This document discusses twist-averaged boundary conditions for calculating the properties of layered materials with a twist. It introduces the twist angle parameters θx, θy, θz and reciprocal lattice vectors k and G. It also discusses how the wavefunction and energy bands are affected by the twist and reciprocal lattice vectors. It then presents an approach for calculating the density matrix with a fixed phase between slices, and shows that this results in an effective potential that depends on the twist angles and reciprocal lattice vectors.

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
))