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

3.5 finite size scaling and self-duality 37
3.5 finite size scaling and self-duality
As explained in Sec. 3.3.1, the center vortex free energy is determined about criticality
by the universal scaling function f I (L 1/ν t) for interfaces in the 2d Ising universality
class. Different lattice realizations simply require a rescaling of the correlation
length in order to match their FSS variables, x = L 1/ν t ∝ L/ξ.
We may determine f I in terms of the FSS variable on a square N × N 2d Ising
model,
x I = N 1/ν t, with t = K/K c − 1, ν = 1. (3.55)
Then, in SU(2), T c may be used as a length scale to form a dimensionless FSS
variable that differs by a non-universal coefficient λ from x I in the 2d Ising model,
x SU(2) = T c L 1/ν t, with t = T/T c − 1, L = aN s , (3.56)
x I = −λx SU(2) . (3.57)
The minus sign is required because the disordered, Z 2 center symmetric phase is
at low temperature for SU(2) whilst it is at high temperature for the Ising model.
We will suppress the subscript where it is not likely to cause confusion.
A full determination of the vortex free energies F k (⃗ k) at criticality follows after
fitting for λ,
F k (x SU(2) ) = f I (−λx SU(2) ), (3.58)
where we can obtain the scaling function f I to arbitrary accuracy using the exact
solutions for interface free energies on finite 2d Ising lattices from Ref. [54].
In Fig. 3.11 we plot N t = 4 results for Z k (1, 0)/Z k (0, 0) = e −F k for one unit of
center flux as a function of the FSS variable x SU(2) , fitted by λ to the universal
scaling function f I computed in the 2d Ising model. Included also is the partition
function ratio Z e (1, 0)/Z e (0, 0) = e −F e
for one unit of electric flux.
The colored data represents various 4 × Ns
2 lattices with up to N s = 96 spatial
points. This impressive collapse of the data onto the universal curve is typical of
each of our fixed N t data sets. Even more striking is the way in which the vortex
and electric flux ensembles collapse onto each other as perfect mirror images under
x → −x. SU(2) in 2 + 1 dimensions exhibits a self-duality at criticality.
This remarkable fact stems from self-duality in the 2d Ising model. It is well
known that the 2d Ising model and, indeed, all 2d N-state Potts models are selfdual
for infinite volumes via Kramers-Wannier duality [69]. Dualities are less well
studied in finite volumes, however, where the boundary conditions become a nuisance.
The behavior of SU(2) fluxes at criticality indicates that Kramers-Wannier duality
for the 2d Ising model on a torus takes the form of ’t Hooft’s discrete Fourier
transform over twists. Indeed, the 2d Ising model is self-dual on a rectangular
lattice with N sites sites under a 2d Z 2 - Fourier transform