Fractional topological insulators

a) b)
Quantum Hall effect w/o Landau levels
A i
c)
d)
t 2 /t 1 =
ε/t 1
2
0
ε/t 1
√
43
Two examples of Hamiltonians of the the form (1a)
12 √ 3 ≈ 0.315495 cos Φ = 1 are
t 1
the following. Example 1: The honeycomb lattice. We
introduce the vectors a A t
1 =(0, −1), a t 2 = √ 3/2, 41/2 i t 2 ,
a t 3 = − √ 3/2, 1/2 connecting B i
NN and the vectors b t 1 =
a t 2−a t 3, b t 2 = a t 3−a t 1, b t 3 = a t 1−a t 2 connecting NNN from
the honeycomb lattice depicted in Fig. 1(a). We denote
with k a wave vector from the BZ of the reciprocal lattice
dual to the triangular lattice spanned by b 1 and b 2 ,say.
The model is then defined by the Bloch Hamiltonian [1]
δ − /∆ =1/7
B 0,k := 2t 2 cos Φ
B k :=
3
cos k · b i ,
i=1
⎛
3
⎝
t ⎞
1 cos k · a i
t 1 sin k · a i
⎠ ,
−2t 2 sin Φ sin k · b i
i=1
(2a)
(2b)
where t 1 ≥ 0 and t 2 ≥ 0 are NN and NNN hopping
amplitudes, respectively, and the real numbers ±Φ
are
− 4π
k
3 √ the magnetic fluxes penetrating the two halves
3 x
3 √ − 2π of
the hexagonal unit cell. For t 1 t 2 , the gap ∆ ≡
min k ε +,k
− max k
ε −,k
is proportional to t 2
3. The3width
of the lower band is δ − ≡ max k
ε −,k
− min k ε −,k
. The
flatness ratio δ − /∆ is extremal for the choice cos Φ =
Thursday, March 22, 2012
0 4π
0
a) b)
c)
k
0
y
–2
B i
ε/t 1
2
0
σ xy = −1
d)
σ xy =+1
ε/t 1
2
− 4π
3 √ 3
− π
2π
3
k x
k x
0
0
A i
B i
4π
3 √