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

48 universal aspects of confinement in 2 + 1 dimensions
4
3
2
1
0.5 1.0 1.5 2.0
Figure 3.18: The exact interface tensions per coupling, Kσ I over temperature K = T/J,
for straight (1, 0) and diagonal (1, 1) interfaces. They differ by factor √ 2 at
criticality. This implies string formation at criticality for all models in the 2d
Ising universality class.
It is here that the interface tension amplitude σ 0 = 2 ln(1 + √ 2) is relevant to all
models in the 2d Ising universality class, irrespective of the microscopic physics.
String formation in 2 + 1 d SU(2) gauge theory for electric fluxes in the scaling
window below T c , and for spacelike center vortices above T c , then follows from
universality and this square root signature for diagonal interfaces at criticality in
the 2d Ising model.
In the continuum limit of the gauge theory, we also expect isotropy and hence
square root scaling for center and electric fluxes away from criticality. This is verified
in Appendix A.
3.6.1.3 A note for 3 + 1 d
We have used self-duality to fix the relationship between the electric string and
dual center vortex tension in 2 + 1 d. More generally, this is fixed by the universal
amplitude ratio that relates the relevant scales on each side of criticality. For
Ising models, this is the interface tension σ I in the ordered phase and the exponential
correlation length, i.e., inverse mass gap ξ + gap, in the disordered phase. The
hyperscaling relation µ = (d − 1)ν gives the universal amplitude ratio [52]
σ I (ξ gap) + d−1 = R + σ gap , where (3.85)
R + σ gap = {
1 , for q = 2, 3, 4 , in d = 2 ,
0.40(1) , for q = 2 , in d = 3 .
For the 2, 3 and 4-state Potts models with second order phase transitions, selfduality
gives unity for the amplitude ratio. The value for the 3d Ising model is
determined numerically. It can be applied to SU(2) gauge theory in 3 + 1 d, where
the mass gap in the disordered phase comes from the electric string tension σ/T =