TKNN-equations
for
Dirac-like operators
Giuseppe De Nittis
(LAGA,Université Paris 13)——————————————————————————–
IVrd MathematicalMethods inQuantumMechanics
Bressanone, Italy. February 14-19, 2011
——————————————————————————–
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
Hθ(1):=Dθ+K
•The spectrum ofHθ(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(Pj) +M sj=Rj, 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:
Hθ(1):=Dθ+K on H1:=L2(R).
•The (2-dimensional,1-twisted)θ-Harper Hamiltonian:
Hθ(2,1):= K Dθ−12 Dθ−1
2
−K
!
on H2:=L2(R)⊗C2.
•Hθ(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).
•Hθ(1)andHθ(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:
Hθ(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∗-algebraAθ
u∗=u−1, v∗=v−1, uv=ei2πθvu,
θ∈R,
withuniversal normkak:=sup{kπ(a)kH |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∗∈Aθ (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(Aθ)→Z
C1(p) := i21π
R
− −
•Letd∈Aθ (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(AM/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(AM/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⊕. . .⊕pN=1then Rk(pj) =1 and|C1(pj)|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(gj) +Msj=j
Colored butterflies (Hofstadter, 2004)
θ=NM =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(Aθ)→Z+θZ∩[0,1]
(Pimsner-Voiculescu, 1980) R
−−−(p) =M(p)−θ C1(p)
whereM :Proj(Aθ)→Zuniquelyspecified by 06R
− −−61.
•Letdθ =f(u,v)where the dependence inθ is only in the
commutation relation betweenuandv.dθ 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)
Letdθ ∈Aθ 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
AM/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(AM/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(jq,r)(k) where ζ(jq,r)(k) :=
ζ(jq,r)(k)|(·;0) ζ(jq,r)(k)|(·;1)
.. .
ζ(jq,r)(k)|(·;q−1)
•Explicit formula for the components:
ζ(jq,r)(k)|(·;`) :=
r
|M0| N m
∑
∈Ze−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(jq,r)(k)ihζ(jq,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