Fractional topological insulators

Fractional topological insulators 

Claudio Chamon 

Christopher Mudry - PSI 

Titus Neupert - PSI 

Shinsei Ryu - Berkeley 

Luiz Santos - Harvard 

PRL 106, 236804 (2011) 

PRB 84, 165107 (2011) 

PRB 84, 165138 (2011) 

PRL 108, 046806 (2012) 

arXiv:1202.5188 

support: DOE 

Thursday, March 22, 2012

Christopher Mudry 

Shinsei Ryu 

Luiz Santos 

Titus Neupert 


Fractional topological insulators 

Claudio Chamon 

Christopher Mudry - PSI 

Titus Neupert - PSI 

Shinsei Ryu - Berkeley 

Luiz Santos - Harvard 

PRL 106, 236804 (2011) 

PRB 84, 165107 (2011) 

PRB 84, 165138 (2011) 

PRL 108, 046806 (2012) 

arXiv:1202.5188 

support: DOE 


Landau flat bands 

Landau levels 

E 

ω c 

ω c 

ω c 

B 

degeneracy N φ = B A/φ 0 


Landau flat bands are interesting! 

Partially filled Landau levels 

fractional quantum Hall effect (FQHE) 

source: 

Willett et al, PRL 1987 


Are there other ways to get flat bands? 

Are they as interesting as Landau levels? 


Early flat bands 

D. Weire and M. F. Thorpe, PRB 4, 2508 (1971) 

J. P. Straley, PRB 6, 4086 (1972) 


More early flat bands E. Dagotto, E. Fradkin, and A. Moreo, Phys. Lett. B 172, 383 (1986). 

Volume 172, number 3,4 

PHYSICS L 

PHYSICS LETTERS B 22 May 1986 

a 

I 

~E 

C2 

C-I 

= C- 2 tx~, 

Fig. 2. A two-dimensional example showing the position of 

the variables used in eq. (1). 

Of the generic form in 

sites. U 1 represents an on-site coupling while U 2 

couples the variables belonging to the same line. i and 

J. P. Straley, PRB 6, 4086 (1972) 

j take the values -+1, +2, +3. A related hamiltonian 

was originally proposed in ref. [8] in the context of 

topological disorder in a tetrahedrally bonded solid. 

(A similar model was analyzed in ref. [9] in writing 

the classical path integral version of two-dimensional 

Ising model). It can be shown that by choosing a special 

ratio between U 1 and U2, there is a band crossing 

at zero momentum. By explicit diagonalization in mo- 

Thursday, March mentum 22, 2012space 

or using the method presented in ref. 

I 

P,: py=o 

b 

Motivation was to get Weyl fermions on the 

lattice, with a single cone 

E 

Z 

I ~-

Possible flat band energies in the Kagome model 

D. Green, L. Santos, and C. Chamon, PRB 2010 

E = −2g 

E =+2g 

E =0 

Flat very bottom band 

Flat very top band 

φ ± =2π n ± 

φ ± = π (2n ± + 1) 

Flat middle band φ + + φ − = π (mod 2π) 


Flat band as a “critical point” 

φ + = φ − =3(π/2 − ) 

< 0 =0 > 0 


Can go on and on... 

Parallel flat bands- honeycomb lattice w/ 3 flavors 









Up to that point, had examples of flat bands with 

zero Chern number 


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

Can one flatten bands with a Chern number? 

YES! 

T. Neupert, L. Santos, C. Chamon, and C. Mudry, PRL 2011 

E. Tang, J. W. Mei, and X. G. Wen, PRL 2011 

K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, PRL 2011 


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 


i=1 

⎛ 

3 

⎝ 

t ⎞ 



⎠ , 


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 √

B i 

c) 

ε/t 1 

Quantum Hall effect w/o Landau levels 2 

ples of Hamiltonians 

2 

t 2 /t 1 = 1 

of the the form (1a) are 

ing. Example 1: The honeycomb lattice. We 

√ 

the vectors a t 1 =(0, −1), 2 

a t 2 = √ 3/2, 1/2 , 

3/2, 1/2 0 connecting NN and the vectors b t 1 = 

= a t 3−a t 1, b t 3 = a t 1−a t 2 connecting NNN from 

omb lattice δ depicted − /∆ ≈ 1/5 in Fig. 1(a). We denote 

ave vector 

− 4π from the BZ of the reciprocal lattice 

e triangular lattice kspanned x 

by 0 b 1 and b 2 ,say. 4π c) ε/t 

0 

1 

l is then defined by the Bloch Hamiltonian [1] 

d) 

ε/t 1 

2 

0 

3 √ 3 

3 √ − 2π 3 3 

B 0,k := 2t 2 cos Φ 

B k := 

3 


i=1 

⎛ 

3 

⎝ 

t ⎞ 



⎠ , 


i=1 

(2a) 

(2b) 

≥ 0–2and t 2 ≥ 0 are NN and NNN hopitudes, 

respectively, and the real numbers ±Φ 

2 

agnetic fluxes penetrating the two halves of 

0 

onal unit − πcell. For k t 

x 

1 t0 2 , the gap ∆ ≡ 

π 

max 

− π 

k 

ε −,k 

is proportional to t 2 . The width 

–2 

0 

er band is δ − ≡ max k 

ε −,k 


. The 

tio 

Thursday, 

δ − /∆ is extremal for the choice cos Φ = 

March 22, 2012 

− π 

a) b) 

. 1. (Color online) (a) Unit cell of Haldane’s model on the 

d) 

2 

0 

k y 

k y 

2π 

3 

π 

σ xy = −1 

− 4π 

k x 

4π 

3 √ 3 

3 √ − 2π 3 3 

ε/t 1 

k x 

A i 

B i 

A i 

0 0 

σ xy =+1 

0 0 

π − π 

k y 

k y 

B i 

2π 

3 

π

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 ⎞ 


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 


. 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


Is there a FQHE when flat bands with 

non-zero Chern number are partially fixed?

Add interactions 2 

b) 

A i 

B i 

A i 

B i 

ε/t 1 

2 

H int := 1 2 

 

i,j 

ρ i V i,j ρ j ≡ V ij 

ρ i ρ j , V > 0 

0 

− 4π 

2π 

k x 0 0 k 

3 √ 3 − 2π 3 


y 

4π √

Chern insulator: exact diagonalization 

inverse compressibility 

6x4 la/ce 

1/3 

filling 

2/3 

filling 



6x4 la/ce 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10 

Q 

9.9 

10.0 

10.1 

10.2 

10.3 

10.4 

10.5 

E 

3x4 la/ce 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10 12 14 

Q 

12.3 

12.4 

12.5 

12.6 

12.7 

12.8 

E 

3x5 la/ce 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 5 10 15 

Q 

14.8 

14.9 

15.0 

15.1 

15.2 

E 

3x6 la/ce 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 5 10 15 20 

Q 

19.7 

19.8 

19.9 

20.0 

E 


!& 

"#( 

"#"$ 

"#"$ 



" 

3 × 6 plaquettes (36 sites) 

" "#$ %& ' 

%()(* ' ( + 

!"! ! 

"#( 

- 

%& ' 

%(,(* ' 

*& 

+(.(" +(.(( 

# $ 

)- " - 

)- 

!"! ! 

FIG. 2. (Color Nonline) (a) The lowest eigenvalues of H flat 

GS =3 N GS =1 0 + 

H int for the chiral π-flux phase obtained from exact diagonalization 

tfor 2 6 C particles = 1 on a 3 × 6sublatticeA μ C = 0 (1/3 filling), 

normalized by the bandwidth E b . The parameters t 2 

and µ s of H0 flat interpolate between topological (|t 2 | > |µ s |) 

and nontopological (|t 2 | < |µ s |) single-particle bands. Here, 

"#( 

"#"$ 

# $ 

bands o 

ing ma 

interact 

that a 

1/3 filli 

by Lau 

in futur 

We 

Storni, 

work w 

06ER46 

Theory 

port. 

Note 

19] in w 

cussed. 

onaliza 

[1] F.

Is there a fractional Hall effect? 

Other recent works 

• D. N. Sheng, Z. Gu, K. Sun, and L. Sheng, Nature Comm. 2, 389 (2011) 

• N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 (2011) 

• Yang-Le Wu; B. A. Bernevig, N. Regnault, arXiv:1111.1172 

• S. Parameswaran, R. Roy, and S. Sondhi, arXiv:1106.4025 

• B. A. Bernevig, N. Regnault arXiv:1110.4488 

• M. Goerbig, Eur. Phys. J. B 85(1), 15 (2012) 

• R. Shankar and G. Murthy arXiv:1108.5501 








What about time-reversal symmetric systems? 

T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry 

PRB 84, 165107 (2011) 

PRB 84, 165138 (2011) 

PRL 108, 046806 (2011) 


Spin quantum Hall effect - two FQHE layers 

S = S ↑ CS + S↓ CS 

Doubled model with opposite FQHE for each spin species 

B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006). 

M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803 (2009). 


Time-reversal symmetric Abelian fractional liquids 

T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry PRB (2011) 

“ ” 

ν = 0 

FQHE 

constraints from TRS 

S ≡ 1 

4π 

K = 

κ 

∆ T 

 

Abelian Chern-Simons action 

dt d 2 x µνρ K ij a i µ ∂ ν a j ρ 

∆ 

−κ 

κ T = κ, 

∆ T = −∆ 

degeneracy on torus 

for this subclass 

# GS = 

= 

κ 

det 

∆ 

T 

Pf 

κ 

∆ T 

∆ = 

∆ 

−κ det T 

−κ 

κ ∆ 

2 

−κ 

= [integer] 2 

∆ 


Lattice realization of time-reversal symmetric 

fractional topological liquids 

2 

T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry PRB (2011); PRL (2011). 

i 

b) 2 copies of flatband models, 

with opposite chirality for 

flattened Kane+Mele model 

spins 

A i 

B i 

i 

Add interactions 

H int := U i∈Λ 

ρ i,↑ ρ i,↓ + V 

 

ij∈Λ 

 

ρi,↑ ρ j,↑ + ρ i,↓ ρ j,↓ +2λρ i,↑ ρ j,↓ 

 


Topological insulator: 1/2 filling 

1/2 filling of lower bands, 

4x3 la/ce, 

12 parCcles 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 




4x3 la/ce, 

12 parCcles 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

flat band 

ferromagneCsm 




4x3 la/ce, 

12 parCcles 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

flat band 

ferromagneCsm 

t 2 

μ 




4x3 la/ce, 

12 parCcles 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

flat band 

ferromagneCsm 

t 2 

μ 




4x3 la/ce, 

12 parCcles 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

flat band 

ferromagneCsm 

Spontaneous 

QHE 

σ xy = e2 

h 

t 2 

μ 



Restoring bandwidth: 

full polarizaCon stable up to W ≈ 0.7 U 




! 

% 

$./ 

!" 

'()*+, 

-()*+, 

$ %$ &$ 

B 0,k := 0, 

$ 

$ % & 

B 

' 1,k +iB 2, 

!"# 

#" 

2" ," 

!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"% 

B 3,k := 2t 

H 2 

int := U ρ i,↑ ρ i,↓ + V 

 


i∈Λ 

ij∈Λ 

and the three Pau 

sublattice index. 

neighbor (NN) an 

ping amplitudes. 

$"# 

$.& 

$.% 

& 

lattice momentum 

α = A, B and c 

ψ † k,σ := c † k,A,σ , 

single-particle Ha 

H 0 := 

k∈BZ 

ψ † k 

where the 3-vecto

= 0 

γ x 

3-fold 

9-fold 

0 10 20 

Topological insulator: 2/3 0filling 

1 2 3 U/V 

c) a) 

d) 

U/V = 0, λ = 1 U/V = 3, λ = 1 

1 

λ 

0.5 

2ϖ 

γ x 

! 

% 

$./ 

!" 

3-fold 

9-fold 

0 ϖ 2ϖ 0 ϖ γ 2ϖ 

x 

'()*+, 

-()*+, 

0 

0 1 2 3 U/V 

b) 

c) 

E/V 

0.2 

0.1 

0 

U/V = 0, λ = 0 

0 ϖ 

γ x 


2ϖ 

U/V = 0, λ = 1 

d) 

U/V = 3, λ = 1 

$ %$ &$ 

B 0,k := 0, 

$ 

$ % & ' !"# 

B 1,k +iB 2, 

#" 

2" ," 

!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"% 

B 3,k := 2t 

H 2 

int := U ρ i,↑ ρ i,↓ + V 

 


i∈Λ 

ij∈Λ 



γ x 

2ϖ 0 ϖ 


ping amplitudes. 

$"# 

$.& 

$.% 

∋ 

0 10 20 

E/V 

0 ϖ γ x 

2ϖ 

∋ 

0.2 

0.1 

0 

3-fold 

9-fold 

0 10 20 

0 1 2 3 U/V 

b) 

c) 

U/V = 0, λ = 0 

0 ϖ 

γ x 

2ϖ 

d) 

U/V = 0, λ = 1 U/V = 3, λ = 1 

& 



ψ † k,σ := c † k,A,σ , 


H 0 := 

0 ϖ γ 2ϖ 0 ϖ 

x 

γ 2ϖ 

x 

k∈BZ 

ψ † k 

where the 3-vecto 

∋

λ 

1 

0.5 


a) 

! 

% 

$./ 

$"# 

$.& 

$.% 

!" 

3-fold 

9-fold 

0 10 20 

'()*+, 

-()*+, 

0 

0 1 2 3 U/V 

b) 

c) 

E/V 

0.2 

0.1 

0 

U/V = 0, λ = 0 

0 ϖ 

γ x 


2ϖ 

∋ 

& 



ψ † k,σ := c † k,A,σ , 


H 0 := 

k∈BZ 

ψ † k 

where the 3-vecto 

U/V = 0, λ = 1 

d) 

U/V = 3, λ = 1$ %$ &$ 

B 0,k := 0, 

$ 

$ % & ' !"# 

B 1,k +iB 2, 

#" 

!"#()($01!"#"$ 

2" 

!"#()($01!"#"% 

," 

!"#()('01!"#"% 

lattice realization of a ∆ =0 

B 3,k := 2t 

(Bernevig+Zhang+Levin+Stern) 2 

TRS fractional topological insulator 



0 ϖ γ x 

2ϖ 0 ϖ γ x 

2ϖ 


ping amplitudes.

Topological insulator: 2/3 filling a) 

λ 

1 


! 

% 

$./ 

!" 

'()*+, 

-()*+, 

$ %$ &$ 

$ 

$ % & ' !"# 

#" 

2" ," 

!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"% 

$"# 

↑ ↑0.5 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ 3-fold ↓ ↓ ↓ ↓ ↓ ↓ 

9-fold 

0 10 20 

& 


α = A, B and co 

0 

0 1 2 3 U/V 

b) 

c) 

d) 

E/V 

0.2 

0.1 

0 

U/V = 0, λ = 0 

0 ϖ 

γ x 

2ϖ 

ψ † k,σ := c † k,A,σ ,c 

U/V = 0, λ = 1 U/V = 3, λ = 1 

γ x 

single-particle Ham 

H 0 := 

k∈BZ 

ψ † k 

where the 3-vector 

B 0,k := 0, 

0 ϖ 2ϖ 0 ϖ γ 2ϖ 

x 

B 1,k +iB 2,k 

B 

∋ 

:= 2t

Topological insulator: 2/3 filling a) 

spontaneous TRS breaking 


spontaneous FQHE 

! 

% 

$./ 

!" 

'()*+, 

-()*+, 

$ %$ &$ 

$ 

$ % & ' !"# 

#" 

2" ," 

!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"% 

$"# 

λ 

1 

↑ ↑0.5 

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ 

↓ ↓ ↓ ↓ ↓ ↓ 3-fold ↓ ↓ ↓ ↓ ↓ ↓ 

9-fold 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ν ↓ =1 

0 10 20 

& 


α = A, B and co 

0 

0 1 2 3 U/V 

b) 

c) 

d) 

E/V 

0.2 

0.1 

0 

U/V = 0, λ = 0 

0 ϖ 

γ x 

2ϖ 

γ x 

ψ † k,σ := c † k,A,σ ,c 

U/V = 0, λ = 1 U/V = 3, λ = 1 

single-particle Ham 

H 0 := 

k∈BZ 

ψ † k 

where the 3-vector 

B 0,k := 0, 

0 ϖ 2ϖ 0 ϖ γ 2ϖ 

x 

B 1,k +iB 2,k 

B 

ν ↑ =1/3 

∋ 

:= 2t

% 

!" 

! 

Perspective - bulk vs. edge 

$./ 

'()*+, 

-()*+, 

$ %$ &$ 

$ 

$ % & ' !"# 

#" 

2" 

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$ ' % &' $ ' 

& 

% &' 

& 

FIG. 1. (color online). Numerical exact diagonalization results 

of Hamiltonian (2.1) for 16 electrons when sublattice A 

is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state 

degeneracies. Denote with E n the n-th lowest energy eigenvalue 

of the many-body spectrum where E 1 is the many-body 

ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter 

by n := (E n+1 − E n )/(E n − E 1 ). If a large gap 

& 

lattice momentum k and spin σ =↑, ↓ in t 

α = A, B and combine them in the subl 

ψ † k,σ 

c := † k,A,σ ,c† k,B,σ 

. Then, the secon 

single-particle Hamiltonian reads 

H 0 := 

B k · τ 

|B k 

| ψ k,↑ + ψ † B −k · 

 

k,↓ B − 

ψ † k,↑ 

k∈BZ 

Edge modes? 

where the 3-vector B k 

is defined by 

M. Levin and A. Stern, PRL (2009) 

B 0,k := 0, 

T. Neupert et al. PRB (2011) 

K = 

B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y − 

⎛ 

+1 +2 0 0 

⎜+2 +1 0 0 

⎝ 

+ t 1 e +iπ/4 e −ik x + e 

B 3,k := 2t 2 

 

cos kx − cos k y 

 

, 

and the three 0 Pauli-matrices 0 −1 −2τ =(τ 1 , τ 2 , τ 3 

sublattice 

0index. 0Here, −2t 1 −1 

and t 2 represent 

neighbor (NN) and next-nearest neighbor 

ping amplitudes. The Hamiltonian (2.2a) 

defined if t 2 = 0. 

One verifies that H 0 is both time-reversa 

and invariant under U(1) spin-1/2 rotations 

time-reversal operation T acts on numbers 

conjugation and on the electron-operators a 

ψ † k,↑ 

T 

−→ +ψ † −k,↓ , 

ψ† k,↓ 

T 

−→ −ψ † −k 

The action of the U(1) spin-1/2 rotation R γ 

0 ≤ γ < 2π is given by 

ψ † k,↑ 

R γ 

−→ e +iγ ψ † k,↑ , 

⎞ 

⎟ 

⎠ 

ψ† k,↓ 

R γ 

−→ e −iγ ψ † k

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& 



ψ † k,σ 

c := † k,A,σ ,c† k,B,σ 



H 0 := 

B k · τ 

|B k 

| ψ k,↑ + ψ † B −k · 

 

k,↓ B − 

ψ † k,↑ 

k∈BZ 

Edge modes? 


is defined by 


B 0,k := 0, 



+ t 1 e +iπ/4 e −ik x + e 

B 3,k := 2t 2 

 


 

, 

and the three Pauli-matrices τ =(τ 1 , τ 2 , τ 3 

sublattice index. Here, t 1 and t 2 represent 








ψ † k,↑ 

T 

−→ +ψ † −k,↓ , 

ψ† k,↓ 

T 

−→ −ψ † −k 



ψ † k,↑ 

NO 

R γ 

−→ e +iγ ψ † k,↑ , 

ψ† k,↓ 

R γ 

−→ e −iγ ψ † k

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edge poor % &' $ ' 

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No propagating edge modes, but... 








& 



ψ † k,σ 

c := † k,A,σ ,c† k,B,σ 



H 0 := 

B k · τ 

|B k 

| ψ k,↑ + ψ † B −k · 

 

k,↓ B − 

ψ † k,↑ 

k∈BZ 

Edge modes? 


is defined by 


B 0,k := 0, 



+ t 1 e +iπ/4 e −ik x + e 

B 3,k := 2t 2 

 


 

, 










ψ † k,↑ 

T 

−→ +ψ † −k,↓ , 

ψ† k,↓ 

T 

−→ −ψ † −k 



ψ † k,↑ 

NO 

R γ 

−→ e +iγ ψ † k,↑ , 

ψ† k,↓ 

R γ 

−→ e −iγ ψ † k

% 

!" 

! 


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More structure than captured by a 








& 

Z 2 



ψ † k,σ 

c := † k,A,σ ,c† k,B,σ 



H 0 := 

B k · τ 

|B k 

| ψ k,↑ + ψ † B −k · 

 

k,↓ B − 

ψ † k,↑ 

k∈BZ 

Edge modes? 


is defined by 


B 0,k := 0, 



+ t 1 e +iπ/4 e −ik x + e 

B 3,k := 2t 2 

 


 

, 










ψ † T 

classification 

k,↑ 

−→ +ψ alone. 

† −k,↓ , 

ψ† k,↓ 

T 

−→ −ψ † −k 



ψ † k,↑ 

NO 

R γ 

−→ e +iγ ψ † k,↑ , 

ψ† k,↓ 

R γ 

−→ e −iγ ψ † k

% 

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! 


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& 

% &' 

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More structure than captured by a 








& 

Z 2 



ψ † k,σ 

c := † k,A,σ ,c† k,B,σ 



H 0 := 

B k · τ 

|B k 

| ψ k,↑ + ψ † B −k · 

 

k,↓ B − 

ψ † k,↑ 

k∈BZ 

Edge modes? 


is defined by 


B 0,k := 0, 



+ t 1 e +iπ/4 e −ik x + e 

B 3,k := 2t 2 

 


 

, 










ψ † T 

classification 

k,↑ 

−→ +ψ alone. 

† −k,↓ , 

“Either you are with us, or you are with the terrorists." - G.W. Bush 

ψ† k,↓ 

T 

−→ −ψ † −k 



ψ † k,↑ 

NO 

R γ 

−→ e +iγ ψ † k,↑ , 

ψ† k,↓ 

R γ 

−→ e −iγ ψ † k

3D 

T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry arXiv:1202.5188 


Classification of non-interacting states of matter 

“Periodic Table of Topological Insulators” 

Symmetry 

d 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 

d 

E Conduction Band 

2 

Conventional 

!=0 

Insulator 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

(a) 

(b) 

E F 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 

Quantum spin 

1 Z 0 Z 2 Valence ZBand 

2 Z 0 0 0 

C 0 

"#/a 

0 k "#/a 

−1 0 0 Z 0 Z 2 Z 

FIG. 5 Edge states in the quantum spin Hall insulator. (a) 2 Z 0 0 

CI 1 

shows the interface between a QSHI and an ordinary insulator, 

and 1(b) shows 0the edge0state dispersion Z in0the graphene Z 2 −1 Z 2 Z 0 

Symmetry 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0 

C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0 

CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0 

TABLE I Periodic (2001) table corresponds of totopological the d = 4 entry in class insulators A AII. and superconductors. 

The have 10yet symmetry to be filled by realistic classes systems. are The search labeled is using the 

There are also other entries in physical dimensions that 

on to discover 

notation of Altland andNew such 

Zirnbauer 

J. phases. Phys. 12, 

(1997) 

065010 

(AZ) 

(2010) 

and are specified 

by presence or absence of T symmetry Θ, particle-hole 

III. QUANTUM SPIN HALL INSULATOR 

symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes 

TABLE I Periodic table of topological insulators and superconductors. 

The 10 symmetry classes are labeled using the 

notation of Altland and Zirnbauer (1997) (AZ) and are specified 



the presence and absence of symmetry, with ±1 specifying 

the value of Θ 2 and Ξ 2 . As a function of symmetry and space 

dimensionality, d, thetopologicalclassifications(Z, Z 2 and 0) 

show a regular pattern that repeats when d → d +8. 


model, in which up and down spins propagate in opposite 

directions. 

M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) 

Ryu, S., A. Schnyder, A. Furusaki, A. W. W. Ludwig, 

Kitaev, A., 2009, AIP Conf. Proc. 1134, 22; arXiv:0901.2686 

The 2D topological insulator is known as a quantum 

Conven 

Insulato 

(a) 

Quantum 

Hall insu 

FIG. 5 Edge 

shows the int 

tor, and (b) 

model, in wh 

directions. 

(2001) corre 

There are a 

have yet to 

on to discov



Symmetry 

IQHE d 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 

d 


2 

Conventional 

!=0 

Insulator 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

(a) 

(b) 

E F 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 

Quantum spin 


2 Z 0 0 0 

C 0 

"#/a 

0 k "#/a 

−1 0 0 Z 0 Z 2 Z 


CI 1 



Symmetry 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0 

C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0 

CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0 






Zirnbauer 


(1997) 

065010 

(AZ) 

(2010) 
















directions. 





Conven 

Insulato 

(a) 

Quantum 

Hall insu 

FIG. 5 Edge 

shows the int 

tor, and (b) 

model, in wh 

directions. 

(2001) corre 

There are a 

have yet to 

on to discov



Symmetry 

IQHE d 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 

d 


2 

Conventional 

!=0 

Insulator 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

(a) 

(b) 

E F 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 

Quantum spin 


2 Z 0 0 0 

C 0 

"#/a 

0 k "#/a 

−1 0 0 Z 0TI 

Z 2 Z 


CI 1 



Symmetry 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0 

C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0 

CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0 






Zirnbauer 


(1997) 

065010 

(AZ) 

(2010) 
















directions. 





Conven 

Insulato 

(a) 

Quantum 

Hall insu 

FIG. 5 Edge 

shows the int 

tor, and (b) 

model, in wh 

directions. 

(2001) corre 

There are a 

have yet to 

on to discov



Symmetry 

IQHE d 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

chiral 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 

d 


2 

Conventional 

!=0 

Insulator 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

(a) 

(b) 

E F 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 

Quantum spin 


2 Z 0 0 0 

C 0 

"#/a 

0 k "#/a 

−1 0 0 Z 0TI 

Z 2 Z 


CI 1 



Symmetry 

AZ Θ Ξ Π 1 2 3 4 5 6 7 8 

A 0 0 0 0 Z 0 Z 0 Z 0 Z 

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0 

AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z 

BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2 

D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2 

DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0 

AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z 

CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0 

C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0 

CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0 






Zirnbauer 


(1997) 

065010 

(AZ) 

(2010) 
















directions. 





Conven 

Insulato 

(a) 

Quantum 

Hall insu 

FIG. 5 Edge 

shows the int 

tor, and (b) 

model, in wh 

directions. 

(2001) corre 

There are a 

have yet to 

on to discov

3D Lattice model 

3 orbital model on the cubic lattice 

3 

H k = λ 3+i sin k i + λ 7 

M − 

i=1 

3 

i=1 

cos k i 

 

Gell-Mann matrices 

flat band 

a) M = 0 

3 

2 

1 

0 

1 

2 

3 

4 

2 

0 

2 

4 

X M R X R M 

X M R X R M 

X M R X R M 

X M R X R M 

b) M = 1 

X M R X R M 

4 

2 

0 

2 

4 

X M R X R M 

c) M = 2 d) M = 3 

6 

4 

2 

0 

2 

4 

6 

X M R X R M 

X M R X R M 




3 

H k = λ 3+i sin k i + λ 7 

M − 

i=1 

3 

i=1 

cos k i 

 

Gell-Mann matrices 

flat band 

 

02×2 q 

H k = k 

q † k 

0 

chiral class (AIII) 




3 

H k = λ 3+i sin k i + λ 7 

M − 

i=1 

3 

i=1 

cos k i 

 

θ(M) = 

⎧ 

⎪⎨ +2π, |M| < 1, 

−π, 1 < |M| < 3, 

⎪⎩ 

0, 3 < |M|. 


Non-commuting projected position operators 

2D: Projection onto Landau level 

X = P n 

R Pn 

= R − 2 B 

e 3 × Π 

 

X1 , X 2 

 

=+i 2 B 




X = P n 

R Pn 

= R − 2 B 

e 3 × Π 

 

X1 , X 2 

 

=+i 2 B 

X 2 

∆X 1 

∆X 2 

X 1 




X = P n 

R Pn 

= R − 2 B 

e 3 × Π 

 

X1 , X 2 

 

=+i 2 B 

X 2 

∆X 1 

∆X 2 

non-commuting coordinates 

“fuzziness” 

X 1 



2D: Coordinate transformation 

r µ → f µ (r) 

 

X1 , X 2 

 

=+i 2 B 




r µ → f µ (r) 

 

X1 , X 2 

 

=+i 2 B 

 

f 1 ( X),f 2 ( X) 

=+i 2 B {f 1 ,f 2 } P ( X)+O 4 

B 




r µ → f µ (r) 

 

X1 , X 2 

 

=+i 2 B 

 

f 1 ( X),f 2 ( X) 

=+i 2 B {f 1 ,f 2 } P ( X)+O 4 

B 

Poisson bracket 

 

{f 1 ,f 2 } P 

(r) = µν ∂f1 ∂f 2 

∂r µ ∂r ν 




r µ → f µ (r) 

 

X1 , X 2 

 

=+i 2 B 

 

f 1 ( X),f 2 ( X) 

=+i 2 B {f 1 ,f 2 } P ( X)+O 4 

B 

Poisson bracket 

 

{f 1 ,f 2 } P 

(r) = µν ∂f1 ∂f 2 

∂r µ ∂r ν 

transformation preserves area 

⇒ 

commutators are unchanged 



3D: what would be a “pristine” extension? 

 

X1 , X 2 , X 3 

 

=+i 3 

Nambu (quantum) 3-bracket 

[ A 1 , A 2 , A 3 ]= ijk 

Ai Aj Ak 




r µ → f µ (r) 

 

f 1 ( X),f 2 ( X),f 2 ( X) 

 

X1 , X 2 , X 3 

 

=+i 3 


[ A 1 , A 2 , A 3 ]= ijk 

Ai Aj Ak 

=i 3 {f 1 ,f 2 ,f 3 } N ( X)+O( 5 ) 

 

{f 1 ,f 2 ,f 3 } N 

(r) = µνλ ∂f1 ∂f 2 ∂f 3 

(r) 

∂r µ ∂r ν ∂r λ 

Nambu (classical) 3-bracket 




r µ → f µ (r) 

 

f 1 ( X),f 2 ( X),f 2 ( X) 

 

X1 , X 2 , X 3 

 

=+i 3 


[ A 1 , A 2 , A 3 ]= ijk 

Ai Aj Ak 

=i 3 {f 1 ,f 2 ,f 3 } N ( X)+O( 5 ) 

 

{f 1 ,f 2 ,f 3 } N 

(r) = µνλ ∂f1 ∂f 2 ∂f 3 

(r) 

∂r µ ∂r ν ∂r λ 

Nambu (classical) 3-bracket 

transformation preserves volume 

⇒ 

3-brackets are unchanged 



3D: non-pristine 

 

X1 , X 2 , X 3 

 

=+i 3 + ... 

X µ = 

 

d d k χ † ã (k) iD ã ˜b 

µ (k) χ˜b(k) 

D µ (k) =∂ µ + A µ (k) 

when 

Tr D µ ,D ν 

 

= V 

 

d d k tr F µν (k) =0 

 

: 

 

Xµ , X ν , X ρ 

 

 

: = − iTr 

D µ ,D ν ,D ρ 




 

: 

 


 

 

: = − iTr 


A. P. Polychronakos (2007) 




 

: 

 


 

 

: = − iTr 


1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

= 12π2 i 

¯ρ 


CS (3) 




 

: 

 


 

 

: = − iTr 


1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

= 12π2 i 

¯ρ 


CS (3) 

¯ρ = N p 

V 

CS (3) = π 2 

 

d 3 k 

(2π) 3 

ijk tr 

A i F jk − 2 3 A iA j A k 

 




 

: 

 


 

 

: = − iTr 


1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

= 12π2 i 

¯ρ 


CS (3) 

1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

=+i 3 



3D: fuzzy coordinates 

1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

=+i 3 




1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

=+i 3 

X 2 

X 1 




1 

 

: 

N p 

 

X1 , X 2 , X 3 

 

 

: 

=+i 3 

X 2 

X 3 

X 1 


Summary 

It is possible to have dispersionless bands (flat bands) that are isolated from all 

others by an energy gap. 

The isolated flat bands can be non-trivial in the sense of having a non-zero 

Chern number, and thus sustain an IQHE when fully occupied. 

When partially filled, electron-electron interactions can lead to the FQHE and 

other topological phases in time-reversal invariant systems. 

Topological Hubbard models have to ferromagnetic ground states: symmetry 

breaking simultaneously with IQHE or FQHE. 

Fractional TI in 3D: non-commuting coordinates

Fractional topological insulators

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