A wzw term which is the case when d g h d1 g h

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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 p-wave DIII supercond AII uctor CII C CI 11 ... New Kane-Mele 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 Non-chiral 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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