Long range order in gauge theories: Deformed QCD as a toy model

We study a number of different ingredients related to long range order observed in lattice QCD simulations, using a simple “deformed QCD” model. This model is a weakly coupled gauge theory, which, however, has all the relevant crucial elements, allowing us to study difficult and nontrivial problems which are known to be present in real, strongly coupled QCD. In the present study, we want to understand the physics of long range order in the form of coherent low-dimensional vacuum configurations observed in Monte Carlo lattice simulations. We demonstrate the presence of double-layer domain wall structures in the deformed QCD, and study their interaction with localized topological monopoles. Furthermore, we show that there is in fact an attractive interaction between the two, such that the monopole favors a position within the domain wall.

Figure 1

The transition between paths corresponding to the decay of some domain wall state to a domain wall free ground state. The path wrapping the peg represents a state with some domain walls, while the path that does not denotes a state with no domain walls. We can deform the DW path by lifting it over the obstacle so that we can unwind it and deform it into the DW-free path. If the path describes domain walls with some weight, then it would require some energy to lift over the obstacle. If this energy is not available, then classically, the configurations that wind around the peg are stable. Quantum mechanically, however, the domain wall could still tunnel through the peg, and so the configurations are unstable quantum mechanically; see an estimate for this probability in the Appendix. Picture adapted from Ref. 28.

Figure 2

The two-layer structure of the topological charge density plotted against one direction across the domain wall and the other one of the two dimensions along it.

Figure 3

A plot of the numerical result for the binding energy at various separation distances between domain wall and monopole. Notice that for $z_0<0$, the monopole to the right of the domain wall, there is an “attractive” potential with a minimum very near $z_0=0$.

Figure 4

This plot is a close up of the points near the minimum in Fig. 3 showing that the minimum is slightly to the $z_0<0$ side of the center.

Figure 5

A close up of the points to the right of Fig. 3 showing the small barrier present on the $z_0>0$ side. Notice the much finer vertical scale.