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

130 more on c-periodic boundary conditions
b.2 mixed c-periodic boundary conditions
b.2.1
x direction C-periodic, y, z directions periodic
Suppose that we employ boundary conditions with a single C-periodic direction,
chosen to be the x direction. These boundary conditions may be written as
Φ(x + L ˆx) = Ω † xΦ ∗ (x)Ω x , U µ (x + L ˆx) = Ω † xU ∗ µ(x)Ω x ,
Φ(x + Lŷ) = Ω † yΦ(x)Ω y , U µ (x + Lŷ) = Ω † yU µ (x)Ω y ,
Φ(x + Lẑ) = Ω † zΦ(x)Ω z , U µ (x + Lẑ) = Ω † zU µ (x)Ω z .
Again assuming constant transition funcitons, consistency of the boundary conditions
now requires
Ω x Ω y = z 12 Ω ∗ yΩ x ,
Ω x Ω z = z 13 Ω ∗ zΩ x ,
Ω y Ω z = z 23 Ω z Ω y , (B.6)
where z ij = e in ij
, n ij = (2π/N) ɛ ijk m k with m k ∈ Z N , are center elements as before.
Note that charge conjugation only ever happens on one side of the equation.
The gauge transformations to diagonalise the Higgs field have the following
genuine boundary conditions,
R(x + L ˆx) = Ω † xR ∗ (x), (B.7)
R(x + Lŷ) = Ω † yR(x),
R(x + Lẑ) = Ω † z R(x),
and we define the following doubly translated R’s at the far edges and corner by
lexicographic order,
for r = 0, . . . L − 1, and
R(L, L, r) ≡ Ω † yΩ † xR ∗ (0, 0, r), (B.8)
R(L, r, L) ≡ Ω † zΩ † xR ∗ (0, r, 0),
R(r, L, L) ≡ Ω † zΩ † yR(r, 0, 0),
R(L, L, L) ≡ Ω † zΩ † yΩ † xR ∗ (0, 0, 0).
(B.9)
It is straightforward to derive the corresponding boundary conditions for the Abelian
projected fields (4.31),
with the following exceptions,
α a i (x + L ˆx) = −αa i (x),
α a i (x + Lŷ) = αa i (x),
αi a(x + Lẑ) = αa i (x), (B.10)
α a y(L, L − 1, r) = −α a y(0, L − 1, r) − 2π N m 3,
α a z(L, r, L − 1) = −α a z(0, r, L − 1) + 2π N m 2,
α a z(r, L, L − 1) = α a z(r, 0, L − 1) − 2π N m 1, (B.11)