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

3.6 exploiting self-duality 47
with reduced χ 2 /dof.’s of 1.5 and 2.5. They are consistent within the rather generous
fitting errors. Our λ(N t ) points actually agree within ±0.01 for N t ≥ 7. 7
After rescaling the FSS variable x by λ(N t ), our data for vortex ensembles on
various N t × Ns
2 lattices collapses beautifully onto the universal Ising curve in a
large window around x = 0. Compare the data in Fig. 3.17 before and after renormalization
of the FSS variable x = λT c Lt. The electric flux partition function ratios
obtained from vortex ensembles by a discrete Fourier transform scale equally well
(not shown).
With our determination of the non-universal constant λ ∞ , the scaling of the continuum
string tension σ and dual string tension ˜σ at criticality is now completely
specified in units of T c . Repeating Eq. (3.74) for convenience,
σ(t) = λ ∞ T 2 c 2 ln(1 + √ 2) |t| + · · · , t → 0 − , and
˜σ(t) = λ ∞ T c 2 ln(1 + √ 2) t + · · · , t → 0 + , λ ∞ = 1.354(35). (3.81)
One may convert the length scale T c to units of the continuum coupling g 3 or, if
preferred, the zero temperature string tension using our earlier determinations,
√
T c /g3 2 = 0.3732(6), T c / σ(T = 0) = 1.1150(24) . (3.82)
3.6.1.2 Diagonal string formation
Our determination of the string tensions Eq. (3.74) is also valid for pairs of orthogonal
fluxes under the assumption that they minimize the total length by forming a
diagonal string. A string tension implies that the free energy for a pair of perpendicular
fluxes on a symmetric volume should be √ 2 that of a single flux.
This can also be coaxed out of the 2d Ising model. The free energy per N for a diagonal
interface (anti-periodic boundary conditions in both directions) is provided
by Baxter in Ref. [79],
F (1,1)
IN = σ(1,1) I
= 2 ln sinh(2K), N → ∞. (3.83)
Multiplying by K = T/J gives the tension per coupling J, which we plot in Fig.
3.18 together with the the result for a straight interface σ (1,0)
I
.
On a rectangular lattice, discretization breaks rotational invariance such that
straight and diagonal fluxes are not on an equal footing. For the Ising model at
zero temperature, the system is in a state of maximum order. In that case a diagonal
spin interface is forced to take a step-like path through the lattice and is twice
the length of a straight interface. Isotropy only emerges when thermal fluctuations
become large and the correlation length dwarfs the finite lattice spacing. At criticality,
the underlying lattice geometry becomes irrelevant and a √ 2 factor between
straight and diagonal interfaces emerges,
σ (1,1)
I
= 2 √ 2 ln(1 + √ 2)|t| = √ 2 σ (1,0)
I
, K = K c . (3.84)
7 The data is better described by a power law fit with extrapolated values λ ∞ = 1.337(5) and 1.338(5)
for the horizontal and vertical N s offset corrections, with reduced χ 2 /dof. of 1 and 1.9 respectively.
Without a justification for the fitting function we prefer to quote the less constrained results that
better accommodate any remaining systematic uncertainties.