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

twist away from criticality 125
3.5
3
2.5
a˜σ/T
2
1.5
1
0.5
10 20 30 40 50 60 70 80 90
i
Figure A.4: Cost of flipping one additional plaquette on a 4 × 96 2 lattice for β = 50.
lengthening the vortex is then expected to decrease more rapidly for n ≈ N s , where
the production of a closed vortex becomes viable. This prediction is borne out for
SU(2) in 2 + 1 d in Fig. A.4.
In the region of the plateau, the leading finite size corrections originate from
Gaussian fluctuations of the vortex. These may be estimated by considering the
vortex as the worldline of a Euclidean particle propagator [178]. The effective string
tension is, then,
a˜σ e f f (T) = − ln Z (n+1)
k
/Z n k
= a˜σ(T) + ln ((n + 1)/n). (A.12)
The second term is the 1 + 1 d string worldsheet analog of the Lüscher term. We
extracted a˜σ(T) after the removal of this correction for N t = 4 and N s = 96. The
results are plotted against β in Fig. A.5.
The high temperature perturbative result for the dual string tension was determined
in Ref. [179],
˜σ = α g 3
T 1.5 ,
(A.13)
where the factor α is an N t dependent coefficient. Since aN t = 1/T, we have a˜σ =
˜σ/(N t T). So our plot of a˜σ versus T should go like T 0.5 for large T. Our results
match this perturbative prediction quite well.
a.0.3
Electric string formation at zero temperature
We have also studied string formation for electric flux free energies at zero temperature
for SU(2) in 2 + 1 d.
As L t = aN t → ∞, the free energy F e (⃗e) becomes the zero temperature energy or
mass E(⃗e) of a system with electric flux ⃗e,
Z e (⃗e)
lim
L t →∞ Z e (⃗0) ≡ e−L tE(⃗e;L i ) .
(A.14)