Fractional topological insulators

Classification of non-interacting states of matter
“Periodic Table of Topological Insulators”
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
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 0TI
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
by presence or absence of T symmetry Θ, particle-hole
symmetry Ξ and chiral symmetry Π = ΞΘ. ±1 and0denotes
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.
Thursday, March 22, 2012
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