in other words in such a limit a theory in the adjoint representation is equivalent to the fundamental one and a real center-blind theory cannot be studied. In our case such a duality is not evident and in fact we get a disconnected phase diagram.
This is why we should not identify the chemical potentialλV with λ. Moreover this kind of suppression term can be used with every representation of the gauge group SU(2) and not only with the Villain discretization, so in this sense it is more general.
We can also define a twist observable zxt like in Eq. (4.17), which is again a truly SO(3) observable, sincezxt→zxt ifUµ(x)→ −Uµ(x), and can be measured in every representation.
Mostly we shall be interested in the pure adjoint case βF = 0. In this special case we have analyzed the model with the link variables represented both bySO(3) matrices and by the fundamental representation 2×2 matrices, exploiting the prop-erty TrA = Tr2F −1. Nothing changes in the phase diagram but in the latter case simulations become much simpler and faster. This is the reason why we favored the 2-dimensional representation. A standard Metropolis algorithm has been used to update the links (for details see Appendix A).
5.2 The bulk transition
In this section we investigated the effect of a varying chemical potential (0.0≤λ ≤ 1.0) on some simple observables, like the adjoint plaquette P, the adjoint Polyakov loop LA and the density of cubes M= (
cσc)/Nc. We used rather small volumes (V = 4×123) in order to understand what happens by varyingλ from zero to one.
The volume is chosen asymmetric for convenience but the bulk effects are present also in symmetric volumes.
0 0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
βA
λ
<zit>=0
<M><1 I
II
<M>=1
<zit>=/ 0
1st order bulk 2nd order bulk
Figure 5.2: Phase diagram in theβA−λplane at zero temperature.
As we will show more in detail in the next section, in the region of the phase diagram called I in Fig. 5.2, the twist is not well defined since there are fluctuations between the different twist sectors and it averages to zero. In region II the energy barriers between the different twist sectors become higher and tunneling is strongly suppressed. The system will stay in a fixed twist sector that can be chosen with appropriate initial conditions, and in this section, for large enoughλand βA, we will study the trivial twist sector.
The plaquette shows a clear gap, signalling the presence of a first order phase transition, for λ ≤ 0.7. For λ > 0.7 such a signal is strongly suppressed. This is indeed what we wanted and what we expected, since the chemical potential should suppress the lattice artifacts which produce the bulk phase transition and as a con-sequence the transition, by varying smoothly λ, should slowly become weaker and for large enough λ eventually disappear. For such values of the chemical potential (λ > 0.7) the observables seem quite smooth, but a second order bulk phase tran-sition could still be present, as the analysis of the twist in the next section seems to suggest. At βA = 0.0 and λ = 0.953 it is in fact expected a second order phase transition, being the theory dual to Ising 4d; this transition could eventually enter the βA−λ plane and join the first order one. A more detailed analysis must be per-formed to understand if the bulk phase transition is always there and in the positive case its order. At λ= 1.0 the bulk phase transition seems instead to be completely absent for βA ≥ 0.0, thus suggesting that the bulk phase transition crosses the λ axis around λ = 0.953.
The Z2 cube density shows a behavior similar to the plaquette, with a jump for 0.0≤λ≤0.7, clear signal of a first order phase transition. Above λ= 0.7 it goes to 1, thus signalling the absence of Z2 monopoles, but in a smooth way, without any discontinuity. It is worth saying that the region in the βA−λ plane where the cube density goes to one coincide with the region where the twist becomes non-zero and eventually reaches one. At λ= 1.0 for each value of βA the cube density is fixed to one and there is no presence of any bulk phase transition.
The Polyakov loop shows again an abrupt jump for 0.0 ≤ λ ≤ 0.7. In such a range the Polyakov loop averages to zero below the bulk transition and becomes different from zero above it. The non-zero value of the Polyakov loop is always positive since we are fixed in the trivial twist sector; in the non-trivial twist sector the jump would be again present, but then the Polyakov loop would be negative. It is essential to note its behavior for λ >0.7. At λ= 0.8,0.85,1.0 it takes a non-zero value no more above the bulk phase transition, but in a different region in theβA−λ plane. It means that above a certainλits behavior is completely decoupled from the dynamics of the lattice artifacts. If we can interpretLAas the free energy of a static adjoint source, then LA= 0 should be interpreted as a signal for confinement. It is of course true, in principle, that an adjoint static charge could always be screened by a gluon, but this effect of string breaking is expected to occur, in practice, only for for very large volumes [137,138], so we can safely take the Polyakov loop as a signature of the presence of two phases, characterized respectively by LA= 0 and LA = 0. Whether these two phases are separated by a real phase transition or rather a cross-over is not easy to understand and we will address the question later on.
5.2 The bulk transition 53
0
Figure 5.4: Adjoint Polyakov loop as a function of βA for different values of λ (V = 4×123).
5.2 The bulk transition 55