Unformatted text preview: 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
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 ...
...
...
...
...
...
...
... 11 ... symmetry
¯
Kitaev, 2009;
ensions if and only if the target space of the NLσ M on the d dimensional boundary allows
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 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 1 2 3 4 5 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 IQHE 0 d space dimensionality
6
7
8
9
10 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 ...
...
...
...
...
...
...
... 11 ... symmetry
¯
Kitaev, 2009;
ensions if and only if the target space of the NLσ M on the d dimensional boundary allows
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 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
DIII
AII
CII
C
CI 11 ... symm...
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This document was uploaded on 03/12/2014 for the course PHYSICS 240C at UC Davis.
 Spring '12
 WarrenE.Pickett
 Physics

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