Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Bulk-Boundary Correspondence in Disordered Topological Insulators and Superconductors Christopher Max 04.09.2018 Supervisor: PD Alexander Alldridge Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Topics 1 Construction of the C ∗ -algebra of observables. 2 Classification of gapped bulk systems in Van Daele KR-theory. 3 Systematic pseudo-symmetry picture for the corresponding boundary classes in Kasparov’s KR-theory. Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Controlled Lattice Operators Tight-binding model over the lattice L . bulk lattice : L = | Z d | , half-space: L = | Z d − 1 × N | Localized lattice states: ℓ 2 ( L ) := ℓ 2 ( L , C ) → Complex Hilbert space with real structure c of point-wise complex conjugation. Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Controlled Lattice Operators Tight-binding model over the lattice L . bulk lattice : L = | Z d | , half-space: L = | Z d − 1 × N | Localized lattice states: ℓ 2 ( L ) := ℓ 2 ( L , C ) → Complex Hilbert space with real structure c of point-wise complex conjugation. Definition (Controlled operators) T ∈ B ( ℓ 2 ( L )) is controlled or has finite propagation , if there is some R > 0 such that � x | T | y � = 0 for all x , y ∈ L with | x − y | > R . Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Controlled Lattice Operators Tight-binding model over the lattice L . bulk lattice : L = | Z d | , half-space: L = | Z d − 1 × N | Localized lattice states: ℓ 2 ( L ) := ℓ 2 ( L , C ) → Complex Hilbert space with real structure c of point-wise complex conjugation. Definition (Controlled operators) T ∈ B ( ℓ 2 ( L )) is controlled or has finite propagation , if there is some R > 0 such that � x | T | y � = 0 for all x , y ∈ L with | x − y | > R . Definition (Real C ∗ -algebra) A complex C ∗ -algebra A is a complex Banach algebra with an anti-linear anti-involution ∗ : A → A s.th. � a ∗ a � = � a � 2 ∀ a ∈ A . A Real C ∗ -algebra is a complex C ∗ -algebra A with a real involution, i.e. a ∗ -isometric anti-linear involution ¯ · : A → A . Definition (Uniform Roe C ∗ -algebra) �·� defines a Real C ∗ -algebra with real C ∗ u ( L ) := { T ∈ B ( ℓ 2 ( L )) | T controlled } involution Ad c . Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Nambu Space of Internal Degrees of Freedom Single particle picture V : complex vector space of internal d.o.f.; inner product �· , ·� . Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Nambu Space of Internal Degrees of Freedom Single particle picture V : complex vector space of internal d.o.f.; inner product �· , ·� . Many particle space without interactions: Nambu space of fields: W = V ⊕ V ∗ Choice of basis e 1 , . . . , e n of V : V ⊕ V ∗ ∼ = span C ( c † 1 , . . . , c † n , c 1 , . . . , c n ) Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Nambu Space of Internal Degrees of Freedom Single particle picture V : complex vector space of internal d.o.f.; inner product �· , ·� . Many particle space without interactions: Nambu space of fields: W = V ⊕ V ∗ Choice of basis e 1 , . . . , e n of V : V ⊕ V ∗ ∼ = span C ( c † 1 , . . . , c † n , c 1 , . . . , c n ) Anti-linear Riesz isomorphism R : V → V ∗ : R ( v ) = � v , ·� . � � 0 R ∗ , γ 2 W = 1 , γ W ( λ w ) = ¯ Real structure on W : γ W = λγ W ( w ). R 0 γ W induced by fermionic anti-commutation relations {· , ·} and the inner product �· , ·� on V and V ∗ . ¯ M = Ad γ W ( M ) = γ W M γ W real structure on End ( W ). Hamiltonian without interaction: c † i A ij c j + c † i B ij c † j + c i C ij c j + c i D ij c † � H = j i , j � A � � A � � A � B B B → ∈ End ( W ) , = − . C D C D C D Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Homogeneous Disorder Definition (Dynamical system of disorder) A dynamical system (Ω , τ, Z d ) describing homogeneous disorder is given by a compact Hausdorff space Ω = (Ω 0 ) Z d , the Z d -action on Ω: τ : Z d → Homeo (Ω), τ x ( ω y ) = ω y − x . Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Homogeneous Disorder Definition (Dynamical system of disorder) A dynamical system (Ω , τ, Z d ) describing homogeneous disorder is given by a compact Hausdorff space Ω = (Ω 0 ) Z d , the Z d -action on Ω: τ : Z d → Homeo (Ω), τ x ( ω y ) = ω y − x . Disorder on the level of operators: Definition (The disordered bulk C ∗ -algebra) Let U : Z d → B ℓ 2 ( | Z d | ) be the action via translations. The Real C ∗ -algebra of bulk � � observables is given by � �·� � A W | T ( τ x ( ω )) = U x T ( ω ) U − 1 � Ω , C ∗ u ( | Z d | ) ⊗ End ( W ) � ∀ x ∈ Z d = T ∈ C x d ⊂ C (Ω) ⊗ C ∗ u ( | Z d | ) ⊗ End ( W ) Real structure on A W induced by real structures on End ( W ) and C ∗ u ( | Z d | ). d Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Bulk C ∗ -algebra as crossed product C ∗ -algebra Theorem (Crossed product form of bulk C ∗ -algebra) A W � � ⋊ Z d . = C (Ω) ⊗ End ( W ) d The crossed product C ∗ -algebra is the norm-closure of the non-commutative polynomials � � | M x ∈ C (Ω) ⊗ End ( W ) , M x = 0 for almost all x ∈ Z d � M x u x 1 1 · · · u x d , d x ∈ Z d where i = u − 1 u i M ( ω ) u ∗ , u ∗ i = M � τ e i ( ω ) � , u i u j = u j u i i for all i , j ∈ { 1 , . . . , d } and M ∈ C (Ω) ⊗ End ( W ) = C (Ω , End ( W )). Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Bulk C ∗ -algebra as crossed product C ∗ -algebra Theorem (Crossed product form of bulk C ∗ -algebra) A W � � ⋊ Z d . = C (Ω) ⊗ End ( W ) d The crossed product C ∗ -algebra is the norm-closure of the non-commutative polynomials � � | M x ∈ C (Ω) ⊗ End ( W ) , M x = 0 for almost all x ∈ Z d � M x u x 1 1 · · · u x d , d x ∈ Z d where i = u − 1 u i M ( ω ) u ∗ , u ∗ i = M � τ e i ( ω ) � , u i u j = u j u i i for all i , j ∈ { 1 , . . . , d } and M ∈ C (Ω) ⊗ End ( W ) = C (Ω , End ( W )). Clean system: Ω 0 = { pt } . Trivial action of Z d on Ω → translational invariance, A W = End ( W ) ⊗ C ∗ ( Z d ) . d Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Construction of C ∗ -algebra of observables Bulk Classification in Van Daele KR-theory Systematic boundary classification and bulk-boundary correspondence Half-space and boundary C ∗ -algebra � �·� �� Half-space C ∗ -algebra: ˆ ∼ u ∗ A W u d ) n 1 (ˆ d ) n 2 , for p n 1 , n 2 ∈ A W = n 1 , n 2 ∈ N p n 1 , n 2 (ˆ d d − 1 and u ∗ u ∗ ˆ d ˆ u d = 1 , ˆ u d ˆ d = 1 − P 0 , u ∗ u ∗ ˆ u d M ( ω ) = M ( τ e d ( ω ))ˆ u d , ˆ d M ( ω ) = M ( τ − e d ( ω ))ˆ d , where P 0 is a 1-dim. projection ( P 0 ˆ = | 0 �� 0 | ). Christopher Max Bulk-boundary correspondence in disordered TIs and SCs

Recommend

More recommend