468
A. Martín-Ruiz et al.
interface at constant ρ. In all this cases the characteristic function θ (x) defined in
(17.9) can be written in terms of the Heaviside function H of one coordinate like
H (z − a), H (r − a) and H (ρ − a), respectively, with the associated unit vectors ˆ
n
given by ˆ
k, ˆ
r and ˆ
ρ, in each of the adapted coordinate systems. Then (17.17) reduces
to
η
μ
ν ∂
2
− ˜
θδ (ξ − ξ 0 )
ξμα
ν ∂ α
G
ν
σ
x, x
= 4πη
μ
σ δ
4
x − x
,
(17.19)
where ξ denotes the coordinate defining the interface at ξ = ξ 0 and the coupling of
the θ -term is given by a one dimensional delta function with support only at ξ 0 . Also,
the unit vector ˆ
n will have a component only in the direction ξ .
Let us illustrate the procedure sketched above by taking the static case of a planar
interface located at z = a, separating two semi-infinite TIs, such that the MEP is
θ(z) = θ 1 H (a − z) + θ 2 H (z − a). In the Coulomb gauge the GF G
ν σ
x, x
satisfies
−η
μ
ν ∇
2
− ˜
θδ (z − a))
3μα
ν ∂ α
G
ν
σ
x, x
= 4πη
μ
σ δ
3
x − x
,
(17.20)
together with the boundary conditions (17.16). The coordinates are separated according to ξ = z, ξ 0 = a, plus the two remaining x and y defining the plane parallel to
the interface (i.e., perpendicular to the z-axis). The separation of the Laplacian is
direct
∇
2
=
∂
2
∂z 2 + ∇
2
,
∇
2
=
∂
2
∂ x 2 +
∂
2
∂ y 2 ,
(17.21)
with the operator ∇
2
having the eigenfunctions
p (x, y) =
1
2π
e
ip ·x ,
(17.22)
labelled by the momentum p = ( p x , p y ) parallel to and where x = (x, y). Let
us emphasize that the subindex denotes objects living in the x − y plane, parallel
to the interface z = a. The eigenfunctions p (x, y) are a complete and orthonormal
set in the x − y plane, satisfying [41]
dxdy ∗
p (x, y)) p
(x, y) = δ 2
p ,p
,
d 2 p p (x, y)) ∗
p (x, y) = δ 2 (x − x ).
(17.23)
We recall that in the full 3D-space we have d
3 x = dx dz and δ
3
x − x
=
δ
2
(x − x
)δ(z − z
). Invariance under translation in the x − y plane together with
the the properties in (17.23) allow us to simplify (17.20) introducing the reduced GF
g
μ
ν
z, z
, p
, such that
G
μ
ν
x, x
= 4π
d
2 p
(2π )
2
e
ip ·(x−x
) g
μ
ν
z, z
, p
.
(17.24)
A. Martín-Ruiz et al.
interface at constant ρ. In all this cases the characteristic function θ (x) defined in
(17.9) can be written in terms of the Heaviside function H of one coordinate like
H (z − a), H (r − a) and H (ρ − a), respectively, with the associated unit vectors ˆ
n
given by ˆ
k, ˆ
r and ˆ
ρ, in each of the adapted coordinate systems. Then (17.17) reduces
to
η
μ
ν ∂
2
− ˜
θδ (ξ − ξ 0 )
ξμα
ν ∂ α
G
ν
σ
x, x
= 4πη
μ
σ δ
4
x − x
,
(17.19)
where ξ denotes the coordinate defining the interface at ξ = ξ 0 and the coupling of
the θ -term is given by a one dimensional delta function with support only at ξ 0 . Also,
the unit vector ˆ
n will have a component only in the direction ξ .
Let us illustrate the procedure sketched above by taking the static case of a planar
interface located at z = a, separating two semi-infinite TIs, such that the MEP is
θ(z) = θ 1 H (a − z) + θ 2 H (z − a). In the Coulomb gauge the GF G
ν σ
x, x
satisfies
−η
μ
ν ∇
2
− ˜
θδ (z − a))
3μα
ν ∂ α
G
ν
σ
x, x
= 4πη
μ
σ δ
3
x − x
,
(17.20)
together with the boundary conditions (17.16). The coordinates are separated according to ξ = z, ξ 0 = a, plus the two remaining x and y defining the plane parallel to
the interface (i.e., perpendicular to the z-axis). The separation of the Laplacian is
direct
∇
2
=
∂
2
∂z 2 + ∇
2
,
∇
2
=
∂
2
∂ x 2 +
∂
2
∂ y 2 ,
(17.21)
with the operator ∇
2
having the eigenfunctions
p (x, y) =
1
2π
e
ip ·x ,
(17.22)
labelled by the momentum p = ( p x , p y ) parallel to and where x = (x, y). Let
us emphasize that the subindex denotes objects living in the x − y plane, parallel
to the interface z = a. The eigenfunctions p (x, y) are a complete and orthonormal
set in the x − y plane, satisfying [41]
dxdy ∗
p (x, y)) p
(x, y) = δ 2
p ,p
,
d 2 p p (x, y)) ∗
p (x, y) = δ 2 (x − x ).
(17.23)
We recall that in the full 3D-space we have d
3 x = dx dz and δ
3
x − x
=
δ
2
(x − x
)δ(z − z
). Invariance under translation in the x − y plane together with
the the properties in (17.23) allow us to simplify (17.20) introducing the reduced GF
g
μ
ν
z, z
, p
, such that
G
μ
ν
x, x
= 4π
d
2 p
(2π )
2
e
ip ·(x−x
) g
μ
ν
z, z
, p
.
(17.24)