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

24 universal aspects of confinement in 2 + 1 dimensions
Polyakov loop this amounts to undoing the effect of the transition function Ω t (0,⃗x)
in the time direction,
P(⃗x) = Ω t (0,⃗x)U t (⃗x, 1)U(⃗x, 2) . . . U t (⃗x, 1/T − 1) ∈ SU(N), (3.23)
where Gothic notation is used to distinguish this from the usual lattice definition.
It is then gauge invariant under possibly non-periodic gauge transformations that
may change the transition functions but not the physical twist sector.
When forming a correlator with separation Lî, the trace in color space must be
taken after forming a product of untraced Polyakov loops (3.23), so that the spatial
transition functions may be taken into account. Carefully applying the cocycle condition
(3.17) and the boundary conditions for the gauge fields (3.16), one finds that
the phases e 2πi k i/N are picked up in the product P(⃗x)P † (⃗x + îL). Parallel transport
picks up the center flux, as expected. Taking the trace and forming an average over
the twisted ensembles {Z k (⃗ k)} gives,
1
〈 (
)〉
tr P(⃗x)P † (⃗x + îL)
N
=
no-flux
∑
⃗ k∈Z d
N
e 2πi k i/N Z k (⃗ k)/ ∑
⃗ k∈Z d
N
Z k (⃗ k). (3.24)
When performing successive translations to create a diagonal separation L⃗e ∈ LZ d N ,
one picks up the twists along the way and obtains a discrete Fourier transform over
the twist vector⃗ k ,
1
〈 (
)〉
tr P(⃗x)P † (⃗x +⃗eL)
N
=
no-flux
∑
⃗ k∈Z d
N
e 2πi ⃗e·⃗ k/N
Z k (⃗ k)/ ∑
⃗ k∈Z d
N
Z k (⃗ k). (3.25)
The correlator in Eq. (3.25) creates the color electric flux corresponding to a color
charge in the direction of a mirror charge displaced by⃗eL. If e i > 1, then the electric
flux wraps several times around the lattice to give multiple units of flux in the same
direction. Ratios of the electric flux partition functions Z e (⃗e) ∝ ∑ ⃗k e 2πi ⃗e·⃗ k/N
Z k (⃗ k)
subsequently define the excess free energy of electric fluxes via
Z e (⃗e)
Z e (⃗0) ≡ e−F e(⃗e;T,L) . (3.26)
The subscript ‘no-flux’ is a reminder that the expectation value is normalized with
respect to the sum ∑ ⃗k∈Z d
N
Z k (⃗ k) with ⃗e = ⃗0.
Note that there is no explicit computation of loop operators, so the self-energy of
static color charges does not contribute to the free energy. The dynamic spreading
of center flux in the twisted ensembles Z k (⃗ k) prevents any singular behavior.
The spreading of center flux in Euclidean space also provides a intuitive picture
for the deconfinement transition. At low temperatures the compact temporal direction
is large and does not prevent spacelike center vortices from spreading their
flux over a large area. As temperature is increased, however, they are gradually
squeezed by the finite extent 1/T . The system, must more abruptly rotate between
center sectors. At some critical temperature T c , the increased cost of interfaces overwhelms
the entropic contribution to their free energy and spacelike center vortices
are completely suppressed. One is stuck with systems in a given center sector and
only the strictly periodic ensemble Z k (⃗0) contributes to the electric flux partition