On the topological content of SU(2) gauge fields below T_c

TL;DR

The paper uses finite-temperature SU(2) lattice gauge theory in the confinement phase, cooled to reveal long-lived topological excitations carrying non-integer charge. It demonstrates that lattice configurations naturally host Kraan-van Baal calorons and dyonic lumps (DD, CAL, D\bar D) whose spatial profiles, Polyakov-loop structure, and fermionic zero modes align with analytic KvB solutions, for both periodic and fixed-holonomy boundary conditions. At higher action plateaus, it uncovers multi-dyon configurations with holonomy extending beyond the lumps, indicating that a semi-classical picture of the finite-temperature vacuum should incorporate nontrivial holonomy and strong color correlations (Taubes rotation). These results support a semi-classical framework for the non-zero-T path integral near the deconfinement transition, with potential implications for understanding confinement, though constructing a complete analytic description remains a challenge.

Abstract

Finite temperature Euclidean SU(2) lattice gauge fields generated in the confinement phase close to the deconfinement phase transition are subjected to cooling. The aim is to identify long-living, almost-classical local excitations which carry (generically non-integer) topological charge. Two kinds of spatial boundary conditions (fixed holonomy and standard periodic boundary conditions) are applied. For the lowest-action almost-classical configurations we find that their relative probability semi-quantitatively agrees for both types of boundary conditions. We find calorons with unit topological charge as well as (anti-)selfdual lumps (BPS-monopoles or dyons) combined in pairs of non-integer (equal or opposite sign) topological charge. For calorons and separated pairs of equal-sign dyons obtained by cooling we have found that (i) the gluon field is well-described by Kraan-van Baal solutions of the Euclidean Yang-Mills field equations and (ii) the lowest Wilson-fermion modes are well-described by analytic solutions of the corresponding Dirac equation. For metastable configurations found at higher action, the multi-center structure can be interpreted in terms of dyons and antidyons, using the gluonic and fermionic indicators as in the dyon-pair case. Additionally, the Abelian monopole structure and field strength correlators between the centers are useful to analyse the configurations in terms of dyonic constituents. We argue that a semi-classical approximation of the non-zero temperature path integral should be built on superpositions of solutions with non-trivial holonomy.

Figures (13)

• Figure 1: Various portraits of a selfdual $DD$ pair obtained by cooling under periodic gluonic boundary conditions. The sub-panels show: appropriate $2D$ cuts of the topological charge density (a) and of the Polyakov loop (b), the plot of lowest fermionic eigenvalues (c,d) and the $2D$ cut of the real-mode fermion densities (e,f), for the cases of time-periodic (c,e) and time-antiperiodic (d,f) fermionic boundary conditions, respectively ($\beta=2.2$ and lattice size $16^3\times4$).
• Figure 2: The $DD$ lattice configuration of Fig. 1 (dashed lines), fitted by the KvB solution (solid lines) according to the action density, shown in three spatial views ($1D$ projections). In (a1),(a2),(a3) the topological charge density is summed over the two unspecified spatial coordinates. In (b1),(b2),(b3) the Polyakov loop is averaged over the two unspecified spatial coordinates. In the time-antiperiodic case (c1,c2,c3) or the time-periodic case (d1,d2,d3), respectively, the fermion density it is summed over the two unspecified spatial coordinates.
• Figure 3: Various portraits of a selfdual $CAL$ configuration obtained by cooling under periodic gluonic boundary conditions. The sub-panels show: appropriate $2D$ cuts of the topological charge density (a) and of the Polyakov loop (b), the plot of lowest fermionic eigenvalues (c,d) and the $2D$ cut of the real-mode fermion densities (e,f), for the cases of time-periodic (c,e) and time-antiperiodic (d,f) fermionic boundary conditions, respectively ($\beta=2.2$ and lattice size $16^3\times4$).
• Figure 4: Various portraits of a mixed-duality $D \bar{D}$ pair obtained by cooling under periodic gluonic boundary conditions. The sub-panels show: appropriate $2D$ cuts of the topological charge density (a) and of the Polyakov loop (b), the plot of lowest fermionic eigenvalues (c,d) for the cases of time-periodic (c) and time-antiperiodic (d) fermionic boundary conditions, respectively ($\beta=2.2$ and lattice size $16^3\times4$). A $2D$ cut of the fermionic mode density related to the two distinct almost real eigenvalues in (c) is shown in (e).
• Figure 5: Part of the cooling history for a gauge field configuration taken from the Monte Carlo sample generated at $\beta=2.2$ on a $24^3 \times 4$ lattice, with f.h.b.c. of $L_{\vec{x} \in \Omega}= 0$. The sub-panels show: (a) full action $S/S_{inst}$, (b) non-stationarity $\delta_t$ and (c) mean violation $\Delta$ per link of the lattice field equations, vs. the number of cooling steps. The vertical dotted lines indicate the passages of $\Delta$ through local minima having occured at 800 (A), 1650 (B) and 7000 (C) cooling steps for which the configurations will be portrayed in Figures \ref{['fig:cool_q_pol']}, \ref{['fig:cool_fermion_1']}, \ref{['fig:cool_fermion_2']} and \ref{['fig:cool_fermion_3']}.