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

38 universal aspects of confinement in 2 + 1 dimensions
Figure 3.11: The ratio Z k (1, 0)/Z k (0, 0) of the partition function with twist ⃗ k = (1, 0) over
the periodic ensemble in the 2 + 1 dimensional SU(2) gauge theory compared
to Z e (1, 0)/Z e (0, 0) for one unit of electric flux ⃗e = (1, 0) relative to the no-flux
ensemble (left), and to its mirror image (right). The data here is obtained for
N t = 4 and spatial volumes with up to N s = 96 [17]. The dashed lines represent
the universal scaling function from the 2d Ising model, exp{− f I (−λx)} for Z k
and its mirror exp{− f I (λx)} for Z e from self-duality.
⎛ ⎞
⎛
⎞ ⎛ ⎞
Z (0,0) ( ˜K)
1 1 1 1 Z (0,0) (K)
Z (1,0) ( ˜K)
⎜
⎝Z (0,1) ⎟
( ˜K)
= 1 ⎠ 2 sinh(2K)−N sites
1 1 −1 −1
Z (1,0) (K)
⎜
⎟ ⎜
⎝1 −1 1 −1⎠
⎝Z (0,1) ⎟
(K) ⎠
Z (1,1) ( ˜K)
1 −1 −1 1 Z (1,1) (K)
(3.59)
with dual coupling
˜K = − 1 ln tanh K, K = J/T. (3.60)
2
This is easy to verify using the analytic solutions in Ref. [54] for partition functions
with mixtures of periodic and antiperiodic boundary conditions. A proof of the
result is also given in Ref. [70].
The couplings ˜K and K have a linear relationship at criticality,
˜K − ˜K c = −(K − K c ) + O(K 2 ), K c = ˜K c = 1 2 ln(1 + √ 2), (3.61)
which ensures that the dual partition functions are mirror images of the originals
when plotted as functions of the reduced temperature t or FSS variable x.
The duality relation generalizes to the N-state Potts model. For partition functions
Z (m,n) with cyclically shifted spins, s(⃗x + Lî) = e i2πn/N s(⃗x), in orthogonal
directions on a torus, the duality transformation (3.59) becomes [17],
Z (−s,r) ( ˜K) =
with the dual coupling ˜K obtained from
(
e ˜K Nsites
− 1 1
e K − 1)
N ∑ e 2πi (rm+sn)/N Z (m,n) (K) ,
m, n
m, n, r, s = 0, 1, . . . N − 1, (3.62)
(
e ˜K − 1 )( e K − 1 ) = N . (3.63)