Fractional topological insulators
Fractional topological insulators 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
- Page 2 and 3: Christopher Mudry Shinsei Ryu Luiz
- Page 4 and 5: Landau flat bands Landau levels E
- Page 6 and 7: Are there other ways to get flat ba
- Page 8 and 9: More early flat bands E. Dagotto, E
- Page 10 and 11: Flat band as a “critical point”
- Page 12 and 13: Are there other ways to get flat ba
- Page 14 and 15: Are they as interesting as Landau l
- Page 16 and 17: Can one flatten bands with a Chern
- Page 18 and 19: B i c) ε/t 1 Quantum Hall effect w
- Page 20 and 21: Thursday, March 22, 2012 Is there a
- Page 22 and 23: Chern insulator: exact diagonalizat
- Page 24 and 25: !& "#( "#"$ "#"$ Thursday, March 22
- Page 26 and 27: Thursday, March 22, 2012 Is there a
- Page 28 and 29: What about time-reversal symmetric
- Page 30 and 31: Time-reversal symmetric Abelian fra
- Page 32 and 33: Topological insulator: 1/2 filling
- Page 34 and 35: Topological insulator: 1/2 filling
- Page 36 and 37: Topological insulator: 1/2 filling
- Page 38 and 39: Topological insulator: 2/3 filling
- Page 40 and 41: λ 1 0.5 Topological insulator: 2/3
- Page 42 and 43: Topological insulator: 2/3 filling
- Page 44 and 45: % !" ! Perspective - bulk vs. edge
- Page 46 and 47: % !" ! Perspective - bulk vs. edge
- Page 48 and 49: 3D T. Neupert, L. Santos, S. Ryu C.
- Page 50 and 51: Classification of non-interacting s
<strong>Fractional</strong> <strong>topological</strong> <strong>insulators</strong><br />
Claudio Chamon<br />
Christopher Mudry - PSI<br />
Titus Neupert - PSI <br />
Shinsei Ryu - Berkeley<br />
Luiz Santos - Harvard<br />
PRL 106, 236804 (2011)<br />
PRB 84, 165107 (2011)<br />
PRB 84, 165138 (2011)<br />
PRL 108, 046806 (2012)<br />
arXiv:1202.5188<br />
support: DOE<br />
Thursday, March 22, 2012
Christopher Mudry<br />
Shinsei Ryu<br />
Luiz Santos<br />
Titus Neupert <br />
Thursday, March 22, 2012
<strong>Fractional</strong> <strong>topological</strong> <strong>insulators</strong><br />
Claudio Chamon<br />
Christopher Mudry - PSI<br />
Titus Neupert - PSI <br />
Shinsei Ryu - Berkeley<br />
Luiz Santos - Harvard<br />
PRL 106, 236804 (2011)<br />
PRB 84, 165107 (2011)<br />
PRB 84, 165138 (2011)<br />
PRL 108, 046806 (2012)<br />
arXiv:1202.5188<br />
support: DOE<br />
Thursday, March 22, 2012
Landau flat bands<br />
Landau levels<br />
E<br />
ω c<br />
ω c<br />
ω c<br />
B<br />
degeneracy N φ = B A/φ 0<br />
Thursday, March 22, 2012
Landau flat bands are interesting!<br />
Partially filled Landau levels<br />
fractional quantum Hall effect (FQHE)<br />
source:<br />
Willett et al, PRL 1987<br />
Thursday, March 22, 2012
Are there other ways to get flat bands?<br />
Are they as interesting as Landau levels?<br />
Thursday, March 22, 2012
Early flat bands<br />
D. Weire and M. F. Thorpe, PRB 4, 2508 (1971)<br />
J. P. Straley, PRB 6, 4086 (1972)<br />
Thursday, March 22, 2012
More early flat bands E. Dagotto, E. Fradkin, and A. Moreo, Phys. Lett. B 172, 383 (1986).<br />
Volume 172, number 3,4<br />
PHYSICS L<br />
PHYSICS LETTERS B 22 May 1986<br />
a<br />
I<br />
~E<br />
C2<br />
C-I<br />
= C- 2 tx~,<br />
Fig. 2. A two-dimensional example showing the position of<br />
the variables used in eq. (1).<br />
Of the generic form in<br />
sites. U 1 represents an on-site coupling while U 2<br />
couples the variables belonging to the same line. i and<br />
J. P. Straley, PRB 6, 4086 (1972)<br />
j take the values -+1, +2, +3. A related hamiltonian<br />
was originally proposed in ref. [8] in the context of<br />
<strong>topological</strong> disorder in a tetrahedrally bonded solid.<br />
(A similar model was analyzed in ref. [9] in writing<br />
the classical path integral version of two-dimensional<br />
Ising model). It can be shown that by choosing a special<br />
ratio between U 1 and U2, there is a band crossing<br />
at zero momentum. By explicit diagonalization in mo-<br />
Thursday, March mentum 22, 2012space<br />
or using the method presented in ref.<br />
I<br />
P,: py=o<br />
b<br />
Motivation was to get Weyl fermions on the<br />
lattice, with a single cone<br />
E<br />
Z<br />
I ~-
Possible flat band energies in the Kagome model<br />
D. Green, L. Santos, and C. Chamon, PRB 2010<br />
E = −2g<br />
E =+2g<br />
E =0<br />
Flat very bottom band<br />
Flat very top band<br />
φ ± =2π n ±<br />
φ ± = π (2n ± + 1)<br />
Flat middle band φ + + φ − = π (mod 2π)<br />
Thursday, March 22, 2012
Flat band as a “critical point”<br />
φ + = φ − =3(π/2 − )<br />
< 0 =0 > 0<br />
Thursday, March 22, 2012
Can go on and on...<br />
Parallel flat bands- honeycomb lattice w/ 3 flavors<br />
Thursday, March 22, 2012
Are there other ways to get flat bands?<br />
Are they as interesting as Landau levels?<br />
Thursday, March 22, 2012
Are there other ways to get flat bands?<br />
Are they as interesting as Landau levels?<br />
Thursday, March 22, 2012
Are they as interesting as Landau levels?<br />
Up to that point, had examples of flat bands with<br />
zero Chern number<br />
Thursday, March 22, 2012
Quantum Hall effect w/o Landau levels<br />
wo examples of Hamiltonians of the the form (1a) are<br />
he following. F. D. M. Haldane, Example Phys. 1: Rev. The Lett. honeycomb 61, 2015 lattice. (1988). We<br />
troduce the vectors a t 1 =(0, −1), a t 2 = √ 3/2, 1/2 ,<br />
t<br />
3 = − √ 3/2, 1/2 connecting NN and the vectors b t 1 =<br />
t<br />
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<br />
he honeycomb lattice depicted in Fig. 1(a). We denote<br />
ith k a wave vector from the BZ of the reciprocal lattice<br />
ual to the triangular lattice spanned by b 1 and b 2 ,say.<br />
he model is then defined by the Bloch Hamiltonian [1]<br />
B 0,k := 2t 2 cos Φ<br />
B k :=<br />
3<br />
cos k · b i ,<br />
i=1<br />
⎛<br />
3<br />
⎝<br />
t ⎞<br />
1 cos k · a i<br />
t 1 sin k · a i<br />
⎠ ,<br />
−2t 2 sin Φ sin k · b i<br />
i=1<br />
σ xy = −1<br />
(2a)<br />
σ xy =+1<br />
(2b)<br />
here t 1 ≥ 0 and t 2 ≥ 0 are NN and NNN hoping<br />
amplitudes, respectively, and the real numbers ±Φ<br />
re the magnetic fluxes penetrating the two halves of<br />
he Thursday, hexagonal March 22, 2012 unit cell. For t 1 t 2 , the gap ∆ ≡<br />
a) b)<br />
c)<br />
d)<br />
ε/t 1<br />
2<br />
0<br />
ε/t 1<br />
2<br />
0<br />
− 4π<br />
3 √ 3<br />
t 2<br />
k x<br />
0<br />
A i<br />
B i
Can one flatten bands with a Chern number?<br />
YES!<br />
T. Neupert, L. Santos, C. Chamon, and C. Mudry, PRL 2011<br />
E. Tang, J. W. Mei, and X. G. Wen, PRL 2011<br />
K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, PRL 2011<br />
Thursday, March 22, 2012
a) b)<br />
Quantum Hall effect w/o Landau levels<br />
A i<br />
c)<br />
d)<br />
t 2 /t 1 =<br />
ε/t 1<br />
2<br />
0<br />
ε/t 1<br />
√<br />
43<br />
Two examples of Hamiltonians of the the form (1a)<br />
12 √ 3 ≈ 0.315495 cos Φ = 1 are<br />
t 1<br />
the following. Example 1: The honeycomb lattice. We<br />
introduce the vectors a A t<br />
1 =(0, −1), a t 2 = √ 3/2, 41/2 i t 2 ,<br />
a t 3 = − √ 3/2, 1/2 connecting B i<br />
NN and the vectors b t 1 =<br />
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<br />
the honeycomb lattice depicted in Fig. 1(a). We denote<br />
with k a wave vector from the BZ of the reciprocal lattice<br />
dual to the triangular lattice spanned by b 1 and b 2 ,say.<br />
The model is then defined by the Bloch Hamiltonian [1]<br />
δ − /∆ =1/7<br />
B 0,k := 2t 2 cos Φ<br />
B k :=<br />
3<br />
cos k · b i ,<br />
i=1<br />
⎛<br />
3<br />
⎝<br />
t ⎞<br />
1 cos k · a i<br />
t 1 sin k · a i<br />
⎠ ,<br />
−2t 2 sin Φ sin k · b i<br />
i=1<br />
(2a)<br />
(2b)<br />
where t 1 ≥ 0 and t 2 ≥ 0 are NN and NNN hopping<br />
amplitudes, respectively, and the real numbers ±Φ<br />
are<br />
− 4π<br />
k<br />
3 √ the magnetic fluxes penetrating the two halves<br />
3 x<br />
3 √ − 2π of<br />
the hexagonal unit cell. For t 1 t 2 , the gap ∆ ≡<br />
min k ε +,k<br />
− max k<br />
ε −,k<br />
is proportional to t 2<br />
3. The3width<br />
of the lower band is δ − ≡ max k<br />
ε −,k<br />
− min k ε −,k<br />
. The<br />
flatness ratio δ − /∆ is extremal for the choice cos Φ =<br />
<br />
Thursday, March 22, 2012<br />
0 4π<br />
0<br />
a) b)<br />
c)<br />
k<br />
0<br />
y<br />
–2<br />
B i<br />
ε/t 1<br />
2<br />
0<br />
σ xy = −1<br />
d)<br />
σ xy =+1<br />
ε/t 1<br />
2<br />
− 4π<br />
3 √ 3<br />
− π<br />
2π<br />
3<br />
k x<br />
k x<br />
0<br />
0<br />
A i<br />
B i<br />
4π<br />
3 √
B i<br />
c)<br />
ε/t 1<br />
Quantum Hall effect w/o Landau levels 2<br />
ples of Hamiltonians<br />
2<br />
t 2 /t 1 = 1<br />
of the the form (1a) are<br />
ing. Example 1: The honeycomb lattice. We<br />
√<br />
the vectors a t 1 =(0, −1), 2<br />
a t 2 = √ 3/2, 1/2 ,<br />
3/2, 1/2 0 connecting NN and the vectors b t 1 =<br />
= a t 3−a t 1, b t 3 = a t 1−a t 2 connecting NNN from<br />
omb lattice δ depicted − /∆ ≈ 1/5 in Fig. 1(a). We denote<br />
ave vector<br />
− 4π from the BZ of the reciprocal lattice<br />
e triangular lattice kspanned x<br />
by 0 b 1 and b 2 ,say. 4π c) ε/t<br />
0<br />
1<br />
l is then defined by the Bloch Hamiltonian [1]<br />
d)<br />
ε/t 1<br />
2<br />
0<br />
3 √ 3<br />
3 √ − 2π 3 3<br />
B 0,k := 2t 2 cos Φ<br />
B k :=<br />
3<br />
cos k · b i ,<br />
i=1<br />
⎛<br />
3<br />
⎝<br />
t ⎞<br />
1 cos k · a i<br />
t 1 sin k · a i<br />
⎠ ,<br />
−2t 2 sin Φ sin k · b i<br />
i=1<br />
(2a)<br />
(2b)<br />
≥ 0–2and t 2 ≥ 0 are NN and NNN hopitudes,<br />
respectively, and the real numbers ±Φ<br />
2<br />
agnetic fluxes penetrating the two halves of<br />
0<br />
onal unit − πcell. For k t<br />
x<br />
1 t0 2 , the gap ∆ ≡<br />
π<br />
max<br />
− π<br />
k<br />
ε −,k<br />
is proportional to t 2 . The width<br />
–2<br />
0<br />
er band is δ − ≡ max k<br />
ε −,k<br />
− min k ε −,k<br />
. The<br />
tio<br />
Thursday,<br />
δ − /∆ is extremal for the choice cos Φ =<br />
March 22, 2012<br />
− π<br />
a) b)<br />
. 1. (Color online) (a) Unit cell of Haldane’s model on the<br />
d)<br />
2<br />
0<br />
k y<br />
k y<br />
2π<br />
3<br />
π<br />
σ xy = −1<br />
− 4π<br />
k x<br />
4π<br />
3 √ 3<br />
3 √ − 2π 3 3<br />
ε/t 1<br />
k x<br />
A i<br />
B i<br />
A i<br />
0 0<br />
σ xy =+1<br />
0 0<br />
π − π<br />
k y<br />
k y<br />
B i<br />
2π<br />
3<br />
π
c)<br />
ε/t 1<br />
2<br />
0<br />
− 4π<br />
k x<br />
4π<br />
3 √ 3<br />
3 √ − 2π 3 3<br />
Can one get perfectly flat bands?<br />
0 0<br />
les d) of Hamiltonians of the the form (1a) are<br />
ε/t 1<br />
g. Example 1: The honeycomb lattice. We<br />
he vectors 2 a t 1 =(0, −1), a t 2 = √ 3/2, 1/2 ,<br />
/2, 1/2 0 connecting NN and the vectors b t 1 =<br />
= a t 3−a –2<br />
t 1, b t 3 = a t 1−a t 2 connecting NNN from<br />
omb lattice depicted in Fig. 1(a). We denote<br />
− π<br />
ve vector fromk x the BZ 0 of the reciprocal 0 lattice k y<br />
triangular lattice spanned by b 1 and b 2 ,say.<br />
is then defined by the Bloch Hamiltonian [1]<br />
FIG. 1. (Color online) (a) Unit cell of Haldane’s model on the<br />
honeycomb lattice: The ψ † kNN H kψ hopping k , amplitudes t 1 are real<br />
(solid lines) and the NNN hopping amplitudes are t 2 e i2πΦ/Φ 0<br />
k∈BZ 3<br />
in the direction of the arrow (dotted lines). The flux 3Φ<br />
Band 0,k −Φ<br />
:=<br />
penetrate<br />
2t 2 cos Φthe dark<br />
cosshaded k · b i ,<br />
region and each<br />
(2a)<br />
of the<br />
light shaded regions, i=1 respectively. For Φ = π/3, the model<br />
⎛<br />
is gauge equivalent to having one flux quantum per unit cell.<br />
(b) The chiral-π-flux<br />
3 on the square lattice, where the unit cell<br />
corresponds to Hthe k flat<br />
shaded := H k<br />
B k := ⎝<br />
t ⎞<br />
1 cos k · a i<br />
t 1 sinarea. k · aThe NN hopping amplitudes<br />
are t 1<br />
|ε i<br />
⎠ , (2b)<br />
e iπ/4 in the direction of −,k<br />
|<br />
i=1 −2t the arrow (solid lines) and the<br />
2 sin Φ k · b i<br />
NNN hopping amplitudes are t 2 and −t 2 along the dashed and<br />
≥dotted 0 and<br />
lines, respectively.<br />
t<br />
(c) The band structure of Haldane’s<br />
2 ≥ 0 are<br />
model for cos Φ =1/(4t 2 )=3 NN and NNN hoptudes,<br />
3/43 with the flatness ratio<br />
1/7. √ (d)<br />
respectively,<br />
The band structure<br />
and<br />
of<br />
the<br />
thereal chiral-π-flux<br />
numbers<br />
for<br />
±Φ<br />
t 1 /t 2 =<br />
gnetic 2 with the fluxes flatness penetrating ratio 1/5. The the lower two bands halves can be of made<br />
exactly flat by adding longer range hoppings.<br />
H 0 = <br />
To enhance the effect of interactions, highly degenerate<br />
(i.e., flat) bands are desirable. It is always possible to<br />
π<br />
− π<br />
k y<br />
2π<br />
3<br />
π<br />
H k = B k · σ<br />
nal unit cell. For t 1 t 2 , the gap ∆ ≡<br />
max k<br />
ε −,k<br />
is proportional to t 2 . The width<br />
r band is δ − ≡ max k<br />
ε −,k<br />
− min k ε −,k<br />
. The<br />
io Thursday, δ − /∆ is extremal for the choice cos Φ =<br />
March 22, 2012<br />
a) b)<br />
c)<br />
ε/t 1<br />
2<br />
0<br />
ψ † k = c † k,A ,c† k,B<br />
⇒ In real space, hoppings decay exponentially with distance<br />
d)<br />
− 4π<br />
k x<br />
4π<br />
3 √ 3<br />
3 √ − 2π 3 3<br />
ε/t 1<br />
2<br />
0<br />
–2<br />
− π<br />
k x<br />
A i<br />
B i<br />
A i<br />
0 0<br />
0 0<br />
π − π<br />
k y<br />
k y<br />
B i<br />
<br />
2π<br />
3<br />
π<br />
2
Thursday, March 22, 2012<br />
Is there a FQHE when flat bands with<br />
non-zero Chern number are partially fixed?
Add interactions 2<br />
b)<br />
A i<br />
B i<br />
A i<br />
B i<br />
ε/t 1<br />
2<br />
H int := 1 2<br />
<br />
i,j<br />
ρ i V i,j ρ j ≡ V ij<br />
ρ i ρ j , V > 0<br />
0<br />
− 4π<br />
2π<br />
k x 0 0 k<br />
3 √ 3 − 2π 3<br />
Thursday, March 22, 2012<br />
y<br />
4π √
Chern insulator: exact diagonalization<br />
inverse compressibility<br />
6x4 la/ce<br />
1/3 <br />
filling<br />
2/3 <br />
filling<br />
Thursday, March 22, 2012
Chern insulator: exact diagonalization<br />
6x4 la/ce<br />
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0 2 4 6 8 10<br />
Q<br />
9.9<br />
10.0<br />
10.1<br />
10.2<br />
10.3<br />
10.4<br />
10.5<br />
E<br />
3x4 la/ce<br />
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0 2 4 6 8 10 12 14<br />
Q<br />
12.3<br />
12.4<br />
12.5<br />
12.6<br />
12.7<br />
12.8<br />
E<br />
3x5 la/ce<br />
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0 5 10 15<br />
Q<br />
14.8<br />
14.9<br />
15.0<br />
15.1<br />
15.2<br />
E<br />
3x6 la/ce<br />
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0 5 10 15 20<br />
Q<br />
19.7<br />
19.8<br />
19.9<br />
20.0<br />
E<br />
Thursday, March 22, 2012
!&<br />
"#(<br />
"#"$<br />
"#"$<br />
Thursday, March 22, 2012<br />
Chern insulator: exact diagonalization<br />
"<br />
3 × 6 plaquettes (36 sites)<br />
" "#$ %& '<br />
%()(* ' ( +<br />
!"! !<br />
"#(<br />
-<br />
%& '<br />
%(,(* '<br />
*&<br />
+(.(" +(.((<br />
# $<br />
)- " -<br />
)-<br />
!"! !<br />
FIG. 2. (Color Nonline) (a) The lowest eigenvalues of H flat<br />
GS =3 N GS =1 0 +<br />
H int for the chiral π-flux phase obtained from exact diagonalization<br />
tfor 2 6 C particles = 1 on a 3 × 6sublatticeA μ C = 0 (1/3 filling),<br />
normalized by the bandwidth E b . The parameters t 2<br />
and µ s of H0 flat interpolate between <strong>topological</strong> (|t 2 | > |µ s |)<br />
and non<strong>topological</strong> (|t 2 | < |µ s |) single-particle bands. Here,<br />
"#(<br />
"#"$<br />
# $<br />
bands o<br />
ing ma<br />
interact<br />
that a<br />
1/3 filli<br />
by Lau<br />
in futur<br />
We<br />
Storni,<br />
work w<br />
06ER46<br />
Theory<br />
port.<br />
Note<br />
19] in w<br />
cussed.<br />
onaliza<br />
[1] F.
Is there a fractional Hall effect?<br />
Other recent works<br />
• D. N. Sheng, Z. Gu, K. Sun, and L. Sheng, Nature Comm. 2, 389 (2011)<br />
• N. Regnault and B. A. Bernevig, Phys. Rev. X 1, 021014 (2011)<br />
• Yang-Le Wu; B. A. Bernevig, N. Regnault, arXiv:1111.1172<br />
• S. Parameswaran, R. Roy, and S. Sondhi, arXiv:1106.4025<br />
• B. A. Bernevig, N. Regnault arXiv:1110.4488<br />
• M. Goerbig, Eur. Phys. J. B 85(1), 15 (2012)<br />
• R. Shankar and G. Murthy arXiv:1108.5501<br />
Thursday, March 22, 2012
Thursday, March 22, 2012<br />
Is there a FQHE when flat bands with<br />
non-zero Chern number are partially fixed?
Thursday, March 22, 2012<br />
Is there a FQHE when flat bands with<br />
non-zero Chern number are partially fixed?
What about time-reversal symmetric systems?<br />
T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry<br />
PRB 84, 165107 (2011)<br />
PRB 84, 165138 (2011)<br />
PRL 108, 046806 (2011)<br />
Thursday, March 22, 2012
Spin quantum Hall effect - two FQHE layers<br />
S = S ↑ CS + S↓ CS<br />
Doubled model with opposite FQHE for each spin species<br />
B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, 106802 (2006).<br />
M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803 (2009).<br />
Thursday, March 22, 2012
Time-reversal symmetric Abelian fractional liquids<br />
T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry PRB (2011)<br />
“ ”<br />
ν = 0<br />
FQHE<br />
constraints from TRS<br />
S ≡ 1<br />
4π<br />
K =<br />
κ<br />
∆ T<br />
<br />
Abelian Chern-Simons action<br />
dt d 2 x µνρ K ij a i µ ∂ ν a j ρ<br />
∆<br />
−κ<br />
κ T = κ,<br />
∆ T = −∆<br />
degeneracy on torus<br />
for this subclass<br />
# GS =<br />
=<br />
κ<br />
det <br />
∆<br />
T<br />
Pf<br />
κ<br />
∆ T<br />
∆ =<br />
∆<br />
−κ det T<br />
−κ <br />
κ ∆<br />
2<br />
−κ<br />
= [integer] 2<br />
∆<br />
Thursday, March 22, 2012
Lattice realization of time-reversal symmetric<br />
fractional <strong>topological</strong> liquids<br />
2<br />
T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry PRB (2011); PRL (2011).<br />
i<br />
b) 2 copies of flatband models,<br />
with opposite chirality for<br />
flattened Kane+Mele model<br />
spins<br />
A i<br />
B i<br />
i<br />
Add interactions<br />
H int := U i∈Λ<br />
ρ i,↑ ρ i,↓ + V<br />
<br />
ij∈Λ<br />
<br />
ρi,↑ ρ j,↑ + ρ i,↓ ρ j,↓ +2λρ i,↑ ρ j,↓<br />
<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
1/2 filling of lower bands,<br />
4x3 la/ce, <br />
12 parCcles<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
1/2 filling of lower bands,<br />
4x3 la/ce, <br />
12 parCcles<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
flat band <br />
ferromagneCsm<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
1/2 filling of lower bands,<br />
4x3 la/ce, <br />
12 parCcles<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
flat band <br />
ferromagneCsm<br />
t 2<br />
μ<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
1/2 filling of lower bands,<br />
4x3 la/ce, <br />
12 parCcles<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
flat band <br />
ferromagneCsm<br />
t 2<br />
μ<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
1/2 filling of lower bands,<br />
4x3 la/ce, <br />
12 parCcles<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓<br />
flat band <br />
ferromagneCsm<br />
Spontaneous <br />
QHE<br />
σ xy = e2<br />
h<br />
t 2<br />
μ<br />
Thursday, March 22, 2012
Topological insulator: 1/2 filling<br />
Restoring bandwidth: <br />
full polarizaCon stable up to W ≈ 0.7 U<br />
Thursday, March 22, 2012
Topological insulator: 2/3 filling<br />
Thursday, March 22, 2012<br />
!<br />
%<br />
$./<br />
!"<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
B 0,k := 0,<br />
$<br />
$ % &<br />
B<br />
' 1,k +iB 2,<br />
!"#<br />
#"<br />
2" ,"<br />
!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"%<br />
B 3,k := 2t<br />
H 2<br />
int := U ρ i,↑ ρ i,↓ + V<br />
<br />
ρi,↑ ρ j,↑ + ρ i,↓ ρ j,↓ +2λρ i,↑ ρ j,↓<br />
i∈Λ<br />
ij∈Λ<br />
and the three Pau<br />
sublattice index.<br />
neighbor (NN) an<br />
ping amplitudes.<br />
$"#<br />
$.&<br />
$.%<br />
&<br />
lattice momentum<br />
α = A, B and c<br />
ψ † k,σ := c † k,A,σ ,<br />
single-particle Ha<br />
H 0 := <br />
k∈BZ<br />
ψ † k<br />
where the 3-vecto
= 0<br />
γ x<br />
3-fold<br />
9-fold<br />
0 10 20<br />
Topological insulator: 2/3 0filling<br />
1 2 3 U/V<br />
c) a)<br />
d)<br />
U/V = 0, λ = 1 U/V = 3, λ = 1<br />
1<br />
λ<br />
0.5<br />
2ϖ<br />
γ x<br />
!<br />
%<br />
$./<br />
!"<br />
3-fold<br />
9-fold<br />
0 ϖ 2ϖ 0 ϖ γ 2ϖ<br />
x<br />
'()*+,<br />
-()*+,<br />
0<br />
0 1 2 3 U/V<br />
b)<br />
c)<br />
E/V<br />
0.2<br />
0.1<br />
0<br />
U/V = 0, λ = 0<br />
0 ϖ<br />
γ x<br />
Thursday, March 22, 2012<br />
2ϖ<br />
U/V = 0, λ = 1<br />
d)<br />
U/V = 3, λ = 1<br />
$ %$ &$<br />
B 0,k := 0,<br />
$<br />
$ % & ' !"#<br />
B 1,k +iB 2,<br />
#"<br />
2" ,"<br />
!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"%<br />
B 3,k := 2t<br />
H 2<br />
int := U ρ i,↑ ρ i,↓ + V<br />
<br />
ρi,↑ ρ j,↑ + ρ i,↓ ρ j,↓ +2λρ i,↑ ρ j,↓<br />
i∈Λ<br />
ij∈Λ<br />
and the three Pau<br />
sublattice index.<br />
γ x<br />
2ϖ 0 ϖ<br />
neighbor (NN) an<br />
ping amplitudes.<br />
$"#<br />
$.&<br />
$.%<br />
∋<br />
0 10 20<br />
E/V<br />
0 ϖ γ x<br />
2ϖ<br />
∋<br />
0.2<br />
0.1<br />
0<br />
3-fold<br />
9-fold<br />
0 10 20<br />
0 1 2 3 U/V<br />
b)<br />
c)<br />
U/V = 0, λ = 0<br />
0 ϖ<br />
γ x<br />
2ϖ<br />
d)<br />
U/V = 0, λ = 1 U/V = 3, λ = 1<br />
&<br />
lattice momentum<br />
α = A, B and c<br />
ψ † k,σ := c † k,A,σ ,<br />
single-particle Ha<br />
H 0 := <br />
0 ϖ γ 2ϖ 0 ϖ<br />
x<br />
γ 2ϖ<br />
x<br />
k∈BZ<br />
ψ † k<br />
where the 3-vecto<br />
∋
λ<br />
1<br />
0.5<br />
Topological insulator: 2/3 filling<br />
a)<br />
!<br />
%<br />
$./<br />
$"#<br />
$.&<br />
$.%<br />
!"<br />
3-fold<br />
9-fold<br />
0 10 20<br />
'()*+,<br />
-()*+,<br />
0<br />
0 1 2 3 U/V<br />
b)<br />
c)<br />
E/V<br />
0.2<br />
0.1<br />
0<br />
U/V = 0, λ = 0<br />
0 ϖ<br />
γ x<br />
Thursday, March 22, 2012<br />
2ϖ<br />
∋<br />
&<br />
lattice momentum<br />
α = A, B and c<br />
ψ † k,σ := c † k,A,σ ,<br />
single-particle Ha<br />
H 0 := <br />
k∈BZ<br />
ψ † k<br />
where the 3-vecto<br />
U/V = 0, λ = 1<br />
d)<br />
U/V = 3, λ = 1$ %$ &$<br />
B 0,k := 0,<br />
$<br />
$ % & ' !"#<br />
B 1,k +iB 2,<br />
#"<br />
!"#()($01!"#"$<br />
2"<br />
!"#()($01!"#"%<br />
,"<br />
!"#()('01!"#"%<br />
lattice realization of a ∆ =0<br />
B 3,k := 2t<br />
(Bernevig+Zhang+Levin+Stern) 2<br />
TRS fractional <strong>topological</strong> insulator<br />
and the three Pau<br />
sublattice index.<br />
0 ϖ γ x<br />
2ϖ 0 ϖ γ x<br />
2ϖ<br />
neighbor (NN) an<br />
ping amplitudes.
Topological insulator: 2/3 filling a)<br />
λ<br />
1<br />
Thursday, March 22, 2012<br />
!<br />
%<br />
$./<br />
!"<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2" ,"<br />
!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"%<br />
$"#<br />
↑ ↑0.5<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ 3-fold ↓ ↓ ↓ ↓ ↓ ↓<br />
9-fold<br />
0 10 20<br />
&<br />
lattice momentum<br />
α = A, B and co<br />
0<br />
0 1 2 3 U/V<br />
b)<br />
c)<br />
d)<br />
E/V<br />
0.2<br />
0.1<br />
0<br />
U/V = 0, λ = 0<br />
0 ϖ<br />
γ x<br />
2ϖ<br />
ψ † k,σ := c † k,A,σ ,c<br />
U/V = 0, λ = 1 U/V = 3, λ = 1<br />
γ x<br />
single-particle Ham<br />
H 0 := <br />
k∈BZ<br />
ψ † k<br />
where the 3-vector<br />
B 0,k := 0,<br />
0 ϖ 2ϖ 0 ϖ γ 2ϖ<br />
x<br />
B 1,k +iB 2,k<br />
B<br />
∋<br />
:= 2t
Topological insulator: 2/3 filling a)<br />
spontaneous TRS breaking<br />
Thursday, March 22, 2012<br />
spontaneous FQHE<br />
!<br />
%<br />
$./<br />
!"<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2" ,"<br />
!"#()($01!"#"$ !"#()($01!"#"% !"#()('01!"#"%<br />
$"#<br />
λ<br />
1<br />
↑ ↑0.5<br />
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑<br />
↓ ↓ ↓ ↓ ↓ ↓ 3-fold ↓ ↓ ↓ ↓ ↓ ↓<br />
9-fold<br />
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ν ↓ =1<br />
0 10 20<br />
&<br />
lattice momentum<br />
α = A, B and co<br />
0<br />
0 1 2 3 U/V<br />
b)<br />
c)<br />
d)<br />
E/V<br />
0.2<br />
0.1<br />
0<br />
U/V = 0, λ = 0<br />
0 ϖ<br />
γ x<br />
2ϖ<br />
γ x<br />
ψ † k,σ := c † k,A,σ ,c<br />
U/V = 0, λ = 1 U/V = 3, λ = 1<br />
single-particle Ham<br />
H 0 := <br />
k∈BZ<br />
ψ † k<br />
where the 3-vector<br />
B 0,k := 0,<br />
0 ϖ 2ϖ 0 ϖ γ 2ϖ<br />
x<br />
B 1,k +iB 2,k<br />
B<br />
ν ↑ =1/3<br />
∋<br />
:= 2t
%<br />
!"<br />
!<br />
Perspective - bulk vs. edge<br />
$./<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2"<br />
$"#<br />
$.&<br />
$.%<br />
$<br />
!"#()($01!"#"$<br />
$ '<br />
Thursday, March 22, 2012<br />
% &<br />
&'<br />
,"<br />
!"#()($01!"#"% !"#()('01!"#"%<br />
$ ' % &' $ '<br />
&<br />
% &'<br />
&<br />
FIG. 1. (color online). Numerical exact diagonalization results<br />
of Hamiltonian (2.1) for 16 electrons when sublattice A<br />
is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state<br />
degeneracies. Denote with E n the n-th lowest energy eigenvalue<br />
of the many-body spectrum where E 1 is the many-body<br />
ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter<br />
by n := (E n+1 − E n )/(E n − E 1 ). If a large gap<br />
&<br />
lattice momentum k and spin σ =↑, ↓ in t<br />
α = A, B and combine them in the subl<br />
ψ † k,σ<br />
c := † k,A,σ ,c† k,B,σ<br />
. Then, the secon<br />
single-particle Hamiltonian reads<br />
H 0 := <br />
B k · τ<br />
|B k<br />
| ψ k,↑ + ψ † B −k ·<br />
<br />
k,↓ B −<br />
ψ † k,↑<br />
k∈BZ<br />
Edge modes?<br />
where the 3-vector B k<br />
is defined by<br />
M. Levin and A. Stern, PRL (2009)<br />
B 0,k := 0,<br />
T. Neupert et al. PRB (2011)<br />
K =<br />
B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y −<br />
⎛<br />
+1 +2 0 0<br />
⎜+2 +1 0 0<br />
⎝<br />
+ t 1 e +iπ/4 e −ik x + e<br />
B 3,k := 2t 2<br />
<br />
cos kx − cos k y<br />
<br />
,<br />
and the three 0 Pauli-matrices 0 −1 −2τ =(τ 1 , τ 2 , τ 3<br />
sublattice<br />
0index. 0Here, −2t 1 −1<br />
and t 2 represent<br />
neighbor (NN) and next-nearest neighbor<br />
ping amplitudes. The Hamiltonian (2.2a)<br />
defined if t 2 = 0.<br />
One verifies that H 0 is both time-reversa<br />
and invariant under U(1) spin-1/2 rotations<br />
time-reversal operation T acts on numbers<br />
conjugation and on the electron-operators a<br />
ψ † k,↑<br />
T<br />
−→ +ψ † −k,↓ ,<br />
ψ† k,↓<br />
T<br />
−→ −ψ † −k<br />
The action of the U(1) spin-1/2 rotation R γ<br />
0 ≤ γ < 2π is given by<br />
ψ † k,↑<br />
R γ<br />
−→ e +iγ ψ † k,↑ ,<br />
⎞<br />
⎟<br />
⎠<br />
ψ† k,↓<br />
R γ<br />
−→ e −iγ ψ † k
%<br />
!"<br />
!<br />
Perspective - bulk vs. edge<br />
$./<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2"<br />
$"#<br />
$.&<br />
$.%<br />
$<br />
!"#()($01!"#"$<br />
$ '<br />
Thursday, March 22, 2012<br />
% &<br />
&'<br />
,"<br />
!"#()($01!"#"% !"#()('01!"#"%<br />
$ ' % &' $ '<br />
&<br />
% &'<br />
&<br />
FIG. 1. (color online). Numerical exact diagonalization results<br />
of Hamiltonian (2.1) for 16 electrons when sublattice A<br />
is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state<br />
degeneracies. Denote with E n the n-th lowest energy eigenvalue<br />
of the many-body spectrum where E 1 is the many-body<br />
ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter<br />
by n := (E n+1 − E n )/(E n − E 1 ). If a large gap<br />
&<br />
lattice momentum k and spin σ =↑, ↓ in t<br />
α = A, B and combine them in the subl<br />
ψ † k,σ<br />
c := † k,A,σ ,c† k,B,σ<br />
. Then, the secon<br />
single-particle Hamiltonian reads<br />
H 0 := <br />
B k · τ<br />
|B k<br />
| ψ k,↑ + ψ † B −k ·<br />
<br />
k,↓ B −<br />
ψ † k,↑<br />
k∈BZ<br />
Edge modes?<br />
where the 3-vector B k<br />
is defined by<br />
M. Levin and A. Stern, PRL (2009)<br />
B 0,k := 0,<br />
T. Neupert et al. PRB (2011)<br />
B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y −<br />
+ t 1 e +iπ/4 e −ik x + e<br />
B 3,k := 2t 2<br />
<br />
cos kx − cos k y<br />
<br />
,<br />
and the three Pauli-matrices τ =(τ 1 , τ 2 , τ 3<br />
sublattice index. Here, t 1 and t 2 represent<br />
neighbor (NN) and next-nearest neighbor<br />
ping amplitudes. The Hamiltonian (2.2a)<br />
defined if t 2 = 0.<br />
One verifies that H 0 is both time-reversa<br />
and invariant under U(1) spin-1/2 rotations<br />
time-reversal operation T acts on numbers<br />
conjugation and on the electron-operators a<br />
ψ † k,↑<br />
T<br />
−→ +ψ † −k,↓ ,<br />
ψ† k,↓<br />
T<br />
−→ −ψ † −k<br />
The action of the U(1) spin-1/2 rotation R γ<br />
0 ≤ γ < 2π is given by<br />
ψ † k,↑<br />
NO<br />
R γ<br />
−→ e +iγ ψ † k,↑ ,<br />
ψ† k,↓<br />
R γ<br />
−→ e −iγ ψ † k
%<br />
!"<br />
!<br />
Perspective - bulk vs. edge<br />
$./<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2"<br />
$"#<br />
$.&<br />
$.%<br />
!"#()($01!"#"$<br />
,"<br />
!"#()($01!"#"% !"#()('01!"#"%<br />
$<br />
fractionalized excitations in the bulk!<br />
Bulk $ rich '...% &' $ '<br />
&<br />
edge poor % &' $ '<br />
&<br />
% &'<br />
&<br />
Thursday, March 22, 2012<br />
No propagating edge modes, but...<br />
FIG. 1. (color online). Numerical exact diagonalization results<br />
of Hamiltonian (2.1) for 16 electrons when sublattice A<br />
is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state<br />
degeneracies. Denote with E n the n-th lowest energy eigenvalue<br />
of the many-body spectrum where E 1 is the many-body<br />
ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter<br />
by n := (E n+1 − E n )/(E n − E 1 ). If a large gap<br />
&<br />
lattice momentum k and spin σ =↑, ↓ in t<br />
α = A, B and combine them in the subl<br />
ψ † k,σ<br />
c := † k,A,σ ,c† k,B,σ<br />
. Then, the secon<br />
single-particle Hamiltonian reads<br />
H 0 := <br />
B k · τ<br />
|B k<br />
| ψ k,↑ + ψ † B −k ·<br />
<br />
k,↓ B −<br />
ψ † k,↑<br />
k∈BZ<br />
Edge modes?<br />
where the 3-vector B k<br />
is defined by<br />
M. Levin and A. Stern, PRL (2009)<br />
B 0,k := 0,<br />
T. Neupert et al. PRB (2011)<br />
B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y −<br />
+ t 1 e +iπ/4 e −ik x + e<br />
B 3,k := 2t 2<br />
<br />
cos kx − cos k y<br />
<br />
,<br />
and the three Pauli-matrices τ =(τ 1 , τ 2 , τ 3<br />
sublattice index. Here, t 1 and t 2 represent<br />
neighbor (NN) and next-nearest neighbor<br />
ping amplitudes. The Hamiltonian (2.2a)<br />
defined if t 2 = 0.<br />
One verifies that H 0 is both time-reversa<br />
and invariant under U(1) spin-1/2 rotations<br />
time-reversal operation T acts on numbers<br />
conjugation and on the electron-operators a<br />
ψ † k,↑<br />
T<br />
−→ +ψ † −k,↓ ,<br />
ψ† k,↓<br />
T<br />
−→ −ψ † −k<br />
The action of the U(1) spin-1/2 rotation R γ<br />
0 ≤ γ < 2π is given by<br />
ψ † k,↑<br />
NO<br />
R γ<br />
−→ e +iγ ψ † k,↑ ,<br />
ψ† k,↓<br />
R γ<br />
−→ e −iγ ψ † k
%<br />
!"<br />
!<br />
Perspective - bulk vs. edge<br />
$./<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2"<br />
$"#<br />
$.&<br />
$.%<br />
!"#()($01!"#"$<br />
,"<br />
!"#()($01!"#"% !"#()('01!"#"%<br />
$<br />
fractionalized excitations in the bulk!<br />
Bulk $ rich '...% &' $ '<br />
&<br />
edge poor % &' $ '<br />
&<br />
% &'<br />
&<br />
Thursday, March 22, 2012<br />
No propagating edge modes, but...<br />
More structure than captured by a<br />
FIG. 1. (color online). Numerical exact diagonalization results<br />
of Hamiltonian (2.1) for 16 electrons when sublattice A<br />
is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state<br />
degeneracies. Denote with E n the n-th lowest energy eigenvalue<br />
of the many-body spectrum where E 1 is the many-body<br />
ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter<br />
by n := (E n+1 − E n )/(E n − E 1 ). If a large gap<br />
&<br />
Z 2<br />
lattice momentum k and spin σ =↑, ↓ in t<br />
α = A, B and combine them in the subl<br />
ψ † k,σ<br />
c := † k,A,σ ,c† k,B,σ<br />
. Then, the secon<br />
single-particle Hamiltonian reads<br />
H 0 := <br />
B k · τ<br />
|B k<br />
| ψ k,↑ + ψ † B −k ·<br />
<br />
k,↓ B −<br />
ψ † k,↑<br />
k∈BZ<br />
Edge modes?<br />
where the 3-vector B k<br />
is defined by<br />
M. Levin and A. Stern, PRL (2009)<br />
B 0,k := 0,<br />
T. Neupert et al. PRB (2011)<br />
B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y −<br />
+ t 1 e +iπ/4 e −ik x + e<br />
B 3,k := 2t 2<br />
<br />
cos kx − cos k y<br />
<br />
,<br />
and the three Pauli-matrices τ =(τ 1 , τ 2 , τ 3<br />
sublattice index. Here, t 1 and t 2 represent<br />
neighbor (NN) and next-nearest neighbor<br />
ping amplitudes. The Hamiltonian (2.2a)<br />
defined if t 2 = 0.<br />
One verifies that H 0 is both time-reversa<br />
and invariant under U(1) spin-1/2 rotations<br />
time-reversal operation T acts on numbers<br />
conjugation and on the electron-operators a<br />
ψ † T<br />
classification<br />
k,↑<br />
−→ +ψ alone.<br />
† −k,↓ ,<br />
ψ† k,↓<br />
T<br />
−→ −ψ † −k<br />
The action of the U(1) spin-1/2 rotation R γ<br />
0 ≤ γ < 2π is given by<br />
ψ † k,↑<br />
NO<br />
R γ<br />
−→ e +iγ ψ † k,↑ ,<br />
ψ† k,↓<br />
R γ<br />
−→ e −iγ ψ † k
%<br />
!"<br />
!<br />
Perspective - bulk vs. edge<br />
$./<br />
'()*+,<br />
-()*+,<br />
$ %$ &$<br />
$<br />
$ % & ' !"#<br />
#"<br />
2"<br />
$"#<br />
$.&<br />
$.%<br />
!"#()($01!"#"$<br />
,"<br />
!"#()($01!"#"% !"#()('01!"#"%<br />
$<br />
fractionalized excitations in the bulk!<br />
Bulk $ rich '...% &' $ '<br />
&<br />
edge poor % &' $ '<br />
&<br />
% &'<br />
&<br />
Thursday, March 22, 2012<br />
No propagating edge modes, but...<br />
More structure than captured by a<br />
FIG. 1. (color online). Numerical exact diagonalization results<br />
of Hamiltonian (2.1) for 16 electrons when sublattice A<br />
is made of 3 × 4sitesandwitht 2 /t 1 =0.4. (a) Ground state<br />
degeneracies. Denote with E n the n-th lowest energy eigenvalue<br />
of the many-body spectrum where E 1 is the many-body<br />
ground state, i.e., E n+1 ≥ E n for n =1, 2, ···. Define the parameter<br />
by n := (E n+1 − E n )/(E n − E 1 ). If a large gap<br />
&<br />
Z 2<br />
lattice momentum k and spin σ =↑, ↓ in t<br />
α = A, B and combine them in the subl<br />
ψ † k,σ<br />
c := † k,A,σ ,c† k,B,σ<br />
. Then, the secon<br />
single-particle Hamiltonian reads<br />
H 0 := <br />
B k · τ<br />
|B k<br />
| ψ k,↑ + ψ † B −k ·<br />
<br />
k,↓ B −<br />
ψ † k,↑<br />
k∈BZ<br />
Edge modes?<br />
where the 3-vector B k<br />
is defined by<br />
M. Levin and A. Stern, PRL (2009)<br />
B 0,k := 0,<br />
T. Neupert et al. PRB (2011)<br />
B 1,k +iB 2,k := t 1 e −iπ/4 1+e +i (k y −<br />
+ t 1 e +iπ/4 e −ik x + e<br />
B 3,k := 2t 2<br />
<br />
cos kx − cos k y<br />
<br />
,<br />
and the three Pauli-matrices τ =(τ 1 , τ 2 , τ 3<br />
sublattice index. Here, t 1 and t 2 represent<br />
neighbor (NN) and next-nearest neighbor<br />
ping amplitudes. The Hamiltonian (2.2a)<br />
defined if t 2 = 0.<br />
One verifies that H 0 is both time-reversa<br />
and invariant under U(1) spin-1/2 rotations<br />
time-reversal operation T acts on numbers<br />
conjugation and on the electron-operators a<br />
ψ † T<br />
classification<br />
k,↑<br />
−→ +ψ alone.<br />
† −k,↓ ,<br />
“Either you are with us, or you are with the terrorists." - G.W. Bush<br />
ψ† k,↓<br />
T<br />
−→ −ψ † −k<br />
The action of the U(1) spin-1/2 rotation R γ<br />
0 ≤ γ < 2π is given by<br />
ψ † k,↑<br />
NO<br />
R γ<br />
−→ e +iγ ψ † k,↑ ,<br />
ψ† k,↓<br />
R γ<br />
−→ e −iγ ψ † k
3D<br />
T. Neupert, L. Santos, S. Ryu C. Chamon, and C. Mudry arXiv:1202.5188<br />
Thursday, March 22, 2012
Classification of non-interacting states of matter<br />
“Periodic Table of Topological Insulators”<br />
Symmetry<br />
d<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z<br />
d<br />
E Conduction Band<br />
2<br />
Conventional<br />
!=0<br />
Insulator<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
(a)<br />
(b)<br />
E F<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1<br />
Quantum spin<br />
1 Z 0 Z 2 Valence ZBand<br />
2 Z 0 0 0<br />
C 0<br />
"#/a<br />
0 k "#/a<br />
−1 0 0 Z 0 Z 2 Z<br />
FIG. 5 Edge states in the quantum spin Hall insulator. (a) 2 Z 0 0<br />
CI 1<br />
shows the interface between a QSHI and an ordinary insulator,<br />
and 1(b) shows 0the edge0state dispersion Z in0the graphene Z 2 −1 Z 2 Z 0<br />
Symmetry<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0<br />
C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0<br />
CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0<br />
TABLE I Periodic (2001) table corresponds of to<strong>topological</strong> the d = 4 entry in class <strong>insulators</strong> A AII. and superconductors.<br />
The have 10yet symmetry to be filled by realistic classes systems. are The search labeled is using the<br />
There are also other entries in physical dimensions that<br />
on to discover<br />
notation of Altland andNew such<br />
Zirnbauer<br />
J. phases. Phys. 12,<br />
(1997)<br />
065010<br />
(AZ)<br />
(2010)<br />
and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
III. QUANTUM SPIN HALL INSULATOR<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
TABLE I Periodic table of <strong>topological</strong> <strong>insulators</strong> and superconductors.<br />
The 10 symmetry classes are labeled using the<br />
notation of Altland and Zirnbauer (1997) (AZ) and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
the presence and absence of symmetry, with ±1 specifying<br />
the value of Θ 2 and Ξ 2 . As a function of symmetry and space<br />
dimensionality, d, the<strong>topological</strong>classifications(Z, Z 2 and 0)<br />
show a regular pattern that repeats when d → d +8.<br />
Thursday, March 22, 2012<br />
model, in which up and down spins propagate in opposite<br />
directions.<br />
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)<br />
Ryu, S., A. Schnyder, A. Furusaki, A. W. W. Ludwig,<br />
Kitaev, A., 2009, AIP Conf. Proc. 1134, 22; arXiv:0901.2686<br />
The 2D <strong>topological</strong> insulator is known as a quantum<br />
Conven<br />
Insulato<br />
(a)<br />
Quantum<br />
Hall insu<br />
FIG. 5 Edge<br />
shows the int<br />
tor, and (b)<br />
model, in wh<br />
directions.<br />
(2001) corre<br />
There are a<br />
have yet to<br />
on to discov
Classification of non-interacting states of matter<br />
“Periodic Table of Topological Insulators”<br />
Symmetry<br />
IQHE d<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z<br />
d<br />
E Conduction Band<br />
2<br />
Conventional<br />
!=0<br />
Insulator<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
(a)<br />
(b)<br />
E F<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1<br />
Quantum spin<br />
1 Z 0 Z 2 Valence ZBand<br />
2 Z 0 0 0<br />
C 0<br />
"#/a<br />
0 k "#/a<br />
−1 0 0 Z 0 Z 2 Z<br />
FIG. 5 Edge states in the quantum spin Hall insulator. (a) 2 Z 0 0<br />
CI 1<br />
shows the interface between a QSHI and an ordinary insulator,<br />
and 1(b) shows 0the edge0state dispersion Z in0the graphene Z 2 −1 Z 2 Z 0<br />
Symmetry<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0<br />
C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0<br />
CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0<br />
TABLE I Periodic (2001) table corresponds of to<strong>topological</strong> the d = 4 entry in class <strong>insulators</strong> A AII. and superconductors.<br />
The have 10yet symmetry to be filled by realistic classes systems. are The search labeled is using the<br />
There are also other entries in physical dimensions that<br />
on to discover<br />
notation of Altland andNew such<br />
Zirnbauer<br />
J. phases. Phys. 12,<br />
(1997)<br />
065010<br />
(AZ)<br />
(2010)<br />
and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
III. QUANTUM SPIN HALL INSULATOR<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
TABLE I Periodic table of <strong>topological</strong> <strong>insulators</strong> and superconductors.<br />
The 10 symmetry classes are labeled using the<br />
notation of Altland and Zirnbauer (1997) (AZ) and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
the presence and absence of symmetry, with ±1 specifying<br />
the value of Θ 2 and Ξ 2 . As a function of symmetry and space<br />
dimensionality, d, the<strong>topological</strong>classifications(Z, Z 2 and 0)<br />
show a regular pattern that repeats when d → d +8.<br />
Thursday, March 22, 2012<br />
model, in which up and down spins propagate in opposite<br />
directions.<br />
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)<br />
Ryu, S., A. Schnyder, A. Furusaki, A. W. W. Ludwig,<br />
Kitaev, A., 2009, AIP Conf. Proc. 1134, 22; arXiv:0901.2686<br />
The 2D <strong>topological</strong> insulator is known as a quantum<br />
Conven<br />
Insulato<br />
(a)<br />
Quantum<br />
Hall insu<br />
FIG. 5 Edge<br />
shows the int<br />
tor, and (b)<br />
model, in wh<br />
directions.<br />
(2001) corre<br />
There are a<br />
have yet to<br />
on to discov
Classification of non-interacting states of matter<br />
“Periodic Table of Topological Insulators”<br />
Symmetry<br />
IQHE d<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z<br />
d<br />
E Conduction Band<br />
2<br />
Conventional<br />
!=0<br />
Insulator<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
(a)<br />
(b)<br />
E F<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1<br />
Quantum spin<br />
1 Z 0 Z 2 Valence ZBand<br />
2 Z 0 0 0<br />
C 0<br />
"#/a<br />
0 k "#/a<br />
−1 0 0 Z 0TI<br />
Z 2 Z<br />
FIG. 5 Edge states in the quantum spin Hall insulator. (a) 2 Z 0 0<br />
CI 1<br />
shows the interface between a QSHI and an ordinary insulator,<br />
and 1(b) shows 0the edge0state dispersion Z in0the graphene Z 2 −1 Z 2 Z 0<br />
Symmetry<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0<br />
C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0<br />
CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0<br />
TABLE I Periodic (2001) table corresponds of to<strong>topological</strong> the d = 4 entry in class <strong>insulators</strong> A AII. and superconductors.<br />
The have 10yet symmetry to be filled by realistic classes systems. are The search labeled is using the<br />
There are also other entries in physical dimensions that<br />
on to discover<br />
notation of Altland andNew such<br />
Zirnbauer<br />
J. phases. Phys. 12,<br />
(1997)<br />
065010<br />
(AZ)<br />
(2010)<br />
and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
III. QUANTUM SPIN HALL INSULATOR<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
TABLE I Periodic table of <strong>topological</strong> <strong>insulators</strong> and superconductors.<br />
The 10 symmetry classes are labeled using the<br />
notation of Altland and Zirnbauer (1997) (AZ) and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
the presence and absence of symmetry, with ±1 specifying<br />
the value of Θ 2 and Ξ 2 . As a function of symmetry and space<br />
dimensionality, d, the<strong>topological</strong>classifications(Z, Z 2 and 0)<br />
show a regular pattern that repeats when d → d +8.<br />
Thursday, March 22, 2012<br />
model, in which up and down spins propagate in opposite<br />
directions.<br />
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)<br />
Ryu, S., A. Schnyder, A. Furusaki, A. W. W. Ludwig,<br />
Kitaev, A., 2009, AIP Conf. Proc. 1134, 22; arXiv:0901.2686<br />
The 2D <strong>topological</strong> insulator is known as a quantum<br />
Conven<br />
Insulato<br />
(a)<br />
Quantum<br />
Hall insu<br />
FIG. 5 Edge<br />
shows the int<br />
tor, and (b)<br />
model, in wh<br />
directions.<br />
(2001) corre<br />
There are a<br />
have yet to<br />
on to discov
Classification of non-interacting states of matter<br />
“Periodic Table of Topological Insulators”<br />
Symmetry<br />
IQHE d<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
chiral<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 8 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z<br />
d<br />
E Conduction Band<br />
2<br />
Conventional<br />
!=0<br />
Insulator<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
(a)<br />
(b)<br />
E F<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1<br />
Quantum spin<br />
1 Z 0 Z 2 Valence ZBand<br />
2 Z 0 0 0<br />
C 0<br />
"#/a<br />
0 k "#/a<br />
−1 0 0 Z 0TI<br />
Z 2 Z<br />
FIG. 5 Edge states in the quantum spin Hall insulator. (a) 2 Z 0 0<br />
CI 1<br />
shows the interface between a QSHI and an ordinary insulator,<br />
and 1(b) shows 0the edge0state dispersion Z in0the graphene Z 2 −1 Z 2 Z 0<br />
Symmetry<br />
AZ Θ Ξ Π 1 2 3 4 5 6 7 8<br />
A 0 0 0 0 Z 0 Z 0 Z 0 Z<br />
AIII 0 0 1 Z 0 Z 0 Z 0 Z 0<br />
AI 1 0 0 0 0 0 Z 0 Z 2 Z 2 Z<br />
BDI 1 1 1 Z 0 0 0 Z 0 Z 2 Z 2<br />
D 0 1 0 Z 2 Z 0 0 0 Z 0 Z 2<br />
DIII −1 1 1 Z 2 Z 2 Z 0 0 0 Z 0<br />
AII −1 0 0 0 Z 2 Z 2 Z 0 0 0 Z<br />
CII −1 −1 1 Z 0 Z 2 Z 2 Z 0 0 0<br />
C 0 −1 0 0 Z 0 Z 2 Z 2 Z 0 0<br />
CI 1 −1 1 0 0 Z 0 Z 2 Z 2 Z 0<br />
TABLE I Periodic (2001) table corresponds of to<strong>topological</strong> the d = 4 entry in class <strong>insulators</strong> A AII. and superconductors.<br />
The have 10yet symmetry to be filled by realistic classes systems. are The search labeled is using the<br />
There are also other entries in physical dimensions that<br />
on to discover<br />
notation of Altland andNew such<br />
Zirnbauer<br />
J. phases. Phys. 12,<br />
(1997)<br />
065010<br />
(AZ)<br />
(2010)<br />
and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
III. QUANTUM SPIN HALL INSULATOR<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
TABLE I Periodic table of <strong>topological</strong> <strong>insulators</strong> and superconductors.<br />
The 10 symmetry classes are labeled using the<br />
notation of Altland and Zirnbauer (1997) (AZ) and are specified<br />
by presence or absence of T symmetry Θ, particle-hole<br />
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes<br />
the presence and absence of symmetry, with ±1 specifying<br />
the value of Θ 2 and Ξ 2 . As a function of symmetry and space<br />
dimensionality, d, the<strong>topological</strong>classifications(Z, Z 2 and 0)<br />
show a regular pattern that repeats when d → d +8.<br />
Thursday, March 22, 2012<br />
model, in which up and down spins propagate in opposite<br />
directions.<br />
M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)<br />
Ryu, S., A. Schnyder, A. Furusaki, A. W. W. Ludwig,<br />
Kitaev, A., 2009, AIP Conf. Proc. 1134, 22; arXiv:0901.2686<br />
The 2D <strong>topological</strong> insulator is known as a quantum<br />
Conven<br />
Insulato<br />
(a)<br />
Quantum<br />
Hall insu<br />
FIG. 5 Edge<br />
shows the int<br />
tor, and (b)<br />
model, in wh<br />
directions.<br />
(2001) corre<br />
There are a<br />
have yet to<br />
on to discov
3D Lattice model<br />
3 orbital model on the cubic lattice<br />
3<br />
H k = λ 3+i sin k i + λ 7<br />
M −<br />
i=1<br />
3<br />
i=1<br />
cos k i<br />
<br />
Gell-Mann matrices<br />
flat band<br />
a) M = 0<br />
3<br />
2<br />
1<br />
0<br />
1<br />
2<br />
3<br />
4<br />
2<br />
0<br />
2<br />
4<br />
X M R X R M<br />
X M R X R M<br />
X M R X R M<br />
X M R X R M<br />
b) M = 1<br />
X M R X R M<br />
4<br />
2<br />
0<br />
2<br />
4<br />
X M R X R M<br />
c) M = 2 d) M = 3<br />
6<br />
4<br />
2<br />
0<br />
2<br />
4<br />
6<br />
X M R X R M<br />
X M R X R M<br />
Thursday, March 22, 2012
3D Lattice model<br />
3 orbital model on the cubic lattice<br />
3<br />
H k = λ 3+i sin k i + λ 7<br />
M −<br />
i=1<br />
3<br />
i=1<br />
cos k i<br />
<br />
Gell-Mann matrices<br />
flat band<br />
<br />
02×2 q<br />
H k = k<br />
q † k<br />
0<br />
chiral class (AIII)<br />
Thursday, March 22, 2012
3D Lattice model<br />
3 orbital model on the cubic lattice<br />
3<br />
H k = λ 3+i sin k i + λ 7<br />
M −<br />
i=1<br />
3<br />
i=1<br />
cos k i<br />
<br />
θ(M) =<br />
⎧<br />
⎪⎨ +2π, |M| < 1,<br />
−π, 1 < |M| < 3,<br />
⎪⎩<br />
0, 3 < |M|.<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Projection onto Landau level<br />
X = P n<br />
R Pn<br />
= R − 2 B<br />
e 3 × Π<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Projection onto Landau level<br />
X = P n<br />
R Pn<br />
= R − 2 B<br />
e 3 × Π<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
X 2<br />
∆X 1<br />
∆X 2<br />
X 1<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Projection onto Landau level<br />
X = P n<br />
R Pn<br />
= R − 2 B<br />
e 3 × Π<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
X 2<br />
∆X 1<br />
∆X 2<br />
non-commuting coordinates<br />
“fuzziness”<br />
X 1<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Coordinate transformation<br />
r µ → f µ (r)<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Coordinate transformation<br />
r µ → f µ (r)<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
<br />
f 1 ( X),f 2 ( X) <br />
=+i 2 B {f 1 ,f 2 } P ( X)+O 4 <br />
B<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Coordinate transformation<br />
r µ → f µ (r)<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
<br />
f 1 ( X),f 2 ( X) <br />
=+i 2 B {f 1 ,f 2 } P ( X)+O 4 <br />
B<br />
Poisson bracket<br />
<br />
{f 1 ,f 2 } P<br />
(r) = µν ∂f1 ∂f 2<br />
∂r µ ∂r ν<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
2D: Coordinate transformation<br />
r µ → f µ (r)<br />
<br />
X1 , X 2<br />
<br />
=+i 2 B<br />
<br />
f 1 ( X),f 2 ( X) <br />
=+i 2 B {f 1 ,f 2 } P ( X)+O 4 <br />
B<br />
Poisson bracket<br />
<br />
{f 1 ,f 2 } P<br />
(r) = µν ∂f1 ∂f 2<br />
∂r µ ∂r ν<br />
transformation preserves area<br />
⇒<br />
commutators are unchanged<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: what would be a “pristine” extension?<br />
<br />
X1 , X 2 , X 3<br />
<br />
=+i 3<br />
Nambu (quantum) 3-bracket<br />
[ A 1 , A 2 , A 3 ]= ijk<br />
Ai Aj Ak<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: what would be a “pristine” extension?<br />
r µ → f µ (r)<br />
<br />
f 1 ( X),f 2 ( X),f 2 ( X) <br />
<br />
X1 , X 2 , X 3<br />
<br />
=+i 3<br />
Nambu (quantum) 3-bracket<br />
[ A 1 , A 2 , A 3 ]= ijk<br />
Ai Aj Ak<br />
=i 3 {f 1 ,f 2 ,f 3 } N ( X)+O( 5 )<br />
<br />
{f 1 ,f 2 ,f 3 } N<br />
(r) = µνλ ∂f1 ∂f 2 ∂f 3<br />
(r)<br />
∂r µ ∂r ν ∂r λ<br />
Nambu (classical) 3-bracket<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: what would be a “pristine” extension?<br />
r µ → f µ (r)<br />
<br />
f 1 ( X),f 2 ( X),f 2 ( X) <br />
<br />
X1 , X 2 , X 3<br />
<br />
=+i 3<br />
Nambu (quantum) 3-bracket<br />
[ A 1 , A 2 , A 3 ]= ijk<br />
Ai Aj Ak<br />
=i 3 {f 1 ,f 2 ,f 3 } N ( X)+O( 5 )<br />
<br />
{f 1 ,f 2 ,f 3 } N<br />
(r) = µνλ ∂f1 ∂f 2 ∂f 3<br />
(r)<br />
∂r µ ∂r ν ∂r λ<br />
Nambu (classical) 3-bracket<br />
transformation preserves volume<br />
⇒<br />
3-brackets are unchanged<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: non-pristine<br />
<br />
X1 , X 2 , X 3<br />
<br />
=+i 3 + ...<br />
X µ =<br />
<br />
d d k χ † ã (k) iD ã ˜b<br />
µ (k) χ˜b(k)<br />
D µ (k) =∂ µ + A µ (k)<br />
when<br />
Tr D µ ,D ν<br />
<br />
= V<br />
<br />
d d k tr F µν (k) =0<br />
<br />
:<br />
<br />
Xµ , X ν , X ρ<br />
<br />
<br />
: = − iTr <br />
D µ ,D ν ,D ρ<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: non-pristine<br />
<br />
:<br />
<br />
Xµ , X ν , X ρ<br />
<br />
<br />
: = − iTr <br />
D µ ,D ν ,D ρ<br />
A. P. Polychronakos (2007)<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: non-pristine<br />
<br />
:<br />
<br />
Xµ , X ν , X ρ<br />
<br />
<br />
: = − iTr <br />
D µ ,D ν ,D ρ<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
= 12π2 i<br />
¯ρ<br />
A. P. Polychronakos (2007)<br />
CS (3)<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: non-pristine<br />
<br />
:<br />
<br />
Xµ , X ν , X ρ<br />
<br />
<br />
: = − iTr <br />
D µ ,D ν ,D ρ<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
= 12π2 i<br />
¯ρ<br />
A. P. Polychronakos (2007)<br />
CS (3)<br />
¯ρ = N p<br />
V<br />
CS (3) = π 2<br />
<br />
d 3 k<br />
(2π) 3<br />
ijk tr<br />
A i F jk − 2 3 A iA j A k<br />
<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: non-pristine<br />
<br />
:<br />
<br />
Xµ , X ν , X ρ<br />
<br />
<br />
: = − iTr <br />
D µ ,D ν ,D ρ<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
= 12π2 i<br />
¯ρ<br />
A. P. Polychronakos (2007)<br />
CS (3)<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
=+i 3<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: fuzzy coordinates<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
=+i 3<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: fuzzy coordinates<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
=+i 3<br />
X 2<br />
X 1<br />
Thursday, March 22, 2012
Non-commuting projected position operators<br />
3D: fuzzy coordinates<br />
1<br />
<br />
:<br />
N p<br />
<br />
X1 , X 2 , X 3<br />
<br />
<br />
:<br />
=+i 3<br />
X 2<br />
X 3<br />
X 1<br />
Thursday, March 22, 2012
Summary<br />
It is possible to have dispersionless bands (flat bands) that are isolated from all<br />
others by an energy gap.<br />
The isolated flat bands can be non-trivial in the sense of having a non-zero<br />
Chern number, and thus sustain an IQHE when fully occupied.<br />
When partially filled, electron-electron interactions can lead to the FQHE and<br />
other <strong>topological</strong> phases in time-reversal invariant systems.<br />
Topological Hubbard models have to ferromagnetic ground states: symmetry<br />
breaking simultaneously with IQHE or FQHE.<br />
<strong>Fractional</strong> TI in 3D: non-commuting coordinates<br />
Thursday, March 22, 2012