Fractional topological insulators

c)
ε/t 1
2
0
− 4π
k x
4π
3 √ 3
3 √ − 2π 3 3
Can one get perfectly flat bands?
0 0
les d) of Hamiltonians of the the form (1a) are
ε/t 1
g. Example 1: The honeycomb lattice. We
he vectors 2 a t 1 =(0, −1), a t 2 = √ 3/2, 1/2 ,
/2, 1/2 0 connecting NN and the vectors b t 1 =
= a t 3−a –2
t 1, b t 3 = a t 1−a t 2 connecting NNN from
omb lattice depicted in Fig. 1(a). We denote
− π
ve vector fromk x the BZ 0 of the reciprocal 0 lattice k y
triangular lattice spanned by b 1 and b 2 ,say.
is then defined by the Bloch Hamiltonian [1]
FIG. 1. (Color online) (a) Unit cell of Haldane’s model on the
honeycomb lattice: The ψ † kNN H kψ hopping k , amplitudes t 1 are real
(solid lines) and the NNN hopping amplitudes are t 2 e i2πΦ/Φ 0
k∈BZ 3
in the direction of the arrow (dotted lines). The flux 3Φ
Band 0,k −Φ
:=
penetrate
2t 2 cos Φthe dark
cosshaded k · b i ,
region and each
(2a)
of the
light shaded regions, i=1 respectively. For Φ = π/3, the model
⎛
is gauge equivalent to having one flux quantum per unit cell.
(b) The chiral-π-flux
3 on the square lattice, where the unit cell
corresponds to Hthe k flat
shaded := H k
B k := ⎝
t ⎞
1 cos k · a i
t 1 sinarea. k · aThe NN hopping amplitudes
are t 1
|ε i
⎠ , (2b)
e iπ/4 in the direction of −,k
|
i=1 −2t the arrow (solid lines) and the
2 sin Φ k · b i
NNN hopping amplitudes are t 2 and −t 2 along the dashed and
≥dotted 0 and
lines, respectively.
t
(c) The band structure of Haldane’s
2 ≥ 0 are
model for cos Φ =1/(4t 2 )=3 NN and NNN hoptudes,
3/43 with the flatness ratio
1/7. √ (d)
respectively,
The band structure
and
of
the
thereal chiral-π-flux
numbers
for
±Φ
t 1 /t 2 =
gnetic 2 with the fluxes flatness penetrating ratio 1/5. The the lower two bands halves can be of made
exactly flat by adding longer range hoppings.
H 0 =
To enhance the effect of interactions, highly degenerate
(i.e., flat) bands are desirable. It is always possible to
π
− π
k y
2π
3
π
H k = B k · σ
nal unit cell. For t 1 t 2 , the gap ∆ ≡
max k
ε −,k
is proportional to t 2 . The width
r band is δ − ≡ max k
ε −,k
− min k ε −,k
. The
io Thursday, δ − /∆ is extremal for the choice cos Φ =
March 22, 2012
a) b)
c)
ε/t 1
2
0
ψ † k = c † k,A ,c† k,B
⇒ In real space, hoppings decay exponentially with distance
d)
− 4π
k x
4π
3 √ 3
3 √ − 2π 3 3
ε/t 1
2
0
–2
− π
k x
A i
B i
A i
0 0
0 0
π − π
k y
k y
B i
2π
3
π
2