ABSTRACT - DRUM - University of Maryland

can confine ourselves to studying representations of D 2 , D 4 , D 6 , and O(2), which
exhaust all physically relevant possibilities.
2.1.1 BCS Mean Field Theory
Now, we define the superconducting order parameter as
∫
∆ k = ˜f k,k ′〈ĉ −k ′ĉ k ′〉, (2.2)
k ′ ∈BZ
where ˜f k,k ′ = (f k,k ′ ,q=0−f k,−k ′ ,q=0)/2 is the antisymmetrized BCS coupling strength.
Using a Hubbard-Stratonovich decoupling in Eq. (2.1) with q = 0 and ignoring
superconducting fluctuations, we arrive at
∫
Ĥ MF =
(ξ k ĉ † kĉk + 1 2 ∆ kĉ † kĉ† −k + 1 )
2 ∆∗ kĉ −k ĉ k − 1 2
k∈BZ
∫
k,k ′ ∈BZ
∆ ∗ k
˜f
−1
k,k ′ ∆ k ′, (2.3)
with
˜f
−1
k,k ′ being the matrix inverse of ˜fk,k ′. By integrating out the fermions we
find the BCS free energy functional expressed in terms of ∆. It contains two parts,
F [∆ k ] = F I + F II , with
∫
F I [∆ k ] =−T
F II [∆ k ] = − 1 2
k∈BZ
∫
[
ln 2 cosh 1
]
√ξk 2 2T
+ |∆ k| 2
∆ ∗ k
k,k ′ ∈BZ
(2.4)
˜f
−1
k,k ′ ∆ k ′. (2.5)
2.1.2 Topological Classification
As we have reviewed in Chapter 1, 2D superconductors in class D is characterized
by the topological invariant(the Chern number) as follows [70]
C =
∫
k∈BZ
d 2 k
4π d · ∂ k x
d × ∂ ky d, (2.6)
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