1. Twist-averaged boundary conditions
i θx
Ψ ⃗ (r 1 + L x x , r 2 ,⋯, r N )=e Ψ θ (r 1 , r 2 ,⋯, r N )
̂
θ
Ψ θ ( R)=e
i k⋅∑ r j
j
θ x θ y θz
k=
, ,
Lx L y Lz
(
Ψ k ( R)
π
π
)
π
1
̂〉
̂
∫ ∫
〈 O TABC= Ω /2 −π d θ x −π d θ y∫ d θz 〈 Ψ θ∣O∣Ψ θ 〉
θ
0
Half volume
∵ time reversal symmetry

2. eg. 2D electron gas
i k⋅∑ r j
i G⋅∑ r j
Ψ k ∼e
⏟e
⏟
j
twist
j
Ψk
2 π nx 2 π n y
G=
,
Lx
Ly
(
)
reciprocal
lattice vectors
1
2
E k = ∣G +k∣
2
Crystal momenta
Each color band occupies same area.

3. Fixed phase
ρ( R , R ' ;β)=ρ( R , R ' ;β)e
̃
i Φ( R , R ' ; β)
Different Φ
for each slice.
∂ρ
− = [−λ ∇ 2 +V ( R) ] ρ
R
∂β
Re: −λ ∇ ρ+ [ V ( R)+λ∣∇ Φ∣ ] ρ
̃
̃
2
2
2
Im: ρ ∇ Φ+2(∇ ρ)⋅(∇ Φ)=0
̃
̃
2
crucial step
2
∣∇ Φ∣ =( N k )
2
V eff ( R)=V ( R)+λ∣∇ Φ∣
=V ( R)+λ N
2
((
2
2
θx
θy
θz
+G x +
+G y +
+G z
Lx
Ly
Lz
)(
)(
2
))