Unformatted text preview: etry
¯
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
D pwave DIII
supercond AII
uctor
CII
C
CI 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
D pwave DIII
supercond AII
uctor
CII
C
CI 11 ... 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 t...
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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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