Fractional topological insulators

Quantum Hall effect w/o Landau levels
wo examples of Hamiltonians of the the form (1a) are
he following. F. D. M. Haldane, Example Phys. 1: Rev. The Lett. honeycomb 61, 2015 lattice. (1988). We
troduce the vectors a t 1 =(0, −1), a t 2 = √ 3/2, 1/2 ,
t
3 = − √ 3/2, 1/2 connecting NN and the vectors b t 1 =
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
he honeycomb lattice depicted in Fig. 1(a). We denote
ith k a wave vector from the BZ of the reciprocal lattice
ual to the triangular lattice spanned by b 1 and b 2 ,say.
he model is then defined by the Bloch Hamiltonian [1]
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
σ xy = −1
(2a)
σ xy =+1
(2b)
here t 1 ≥ 0 and t 2 ≥ 0 are NN and NNN hoping
amplitudes, respectively, and the real numbers ±Φ
re the magnetic fluxes penetrating the two halves of
he Thursday, hexagonal March 22, 2012 unit cell. For t 1 t 2 , the gap ∆ ≡
a) b)
c)
d)
ε/t 1
2
0
ε/t 1
2
0
− 4π
3 √ 3
t 2
k x
0
A i
B i