Fractional topological insulators

= 0
γ x
3-fold
9-fold
0 10 20
Topological insulator: 2/3 0filling
1 2 3 U/V
c) a)
d)
U/V = 0, λ = 1 U/V = 3, λ = 1
1
λ
0.5
2ϖ
γ x
!
%
$./
!"
3-fold
9-fold
0 ϖ 2ϖ 0 ϖ γ 2ϖ
x
'()*+,
-()*+,
0
0 1 2 3 U/V
b)
c)
E/V
0.2
0.1
0
U/V = 0, λ = 0
0 ϖ
γ x
Thursday, March 22, 2012
2ϖ
U/V = 0, λ = 1
d)
U/V = 3, λ = 1
$ %$ &$
B 0,k := 0,
$
$ % & ' !"#
B 1,k +iB 2,
#"
2" ,"
!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"%
B 3,k := 2t
H 2
int := U ρ i,↑ ρ i,↓ + V
ρi,↑ ρ j,↑ + ρ i,↓ ρ j,↓ +2λρ i,↑ ρ j,↓
i∈Λ
ij∈Λ
and the three Pau
sublattice index.
γ x
2ϖ 0 ϖ
neighbor (NN) an
ping amplitudes.
$"#
$.&
$.%
∋
0 10 20
E/V
0 ϖ γ x
2ϖ
∋
0.2
0.1
0
3-fold
9-fold
0 10 20
0 1 2 3 U/V
b)
c)
U/V = 0, λ = 0
0 ϖ
γ x
2ϖ
d)
U/V = 0, λ = 1 U/V = 3, λ = 1
&
lattice momentum
α = A, B and c
ψ † k,σ := c † k,A,σ ,
single-particle Ha
H 0 :=
0 ϖ γ 2ϖ 0 ϖ
x
γ 2ϖ
x
k∈BZ
ψ † k
where the 3-vecto
∋