$. Lines represent the best fits (42).