Fractional topological insulators
Fractional topological insulators
Fractional topological insulators
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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