Fractional Chern Insulators - Institute of Condensed Matter Theory at ...

Fractional Chern Insulators 

Titus Neupert 

Condensed Matter Theory Group 

Paul Scherrer Institut, Switzerland 

Christopher Mudry, PSI 

Claudio Chamon, Boston U 

Shinsei Ryu, UIUC 

Luiz Santos, Perimeter Institute 

Adolfo G. Grushin, CSIC Madrid

Outline 

1. 

Chern insulators 

Haldane’s model 

noncommutative geometry 

2. Fractional Chern insulators (FCI) 

C=1 exact diagonalization results 

C=2 

3. 

spontaneous formation of FCI in 

Z2 topological insulators

Quantum Hall effect: Landau levels 

energy scale ! c = eB m 

length scale `2B 

= ~c 

eB 

energy degeneracy N = A 

2⇡`2B 

= 

0 

B 

DOS




= ~c 

eB 


2⇡`2B 

= 

0 

B 

µ 

Integer quantum Hall effect 

xy = e2 

h n 

DOS




= ~c 

eB 


2⇡`2B 

= 

0 

B 

µ 

Fractional quantum Hall effect 

xy = e2 

h ⌫ 

DOS

B 

Are there other systems in which a 

quantum Hall effect appears? 

Can we dispose of the large magnetic field?

B 

Answer for integer 

quantum Hall effect: 

F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988) 



Can we dispose of the large magnetic field? 

Yes 

Yes

Citations: Haldane’s model 

fractional 


topological 

insulators 

F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988)

Chern insulator: Haldane’s model 

spinless fermions on honeycomb lattice 

complex NNN hoppings open mass gap 

in graphene spectrum 

“Chern insulator” 

xy = e2 

h C`B 

C = ±1 

! a






xy = e2 

h C`B 

C = ±1 

! a






xy = e2 

h C`B 

C = ±1 

! a






xy = e2 

h C`B 

C = ±1 

! a 

Just like a 

Landau 

level?

3 functions that are constant in the Landau level 

... but not in a generic Chern insulator 

1. 

Dispersion 

"(k) 

2. 

3. 

Quantum metric 

Berry curvature 

g µ⌫ (k) 

F µ⌫ (k)



1. 

Dispersion 

"(k) 

2. 

3. 



g µ⌫ (k) 

F µ⌫ (k) 

Berry connection A µ (k) :=hu(k)|@ µ |u(k)i 

F µ⌫ (k) =@ µ A ⌫ (k) @ ⌫ A µ (k) C = i Z 

2⇡ 

d 2 k F 12 (k)



1. 

Dispersion 

"(k) 

2. 

3. 



g µ⌫ (k) 

F µ⌫ (k) 

Berry connection A µ (k) :=hu(k)|@ µ |u(k)i 

F µ⌫ (k) =@ µ A ⌫ (k) @ ⌫ A µ (k) C = i Z 

2⇡ 

d 2 k F 12 (k) 

h (k)| b X µ 

b X⌫ | (k)i = g µ⌫ (k)+F µ⌫ (k) 

projected 

real, imaginary, 

position operator 

symmetric antisymmetric 

[ X b µ , X b ⌫ ](k) =2F µ⌫ (k) noncommutative geometry

Density algebra 

Landau level 

[b⇢ q , b⇢ q 0]= 

2i sin 

✓`2 

B 

2 q ^ q0 ◆ 

b⇢ q+q 0 

b⇢ q = e 

iq· bX 

S. M. Girvin et al., Phys. Rev. Lett. 54, 581 (1985) 

Chern insulator


Landau level 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

b⇢ q+q 0 

b⇢ q = e 

iq· bX 


Chern insulator 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

e q µg µ⌫ 

q 0 ⌫ b⇢q+q 0 

if F and g const. 

R. Roy, arXiv:1208.2055


Landau level 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

b⇢ q+q 0 

b⇢ q = e 

iq· bX 



✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

e q µg µ⌫ 

q 0 ⌫ b⇢q+q 0 


R. Roy, arXiv:1208.2055 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i 

b⇢ q+q 0 + O(q 3 ) 

if F const. 

S. Parameswaran et al., arXiv:1106.4025 

T. Neupert et al., Phys. Rev B 86, 035125 (2012) 

B. Estienne et al., arXiv:1202.5543


Landau level 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

b⇢ q+q 0 

b⇢ q = e 

iq· bX 



✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i sin 

e q µg µ⌫ 

q 0 ⌫ b⇢q+q 0 


R. Roy, arXiv:1208.2055 

✓`2 

B 

2 q ^ q0 ◆ 

[b⇢ q , b⇢ q 0]= 

2i 

b⇢ q+q 0 + O(q 3 ) 

if F const. 

Z 

S. Parameswaran et al., arXiv:1106.4025 

T. Neupert et al., Phys. Rev B 86, 035125 (2012) 

B. Estienne et al., arXiv:1202.5543 

[b⇢ q , b⇢ q 0]= 

q µ q 0 ⌫ 

d 2 k|u(k)iF µ⌫ (k)hu(k + q + q 0 )| + O(q 3 )

Outline 

1. 






C=2 

3. 



B 



Can we dispose of the large magnetic field?

B 

Answer for fractional 

quantum Hall effect: 

T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011) 



Can we dispose of the large magnetic field? 

Yes 

Yes

Fractional Chern insulator 

Need 


reasonably flat band with C≠0 

(requires some longer range hopping) 

add short-range repulsive interactions 

band gap 

band width 

interaction 

W 

U


Need 





band gap 

band width 

interaction 

W 

U 

with hierarchy of energy scales E. Tang et al., Phys. Rev. Lett., 106, 236802 (2011). 

U W 

K. Sun et al., Phys. Rev. Lett., 106, 236803 (2011). 

T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011)


Need 





band gap 

band width 

interaction 

W 

U 

with hierarchy of energy scales E. Tang et al., Phys. Rev. Lett., 106, 236802 (2011). 

U W 

K. Sun et al., Phys. Rev. Lett., 106, 236803 (2011). 

T. Neupert et al., Phys. Rev. Lett. 106 236804 (2011) 

Strongly correlated many-body problem, no perturbative 

approaches 

exact numerical diagonalization 

of small systems


Model 

square lattice, 

spin 1/2 d.o.f. per site 

it 1 + h 1 3 

h 4 3 it 2 + h 2 3 

m (ij) =( 1) i h 1 +( 1) j h 2 +( 1) i+j 2 h 4 

A. G. Grushin et al., arXiv:1207.4097


Model 

square lattice, 

spin 1/2 d.o.f. per site 

it 1 + h 1 3 

h 4 3 it 2 + h 2 3 

m (ij) =( 1) i h 1 +( 1) j h 2 +( 1) i+j 2 h 4 

C = 1 X 

( 1) i+j sgn m (ij) takes values 0, 1, 2 

2 

i,j=0,1 

repulsive density-density interactions, 

twisted boundary conditions 



C = 1 at 1/3 and 2/3 filling 

Q y 

4 x 6 lattice 

many-body 

c.o.m. BZ 

Q x 




Q y 


many-body 

c.o.m. BZ 

Q x 




5 







5 






Nature of incompressible states: FQH states 

quasi-degeneracy of topologically ordered states 

⌫ g 

featureless 

CDW in thin-torus limit 

A. G. Grushin et al., arXiv:1207.4097 

B. A. Bernevig et al., arXiv:1204.5682


Aside: Detecting broken translational symmetry 


quasi-degenerate ground states | ii, i =1, 

quasi-degeneracy of topologically ordered states ⌫ g ··· ,N 

matrix h i|⇢ r | ji has translational invariant 

featureless spectrum, but r dependent eigenvectors 

v r,i 

map out 



f r = v ⇤ r 0 ,ih i|⇢ r | jiv r0 ,j 

B. A. Bernevig et al., arXiv:1204.5682




⌫ g 




B. A. Bernevig et al., arXiv:1204.5682 

many-body Chern number (Hall conductivity) 1/3 and 2/5 

T. Neupert et al., Phys. Rev. B 86 165133 (2012)


Aside: Computing the many-body Chern number 


Thouless: Average over boundary conditions 

quasi-degeneracy of topologically ordered states ⌫ g 

| ( 1 , 2)i, ( 1 , 2) 2 [0, 2⇡] 2 


C = 

i 

2⇡ 

Z2⇡ 


0 

Z2⇡ 

d 1 

0 

d 2 

apple⌧ @ 

@ 1 

@ 

@ 2 

⌧ @ 


@ 2 



T. Neupert et al., Phys. Rev. B 86 165133 (2012) 

@ 

@ 1


Aside: Computing the many-body Chern number 


Thouless: Average over boundary conditions 

quasi-degeneracy of topologically ordered states ⌫ g 

| ( 1 , 2)i, ( 1 , 2) 2 [0, 2⇡] 2 


C = 

i 

2⇡ 

Z2⇡ 


0 

Z2⇡ 

d 1 

0 

d 2 

apple⌧ @ 

@ 1 

@ 

@ 2 

⌧ @ 


@ 2 


In thermodynamic limit, if 

many-body Chern - only number states in (Hall one band conductivity) occupied1/3 and 2/5 

- no spontaneous symmetry breaking 

one can show 

xy = e2 

h 

Z 

d 2 k n(k) F (k) 


@ 

@ 1 

n(k) :=h | 

† (k) (k)| i




⌫ g 







level counting in the entanglement spectrum 

mapping to Landau level Laughlin state 

with high overlap, continuous deformation 

B.A. Bernevig et al., Phys. Rev. B 85 075128 (2012) 

X.-L. Qi, Phys,Rev. Lett. 107, 126803 (2011) 

Y.-L. Wu, et al., Phys. Rev. B 86, 085129 (2012) 

Z. Liu et al., arXiv:1209.5310. 

Y.-L. Wu, et al., arXiv:1210.6356.


Abelian states compatible with hierarchical CS description 

non-Abelian states (Moore-Read, Zk Read-Rezayi) with 

3-body and 4-body interaction 

B.A. Bernevig et al., Phys. Rev. B 85, 075128 (2012) 

Y.-F. Wang et al., Phys Rev. Lett 108 126805 

Y.-L. Wu et al., Phys. Rev. B 85, 075116 (2012) 

series of states at higher Chern number ⌫ = 

k 

1+2C 

A.M. Läuchli arXiv:1207.6094 

A. Sterdyniak et al., arXiv:1207.6385

h4/t 

h4/t 

(E − E0)/U 

Fractional Chern insulator: Effect of bandwidth 

"(k) g µ⌫ (k) F µ⌫ (k) 

The flatter the better? 

some model parameter 

(a) 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

gap Δ/U 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

0.05 

0 

(b) 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

Δ/δ 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

10 

0 

spectrum at h4 = 1.05 

(c) ------------- 

- - - ------ 

0.07 

- - - 

------ -------- 

- - 

--- -- - - - 

-- 

----- --- - - - 

------ - - 

0.06 

- - - --- - ------ - - - 

0.05 - -------------- ------------ 

- 

- - 

- -- - ------ 0.04 

- 

- 

- - 

- 

0.03 

- Δ 

- 

- 

- 

0.02 

- 

- 

- 

- - 

0.01 

-- - 

- 

- - 

--- --- - - 

δ 

0.00 --- -------------- 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

bandwidth

h4/t 

h4/t 

(E − E0)/U 

Fractional Chern insulator: Effect of bandwidth 

"(k) g µ⌫ (k) F µ⌫ (k) 

The flatter the better? 

some model parameter 

(a) 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

gap Δ/U 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

0.05 

0 

(b) 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

Δ/δ 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

10 

0 

spectrum at h4 = 1.05 

(c) ------------- 

- - - ------ 

0.07 

- - - 

------ -------- 

- - 

--- -- - - - 

-- 

----- --- - - - 

------ - - 

0.06 

- - - --- - ------ - - - 

0.05 - -------------- ------------ 

- 

- - 

- -- - ------ 0.04 

- 

- 

- - 

- 

0.03 

- Δ 

- 

- 

- 

0.02 

- 

- 

- 

- - 

0.01 

-- - 

- 

- - 

--- --- - - 

δ 

0.00 --- -------------- 

0.0 0.2 0.4 0.6 0.8 1.0 

λ 

bandwidth 

Some dispersion can help to stabilize the state!

Fractional Chern insulator: Effect of quantum distance 

Projected interaction: 

X 

H := 

k 1 

,k 2 

,k 3 

,k 4 

V k1 k 2 

k 3 

k 4 

† 

k 1 

† 

k 2 

k 3 

k 4 

overlaps of singleparticle 

states 

V k1 k 2 

k 3 

k 4 

= v k1 k 3 

h k1 

| k 3 

ih k2 

| k 4 

i k1 +k 2 

,k 3 

+k 4



X 

H := 

k 1 

,k 2 

,k 3 

,k 4 

V k1 k 2 

k 3 

k 4 

† 

k 1 

† 

k 2 

k 3 

k 4 



V k1 k 2 

k 3 

k 4 

= v k1 k 3 

h k1 

| k 3 

ih k2 

| k 4 

i k1 +k 2 

,k 3 

+k 4 

q 

Quantum distance 

d(k, k 0 ):= 1 |h (k)| (k 0 )i| 

enters interaction 

Z r 

connects to 

d(k 

Fubini-Study metric 1 , k 2 )=inf 

1,2 

d` 

1,2 

g µ⌫ (k) dkµ 

d` 

dk ⌫ 

d`



X 

H := 

k 1 

,k 2 

,k 3 

,k 4 

V k1 k 2 

k 3 

k 4 

† 

k 1 

† 

k 2 

k 3 

k 4 



V k1 k 2 

k 3 

k 4 

= v k1 k 3 

h k1 

| k 3 

ih k2 

| k 4 

i k1 +k 2 

,k 3 

+k 4 

q 

Quantum distance 

d(k, k 0 ):= 1 |h (k)| (k 0 )i| 

enters interaction 

Z r 

connects to 

d(k 

Fubini-Study metric 1 , k 2 )=inf 

1,2 

d` 

1,2 

g µ⌫ (k) dkµ 

d` 

dk ⌫ 

d` 

Particle-hole transformation + normal ordering 

eH = 

X 

X 

† † 

V k1 , k 2 

, k 3 

, k 4 k 3 

k 4 

k 1 

k 2 

k 1 

,k 2 

,k 3 

,k 4 

k 

" k 

† 

k 

k + constant 

effective dispersion " k = X k 0 v k k 

0 + v k 

0 

k 

⇥ 

1 d 2 (k, k 0 ) ⇤ 2 

.

Outline 

1. 






C=2 

3. 



(2+1)D topological insulator 

topological band insulators in (2+1)D: Z IQHE (class A) 

Z2 TI (class AII) 

Gedanken experiment: filled Landau level with opposite 

chirality (magnetic field) for up-spin and down-spin electrons 

time-reversal symmetric 

with Sz conservation: quantum spin Hall effect 

B 

H =0 

sH = e 

2⇡ 

B 

Physical realization?

(2+1)D topological insulator 

topological band insulators in (2+1)D: Z IQHE (class A) 

Z2 TI (class AII) 

Gedanken experiment: filled Landau level with opposite 

chirality (magnetic field) for up-spin and down-spin electrons 

time-reversal symmetric 

with Sz conservation: quantum spin Hall effect 

B 

H =0 

sH = e 

2⇡ 

B 

Physical realization? 

Yes 

C. L. Kane et al., Phys. Rev. Lett. 95, 226801 & 146802 (2005). 

B. A. Bernevig et al., Science 314, 1757 (2006) 

M. König et al., Science 318, 766 (2007)

Fractional topological insulator 

Gedanken experiment: 

take two fractional quantum Hall layers with opposite field 

with Sz conservation 

B 

⌫ " =1/n 

⌫ # =1/n 

M. Levin et al., Phys. Rev. Lett. 103, 196803 (2009) 

B 

Is this there a physical setting for the fractional 

quantum spin Hall effect to appear?


Gedanken experiment: 

take two fractional quantum Hall layers with opposite field 

with Sz conservation 

B 

⌫ " =1/n 

⌫ # =1/n 

M. Levin et al., Phys. Rev. Lett. 103, 196803 (2009) 

B 

Is this there a physical setting for the fractional 

quantum spin Hall effect to appear? 

?, but ... 

T. Neupert et al., Phys. Rev. B 84, 165107 (2011)


Need 

Quantum spin Hall insulator (TI) 

reasonably flat nontrivial Kramers 

pair of band 


band gap 

band width 

interaction 

W 

U 

with hierarchy of energy scales 

T. Neupert et al., Phys. Rev. B 84, 165107 (2011) 

U 

W


Need 

Quantum spin Hall insulator (TI) 

reasonably flat nontrivial Kramers 

pair of band 


band gap 

band width 

interaction 

W 

U 

with hierarchy of energy scales 


U 

W 

Model 

H =CI # +CI ⇤ " + U X i 

n # i n" i + V X (i,j) 

⇣ 

n # i n# j + n" i n" j +2 n# i n" j 

⌘


Result 

Flatband Stoner magnetism preempts formation of fractional 

topological insulator 

Z 

H = e2 

h 

d 2 k [n " (k) F " (k)+n # (k) F # (k)] 


F " (k) = F # ( k) 

H =CI # +CI ⇤ " + U X i 

n # i n" i + V X (i,j) 

⇣ 

n # i n# j + n" i n" j +2 n# i n" j 

⌘


Result 



Z 

H = e2 

h 

in favor of fractional Chern insulator 


d 2 k [n " (k) F " (k)+n # (k) F # (k)] 

1/3 -1/3 

2/3 0 

F " (k) = F # ( k) 

H =CI # +CI ⇤ " + U X i 

n # i n" i + V X (i,j) 

⇣ 

n # i n# j + n" i n" j +2 n# i n" j 

⌘


Result 



Z 

H = e2 

h 

spin-isotropic 

interaction 



d 2 k [n " (k) F " (k)+n # (k) F # (k)] 

1/3 -1/3 

2/3 0 

F " (k) = F # ( k) 

only intra-species 

repulsion 

H =CI # +CI ⇤ " + U X i 

n # i n" i + V X (i,j) 

⇣ 

n # i n# j + n" i n" j +2 n# i n" j 

⌘


Result 



Z 

H = e2 

h 

spin-isotropic 

interaction 



d 2 k [n " (k) F " (k)+n # (k) F # (k)] 

1/3 -1/3 

2/3 0 

TRS 

fractional TI 

F " (k) = F # ( k) 

spontaneous 

FCI 

only intra-species 

repulsion 

H =CI # +CI ⇤ " + U X i 

n # i n" i + V X (i,j) 

⇣ 

n # i n# j + n" i n" j +2 n# i n" j 

⌘

Flatband topological insulators: Towards a physical system 

... 

models for transition metal 

oxide heterostructures 

2D indium-phenylene 

organometallic framework 

D. Xiao et al., Nature Comm. 2, 596 (2011) 

Zheng Liu et al., arXiv:1210.1826

Fractional Chern insulators: Why do we care? 

What is at the heart of the FQHE? 

many-body gap ≈ U gives high-temperature effect 

at 1/2 filling: full magnetization gives quantum 

anomalous Hall effect. 

low dissipative conductor at room temperature 


Fractional Chern insulators: Why do we care? 

What is at the heart of the FQHE? 

many-body gap ≈ U gives high-temperature effect 

at 1/2 filling: full magnetization gives quantum 

anomalous Hall effect. 

low dissipative conductor at room temperature 

⇢ xx ⇡ R K e 

/T 

gap of ~0.2eV comparable to copper per atomic layer 

gap of ~0.3eV already three orders of magnitude better 


Conclusions/Program 

Fractional Chern insulators in Chern bands 

- numerics, zoo of states and models 

- well understood in close analogy to FQH states 

- wave function mapping 

- departing from FQHE in higher Chern-number bands 

Fractional Chern insulators in Z2 topological insulators 

- more powerful numerical tools needed 

- Most likely condensed matter realization? 

3D fractionalized phases 

- advances in field theory 

- open problem for microscopic lattice models

Conclusions/Program 

Fractional Chern insulators in Chern bands 

- numerics, zoo of states and models 

- well understood in close analogy to FQH states 

- wave function mapping 

- departing from FQHE in higher Chern-number bands 

Fractional Chern insulators in Z2 topological insulators 

- more powerful numerical tools needed 

- Most likely condensed matter realization? 

3D fractionalized phases 

- advances in field theory 

- open problem for microscopic lattice models 

Thank 

you!

Fractional Chern Insulators - Institute of Condensed Matter Theory at ...

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