Deconfinement transition in 2+1-dimensional SU(4) lattice gauge theory

The SU(4) Yang-Mills theory in (2+1) dimensions has a special status: its critical exponents at the finite temperature deconfinement transition are not specified by the Svetitsky-Yaffe (S-Y) conjecture [1]. By measuring them, we can learn about the couplings of the equivalent spin model, i.e. of the effective Polyakov loop model. The S-Y conjecture says that, if the deconfinement transition of the gauge theory is secondorder, then it is in the same universality class as the spin model having the symmetry-breaking pattern of the Polyakov loop. Assume that the (2+1)d SU(4) deconfinement transition is second-order – we will investigate this issue numerically next. Then the equivalent 2d spin model has Z(4) spins, with orientation {0,± 2 , π} corresponding to the 4 Polyakov loop sectors and the 4 perturbative vacua of the gauge theory. A nearest-neighbour Hamiltonian, which gives the same critical properties as the SU(4) theory according to S-Y, will have 3 possible energy levels for each link: E0 for parallel spins ↑↑, E1 for perpendicular spins ↑→, and E2 for antiparallel spins ↑↓. Remarkably, the critical exponents of this spin system are known to vary continuously with the ratio ρ ≡ E2−E1 E1−E0 and the corresponding couplings of the Hamiltonian. Therefore, the S-Y conjecture does not tell us what the SU(4) critical exponents are. Measuring them fixes the couplings of the effective Hamiltonian. A Z(4) spin model with 3 energy levels per link