Topological invariants for disordered solid state systemsclosing+2018/...Topological invariants for...

Topological invariants for disordered solid state systems

Topological invariantsfor disordered solid state systems

Hermann Schulz-Baldes, Erlangen

main collaborators:

Emil Prodan (Yeshiva) and Terry Loring (Alberquerque)

Jena, March 2018


Plan of the talk

• Topological invariant for 1d-chiral Hamiltonian

• Index theorem and physical meaning (bulk-boundary corr.)

• Construction of associated spectral localizer

• Main new result: invariant as signature of spectral localizer

• Proof via spectral flow

• Extension to higher odd dimension

• Implementation of symmetries

• Even dimensional case

• Proof via fuzzy spheres


Model of SSH type (Su-Schriefer-Heeger)

Quasi-1d one-particle H on `2(Z,C2L) with chiral symmetry

J∗HJ = −H J =

(1L 0

0 −1L

)Due to chirality:

H =

(0 A

A∗ 0

)A on `2(Z,CL)

H insulator with µ = 0 Fermi level: H and hence A invertible

Moreover: H periodic, quasiperiodic or random

and short range so that for position X = X ⊗ 12L

‖[X ,H]‖ < ∞


Winding number as topological invariant

If H periodic, partially diagonalize by Bloch-Floquet:

H =

∫ ⊕T1

dk H(k) =

∫ ⊕T1

dk

(0 A(k)

A(k)∗ 0

)Then

Wind(A) =1

2πi

∫T1

dk Tr(A(k)−1∂kA(k)) ∈ Z

Phase label: only changes when gap closes

Theorem (Fritz Noether 1921, Gohberg-Krein 1960)

Let Π = (X > 0) be the Hardy projection onto `2(N,CL)

Then ΠAΠ + (1− Π) is Fredholm and:

Wind(A) = Ind(ΠAΠ + (1− Π)

)


Winding number for disordered system

Use non-commutative differentiation

∇A = i [X ,A]

and non-commutative integration using expectation over disorder

T (A) = EP Tr(〈0|A|0〉

)= lim

Λ→∞

1

|Λ|TrΛ(A)

which is also trace per unit volume. Then

Wind(A) = − i T (A−1∇A)

Theorem (with Prodan 2016)

P-almost surely ΠAΠ + (1− Π) is Fredholm and

Wind(A) = Ind(ΠAΠ + (1− Π)

)


What is the invariant good for?

General theme: bulk invariant leads to non-trivial boundary states

Half-space restriction (Dirichlet or other local boundary condition):

H on `2(N,C2L)

This has chiral boundary states at zero energy


Ker(H) is invariant under J

Let N+ and N− zero modes with positive/negative chirality

N+ − N− = TrKer(H)

(J) = Wind(A)

Recent extension to Anderson localization regime with closed gap

by Graf and Shapiro (arXiv)


Structure: Toeplitz extension (no disorder)

S bilateral shift on `2(Z), then C∗(S) ∼= C (S1)

S unilateral shift on `2(N), only partial isometry with a defect:

S∗S = 1 S S∗ = 1− |0〉〈0|

Then C∗(S) = T Toeplitz algebra with exact sequence:

0 → K i↪→ T π→ C (S1) → 0

K -groups for any C∗-algebra A (only rough definition):

K0(A) = {[P]− [Q] : projections in some Mn(A)}K1(A) = {[U] : unitary in some Mn(A)}

Abelian group operation: Whitney sum

Example: K0(C) = Z = K0(K) with invariant dim(P)

Example: K1(C (S1)) = Z with invariant given by winding number


6-term exact sequence for Toeplitz extension

C∗-algebra short exact sequence =⇒ K -theory 6-term sequence

K0(K) = Z i∗ - K0(T ) = Z π∗ - K0(C (S1)) = Z

K1(C (S1)) = Z

Ind6

� π∗K1(T ) = 0 �

i∗K1(K) = 0

Exp?

Here: [A]1 ∈ K1(C (S1)) and [Pσ3]0 = [P+]0 − [P−]0 ∈ K0(K)

Ind([A]1) = [P+]0 − [P−]0 (bulk-boundary for K -theory)

Tr(Ind(A)) = Wind(A) (bulk-boundary for invariants)

Disordered case and higher-dimensional cases: analogous


New numerical technique: spectral localizer

For tuning parameter κ > 0 introduce spectral localizer:

Lκ =

(κX A

A∗ −κX

)= κX ⊗ σ3 + H

Aρ restriction of A (Dirichlet b.c.) to Dρ = {x ∈ Z : |x | ≤ ρ}

Lκ,ρ =

(κXρ Aρ

A∗ρ −κXρ

)Clearly the latter is a selfadjoint matrix:

(Lκ,ρ)∗ = Lκ,ρ

Fact 1: Lκ,ρ is gapped, namely 0 6∈ Lκ,ρ

Fact 2: Lκ,ρ has spectral asymmetry measured by signature

Fact 3: signature linked to topological invariant


Theorem (with Loring 2017)

Let g = ‖A−1‖−1 be the invertibility gap. Provided that

‖[X ,A]‖ ≤ g3

12 ‖A‖κ(*)

and

2 g

κ≤ ρ (**)

the matrix Lκ,ρ is invertible and

12 Sig(Lκ,ρ) = Ind

(ΠAΠ + (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!


Why it can work:

Proposition

If (*) and (**) hold,

L2κ,ρ ≥

g2

2

Proof:

L2κ,ρ =

(A∗ρAρ 0

0 AρA∗ρ

)+κ2

(X 2ρ 0

0 X 2ρ

)+κ

(0 [Xρ,Aρ]

[Xρ,Aρ]∗ 0

)

Last term is a perturbation controlled by (*)

First two terms positive (indeed: close to origin and away from it)

Now A∗A ≥ g2, but (A∗A)ρ 6= A∗ρAρ

This issue can be dealt with by tapering argument:


Proposition (Bratelli-Robinson)

For f : R→ R with Fourier transform defined without√

2π,

‖[f (X ),A]‖ ≤ ‖f ′‖1 ‖[X ,A]‖

Lemma

∃ even function f : R→ [0, 1] with f (x) = 0 for |x | ≥ ρand f (x) = 1 for |x | ≤ ρ

2 such that ‖f ′‖1 = 8ρ

With this, f = f (X ) = f (|X |) and 1ρ = χ(|X | ≤ ρ):

A∗ρAρ = 1ρA∗1ρA1ρ ≥ 1ρA

∗f 2A1ρ

= 1ρfA∗Af 1ρ + 1ρ

([A∗, f ]fA + fA∗[f ,A]

)1ρ

≥ g2 f 2 + 1ρ([A∗, f ]fA + fA∗[f ,A]

)1ρ

So indeed A∗ρAρ positive close to origin

Then one can conclude... but TEDIOUS 2


Proof by spectral flow

Use Phillips’ result for phase U = A|A|−1 and properties of SF:

Ind(ΠAΠ + 1− Π) = SF(U∗XU,X )

= SF(κU∗XU, κX )

= SF

((U 0

0 1

)∗(κX 0

0 −κX

)(U 0

0 1

),

(κX 0

0 −κX

))

= SF

((U 0

0 1

)∗(κX 1

1 −κX

)(U 0

0 1

),

(κX 0

0 −κX

))

= SF

((κU∗XU U

U∗ −κX

),

(κX 0

0 −κX

))

= SF

((κX U

U∗ −κX

),

(κX 0

0 −κX

))Now localize and use SF = 1

2 Sig on paths of selfadjoint matrices 2


Generalization: odd Fredholm model for invertible

Let A ∈ B(H) be invertible with gap g = ‖A−1‖−1

D = D∗ Dirac with compact resolvent and ‖[D,A]‖ <∞Introduce Π = χ(D < 0) and spectral localizer

Lκ =

(κD A

A∗ −κD

)Restriction Lκ,ρ to Ranχ(|D| ≤ ρ) is finite-dimensional matrix


If

‖[D,A]‖ ≤ g3

12 ‖A‖κ2 g

κ≤ ρ

then Lκ,ρ is invertible and

12 Sig(Lκ,ρ) = Ind

(ΠAΠ + (1− Π)

)


Application to odd-dimensional chiral Hamiltonian

Gapped H = −JHJ on `2(Zd ,C2L) with invertible A on `2(Zd ,CL)

Non-commutative higher winding number (or odd Chern number):

Chd(A) =i(iπ)

d−12

d!!

∑ρ∈Sd

(−1)ρ T

d∏j=1

A−1∇ρjA

where T (A) = EP TrL 〈0|Aω|0〉 and ∇jAω = i [Xj ,Aω]

For periodic H: smooth function k ∈ Td 7→ A(k) ∈ Gl(L,C)

Chd(A) =( 1

2 (d − 1))!

d!

(i

2π

) d+12∫Td

Tr((

A−1dA)d)

For index, use faithful irrep Γ1, . . . , Γd of Clifford Cd on CN

Selfadjoint Dirac operator on `2(Zd ,CL ⊗ CN):

D =∑

j=1,...,d

Xj ⊗ 1L ⊗ Γj


Positive spectral (Hardy) projection Π = χ(D ≥ 0)


Let d be odd. P-almost surely ΠAΠ + (1− Π) is Fredholm and

Chd(A) = Ind(ΠAΠ + (1− Π)

)Theorem (with Prodan 2016)

If Chd(A) 6= 0 there are surface states for H on `2(N× Zd−1,C2L)

For d = 3, QHE for surface states with σH = Chd(A) from bulk


For d odd, Chd(A) can be calculated from spectral localizer

Remark: Ch3(A) of great importance for Floquet top. insulators


Implementation of real symmetries

If H has real symmetry (like PHS or TRS), often Sig(Lκ,ρ) = 0, but

sgn(det(Lκ,ρ)) ∈ Z2 , sgn(Pf(Lκ,ρ)) ∈ Z2

Not general case (paper) but example: Class CII has odd PHS

S∗ AS = A , S = S , S2 = −1

where overline is a real structure on complex Hilbert space

and in d = 3 Dirac D = X1σ1 + X2σ2 + X3σ3 has odd PHS

Σ∗D Σ = −D , Σ = iσ2

Hence with R = Σ⊗ S

R∗ Lκ R = − Lκ , R2 = 1

Thus sgn(Pf(iLκ,ρ)) ∈ Z2


Even dimensional pairings

Consider Hamiltonian on `2(Zd ,CL) with d even, no symmetry

For µ in gap of H consider Fermi projection P = χ(H ≤ µ)

Even-dimensional Dirac operator has grading Γd+1 =(1 0

0 −1

)Thus D =

( 0 D′

(D′)∗ 0

)and Dirac phase F = D ′|D ′|−1

Fredholm operator PFP + (1− P) has index = even Chern number

Spectral localizer

Lκ =

(H − µ κD ′

κ (D ′)∗ −(H − µ)

)Theorem (with Loring 2018)

Suppose ‖[H,D ′]‖ <∞ and D ′ normal, and κ, ρ with (*) and (**)

Chd(P) = Ind(PFP + (1− P)

)= 1

2 Sig(Lκ,ρ)


Elements of proof

Definition

A fuzzy sphere (X1,X2,X3) 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 21 + X 2

2 + X 23 )∥∥∥ < δ ‖[Xj ,Xi ]‖ < δ

Proposition

If δ ≤ 14 , one gets class [L]0 ∈ K0(K) by self-adjoint invertible

L =∑

j=1,2,3

Xj ⊗ σj ∈ M2(K+)

Reason: L invertible and thus has positive spectral projection

Remark: odd-dimensional spheres give elements in K1(K)


For smooth tapering fct Fρ : R→ [0, 1] define fρ : R→ [0, 1] by

Fρ(x)4 + fρ(x)4 = 1

If D ′ = D1 + iD2 with D∗j = Dj , and R = |D|, set

X1 = fρ(R)R−12 D1,ρ R

− 12 fρ(R)

X2 = fρ(R)R−12 D2,ρ R

− 12 fρ(R)

X3 = Fρ(R)Hρ Fρ(R)

Proposition

Lκ,ρ homotopic to L =∑

j=1,2,3 Xj ⊗ σj in invertibles

Theorem

Ind [π(P F P + 1− P)]1 = [Lκ,ρ]0


Theorem (General tool)

0→ K ↪→ B π→ Q→ 0 short exact sequence with Q unital

A ∈ B contraction with π(A) ∈ Q invertible, so [π(A)]1 ∈ K1(Q)

Assume A = A1 + i A2 almost normal, namely ‖[A1,A2]‖ < ε

Choose smooth ψ : [0, 1]→ [0, 1] and φ : [0, 1]→ [−1, 1] such that

φ(1) = 1 = −ψ(0) , x2 ψ(x)4 + φ(x)2 = 1

With B = (A21 + A2

2)12 set

Y1 = ψ(B)A1ψ(B) , Y2 = −ψ(B)A2ψ(B) , Y3 = φ(B)

Then (Y1,Y2,Y3) fuzzy sphere in K giving K -theoretic index map:

Ind[π(A)]1 =[ ∑j=1,2,3

Yj ⊗ σj]

0


Final steps of proof:

Apply this to suitable A homotopic to P F P + 1− P, namely

A1 = P fρ(R)2 R−1 D1 P + (1− P)f 2ρ (1− P)

A2 = P fρ(R)2 R−1 D2 P

Then above Theorem gives fuzzy sphere (Y1,Y2,Y3)

Find classical degree 1 map M : S2 → S2 such that

M(Y1,Y2,Y3) ∼ (X1,X2,X3)


Resume = Plan of the talk

• Topological invariant for 1d-chiral Hamiltonian

• Index theorem and physical meaning

• Construction of associated spectral localizer

• Main result: invariant as signature of spectral localizer

• Proof via spectral flow

• Extension to higher odd dimension

• Implementation of symmetries

• Even dimensional case

• Proof via fuzzy spheres


Implementation of real symmetries

Fix a real structure on complex Hilbert space, denoted by overline

There is irrep Γ1, . . . , Γd and real unitary matrix Σ

d mod 8 1 3 5 7

Σ∗D Σ = D −D D −DΣ2 = 1 −1 −1 1

Σ∗Π Σ = Π 1− Π Π 1− Π

For d = 3: D = X1σ1 + X2σ2 + X3σ3 and Σ = iσ2

Furthermore given real unitary S with [S ,Σ] = [S ,D] = 0:

j mod 8 2 4 6 8

S∗ AS = A∗ A A∗ A

S2 = 1 −1 −1 1


Symmetries of T = ΠAΠ + (1− Π) such that index pairings are:

Ind(2)(T ) j = 2 j = 4 j = 6 j = 8

d = 1 0 2Z Z2 Zd = 3 2Z Z2 Z 0

d = 5 Z2 Z 0 2Zd = 7 Z 0 2Z Z2

where Ind2(T ) = dim(Ker(T ))mod 2 ∈ Z2 (with Großmann 2016)

For spectral localizer follows R∗ Lκ R = s Lκ and R2 = s ′1 with

s = , s ′ = j = 2 j = 4 j = 6 j = 8

d = 1 −1 , −1 1 , −1 −1 , 1 1 , 1

d = 3 1 , −1 −1 , 1 1 , 1 −1 , −1

d = 5 −1 , 1 1 , 1 −1 , −1 1 , −1

d = 7 1 , 1 −1 , −1 1 , −1 −1 , 1


Same pattern!

Thus Ind and Ind2 can be calculated from spectral localizer using:

Proposition (with Loring 2017)

L = L∗ invertible complex matrix. R = R real unitary such

R∗ LR = s L , R2 = s ′ 1

(i) If s = 1 and s ′ = 1, then Sig(L) ∈ Z arbitrary

(ii) If s = 1 and s ′ = −1, then Sig(L) ∈ 2Z arbitrary

(iii) If s = −1 and s ′ = 1, then Sig(L) = 0, but i L is a real

antisymmetric matrix with invariant sgn(Pf(i L)) ∈ Z2

(iv) If s = −1 and s ′ = −1, then Sig(L) = 0

Topological invariants for disordered solid state systemsclosing+2018/...Topological invariants for...

Documents

Transcript of Topological invariants for disordered solid state systemsclosing+2018/...Topological invariants for...