Fractional Chern Insulators - Institute of Condensed Matter Theory at ...
Fractional Chern Insulators - Institute of Condensed Matter Theory at ...
Fractional Chern Insulators - Institute of Condensed Matter Theory at ...
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<strong>Fractional</strong> <strong>Chern</strong> <strong>Insul<strong>at</strong>ors</strong><br />
Titus Neupert<br />
<strong>Condensed</strong> <strong>M<strong>at</strong>ter</strong> <strong>Theory</strong> Group<br />
Paul Scherrer Institut, Switzerland<br />
Christopher Mudry, PSI<br />
Claudio Chamon, Boston U<br />
Shinsei Ryu, UIUC<br />
Luiz Santos, Perimeter <strong>Institute</strong><br />
Adolfo G. Grushin, CSIC Madrid
Outline<br />
1.<br />
<strong>Chern</strong> insul<strong>at</strong>ors<br />
Haldane’s model<br />
noncommut<strong>at</strong>ive geometry<br />
2. <strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors (FCI)<br />
C=1 exact diagonaliz<strong>at</strong>ion results<br />
C=2<br />
3.<br />
spontaneous form<strong>at</strong>ion <strong>of</strong> FCI in<br />
Z2 topological insul<strong>at</strong>ors
Quantum Hall effect: Landau levels<br />
energy scale ! c = eB m<br />
length scale `2B<br />
= ~c<br />
eB<br />
energy degeneracy N = A<br />
2⇡`2B<br />
=<br />
0<br />
B<br />
DOS
Quantum Hall effect: Landau levels<br />
energy scale ! c = eB m<br />
length scale `2B<br />
= ~c<br />
eB<br />
energy degeneracy N = A<br />
2⇡`2B<br />
=<br />
0<br />
B<br />
µ<br />
Integer quantum Hall effect<br />
xy = e2<br />
h n<br />
DOS
Quantum Hall effect: Landau levels<br />
energy scale ! c = eB m<br />
length scale `2B<br />
= ~c<br />
eB<br />
energy degeneracy N = A<br />
2⇡`2B<br />
=<br />
0<br />
B<br />
µ<br />
<strong>Fractional</strong> quantum Hall effect<br />
xy = e2<br />
h ⌫<br />
DOS
B<br />
Are there other systems in which a<br />
quantum Hall effect appears?<br />
Can we dispose <strong>of</strong> the large magnetic field?
B<br />
Answer for integer<br />
quantum Hall effect:<br />
F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)<br />
Are there other systems in which a<br />
quantum Hall effect appears?<br />
Can we dispose <strong>of</strong> the large magnetic field?<br />
Yes<br />
Yes
Cit<strong>at</strong>ions: Haldane’s model<br />
fractional<br />
<strong>Chern</strong> insul<strong>at</strong>ors<br />
topological<br />
insul<strong>at</strong>ors<br />
F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)
<strong>Chern</strong> insul<strong>at</strong>or: Haldane’s model<br />
spinless fermions on honeycomb l<strong>at</strong>tice<br />
complex NNN hoppings open mass gap<br />
in graphene spectrum<br />
“<strong>Chern</strong> insul<strong>at</strong>or”<br />
xy = e2<br />
h C`B<br />
C = ±1<br />
! a
<strong>Chern</strong> insul<strong>at</strong>or: Haldane’s model<br />
spinless fermions on honeycomb l<strong>at</strong>tice<br />
complex NNN hoppings open mass gap<br />
in graphene spectrum<br />
“<strong>Chern</strong> insul<strong>at</strong>or”<br />
xy = e2<br />
h C`B<br />
C = ±1<br />
! a
<strong>Chern</strong> insul<strong>at</strong>or: Haldane’s model<br />
spinless fermions on honeycomb l<strong>at</strong>tice<br />
complex NNN hoppings open mass gap<br />
in graphene spectrum<br />
“<strong>Chern</strong> insul<strong>at</strong>or”<br />
xy = e2<br />
h C`B<br />
C = ±1<br />
! a
<strong>Chern</strong> insul<strong>at</strong>or: Haldane’s model<br />
spinless fermions on honeycomb l<strong>at</strong>tice<br />
complex NNN hoppings open mass gap<br />
in graphene spectrum<br />
“<strong>Chern</strong> insul<strong>at</strong>or”<br />
xy = e2<br />
h C`B<br />
C = ±1<br />
! a<br />
Just like a<br />
Landau<br />
level?
3 functions th<strong>at</strong> are constant in the Landau level<br />
... but not in a generic <strong>Chern</strong> insul<strong>at</strong>or<br />
1.<br />
Dispersion<br />
"(k)<br />
2.<br />
3.<br />
Quantum metric<br />
Berry curv<strong>at</strong>ure<br />
g µ⌫ (k)<br />
F µ⌫ (k)
3 functions th<strong>at</strong> are constant in the Landau level<br />
... but not in a generic <strong>Chern</strong> insul<strong>at</strong>or<br />
1.<br />
Dispersion<br />
"(k)<br />
2.<br />
3.<br />
Quantum metric<br />
Berry curv<strong>at</strong>ure<br />
g µ⌫ (k)<br />
F µ⌫ (k)<br />
Berry connection A µ (k) :=hu(k)|@ µ |u(k)i<br />
F µ⌫ (k) =@ µ A ⌫ (k) @ ⌫ A µ (k) C = i Z<br />
2⇡<br />
d 2 k F 12 (k)
3 functions th<strong>at</strong> are constant in the Landau level<br />
... but not in a generic <strong>Chern</strong> insul<strong>at</strong>or<br />
1.<br />
Dispersion<br />
"(k)<br />
2.<br />
3.<br />
Quantum metric<br />
Berry curv<strong>at</strong>ure<br />
g µ⌫ (k)<br />
F µ⌫ (k)<br />
Berry connection A µ (k) :=hu(k)|@ µ |u(k)i<br />
F µ⌫ (k) =@ µ A ⌫ (k) @ ⌫ A µ (k) C = i Z<br />
2⇡<br />
d 2 k F 12 (k)<br />
h (k)| b X µ<br />
b X⌫ | (k)i = g µ⌫ (k)+F µ⌫ (k)<br />
projected<br />
real, imaginary,<br />
position oper<strong>at</strong>or<br />
symmetric antisymmetric<br />
[ X b µ , X b ⌫ ](k) =2F µ⌫ (k) noncommut<strong>at</strong>ive geometry
Density algebra<br />
Landau level<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
b⇢ q+q 0<br />
b⇢ q = e<br />
iq· bX<br />
S. M. Girvin et al., Phys. Rev. Lett. 54, 581 (1985)<br />
<strong>Chern</strong> insul<strong>at</strong>or
Density algebra<br />
Landau level<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
b⇢ q+q 0<br />
b⇢ q = e<br />
iq· bX<br />
S. M. Girvin et al., Phys. Rev. Lett. 54, 581 (1985)<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
e q µg µ⌫<br />
q 0 ⌫ b⇢q+q 0<br />
if F and g const.<br />
R. Roy, arXiv:1208.2055
Density algebra<br />
Landau level<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
b⇢ q+q 0<br />
b⇢ q = e<br />
iq· bX<br />
S. M. Girvin et al., Phys. Rev. Lett. 54, 581 (1985)<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
e q µg µ⌫<br />
q 0 ⌫ b⇢q+q 0<br />
if F and g const.<br />
R. Roy, arXiv:1208.2055<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i<br />
b⇢ q+q 0 + O(q 3 )<br />
if F const.<br />
S. Parameswaran et al., arXiv:1106.4025<br />
T. Neupert et al., Phys. Rev B 86, 035125 (2012)<br />
B. Estienne et al., arXiv:1202.5543
Density algebra<br />
Landau level<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
b⇢ q+q 0<br />
b⇢ q = e<br />
iq· bX<br />
S. M. Girvin et al., Phys. Rev. Lett. 54, 581 (1985)<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i sin<br />
e q µg µ⌫<br />
q 0 ⌫ b⇢q+q 0<br />
if F and g const.<br />
R. Roy, arXiv:1208.2055<br />
✓`2<br />
B<br />
2 q ^ q0 ◆<br />
[b⇢ q , b⇢ q 0]=<br />
2i<br />
b⇢ q+q 0 + O(q 3 )<br />
if F const.<br />
Z<br />
S. Parameswaran et al., arXiv:1106.4025<br />
T. Neupert et al., Phys. Rev B 86, 035125 (2012)<br />
B. Estienne et al., arXiv:1202.5543<br />
[b⇢ q , b⇢ q 0]=<br />
q µ q 0 ⌫<br />
d 2 k|u(k)iF µ⌫ (k)hu(k + q + q 0 )| + O(q 3 )
Outline<br />
1.<br />
<strong>Chern</strong> insul<strong>at</strong>ors<br />
Haldane’s model<br />
noncommut<strong>at</strong>ive geometry<br />
2. <strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors (FCI)<br />
C=1 exact diagonaliz<strong>at</strong>ion results<br />
C=2<br />
3.<br />
spontaneous form<strong>at</strong>ion <strong>of</strong> FCI in<br />
Z2 topological insul<strong>at</strong>ors
B<br />
Are there other systems in which a<br />
quantum Hall effect appears?<br />
Can we dispose <strong>of</strong> the large magnetic field?
B<br />
Answer for fractional<br />
quantum Hall effect:<br />
T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011)<br />
Are there other systems in which a<br />
quantum Hall effect appears?<br />
Can we dispose <strong>of</strong> the large magnetic field?<br />
Yes<br />
Yes
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Need<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
reasonably fl<strong>at</strong> band with C≠0<br />
(requires some longer range hopping)<br />
add short-range repulsive interactions<br />
band gap<br />
band width<br />
interaction<br />
W<br />
U
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Need<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
reasonably fl<strong>at</strong> band with C≠0<br />
(requires some longer range hopping)<br />
add short-range repulsive interactions<br />
band gap<br />
band width<br />
interaction<br />
W<br />
U<br />
with hierarchy <strong>of</strong> energy scales E. Tang et al., Phys. Rev. Lett., 106, 236802 (2011).<br />
U W<br />
K. Sun et al., Phys. Rev. Lett., 106, 236803 (2011).<br />
T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011)
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Need<br />
<strong>Chern</strong> insul<strong>at</strong>or<br />
reasonably fl<strong>at</strong> band with C≠0<br />
(requires some longer range hopping)<br />
add short-range repulsive interactions<br />
band gap<br />
band width<br />
interaction<br />
W<br />
U<br />
with hierarchy <strong>of</strong> energy scales E. Tang et al., Phys. Rev. Lett., 106, 236802 (2011).<br />
U W<br />
K. Sun et al., Phys. Rev. Lett., 106, 236803 (2011).<br />
T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011)<br />
Strongly correl<strong>at</strong>ed many-body problem, no perturb<strong>at</strong>ive<br />
approaches<br />
exact numerical diagonaliz<strong>at</strong>ion<br />
<strong>of</strong> small systems
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Model<br />
square l<strong>at</strong>tice,<br />
spin 1/2 d.o.f. per site<br />
it 1 + h 1 3<br />
h 4 3 it 2 + h 2 3<br />
m (ij) =( 1) i h 1 +( 1) j h 2 +( 1) i+j 2 h 4<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Model<br />
square l<strong>at</strong>tice,<br />
spin 1/2 d.o.f. per site<br />
it 1 + h 1 3<br />
h 4 3 it 2 + h 2 3<br />
m (ij) =( 1) i h 1 +( 1) j h 2 +( 1) i+j 2 h 4<br />
C = 1 X<br />
( 1) i+j sgn m (ij) takes values 0, 1, 2<br />
2<br />
i,j=0,1<br />
repulsive density-density interactions,<br />
twisted boundary conditions<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
C = 1 <strong>at</strong> 1/3 and 2/3 filling<br />
Q y<br />
4 x 6 l<strong>at</strong>tice<br />
many-body<br />
c.o.m. BZ<br />
Q x<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
C = 1 <strong>at</strong> 1/3 and 2/3 filling<br />
Q y<br />
4 x 6 l<strong>at</strong>tice<br />
many-body<br />
c.o.m. BZ<br />
Q x<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
C = 2 <strong>at</strong> 1/5 and 4/5 filling<br />
5<br />
5 x 4 l<strong>at</strong>tice<br />
5 x 5 l<strong>at</strong>tice<br />
5 x 6 l<strong>at</strong>tice<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
C = 2 <strong>at</strong> 1/5 and 4/5 filling<br />
5<br />
5 x 4 l<strong>at</strong>tice<br />
5 x 5 l<strong>at</strong>tice<br />
5 x 6 l<strong>at</strong>tice<br />
A. G. Grushin et al., arXiv:1207.4097
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es<br />
⌫ g<br />
fe<strong>at</strong>ureless<br />
CDW in thin-torus limit<br />
A. G. Grushin et al., arXiv:1207.4097<br />
B. A. Bernevig et al., arXiv:1204.5682
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Aside: Detecting broken transl<strong>at</strong>ional symmetry<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
quasi-degener<strong>at</strong>e ground st<strong>at</strong>es | ii, i =1,<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es ⌫ g ··· ,N<br />
m<strong>at</strong>rix h i|⇢ r | ji has transl<strong>at</strong>ional invariant<br />
fe<strong>at</strong>ureless spectrum, but r dependent eigenvectors<br />
v r,i<br />
map out<br />
CDW in thin-torus limit<br />
A. G. Grushin et al., arXiv:1207.4097<br />
f r = v ⇤ r 0 ,ih i|⇢ r | jiv r0 ,j<br />
B. A. Bernevig et al., arXiv:1204.5682
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es<br />
⌫ g<br />
fe<strong>at</strong>ureless<br />
CDW in thin-torus limit<br />
A. G. Grushin et al., arXiv:1207.4097<br />
B. A. Bernevig et al., arXiv:1204.5682<br />
many-body <strong>Chern</strong> number (Hall conductivity) 1/3 and 2/5<br />
T. Neupert et al., Phys. Rev. B 86 165133 (2012)
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Aside: Computing the many-body <strong>Chern</strong> number<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
Thouless: Average over boundary conditions<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es ⌫ g<br />
| ( 1 , 2)i, ( 1 , 2) 2 [0, 2⇡] 2<br />
fe<strong>at</strong>ureless<br />
C =<br />
i<br />
2⇡<br />
Z2⇡<br />
CDW in thin-torus limit<br />
0<br />
Z2⇡<br />
d 1<br />
0<br />
d 2<br />
apple⌧ @<br />
@ 1<br />
@<br />
@ 2<br />
⌧ @<br />
A. G. Grushin et al., arXiv:1207.4097<br />
@ 2<br />
B. A. Bernevig et al., arXiv:1204.5682<br />
many-body <strong>Chern</strong> number (Hall conductivity) 1/3 and 2/5<br />
T. Neupert et al., Phys. Rev. B 86 165133 (2012)<br />
@<br />
@ 1
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Aside: Computing the many-body <strong>Chern</strong> number<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
Thouless: Average over boundary conditions<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es ⌫ g<br />
| ( 1 , 2)i, ( 1 , 2) 2 [0, 2⇡] 2<br />
fe<strong>at</strong>ureless<br />
C =<br />
i<br />
2⇡<br />
Z2⇡<br />
CDW in thin-torus limit<br />
0<br />
Z2⇡<br />
d 1<br />
0<br />
d 2<br />
apple⌧ @<br />
@ 1<br />
@<br />
@ 2<br />
⌧ @<br />
A. G. Grushin et al., arXiv:1207.4097<br />
@ 2<br />
B. A. Bernevig et al., arXiv:1204.5682<br />
In thermodynamic limit, if<br />
many-body <strong>Chern</strong> - only number st<strong>at</strong>es in (Hall one band conductivity) occupied1/3 and 2/5<br />
- no spontaneous symmetry breaking<br />
one can show<br />
xy = e2<br />
h<br />
Z<br />
d 2 k n(k) F (k)<br />
T. Neupert et al., Phys. Rev. B 86 165133 (2012)<br />
@<br />
@ 1<br />
n(k) :=h |<br />
† (k) (k)| i
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
N<strong>at</strong>ure <strong>of</strong> incompressible st<strong>at</strong>es: FQH st<strong>at</strong>es<br />
quasi-degeneracy <strong>of</strong> topologically ordered st<strong>at</strong>es<br />
⌫ g<br />
fe<strong>at</strong>ureless<br />
CDW in thin-torus limit<br />
A. G. Grushin et al., arXiv:1207.4097<br />
B. A. Bernevig et al., arXiv:1204.5682<br />
many-body <strong>Chern</strong> number (Hall conductivity) 1/3 and 2/5<br />
T. Neupert et al., Phys. Rev. B 86 165133 (2012)<br />
level counting in the entanglement spectrum<br />
mapping to Landau level Laughlin st<strong>at</strong>e<br />
with high overlap, continuous deform<strong>at</strong>ion<br />
B.A. Bernevig et al., Phys. Rev. B 85 075128 (2012)<br />
X.-L. Qi, Phys,Rev. Lett. 107, 126803 (2011)<br />
Y.-L. Wu, et al., Phys. Rev. B 86, 085129 (2012)<br />
Z. Liu et al., arXiv:1209.5310.<br />
Y.-L. Wu, et al., arXiv:1210.6356.
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or<br />
Abelian st<strong>at</strong>es comp<strong>at</strong>ible with hierarchical CS description<br />
non-Abelian st<strong>at</strong>es (Moore-Read, Zk Read-Rezayi) with<br />
3-body and 4-body interaction<br />
B.A. Bernevig et al., Phys. Rev. B 85, 075128 (2012)<br />
Y.-F. Wang et al., Phys Rev. Lett 108 126805<br />
Y.-L. Wu et al., Phys. Rev. B 85, 075116 (2012)<br />
series <strong>of</strong> st<strong>at</strong>es <strong>at</strong> higher <strong>Chern</strong> number ⌫ =<br />
k<br />
1+2C<br />
A.M. Läuchli arXiv:1207.6094<br />
A. Sterdyniak et al., arXiv:1207.6385
h4/t<br />
h4/t<br />
(E − E0)/U<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or: Effect <strong>of</strong> bandwidth<br />
"(k) g µ⌫ (k) F µ⌫ (k)<br />
The fl<strong>at</strong>ter the better?<br />
some model parameter<br />
(a)<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
gap Δ/U<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
0.05<br />
0<br />
(b)<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
Δ/δ<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
10<br />
0<br />
spectrum <strong>at</strong> h4 = 1.05<br />
(c) -------------<br />
- - - ------<br />
0.07<br />
- - -<br />
------ --------<br />
- -<br />
--- -- - - -<br />
--<br />
----- --- - - -<br />
------ - -<br />
0.06<br />
- - - --- - ------ - - -<br />
0.05 - -------------- ------------<br />
-<br />
- -<br />
- -- - ------ 0.04<br />
-<br />
-<br />
- -<br />
-<br />
0.03<br />
- Δ<br />
-<br />
-<br />
-<br />
0.02<br />
-<br />
-<br />
-<br />
- -<br />
0.01<br />
-- -<br />
-<br />
- -<br />
--- --- - -<br />
δ<br />
0.00 --- --------------<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
bandwidth
h4/t<br />
h4/t<br />
(E − E0)/U<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or: Effect <strong>of</strong> bandwidth<br />
"(k) g µ⌫ (k) F µ⌫ (k)<br />
The fl<strong>at</strong>ter the better?<br />
some model parameter<br />
(a)<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
gap Δ/U<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
0.05<br />
0<br />
(b)<br />
1.4<br />
1.2<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
Δ/δ<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
10<br />
0<br />
spectrum <strong>at</strong> h4 = 1.05<br />
(c) -------------<br />
- - - ------<br />
0.07<br />
- - -<br />
------ --------<br />
- -<br />
--- -- - - -<br />
--<br />
----- --- - - -<br />
------ - -<br />
0.06<br />
- - - --- - ------ - - -<br />
0.05 - -------------- ------------<br />
-<br />
- -<br />
- -- - ------ 0.04<br />
-<br />
-<br />
- -<br />
-<br />
0.03<br />
- Δ<br />
-<br />
-<br />
-<br />
0.02<br />
-<br />
-<br />
-<br />
- -<br />
0.01<br />
-- -<br />
-<br />
- -<br />
--- --- - -<br />
δ<br />
0.00 --- --------------<br />
0.0 0.2 0.4 0.6 0.8 1.0<br />
λ<br />
bandwidth<br />
Some dispersion can help to stabilize the st<strong>at</strong>e!
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or: Effect <strong>of</strong> quantum distance<br />
Projected interaction:<br />
X<br />
H :=<br />
k 1<br />
,k 2<br />
,k 3<br />
,k 4<br />
V k1 k 2<br />
k 3<br />
k 4<br />
†<br />
k 1<br />
†<br />
k 2<br />
k 3<br />
k 4<br />
overlaps <strong>of</strong> singleparticle<br />
st<strong>at</strong>es<br />
V k1 k 2<br />
k 3<br />
k 4<br />
= v k1 k 3<br />
h k1<br />
| k 3<br />
ih k2<br />
| k 4<br />
i k1 +k 2<br />
,k 3<br />
+k 4
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or: Effect <strong>of</strong> quantum distance<br />
Projected interaction:<br />
X<br />
H :=<br />
k 1<br />
,k 2<br />
,k 3<br />
,k 4<br />
V k1 k 2<br />
k 3<br />
k 4<br />
†<br />
k 1<br />
†<br />
k 2<br />
k 3<br />
k 4<br />
overlaps <strong>of</strong> singleparticle<br />
st<strong>at</strong>es<br />
V k1 k 2<br />
k 3<br />
k 4<br />
= v k1 k 3<br />
h k1<br />
| k 3<br />
ih k2<br />
| k 4<br />
i k1 +k 2<br />
,k 3<br />
+k 4<br />
q<br />
Quantum distance<br />
d(k, k 0 ):= 1 |h (k)| (k 0 )i|<br />
enters interaction<br />
Z r<br />
connects to<br />
d(k<br />
Fubini-Study metric 1 , k 2 )=inf<br />
1,2<br />
d`<br />
1,2<br />
g µ⌫ (k) dkµ<br />
d`<br />
dk ⌫<br />
d`
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>or: Effect <strong>of</strong> quantum distance<br />
Projected interaction:<br />
X<br />
H :=<br />
k 1<br />
,k 2<br />
,k 3<br />
,k 4<br />
V k1 k 2<br />
k 3<br />
k 4<br />
†<br />
k 1<br />
†<br />
k 2<br />
k 3<br />
k 4<br />
overlaps <strong>of</strong> singleparticle<br />
st<strong>at</strong>es<br />
V k1 k 2<br />
k 3<br />
k 4<br />
= v k1 k 3<br />
h k1<br />
| k 3<br />
ih k2<br />
| k 4<br />
i k1 +k 2<br />
,k 3<br />
+k 4<br />
q<br />
Quantum distance<br />
d(k, k 0 ):= 1 |h (k)| (k 0 )i|<br />
enters interaction<br />
Z r<br />
connects to<br />
d(k<br />
Fubini-Study metric 1 , k 2 )=inf<br />
1,2<br />
d`<br />
1,2<br />
g µ⌫ (k) dkµ<br />
d`<br />
dk ⌫<br />
d`<br />
Particle-hole transform<strong>at</strong>ion + normal ordering<br />
eH =<br />
X<br />
X<br />
† †<br />
V k1 , k 2<br />
, k 3<br />
, k 4 k 3<br />
k 4<br />
k 1<br />
k 2<br />
k 1<br />
,k 2<br />
,k 3<br />
,k 4<br />
k<br />
" k<br />
†<br />
k<br />
k + constant<br />
effective dispersion " k = X k 0 v k k<br />
0 + v k<br />
0<br />
k<br />
⇥<br />
1 d 2 (k, k 0 ) ⇤ 2<br />
.
Outline<br />
1.<br />
<strong>Chern</strong> insul<strong>at</strong>ors<br />
Haldane’s model<br />
noncommut<strong>at</strong>ive geometry<br />
2. <strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors (FCI)<br />
C=1 exact diagonaliz<strong>at</strong>ion results<br />
C=2<br />
3.<br />
spontaneous form<strong>at</strong>ion <strong>of</strong> FCI in<br />
Z2 topological insul<strong>at</strong>ors
(2+1)D topological insul<strong>at</strong>or<br />
topological band insul<strong>at</strong>ors in (2+1)D: Z IQHE (class A)<br />
Z2 TI (class AII)<br />
Gedanken experiment: filled Landau level with opposite<br />
chirality (magnetic field) for up-spin and down-spin electrons<br />
time-reversal symmetric<br />
with Sz conserv<strong>at</strong>ion: quantum spin Hall effect<br />
B<br />
H =0<br />
sH = e<br />
2⇡<br />
B<br />
Physical realiz<strong>at</strong>ion?
(2+1)D topological insul<strong>at</strong>or<br />
topological band insul<strong>at</strong>ors in (2+1)D: Z IQHE (class A)<br />
Z2 TI (class AII)<br />
Gedanken experiment: filled Landau level with opposite<br />
chirality (magnetic field) for up-spin and down-spin electrons<br />
time-reversal symmetric<br />
with Sz conserv<strong>at</strong>ion: quantum spin Hall effect<br />
B<br />
H =0<br />
sH = e<br />
2⇡<br />
B<br />
Physical realiz<strong>at</strong>ion?<br />
Yes<br />
C. L. Kane et al., Phys. Rev. Lett. 95, 226801 & 146802 (2005).<br />
B. A. Bernevig et al., Science 314, 1757 (2006)<br />
M. König et al., Science 318, 766 (2007)
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Gedanken experiment:<br />
take two fractional quantum Hall layers with opposite field<br />
with Sz conserv<strong>at</strong>ion<br />
B<br />
⌫ " =1/n<br />
⌫ # =1/n<br />
M. Levin et al., Phys. Rev. Lett. 103, 196803 (2009)<br />
B<br />
Is this there a physical setting for the fractional<br />
quantum spin Hall effect to appear?
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Gedanken experiment:<br />
take two fractional quantum Hall layers with opposite field<br />
with Sz conserv<strong>at</strong>ion<br />
B<br />
⌫ " =1/n<br />
⌫ # =1/n<br />
M. Levin et al., Phys. Rev. Lett. 103, 196803 (2009)<br />
B<br />
Is this there a physical setting for the fractional<br />
quantum spin Hall effect to appear?<br />
?, but ...<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Need<br />
Quantum spin Hall insul<strong>at</strong>or (TI)<br />
reasonably fl<strong>at</strong> nontrivial Kramers<br />
pair <strong>of</strong> band<br />
add short-range repulsive interactions<br />
band gap<br />
band width<br />
interaction<br />
W<br />
U<br />
with hierarchy <strong>of</strong> energy scales<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
U<br />
W
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Need<br />
Quantum spin Hall insul<strong>at</strong>or (TI)<br />
reasonably fl<strong>at</strong> nontrivial Kramers<br />
pair <strong>of</strong> band<br />
add short-range repulsive interactions<br />
band gap<br />
band width<br />
interaction<br />
W<br />
U<br />
with hierarchy <strong>of</strong> energy scales<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
U<br />
W<br />
Model<br />
H =CI # +CI ⇤ " + U X i<br />
n # i n" i + V X (i,j)<br />
⇣<br />
n # i n# j + n" i n" j +2 n# i n" j<br />
⌘
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Result<br />
Fl<strong>at</strong>band Stoner magnetism preempts form<strong>at</strong>ion <strong>of</strong> fractional<br />
topological insul<strong>at</strong>or<br />
Z<br />
H = e2<br />
h<br />
d 2 k [n " (k) F " (k)+n # (k) F # (k)]<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
F " (k) = F # ( k)<br />
H =CI # +CI ⇤ " + U X i<br />
n # i n" i + V X (i,j)<br />
⇣<br />
n # i n# j + n" i n" j +2 n# i n" j<br />
⌘
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Result<br />
Fl<strong>at</strong>band Stoner magnetism preempts form<strong>at</strong>ion <strong>of</strong> fractional<br />
topological insul<strong>at</strong>or<br />
Z<br />
H = e2<br />
h<br />
in favor <strong>of</strong> fractional <strong>Chern</strong> insul<strong>at</strong>or<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
d 2 k [n " (k) F " (k)+n # (k) F # (k)]<br />
1/3 -1/3<br />
2/3 0<br />
F " (k) = F # ( k)<br />
H =CI # +CI ⇤ " + U X i<br />
n # i n" i + V X (i,j)<br />
⇣<br />
n # i n# j + n" i n" j +2 n# i n" j<br />
⌘
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Result<br />
Fl<strong>at</strong>band Stoner magnetism preempts form<strong>at</strong>ion <strong>of</strong> fractional<br />
topological insul<strong>at</strong>or<br />
Z<br />
H = e2<br />
h<br />
spin-isotropic<br />
interaction<br />
in favor <strong>of</strong> fractional <strong>Chern</strong> insul<strong>at</strong>or<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
d 2 k [n " (k) F " (k)+n # (k) F # (k)]<br />
1/3 -1/3<br />
2/3 0<br />
F " (k) = F # ( k)<br />
only intra-species<br />
repulsion<br />
H =CI # +CI ⇤ " + U X i<br />
n # i n" i + V X (i,j)<br />
⇣<br />
n # i n# j + n" i n" j +2 n# i n" j<br />
⌘
<strong>Fractional</strong> topological insul<strong>at</strong>or<br />
Result<br />
Fl<strong>at</strong>band Stoner magnetism preempts form<strong>at</strong>ion <strong>of</strong> fractional<br />
topological insul<strong>at</strong>or<br />
Z<br />
H = e2<br />
h<br />
spin-isotropic<br />
interaction<br />
in favor <strong>of</strong> fractional <strong>Chern</strong> insul<strong>at</strong>or<br />
T. Neupert et al., Phys. Rev. B 84, 165107 (2011)<br />
d 2 k [n " (k) F " (k)+n # (k) F # (k)]<br />
1/3 -1/3<br />
2/3 0<br />
TRS<br />
fractional TI<br />
F " (k) = F # ( k)<br />
spontaneous<br />
FCI<br />
only intra-species<br />
repulsion<br />
H =CI # +CI ⇤ " + U X i<br />
n # i n" i + V X (i,j)<br />
⇣<br />
n # i n# j + n" i n" j +2 n# i n" j<br />
⌘
Fl<strong>at</strong>band topological insul<strong>at</strong>ors: Towards a physical system<br />
...<br />
models for transition metal<br />
oxide heterostructures<br />
2D indium-phenylene<br />
organometallic framework<br />
D. Xiao et al., N<strong>at</strong>ure Comm. 2, 596 (2011)<br />
Zheng Liu et al., arXiv:1210.1826
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors: Why do we care?<br />
Wh<strong>at</strong> is <strong>at</strong> the heart <strong>of</strong> the FQHE?<br />
many-body gap ≈ U gives high-temper<strong>at</strong>ure effect<br />
<strong>at</strong> 1/2 filling: full magnetiz<strong>at</strong>ion gives quantum<br />
anomalous Hall effect.<br />
low dissip<strong>at</strong>ive conductor <strong>at</strong> room temper<strong>at</strong>ure<br />
T. Neupert et al., Phys. Rev. Lett. 108 046806 (2011)
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors: Why do we care?<br />
Wh<strong>at</strong> is <strong>at</strong> the heart <strong>of</strong> the FQHE?<br />
many-body gap ≈ U gives high-temper<strong>at</strong>ure effect<br />
<strong>at</strong> 1/2 filling: full magnetiz<strong>at</strong>ion gives quantum<br />
anomalous Hall effect.<br />
low dissip<strong>at</strong>ive conductor <strong>at</strong> room temper<strong>at</strong>ure<br />
⇢ xx ⇡ R K e<br />
/T<br />
gap <strong>of</strong> ~0.2eV comparable to copper per <strong>at</strong>omic layer<br />
gap <strong>of</strong> ~0.3eV already three orders <strong>of</strong> magnitude better<br />
T. Neupert et al., Phys. Rev. Lett. 108 046806 (2011)
Conclusions/Program<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors in <strong>Chern</strong> bands<br />
- numerics, zoo <strong>of</strong> st<strong>at</strong>es and models<br />
- well understood in close analogy to FQH st<strong>at</strong>es<br />
- wave function mapping<br />
- departing from FQHE in higher <strong>Chern</strong>-number bands<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors in Z2 topological insul<strong>at</strong>ors<br />
- more powerful numerical tools needed<br />
- Most likely condensed m<strong>at</strong>ter realiz<strong>at</strong>ion?<br />
3D fractionalized phases<br />
- advances in field theory<br />
- open problem for microscopic l<strong>at</strong>tice models
Conclusions/Program<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors in <strong>Chern</strong> bands<br />
- numerics, zoo <strong>of</strong> st<strong>at</strong>es and models<br />
- well understood in close analogy to FQH st<strong>at</strong>es<br />
- wave function mapping<br />
- departing from FQHE in higher <strong>Chern</strong>-number bands<br />
<strong>Fractional</strong> <strong>Chern</strong> insul<strong>at</strong>ors in Z2 topological insul<strong>at</strong>ors<br />
- more powerful numerical tools needed<br />
- Most likely condensed m<strong>at</strong>ter realiz<strong>at</strong>ion?<br />
3D fractionalized phases<br />
- advances in field theory<br />
- open problem for microscopic l<strong>at</strong>tice models<br />
Thank<br />
you!