On the K -theoretic classification of topological phases of matter - PowerPoint PPT Presentation

On the K -theoretic classification of topological phases of matter arXiv:1406.7366 Guo Chuan Thiang University of Oxford FFP14 — 17 July 2014 1 / 1

Kitaev’s Periodic Table of topological insulators and superconductors Kitaev ’09: Based on K -theory, Bott periodicity. Q: Are interesting features robust/model-independent? Related: Freed–Moore ’13: Twisted equivariant K -theory classification of gapped free-fermion phases. 2 / 1

Classification principles Wanted: some object (group?) classifying gapped free-fermion phases compatible with certain given symmetry G . Existing literature: Differ on many basic definitions! Basic classification principles: ◮ Symmetries of dynamics can preserve/reverse time/charge. ◮ P rojective U nitary- A ntiunitary symmetries ∼ Wigner. ◮ Charged fermionic dynamics, as opposed to neutral dynamics. ◮ Insensitive to “deformations” preserving gap. Strategy: encode symmetry data in a C ∗ -algebra A . ◮ “Topological” invariants are those of A . 3 / 1

Charged gapped free-fermion dynamics Non-degenerate (“gapped”) dynamics Unitary time evolution U t = e i tH on ( H , h ), with 0 in gap of spec ( H ). Define Γ := sgn ( H ), so that U t = e i tH = e i Γ t | H | . ◮ Γ splits H into positive/negative energy subspaces. Important for positive energy (second) quantization 1 . ◮ Kitaev ’09: consider all H with same flattening Γ to be “homotopy” equivalent. Only grading Γ is important. 1 Derezi´ nski–G´ erard ’10, ’13. 4 / 1

Time and charge reversing symmetries Symmetry group G → B ❘ ( H ). Extra data: ◮ Homomorphisms 2 φ, τ : G → {± 1 } encode whether rep. g ≡ θ g is unitary/antiunitary and time preserving/reversing: g i = φ ( g ) i g , g U t = U τ ( g ) t g. ◮ Consequence: g is even/odd according to c := φ ◦ τ , gΓ = c ( g )Γg. ◮ 2-cocycle σ : G × G → ❚ encodes θ g 1 θ g 2 = σ ( g 1 , g 2 ) θ g 1 g 2 . ◮ Summary: Symmetry data is ( G , c , φ, σ ), acting projectively on graded Hilbert space ( H , Γ) as even/odd (anti)unitary operators according to c , φ —— “Graded PUA-rep” 2 c.f. Freed–Moore ’13. 5 / 1

Symmetry algebra: twisted crossed products 1 − 1 Graded PUA-rep of ( G , c , φ, σ ) ← − → non-degenerate ∗ -rep of associated graded twisted crossed product C ∗ -algebra 3 A := ❈ ⋊ ( α,σ ) G . ◮ ❈ is regarded as a real algebra. ⇒ α ( g )( λ ) = ¯ ◮ φ ( g ) = − 1 ⇐ λ , twisted by σ . ◮ c determines ❩ 2 -grading on A . ◮ All symmetry data is in A . Notation: ❈ ⋊ (1 , 1) G − → ❈ ⋊ G . 3 Leptin ’65, Busby ’70, Packer–Raeburn ’89 6 / 1

Example: CT -symmetries, Clifford algebras, tenfold way Let T =“ T ime-reversal”, C =“ C harge-conjugation”. Consider G = P ⊂ { 1 , T } × { 1 , C } =“ CT ”-group. ◮ c , φ are standard, e.g., φ ( T ) = − 1 , c ( C ) = − 1. ◮ σ can be standardised using U(1) phase freedom. ◮ Ten possible “ CT -classes”; each symmetry algebra A = ❈ ⋊ ( α,σ ) P is a Clifford algebra. 7 / 1

Example: CT -symmetries, Clifford algebras, tenfold way 4 Ungraded Associated Generators Graded C 2 T 2 Clifford algebra of P Morita class algebra T +1 M 2 ( ❘ ) ⊕ M 2 ( ❘ ) Cl 1 , 2 Cl 0 , 0 C , T − 1 +1 M 4 ( ❘ ) Cl 2 , 2 Cl 1 , 0 C − 1 M 2 ( ❈ ) Cl 2 , 1 Cl 2 , 0 C , T − 1 − 1 M 2 ( ❍ ) Cl 3 , 1 Cl 3 , 0 T − 1 ❍ ⊕ ❍ Cl 3 , 0 Cl 4 , 0 C , T +1 − 1 M 2 ( ❍ ) Cl 0 , 4 Cl 5 , 0 C +1 M 2 ( ❈ ) Cl 0 , 3 Cl 6 , 0 C , T +1 +1 M 4 ( ❘ ) Cl 1 , 3 Cl 7 , 0 N/A N/A ❈ ⊕ ❈ ❈ l 1 ❈ l 0 S 2 = +1 S M 2 ( ❈ ) ❈ l 2 ❈ l 1 Table: The ten classes CT -symmetries ( P , σ ), and their corresponding Clifford-symmetry-algebras. 4 Dyson ’62, Altland–Zirnbauer–Heinzner–Huckleberry ’97, ’05, Abramovici–Kalugin ’11 etc. 8 / 1

Towards K -theory groups of symmetry algebras What are physically relevant groups for A = ❈ ⋊ ( α,σ ) G ? Start with c ≡ 1 (trivial grading) case. ◮ For compact G : Representation ring/group R ( G ) ∼ = K 0 ( A ). ◮ For general ( G , φ, σ ), define R ( G , φ, σ ) to be K 0 ( A ). E.g commutative case: ❈ ⋊ ❩ d ∼ Serre − Swan = C ( ❚ d )-module ← − − − − − − → Γ ( E → ❚ d ); reminiscent of Bloch theory and band insulators. Q:What is “ K 0 ( A )” for graded symmetry algebras ( c �≡ 1)? A1: “Super-rep group”, super-Brauer group, super-division algebras. . . recovers d = 0 in Table. A2: Use a model for K -theory in Karoubi ’78, roughly: stable homotopy classes of grading operators compatible with A . 9 / 1

A model for K -theory: Difference-groups Consider the set Grad A ( W ) of possible grading operators on an ungraded A -module W . ◮ “symmetry-compatible gapped Hamiltonians on W ”. Note: π 0 ( Grad A ( W )) has no group structure yet! ⇒ Study differences of compatible Hamiltonians. ◮ Triple ( W , Γ 1 , Γ 2 ) represents ordered difference. ◮ Triple is trivial if Γ 1 , Γ 2 are homotopic within Grad A ( W ). ◮ ⊕ gives monoid structure to the set Grad A of all triples; trivial triples form submonoid Grad triv A . The difference-group of symmetry-compatible gapped Hamiltonians, K 0 ( A ), is Grad A / ∼ Grad triv A . 10 / 1

A model for K -theory: Difference-groups Nice properties of K 0 ( · ): ◮ K 0 ( A ) is an abelian group, with [ W , Γ 1 , Γ 2 ] = − [ W , Γ 2 , Γ 1 ]. ◮ Path independence: [ W , Γ 1 , Γ 2 ] + [ W , Γ 2 , Γ 3 ] = [ W , Γ 1 , Γ 3 ]. ◮ [ W , Γ 1 , Γ 2 ] depends only on the homotopy class of Γ i . Special case: for purely-even A ev , our K 0 ( A ev ) is one of Karoubi’s models for the ordinary K 0 ( A ev ). Karoubi ’78, ’08: Clifford “suspension” A �→ A ˆ ⊗ Cl 0 , 1 is compatible with the usual suspension A �→ C 0 ( ❘ , A ), i.e., ∼ K 0 ( C 0 ( ❘ n , A )) . K 0 ( A ˆ ⊗ Cl 0 , n ) = ∼ K n ( A ) , if A = A ev . = 11 / 1

Dimension shifts in Periodic Table Common claim: Classification in d dimensions is the same as d = 0 classification, except for shift in by d . To what extent is this true? ◮ G = G 0 × P , plus mild assumptions, symmetry algebra A ∼ = A ev ❘ ˆ ⊗ Cl r , s . Thus, K 0 ( A ) ∼ ❘ ) “ ∼ = K s − r ( A ev =” KR r − s ( X ) . Suppose ˜ G = ˜ G 0 × P where ˜ G 0 is an extension of G 0 by ❘ d . Then ˜ σ ) ˜ G ∼ ❘ ⋊ ( β,ν ) ❘ d )ˆ = ( A ev A = ❈ ⋊ (˜ ⊗ Cl r , s . α, ˜ Q: How is K 0 ( ˜ A ) related to K 0 ( A )? 12 / 1

Dimension shifts in Periodic Table Powerful results from K -theory of crossed products: Connes–Thom isomorphism, Connes ’81 K n ( A ⋊ ( α, 1) ❘ ) ∼ = K n − 1 ( A ) for any action of ❘ . Packer–Raeburn stabilisation trick, Packer–Raeburn ’89 Twisted crossed products can be untwisted after stabilisation: ( A ⋊ ( α,σ ) G ) ⊗ K ∼ = ( A ⊗ K ) ⋊ ( α ′ , 1) G . Corollary: Dimension shifts K n ( A ⋊ ( α,σ ) ❘ d ) ∼ = K n − d ( A ). 13 / 1

Dimension shifts in Periodic Table Thus, extra ❘ d symmetry shifts degree of the difference group: A ) ∼ ❘ ⋊ ( β,ν ) ❘ d ) ∼ K 0 ( ˜ = K s − r ( A ev = K s − r − d ( A ev ❘ ) . ◮ Note: result does not depend on how ❘ d fits in G 0 → ❘ d → 1. 1 → G 0 → ˜ ◮ Extra ❘ d symmetries may be projectively realised (IQHE). ◮ Some extra assumptions needed for discretised version of this result. 14 / 1

Periodic Table of difference-groups of gapped topological phases K 0 ( A ) ∼ = K n − d ( ❘ ) or K n − d ( ❈ ) C 2 T 2 n d = 0 d = 1 d = 2 d = 3 0 +1 ❩ 0 0 0 1 +1 +1 ❩ 2 ❩ 0 0 2 +1 ❩ 2 ❩ 2 ❩ 0 3 +1 − 1 0 ❩ 2 ❩ 2 ❩ 4 − 1 ❩ 0 ❩ 2 ❩ 2 5 − 1 − 1 0 0 ❩ ❩ 2 6 − 1 0 0 0 ❩ 7 − 1 +1 0 0 0 ❩ 0 N/A 0 0 ❩ ❩ S 2 = +1 1 0 ❩ 0 ❩ Table: Vertical degree shifts — effect on K 0 ( A ) of tensoring with a Clifford algebra. Horizontal shifts — Connes–Thom isomorphism. Twofold and eightfold periodicities — Bott periodicity. Assuming translational symmetry × CT -symmetry. 15 / 1

General remarks ◮ Conceptual advantage: all symmetries are treated on an equal footing. These include T , C , projective symmetries (e.g. IQHE), ❩ d (band insulators), and extra spatial translations ❘ d . ◮ Phenomenon of “dimension shift” is robust and model-independent. ◮ We see why T -symmetry needs to be broken for IQHE, but ❩ 2 -invariant possible for QSHE. ◮ Not restricted to condensed matter applications. 16 / 1