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

3.8 su(4) in 2 + 1 d 55
3.8 su(4) in 2 + 1 d
The deconfinement transition for SU(4) in 2 + 1 d is especially interesting. Z 4
center symmetry is not enough to specify the simplest effective spin model. There
may be different tensions for the interfaces that separate ‘orthogonal’ center sectors
with center flux eiπ /2
∈ Z 4 , and those that separate ‘anti-parallel’ sectors, with
flux −1 ∈ Z 4 . If one performs a strong coupling expansion of the effective action in
terms of Z 4 invariant operators as in Ref. [83], the two fundamental representations
4, ¯4, as well as the double-fundamental representation 6 = 6 ∗ that is obtained from
the antisymmetric product, (4 ⊗ 4) asym , must be taken into account. The relative
weight of the couplings between loops in the 4/ ¯4 representations and loops in the
6 representation is required to pin down the corresponding spin system. It could
be any one from a wide class of symmetric Ashkin-Teller models.
It is conventional to formulate states of the symmetric 9 Ashkin-Teller model in
terms of a pair of Ising models spins (s i , σ i ) = (±1, ±1) coupled via the Hamiltonian
[79],
H = − ∑ ( J s i s j + J σ i σ j + J ′ s i s j σ i σ j ). (3.98)
There is a critical line of second order transitions between the limiting cases of the
4-state Potts model, with J = J ′ , and the vector Potts model, with J ′ = 0. In the
4-state Potts model, it doesn’t cost additional energy to flip an orthogonal pair of
neighboring spin states,
(1, 1), (−1, 1), (3.99)
to an antiparallel pair, e.g.,
(1, 1), (−1, −1). (3.100)
In the vector model, on the other hand, antiparallel pairs cost twice as much energy,
and the system corresponds to decoupled Ising systems. Along the line of second
order transitions the critical exponents vary continuously with the relative weight
J : J ′ between those for the 4−state Potts model, ν = 2/3, β = 1/12, 10 and those
for the 4-state vector Potts model, ν = 1, β = 1/8.
Not only does the Svetitsky-Yaffe conjecture not predict a universality class for
SU(4) in 2 + 1 d, but its finite temperature transition and that of the 4-state Potts
model lie right on the separation line between first and second order transitions
(see Fig. 3.2). Increases in the size of the Z N center or number of dimensions result
in systems with first order transitions.
Numerical determination of the order of the SU(4) transition has proven to be
surprisingly challenging. Simulations are hampered by logarithmic finite size corrections,
present also for the 2d 4-state Potts model, and the existence of a bulk
crossover centered at lattice coupling β ∼ 13.5 [41]. This necessitates large spatial
volumes as well as N t values large enough to ensure that the bulk transition does
not influence the order of the deconfinement transition.
9 Symmetric means that the couplings for the pair Ising spin systems are identical. They may differ in
a generic Ashkin-Teller model.
10 β is the critical exponent for the magnetization, 〈s i 〉 ∼ 〈σ i 〉 ∼ −t β . On the critical line away from the
limiting cases J = {J ′ , 0} this ‘magnetic’ order parameter is distinguished from an additional ‘electric’
one 〈s i σ i 〉 [79].