TKNN-equations for Dirac-like operators

TKNN-equations

for

Dirac-like operators

Giuseppe De Nittis

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IVrd MathematicalMethods inQuantumMechanics

Bressanone, Italy. February 14-19, 2011

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Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

•TheHarper equationis

(

ψ(x−θ) +ψ(x+θ)−[E−2 cos(2πx)]ψ(x) =0

ψ∈L2(R).

Deformation parameterθ∈R;EnergyE∈R.

•Extensively studied in the last six decades (Harper,Rauh, Wilkinson,Bellissard,Helffer - Sjöstrand,...,D. - Panati) as

effective modelfor thequantum Hall effect(QHE) in the limit

θ−1:=B→∞.

•The Harper equation is the eigenvalues equation for

H_θ(1):=Dθ+K

•The spectrum ofH_θ(1)isCantor-typeifθ∈R\Q(ten martini

problem,Avila - Jitomirskaya, 2009).

•ifθ=M/NwithN,M∈Qand co-prime

σ(H(M1)

N

) =I1∪I2∪. . .∪IN∗, N∗= (

N−1 ifN even

N ifN odd.

whereIj⊂Rcompact,Ii∩Ij=/0ifi6=j.

•AnyIj defines aspectral projection

pj:=

1 i2π

I

Cj

1

H(M1) N

−λ

dλ

whereCj⊂C\σ(H(M1) N

)is aJordan curveenclosingIj.

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

•Letgj the gap betweenIj andIj+1. TheHall conductivitydue

to the energy spectrum up togj is

σHall(gj) : =

e2

i} Tr

Pj

[Pj;Λ1]; [Pj;Λ2] (Kubo formula) = e2 h i 2π Z

T2trN

Pj(k)

∂k1Pj(k);∂k2Pj(k)

dk

| {z }

(Chern number)

whereΛi,i=1,2 arestep functions(in position and momentum

!), trNis anN-dimensional trace and

Pj:=

Z ⊕

T2Pj(k)dk, Pj(k)∈B(C

N_{) (Bloch-Floquet)}

•TheKubo-Chern formulahas a long story (Thouless et al., Stˇreda,Bellissard,Avron,Graf,Elgart-Schlein,. . .) still open !

•C(Pj)is an integer (Chern number) and verifies the

Diophantine equation

N C(P_j) +M s_j=R_j, j=1, . . . ,N∗

sj∈Z, |sj|<N, θ=

M

N, Rj=

  

 

j ifj<N/2

j+1otherwise ifNeven

j ifNodd.

•TKNN-equationprovides the (unique) value ofC(Pj).

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

•Dαψ(x) :=ψ(x−α) +ψ(x+α), α∈R, Kψ(x) :=2 cos(2πx).

•The (1-dimensional)θ-Harper Hamiltonian:

H_θ(1):=Dθ+K on H1:=L2(R).

•The (2-dimensional,1-twisted)θ-Harper Hamiltonian:

H_θ(2,1):= K Dθ−12 D_θ−1

2

−K

!

on H2:=L2(R)⊗C2.

•H_θ(2,1)isDirac-like.(i)Model forQHE in graphene(Bellissard). (ii)Effective Hamiltonian for the coupling of Landau levels in strong magnetic field regime (D. - Panati).

•H_θ(1)andH_θ(2,1)areiso-spectral, i.e. same system of energy bands and gaps.

•PROBLEM !How to computeC(Pj)for the spectral

•T1ψ(x) :=ei2πxψ(x), T2αψ(x) :=ψ(x−α), α∈R.

Dα=T2α+ (T2α) †

, K =T1+T1†, T1T2=ei2παT2T1.

•For anyq∈_N\ {0}

Uq:=

     

1 0 . . . 0

0 ei2πq _{. . . 0}

..

. ... . .. ...

0 0 . . . ei 2π(q−1)

q      

, Vq:=



   

0 . . . 0 1 1 . . . 0 0 ..

. . .. ... ... 0 . . . 1 0

     .

•For anyr∈ {±1, . . . ,±(q−1)}coprime with respect toq

Uq:=T1⊗Uq, Vqθ,r :=T2ε⊗V

r

q, ε(θ,q,r) :=θ−

r q

•The (q-dimensional,r-twisted)θ-Harper Hamiltonian:

H_θ(q,r):=Uq+Uq†+Vqε,r+ Vqε,r

†

on Hq:=L2(R)⊗Cq.

•PROBLEM !How to computeC(Pj)for the spectral

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

Non-Commutative Torus (NCT)

“abstract”C∗-algebraA_θ

u∗=u−1, v∗=v−1, uv=ei2πθ_vu,

θ∈R,

withuniversal normkak:=sup{kπ(a)k_H |πrep. ofAθ onH}.

(q,r)-representation:

Observing that UqVqε,r =ei2πθVqε,rUq then

Πq,r :Aθ →B(Hq), Πq,r(u) :=Uq, Πq,r(v) :=Vqε,r.

is afaithfulrepresentation.

THEOREM (isospectrality)

For anyq,r and for anya∈Aθ

Black and white butterfly (Hofstadter, 1976)

h:=u+u∗+v+v∗∈A_θ (universal Harper operator)

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

•Canonical trace (measure structure):

R

−

−−_:Aθ →C

R

−−−₍unvm) =δn,0δm,0.

state(linear, positive, normalized), faithful(R

−−−(a∗a) =0⇔a=0), trace property(R

−−−(ab) =R

− − −(ba)).

•Canonical derivations (differentiable structure):

∂−−−j :Aθ →Aθ j=1,2

∂−−−1(unvm) =i2πn(unvm), ∂−−−2(unvm) =i2πm(unvm).

symmetric(∂−−−j(a∗) =∂−−−j(a)∗), commuting(∂−−−1◦∂−−−2=∂−−−2◦∂−−−1),

trace-compatibility(R

−−−◦∂−−−j=0).

•universal Chern number:

C₁:Proj(A_θ)→_Z

C1(p) := _i21_π

R

− −

•Letd∈A_θ (such thatd=d∗). IfI_$σ(d)is connected then

p:= 1

i2π

I

CI

1

d−λ dλ ∈ Proj(Aθ), p6=0,1.

•Ifθ=M/N then

σ(d) =I1∪. . .∪IN∗, 16N∗6N

Ij⊂Rdisjointintervals. Moreover

Rk:=NR

−

−−_:Proj(A_M/N)−→ {0, . . . ,N},

Rk(p) =0 (resp. =1) iffp=0 (resp.=1).

•SinceΠq,r is faithful for anyq,r, one has that the spectral

projection ofΠq,r(d)∈B(Hq)related to the energy bandIj is

THEOREM (D. - Landi, D. - Panati, weak form)

For anyp∈Proj(A_M/N)letCq,r(p) :=C(Πq,r(p)), then

N Cq,r(p)−(qM−rN) C1(p) =qRk(p).

•IfN is oddh=u+u−1+v+v−1hasNdisjoint energy bands. Sincep₁⊕. . .⊕p_N=1then Rk(p_j) =1 and|C₁(p_j)|₆N/2;

N Cq,r(pj)−(qM−rN) C1(pj) =q ∀ j=1, . . . ,N.

•LetPj:=p1⊕. . .⊕pj (j-th gap projection), then

N Cq,r(Pj)−(qM−rN) C1(Pj) =q j ∀ j=1, . . . ,N.

•Letq=1 andr =0. Setsj:=−C1(Pj)andC(gj) :=C1,0(Pj),

then

NC(g_j) +Ms_j=j

Colored butterflies (Hofstadter, 2004)

θ=N_M =1₄, j=2, P2=p1⊕p2, Rk(P2) =1+2=3

4C(g3) | {z }

=1

+1 −C₁(P₃)

| {z }

=−1

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis Geometric duality formula

•Ifθ∈R\Q(irrational condition)

R

−−−:Proj(A_θ)→_Z+θZ∩[0,1]

(Pimsner-Voiculescu, 1980) R

−−−₍p) =M(p)−θ C1(p)

whereM :Proj(A_θ)→_Zuniquelyspecified by 0₆R

− −−₆1.

•Letd_θ =f(u,v)where the dependence inθ is only in the

commutation relation betweenuandv.d_θ has astable gapif

for allθ∈(a,b)⊂R there exists a ε∗∈R\σ(dθ).

•LetPθ the spectral projection ofdθ related to(−∞,ε∗), then

Ifθ=M/Nandp∈AM/N

NCq,r(p)−(qM−rN) C1(p) =q Rk(p)

↓

Cq,r(p) =q

Rk(p)

N +

qM N −r

C1(p)

↓

Cq,r(p) =q −−R−(p) +θC1(p)

−r C₁(p).

THEOREM (D. - Landi, strong form)

Letd_θ ∈A_θ selfadjoint with astable gapat energyε∗for all

θ∈(a,b)andPθ the spectral projection in(−∞,ε∗). Then

Cq,r(Pθ) =qM(Pθ)−r C1(Pθ)

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis

Geometric duality formula

•Bundle representation:

There exists a rankNHermitian vector bundleEq→T2and a

unitary map

Fq,r :Hq−→L2(Eq), (generalized Bloch - Floquet)

such that

A_M/N Π−→q,r Πq,r(AM/N)

Fq,r...Fq−,r1

−→ Γ(End(Eq))$B(L2(Eq)).

MoreoverEq isnot trivialandC(Eq) =q.

•From universal projections to subbundles:

IfP(·)∈Γ(End(Eq))then

G

k∈T2

RangeP(k) −→ _T2, subbundleofEq,

then for anyq,r

•Untwisting (continuous) mapsof the two-dimensional torus:

T23(eik1_,eik2_)7−→α _(eiNk1_,eik2_)∈T2

T23(eik1_,eik2_)7−→β _(eiM0k1_,eik2_)∈T2

M0:=qM−rN.

•Determinant Harper line bundle:

ι :det(Eq)→T2, C(det(Eq)) =C(Eq) =q

THEOREM (D. - Landi, D. - Panati, geometric duality)

For anyP∈Proj(A_M/N)the exists areferencevector bundle

L0(P)→T2such that

α∗Lq,r(P)'β∗L0(P)⊗det(Eq)

Outline

1 Introduction

Spectrum of the Harper equation Hall conductivity and TKNN-equation “Dirac-like” Harper Hamiltonians

2 Generalized TKNN-equations

Noncommutative Torus and(q,r)-representations The main result (weak form)

The main result (strong form)

3 The geometric analysis

Geometric duality formula

•Simmetries:

For anyq,r andθ =M/N:

S₁(q,r):=T 1

qε

1 ⊗U

a

q, S

(q,r)

2 :=T 1

q

2 ⊗V

−1

q , [ra]q=−1

S₁(q,r);S(₂q,r)

=

S₁(q,r); Πq,r(AM/N)

=

S₂(q,r); Πq,r(AM/N)

=0

•Simultaneous generalized eigenspace:

For anyk∈_R2letΥk ∈Dq0 :=D0(R)⊗Cqsuch that

S₁(q,r)Υk =ei2πk1 Υk, S₂(q,r)Υk=ei2πk2 Υk.

Then

Υk=

N−1

∑

j=0

cjζ₍j_q,r)(k) where ζ₍j_q,r)(k) :=

     

ζ₍j_q,r)(k)|(·;0) ζ₍j_q,r)(k)|(·;1)

.. .

ζ₍j_q,r)(k)|(·;q−1)

•Explicit formula for the components:

ζ₍j_q,r)(k)|(·;`) :=

r

|M₀| N _m

∑

−i2πk1(τ`+mq)

δ

· −M0

N (k2+j)−mM0−τ` M0

q

where for`∈ {0, . . . ,q−1}, the permutationτ:`7→τ`is defined

by`= [τ`rN]qand the Dirac delta functionδ(· −x0)acts as the evaluation inx0.

•Pseudo-periodic conditions:

Let~ζ₍q,r)(·):= (ζ(0q,r)(·), . . . ,ζ

N−1

(q,r)(·)). For alln1,n2∈Z

~_ζ

(q,r)(k1+n1,k2+n2) = GN,q(k1,k2)

n2_·~

ζ₍q,r)(k1,k2)

GN,q(k) :=

     

0 1 . . . 0

0 0 . .. ... ..

. ... . .. 1 ei2πqk1 _{. . . 0 0}

•The vector bundle: Consider the projection

P(k) := N−1

∑

j=0

|ζ₍j_q,r)(k)ihζ₍j_q,r)(k)|.

For anyn= (n1,n2)∈Z2it transforms as

P(k+n) = GN,q(k)

n2

P(k) GN,q(k)

−n2 .

TheK-theory classofP(·)depends only on[k]∈_R2_/Z2_{,Then it}

describesι :EN,q→T2, with the fibers given by

ι−1(k) =P(k)Dq0.

•The curvature:P(·)provides a canonical (Berry,

Grassmannian,Levi-Civita, ...) connection and a curvature

∇(B):=P◦d, (∇(B))2:=P(dP∧dP).

The related Chern class is

c1(EN,q)=

i

2πTrN[(∇

(B)₎2_{] =q dk}