Unformatted text preview: he case when πd (G / H ) = πd¯+1 (G / H ) = Z. By using this rule in
Thursday, January 6, 2011 12
the topologically distinct phases within a given symmetry class of topological
insulators (superconductors) are characterized by an integer invariant (Z) or a
Z2 quantity, respectively. The symbol ‘0’ denotes the case when there exists no
topological insulator (superconductor), i.e. when all quantum ground states are
Table from Ryu, Schnyder, Furusaki, Ludwig, 2010
topologically equivalent to the trivial state. Classiﬁcation table of topological
insulators and superconductors
Cartan IQHE 0 1 2 3 4 5 d space dimensionality
6
7
8
9
10 Complex case:
A
AIII Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z ...
... Z
Z2
Z2
0
2Z
0
0
0 0
Z
Z2
Z2
0
2Z
0
0 0
0
Z
Z2
Z2
0
2Z
0 0
0
0
Z
Z2
Z2
0
2Z 2Z
0
0
0
Z
Z2
Z2
0 0
2Z
0
0
0
Z
Z2
Z2 Z2
0
2Z
0
0
0
Z
Z2 Z2
Z2
0
2Z
0
0
0
Z Z
Z2
Z2
0
2Z
0
0
0 0
Z
Z2
Z2
0
2Z
0
0 0
0
Z
Z2
Z2
0
2Z
0 0
0
0
Z
Z2
Z2
0
2Z ...
...
...
...
...
...
...
... Su,
Real case:
Schrieffer,
Heeger AI
BDI
D
D pwave DIII
supercond AII
uctor
CII
C
CI 11 ... New KaneMele topological insulators
symmetry
3
¯
Kitaev, 2009;
ensions if and only if the target He, phasethe NLσ M on the d dimensional boundary allows
space of B
classes
Ludwig, Ryu, Schnyder, Furusaki, Z2 , or
ither (i) a Z2 topological term, which is the case when πd¯ (G / H ) = πd −1 (G / H ) = 2009. a WZW term, which is the case when πd (G / H ) = πd¯+1 (G / H ) = Z. By using this rule in
Thursday, January 6, 2011 Chiral symmetry Thursday, January 6, 2011 13 Chiral symmetry
Often realized as hopping on a bipartite lattice Thursday, January 6, 2011 13 Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems
All levels come in pairs Thursday, January 6, 2011 Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems
All levels come in pairs {
Thursday, January 6, 2011 right zero modes
left zero modes Chiral symmetry 13 Often realized as hopping on a bipartite lattice Properties of chiral systems
All levels come in pairs { right zero modes
left zero modes #R#L is a topological invariant (index theorem)
Thursday, January 6, 2011 Chiral vs nonchiral systems Nonchiral systems can be characterized by
an integer topological
invariant
in even spacial
dimensions only Thursday, January 6, 2011 14 Chiral systems can be characterized by
an integer topological
invariant
in odd spacial
dimensions only the topologically distinct phases within a given symmetry class of topological
insulators (superconductors) are characterized by an integer invariant (Z) or a
Z2 quantity, respectively. The symbol ‘0’ denotes the case when there exists no
topological insulator (superconductor), i.e. when all quantum ground states are
topologically equivalent to the trivial state. Chiral vs nonchiral systems Cartan 0 1 2 3 4 5 d
6 Complex case:
A
AIII Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z Z
0 0
Z ...
... Real case:
AI
BDI
D
DIII
AII
CII
C
CI Z
Z2
Z2
0
2Z
0
0
0 0
Z
Z2
Z2
0
2Z
0...
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 Spring '12
 WarrenE.Pickett
 Physics, Trigraph, Condensed matter physics, z2, Quantum Hall effect, Quantum spin Hall effect

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