Topological invariants in disordered topological insulators - PowerPoint PPT Presentation

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Hermann Schulz-Baldes, Erlangen collaborators: Terry Loring (Alberquerque) Edgar Lozano (UNAM Cuernavaca, numerics) ICMP, Montreal, July, 2018

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Plan of the talk • Winding number as prototype of an odd index pairing • Construction of associated spectral localizer • Main result: invariant as half-signature of spectral localizer • Proof via spectral flow • Even dimensional case • Proof via fuzzy spheres • Numerical results

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Winding number as odd index pairing For differentiable map k ∈ R / (2 π Z ) = T 1 �→ A ( k ) ∈ C N × N of invertible matrices, set � 1 T 1 dk Tr ( A ( k ) − 1 ∂ k A ( k )) ∈ Z Wind ( A ) = 2 π i View A as multiplication operator on L 2 ( T 1 ) Theorem (Fritz Noether 1921, Gohberg-Krein 1960) Let Π be Hardy projection onto H 2 ⊂ L 2 ( T 1 ) Then Π A Π + ( 1 − Π) is Fredholm and: � � Wind ( A ) = Ind Π A Π + ( 1 − Π)

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Winding number in Fourier space After Fourier F : L 2 ( T 1 ) → ℓ 2 ( Z ): convolution operator A Differentiability of A ∼ = bounded non-commutative derivative ∇ A = i [ D , A ] where D is unbounded position (dual Dirac) operator D | n � = n | n � Theorem Let Π = ( D > 0) be Hardy projection. Then � � Wind ( A ) = Ind Π A Π + ( 1 − Π) Physics: invariant for 1 d disordered chiral topological insulators Mathematically: canonical odd index paring of invertible A on H with an odd Fredholm module specified by a Dirac operator D with compact resolvent and bounded commutator [ D , A ]

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing New numerical technique: spectral localizer For tuning parameter κ > 0 introduce spectral localizer: � � κ D A L κ = A ∗ − κ D A ρ restriction of A (Dirichlet b.c.) to range of χ ( | D | ≤ ρ ) � � κ D ρ A ρ L κ,ρ = A ∗ − κ D ρ ρ Clearly selfadjoint matrix: ( L κ,ρ ) ∗ = L κ,ρ Fact 1: L κ,ρ is gapped, namely 0 �∈ L κ,ρ Fact 2: L κ,ρ has spectral asymmetry measured by signature Fact 3: signature linked to topological invariant

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Theorem (with Loring 2017) Given D = D ∗ with compact resolvent and invertible A with invertibility gap g = � A − 1 � − 1 . Provided that g 3 � [ D , A ] � ≤ (*) 12 � A � κ and 2 g ≤ ρ (**) κ the matrix L κ,ρ is invertible and with Π = χ ( D ≥ 0) � � 1 2 Sig ( L κ,ρ ) = Ind Π A Π + ( 1 − Π) How to use: form (*) infer κ , then ρ from (**) If A unitary, g = � A � = 1 and κ = (12 � [ D , A ] � ) − 1 and ρ = 2 κ Hence small matrix of size ≤ 100 sufficient! Great for numerics!

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Why it can work: Proposition If (*) and (**) hold, κ,ρ ≥ g 2 L 2 2 Proof: � � � � � � D 2 A ρ A ∗ 0 0 0 [ D ρ , A ρ ] L 2 + κ 2 ρ ρ κ,ρ = + κ D 2 0 A ∗ 0 [ D ρ , A ρ ] ∗ 0 ρ A ρ ρ Last term is a perturbation controlled by (*) First two terms positive (indeed: close to origin and away from it) Now A ∗ A ≥ g 2 , but ( A ∗ A ) ρ � = A ∗ ρ A ρ This issue can be dealt with by tapering argument:

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Proposition (Bratelli-Robinson) √ For f : R → R with Fourier transform defined without 2 π , � [ f ( D ) , A ] � ≤ � � f ′ � 1 � [ D , A ] � Lemma ∃ even function f ρ : R → [0 , 1] with f ρ ( x ) = 0 for | x | ≥ ρ 2 such that � � ρ � 1 = 8 and f ρ ( x ) = 1 for | x | ≤ ρ f ′ ρ With this, f = f ρ ( D ) = f ρ ( | D | ) and 1 ρ = χ ( | D | ≤ ρ ): ρ A ρ = 1 ρ A ∗ 1 ρ A 1 ρ ≥ 1 ρ A ∗ f 2 A 1 ρ A ∗ � � = 1 ρ fA ∗ Af 1 ρ + 1 ρ [ A ∗ , f ] fA + fA ∗ [ f , A ] 1 ρ � � ≥ g 2 f 2 + 1 ρ [ A ∗ , f ] fA + fA ∗ [ f , A ] 1 ρ So indeed A ∗ ρ A ρ positive close to origin Then one can conclude... but a bit tedious ✷

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Proof by spectral flow Use Phillips’ result for phase U = A | A | − 1 and properties of SF: Ind (Π A Π + 1 − Π) = SF ( U ∗ DU , D ) = SF ( κ U ∗ DU , κ D ) �� � ∗ � � � � � �� U 0 κ D 0 U 0 κ D 0 = SF , 0 1 0 − κ D 0 1 0 − κ D �� � ∗ � � � � � �� 0 κ D 1 0 κ D 0 U U = SF , 0 1 1 − κ D 0 1 0 − κ D �� � � �� κ U ∗ DU U κ D 0 = SF , U ∗ − κ D 0 − κ D �� � � �� κ D U κ D 0 = SF , U ∗ − κ D 0 − κ D Now localize and use SF = 1 2 Sig on paths of selfadjoint matrices ✷

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Even pairings (in even dimension) Consider gapped Hamiltonian H on H specifying P = χ ( H ≤ 0) � 1 � 0 Dirac operator D on H ⊕ H is odd w.r.t. grading Γ = 0 − 1 � 0 � D ′ and Dirac phase F = D ′ | D ′ | − 1 Thus D = − Γ D Γ = ( D ′ ) ∗ 0 Fredholm operator PFP + ( 1 − P ) has index = Chern number Spectral localizer � � H κ D ′ L κ = = H ⊗ Γ + κ D κ ( D ′ ) ∗ − H Theorem (with Loring 2018) Suppose � [ H , D ′ ] � < ∞ and D ′ normal, and κ , ρ with (*) and (**) � � 1 Ind PFP + ( 1 − P ) = 2 Sig ( L κ,ρ )

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Elements of proof Definition A fuzzy sphere ( X 1 , X 2 , X 3 ) of width δ < 1 in C ∗ -algebra K is a collection of three self-adjoints in K + with spectrum in [ − 1 , 1] and � � � � � 1 − ( X 2 1 + X 2 2 + X 2 3 ) � < δ � [ X j , X i ] � < δ Proposition If δ ≤ 1 4 , one gets class [ L ] 0 ∈ K 0 ( K ) by self-adjoint invertible � X j ⊗ σ j ∈ M 2 ( K + ) L = j =1 , 2 , 3 Reason: L invertible and thus has positive spectral projection Remark: odd-dimensional spheres give elements in K 1 ( K )

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Proposition L κ,ρ homotopic to L = � j =1 , 2 , 3 X j ⊗ σ j in invertibles Construction of that particular fuzzy sphere: Smooth tapering f ρ : R → [0 , 1] with supp( f ρ ) ⊂ [ − ρ, ρ ] as above Define F ρ : R → [0 , 1] by F ρ ( x ) 4 + f ρ ( x ) 4 = 1 If D ′ = D 1 + iD 2 with D ∗ j = D j , and R = | D | , set X 1 = F ρ ( R ) R − 1 2 D 1 ,ρ R − 1 2 F ρ ( R ) X 2 = F ρ ( R ) R − 1 2 D 2 ,ρ R − 1 2 F ρ ( R ) X 3 = f ρ ( R ) H ρ f ρ ( R ) Theorem Ind [ π ( P F P + 1 − P )] 1 = [ L κ,ρ ] 0

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Proof: General tool: Image of K -theoretic index map can be written as fuzzy sphere � � � Ind [ π ( A )] 1 = Y j ⊗ σ j 0 j =1 , 2 , 3 (by choosing an almost unitary lift A ) Formulas for Y 1 , Y 2 , Y 3 are explicit (but long) General tool for P F P + 1 − P provides fuzzy sphere ( Y 1 , Y 2 , Y 3 ) Final step: find classical degree 1 map M : S 2 → S 2 such that M ( Y 1 , Y 2 , Y 3 ) ∼ ( X 1 , X 2 , X 3 )

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Numerics for toy model: p + ip superconductor Hamiltonian on ℓ 2 ( Z 2 , C 2 ) depending on µ and δ � � � � S 1 + S ∗ 1 + S 2 + S ∗ S 1 − S ∗ 1 + ı ( S 2 − S ∗ 2 − µ δ 2 ) H = � � ∗ + λ V dis δ S 1 − S ∗ 1 + ı ( S 2 − S ∗ 2 ) − ( S 1 + S ∗ 1 + S 2 + S ∗ 2 − µ ) and disorder strength λ and i.i.d. uniformly distributed entries in � � � v n , 0 0 V dis = | n �� n | 0 v n , 1 n ∈ Z 2 Build even spectral localizer from D = X 1 σ 1 + X 2 σ 2 = − σ 3 D σ 3 : � � H ρ κ ( X 1 + iX 2 ) ρ L κ,ρ = κ ( X 1 − iX 2 ) ρ − H ρ Calculation of signature by block Chualesky algorithm

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Low-lying spectrum of spectral localizer Energy Levels of the Spectral Localizer with disorder δ =-0.35, µ=0.25, κ =0.1, ρ =15 3.5 3 Level of Disorder ( λ ) 2.5 2 1.5 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Energy Levels

Topological invariants in disordered topological insulators — Subtitle: Spectral localizer of an index pairing Half-signature and gaps for p + ip superconductor Half-Signature for Spectral Localizer with disorder Average of 20 repetitions δ =-0.35, µ=0.25, κ =0.1, ρ =15 1.25 0.3125 Average of Half-Signature Average Gap (SL) Average Gap (Hp) Minimum Gap (Hp) 1 0.25 0.75 0.1875 Half-Signature Gap Size 0.5 0.125 0.25 0.0625 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Level of Disorder ( λ )