Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

universal aspects of confinement in 2 + 1 dimensions 17
(d+1) dimensional SU(Nc)
3 4 5
q, Nc
Ising model
percolation
spatial dimensions d
self-dual for all q
Figure 3.2: Separation line between 1 st and 2 nd order phase transitions in the q-state Potts
models, from [45], and pure SU(N) gauge theories.
in the gauge and spin systems. We then exploit universality to locate the SU(2)
phase transition with extreme precision and perform a finite size scaling analysis
at criticality. This uncovers a remarkable manifestation of Kramers-Wannier duality
between center vortices and the color electric fluxes that they confine, rooted in the
self-duality of the 2d N-state Potts models. 1
Our high precision methods apply equally well to SU(3) in 2 + 1 d, which we
consider briefly before turning to the conundrum of SU(4). There are two riddles
here. Scaling that is consistent with the 4-state Potts model has been observed
on intermediate lattice volumes [41], despite evidence that the SU(4) transition
is first order [39, 40, 41]. Why is the 4-state Potts model favored from the wide
class of Z 4 symmetric Ashkin-Teller models? And, what exactly determines the
order of a deconfinement transition? It has been conjectured [42, 43, 40, 44] that
the change to a first order transition with increasing numbers of colors N develops
from a mismatch of degrees of freedom at criticality, namely the glueballs below
and deconfined gluon states above T c . Note, however, that the 2d N-state Potts have
first order transitions for N > 4 (Fig. 3.2), yet cannot have a mismatch of degrees
of freedom at criticality because they are self-dual.
1 q is conventionally used to label the number of states of the standard Potts model. We will refer to
them as N-state Potts models to match the number of colors in SU(N) gauge theory.