lecture 1

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Models for conductance

in presence of a magnetic field

Giuseppe De Nittis

Mathematical Physics Sector of:

SISSA,InternationalSchool forAdvancedStudies, Trieste

mini-course at:

Mathematisches Institut, Universität Tübingen

January 18-21, 2010

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Overview of the course

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•The study of the conductance in presence of a uniform magnetic field is an old and fascinating problem. In 1879 E. H. Hall inferred from the Maxwell’s equations the existence of the

transverse currents(classical Hall effect)[Hal79].

•At very low temperature (T ∼0 Ko) the quantum effects become prominent. In 1980 K. von Klitzing (et al.) observed the Quantum Hall Effect (QHE), i.e. the quantization of the

transverse conductance[KDP80].

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Experimental setting for the QHE

Gas of 2-dimensional free magnetic-Bloch-electrons(2DMBE),

Γ'_Z2_{crystal lattice,}_B _{uniform orthogonal magnetic field .}

Dimensionless parameter:hB:= _BSΦ0

Γ,SΓ fundamental cell

surface,Φ0:= hce (magnetic flux quantum).

Hofstadter regime:hB1 (B→0, usual experimental setting).

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Quantization of the conductance

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Guiding ideas for a theoretical explanation of the QHE

The pioneering work[TKNN82]

paved the way for the explanation of theQHE in terms of TQN.

•The paper is a collection ofvery interesting ideas.

•The ideas are developedwithout mathematical rigor.

•The mini-course is a“rigorous revisitation”of the[TKNN82].

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1stidea:The relevant physical regime ishB1. The realistic model is reduced to a simpler model (Hofstadter model) via a (non-rigorous) perturbative argument (hB−1small parameter) .

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1stidea:The relevant physical regime ishB1. The realistic model is reduced to a simpler model (Hofstadter model) via a (non-rigorous) perturbative argument (hB−1small parameter) .

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2ndidea:The hypothesis ofrational flux(hB=N/M) implies a

Z2-symmetry. A family of Hilbert spaces emerges by a “simultaneous diagonalization”. Avector bundle structure

“seems” to appear from the underlying symmetry.

Lecture IIAnalysis of the link betweensymmetryand

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2ndidea:The hypothesis ofrational flux(hB=N/M) implies a

Z2-symmetry. A family of Hilbert spaces emerges by a “simultaneous diagonalization”. Avector bundle structure

“seems” to appear from the underlying symmetry.

Lecture IIAnalysis of the link betweensymmetryand

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3rd_idea:_{Kubo formula}_{(linear response) is used to proof}

conductance∝(first) Chern numbers(TQN).

A rigorous justification for the Harper regime is still anopen problem. We need to proof

σH(P) =

e2

2π}Tr(P[[P;Qx]; [P;Qy]]).

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3rd_idea:_{Kubo formula}_{(linear response) is used to proof}

conductance∝(first) Chern numbers(TQN).

A rigorous justification for the Harper regime is still anopen problem. We need to proof

σH(P) =

e2

2π}Tr(P[[P;Qx]; [P;Qy]]).

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4thidea:Duality(algebraic relation) between the values of the transverse conductance in the two limit cases (no proof!).

Lecture IIIDerivation of the geometric duality between the Hofstadter and Harper models. A rigorous proof of the

TKNN-equation

MC_Hof(P) +NCHar(P) =1

hB=

M N

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4thidea:Duality(algebraic relation) between the values of the transverse conductance in the two limit cases (no proof!).

Lecture IIIDerivation of the geometric duality between the Hofstadter and Harper models. A rigorous proof of the

TKNN-equation

MC_Hof(P) +NCHar(P) =1

hB=

M N

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Lecture I

Derivation of the effective dynamics:

Hofstadter and Harper models

joint work with:

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Outline

1 The physical model

The Bloch-Landau Hamiltonian

2 Generalities on SAPT and Hofstadter regime

Some comments Ingredients for SAPT Main steps for SAPT The Hofstadter model

3 SAPT for the Harper regime

Separation of the scales and adiabatic parameter Semiclassical symbol and gap condition

Formal expansion of the semiclassical symbol Super-adiabatic projection and intertwine unitary The Harper model

4 The effect of a periodic magnetic field

Some comments aboutAΓ₌₀

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Bloch-Landau Hamiltonian

On thephysical state spaceHphy:=L2(R2,dx dy)

HBL:= 1 2

"

−i ∂

∂x +A

Γ x− π hB y 2 + −i ∂

∂y +A

Γ y+ π hB x 2#

+VΓ.

•x,y,HBL,VΓ,AΓ:= (AΓx,AΓy),hB,dimensionlessquantities, i.e.

E0:=}2/mSΓ≡1.

•Γ :={γ=n1a+n2b, n1,n2∈Z}periodic crystal structure.

a,b∈_R2_,_a_∧_b₆₌_{0. To simplify}_a_·_b₌₀_and_|_a_|₌_|_b_|₌₁_{, i.e.}

Γ =Z2(orthogonal lattice).

•VΓ:R2→_Ris theΓ-periodicelectrostatic potential.

•AΓ:R2→_R2_{is the}_Γ_-periodic_{vector potential}_.

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Outline

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•SAPT is a very general theory with “its own philosophy” and a robust “mathematical machinery” applicable to any physical system which has “3 relevant ingredients” (cfr.[PST03a]).

•SAPT can be applied to study the Bloch electron in a slowly varying (weak) external magnetic field[PST03b]. The original proof does not include the case of aweak external uniform magnetic field(Hofstadter regime) nor the contribute of a

periodic internal magnetic field(AΓ6=0).

•With few modifies the original proof can be generalized to include the case of the Hofstadter regime and aAΓ6=0[DFP1]

(derivation of theHofstadter model).

•I will use the case of the Hofstadter regime to do a quick review about the philosophy of the SAPT. In the second part I will describe the SAPT for the “new case” of the Harper regime

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Outline

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1stingredient:

Separation intoslowandfast degrees of freedomquantified by means of a dimensionlessadiabatic parameter.

Hofstadter regime

Adiabatic parameterε0:=_h1_B1

H_BL=1

2

h

−i∇+AΓ+e⊥∧ ε0r

i2

+VΓ

r:= (x,y,0),e⊥:= (0,0,π).

PROPOSITION(regularity of the microscopic potentials)

IfAΓ∈C1(R2)andVΓ∈L2_loc(R2)then HBLisessentially

self-adjointon C∞

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1stingredient:

Separation intoslowandfast degrees of freedomquantified by means of a dimensionlessadiabatic parameter.

Adiabatic parameterε0:=_h1_B1

HBL= 1 2

h

−i∇+AΓ+e⊥∧ ε0r

i2

+VΓ

r:= (x,y,0),e⊥:= (0,0,π).

IfAΓ∈C1(R2)andVΓ∈L2_loc(R2)then HBLisessentially

self-adjointon C∞

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2ndingredient:

Unitary decompositionof the “physical” Hilbert space

Hphy→Hs=slow⊗Hf=fast.

Unitary decompositionZak-Bloch-Floquet transformZ

L2(R2)−→Z H_τ:=

n

ψ∈L2_loc(R2;Hf) : ψ(k−γ∗;·) =τ(γ∗)ψ(k;·)

o

τ:Γ∗→U(Hf), τ(γ∗) :=multiplication by eiθ·γ

∗

,Γ∗'2πΓdual lattice.

Hτ'L

2₍

T2B,d2k)

| {z }

Hs

⊗L2(T2V,d2θ)

| {z }

Hf

T2V :=R2/Γ(fundamentalorVoronoicell)

T2B:=R2/Γ

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2ndingredient:

Unitary decompositionof the “physical” Hilbert space

Hphy→Hs=slow⊗Hf=fast.

Unitary decompositionZak-Bloch-Floquet transformZ

L2(R2)−→Z H_τ:=

n

ψ∈L2_loc(R2;Hf) : ψ(k−γ∗;·) =τ(γ∗)ψ(k;·)

o

τ:Γ∗→U(Hf), τ(γ∗) :=multiplication by eiθ·γ

∗

,Γ∗'2πΓdual lattice.

Hτ'L

2₍

T2B,d2k)

| {z }

Hs

⊗L2(T2V,d2θ)

| {z }

Hf

T2V :=R2/Γ(fundamentalorVoronoicell)

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3rdingredient:

The Hamiltonian is realized as the “quantization” of a symbol (operator-valued) on theclassical spaceof the slow degrees of freedom (small commutator). The (principal part of the) symbol has arelevant partof the spectrumadiabatically decoupled

from the rest.

HBLZ ...Z−1

7−→ HZ := 1

2

h

−i∇θ+AΓ(θ) +k+e⊥∧ iε0∇τ_k

i2

+VΓ(θ)

=Op_ε₀(H0) (up to a domain restriction !)

whereD(∇τ_k) =Hτ∩Hloc1 (R2;Hf)and

H0(k,η) := 1 2

h

−i∇θ+A

Γ₍

θ) +k+e⊥∧ η

i2

+VΓ(θ)

withquantization ruleOp_ε

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3rdingredient:

The Hamiltonian is realized as the “quantization” of a symbol (operator-valued) on theclassical spaceof the slow degrees of freedom (small commutator). The (principal part of the) symbol has arelevant partof the spectrumadiabatically decoupled

from the rest.

HBLZ ...Z−1

7−→ HZ := 1

2

h

−i∇θ+AΓ(θ) +k+e⊥∧ iε0∇τ_k

i2

+VΓ(θ)

=Op_ε₀(H0) (up to a domain restriction !)

whereD(∇τ_k) =Hτ∩Hloc1 (R2;Hf)and

H0(k,η) := 1 2

h

−i∇θ+A

Γ₍

θ) +k+e⊥∧ η

i2

+VΓ(θ)

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•D(H0(k,η)) =H2(T2V) =:D is an Hilbert space with respect thegraph-normk(1−∇θ)· kHf.

•H0:T∗R2('R2×R2)→B(D,Hf),operator-valued symbol.

•κ:T∗R2→R2minimal coupling function

κ(k,η) :=k+e⊥∧ η (kinetic momentum). H0isτ-equivariantinκ , i.e.

H0(κ(k,η)) =H0([κ(k,η)]

| {z }

“modulo”Γ∗

−γ∗) =τ(γ∗)H0([κ(k,η)])τ(γ∗)−1

•The spectrum ofH₀(k,η)ispure point,{En(k,η)}n∈N.

T∗_R23(k,η)7−→E∗ E∗(κ(k,η))∈R ∗th_{-energy Bloch band}_{. Smooth (only continuous at the}

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Outline

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Step 1)

Construction of thesuper-adiabatic projectionΠε _{(relevant part} of the spectrum).ΠεH

phyisapproximately invariantunder the evolution generated by the physical HamiltonianH.

Πε0 ₌_Op

ε0(π)with

π(κ(k,η))

∞

∑

j=0

ε0jπj(κ(k,η)) πj∈Sκ,τv (B(Hf,D))∩Sκ,τ1 (B(Hf))

•According to[PST03a],π is builded recursively starting from

π0(κ(k,η)) :=|ϕ∗(κ(k,η))ihϕ∗(κ(k,η))| H0(·)ϕ∗(·) =E∗(·)ϕ∗(·)

•By construction[HZ; Πε0_]₌O₍_ε∞

0)(approximately invariant).

•Letικ:T∗R2→T∗R2defined byικ(k,η) := (κ(k,η),η), then

S_κ,τv :=S_τv◦ικ. It followsSκ,τv ⊂Sv

0

τ withv

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Step 1)

Construction of thesuper-adiabatic projectionΠε _{(relevant part} of the spectrum).ΠεH

phyisapproximately invariantunder the evolution generated by the physical HamiltonianH.

Πε0₌_Op

ε0(π)with

π(κ(k,η))

∞

∑

j=0

ε0jπj(κ(k,η)) πj∈Sκ,τv (B(Hf,D))∩Sκ,τ1 (B(Hf))

•According to[PST03a],π is builded recursively starting from

π0(κ(k,η)) :=|ϕ∗(κ(k,η))ihϕ∗(κ(k,η))| H0(·)ϕ∗(·) =E∗(·)ϕ∗(·)

•By construction[HZ; Πε0_]₌O₍_ε∞

0)(approximately invariant).

•Letικ:T∗R2→T∗R2defined byικ(k,η) := (κ(k,η),η), then

S_κ,τv :=S_τv◦ικ. It followsSκ,τv ⊂Sv

0

τ withv

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Step 2)

Construction of anintertwine unitaryUε _{which maps the} “complicated” subspaceΠεH

phy(ε-dependent) into a “simple”

reference spaceHref(ε-independent).

Uε0₌_Op

ε0(u)with

u(κ(k,η))

∞

∑

j=0

ε0j uj(κ(k,η)) uj∈Sκ,τ1 (B(Hf))

•According to[PST03a],uis builded recursively starting from

u0(κ(k,η)) :=|ϕ∗(κ(k0,η0))

| {z }

:=χ

ihϕ∗(κ(k,η))|+u⊥₀(κ(k,η)).

•Πr:=Uε0Πε0Uε0−1=Opε0(u]π]u

−1_{) =}_Op

ε0(|χihχ|)

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Step 2)

Construction of anintertwine unitaryUε _{which maps the} “complicated” subspaceΠεH

phy(ε-dependent) into a “simple”

reference spaceHref(ε-independent).

Uε0₌_Op

ε0(u)with

u(κ(k,η))

∞

∑

j=0

ε0j uj(κ(k,η)) uj∈Sκ,τ1 (B(Hf))

•According to[PST03a],uis builded recursively starting from

u0(κ(k,η)) :=|ϕ∗(κ(k0,η0))

| {z }

:=χ

ihϕ∗(κ(k,η))|+u⊥₀(κ(k,η)).

•Πr:=Uε0Πε0Uε0−1=Opε0(u]π]u

−1_{) =}_Op

ε0(|χihχ|)

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Step 3)

The dynamics insideΠεH _{is generated by}_Πε_H_Πε _(diagonal part) . The analysis is simpler inHrefwhere theeffective

HamiltonianHε

eff:=UεΠεHΠεUε −1

determines the dynamics (of slow modes).Hε

eff is the quantization of a semiclassical symbol hwhich admits a powers expansion inε.

Hε0

eff:=U

ε0_Πε0_HZ_Πε0_Uε0−1₌_Op

ε0(h)

where (expanding theMoyal product])

h(κ(k,η)) :=u]π]H0]π]u−1

∞

∑

j=0

ε0j hj(κ(k,η)).

•At leading order

h0(·) =|χihϕ∗||ϕ∗ihϕ∗|H0|ϕ∗ihϕ∗||ϕ∗ihχ|=E∗(·)|χihχ|

endHε0

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Step 3)

The dynamics insideΠεH _{is generated by}_Πε_H_Πε _(diagonal part) . The analysis is simpler inHrefwhere theeffective

HamiltonianHε

eff:=UεΠεHΠεUε −1

determines the dynamics (of slow modes).Hε

eff is the quantization of a semiclassical symbol hwhich admits a powers expansion inε.

Hε0

eff:=U

ε0_Πε0_HZ_Πε0_Uε0−1₌_Op

ε0(h)

where (expanding theMoyal product])

h(κ(k,η)) :=u]π]H0]π]u−1

∞

∑

j=0

ε0j hj(κ(k,η)).

•At leading order

h0(·) =|χihϕ∗||ϕ∗ihϕ∗|H0|ϕ∗ihϕ∗||ϕ∗ihχ|=E∗(·)|χihχ|

endHε0

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Outline

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Reference Hilbert space:L2₍_T2

B)withT2B'[0,2π)2(BZ).

Effective Hamiltonian:Hε0

eff:=E∗(K1,K2)where

K1:=k1−iπ ε0

∂

∂k2

, K2:=k2+iπ ε0

∂

∂k1 .

Hofstadter unitaries:

U0:=eiK1, V0:=eiK2, U0V0:=ei2π ε0V0U0.

Peierls substitution:ei(nk1+mk2)7→_ei(nK1+mK2)₌_e−iπnmε0_U

0nV0m.

E∗(k1,k2) =∑an,mei(nk1+mk2)(defined on BZ)

E∗(k1,k2)7−→

∑

an,me−iπnmε0U0nV0m=:Heffε0(U0,V0)

Hofstadter model:LetE∗(k1,k2) =2[cos(k1) +cos(k2)], then

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Outline

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LetPx:=−i∂x,Qx:=multiplication byx (similarly fory),Γ =Z2 and assume thegauge conditions

∇·AΓ=0,

Z

T2 V

AΓ(r)d2r =0.

The Bloch-Landau Hamiltonian becomes

HBL= 1 2

"

Px− π hB Qy 2 +

Py+ π hB

Qx

2#

+VΓ,A+fW

where the potentials are

VΓ,A(Qx,Qy) :=VΓ(Qx,Qy) + 1 2|A

Γ_|2₍_Q_x,_Q y)

f

W(Qx,Qy) :=AΓx(Qx,Qy)

Px− π hB

Qy

+AΓ_y(Qx,Qy)

Py+ π hB

Qx

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Adiabatic parameter:ε∞:=hB1.

Auxiliary parameter:δ :=

q

ε∞

2π1.

New variables:       

K1:=δ

Py+ 1 2δ2Qx

K2:=δ

Px− 1 2δ2Qy

      

G1:=δ2

Qx 2δ2

−Py

G2:=δ2

Qy 2δ2+Px

X_(fast):= (K1,K2)(kinetic momenta),

X_(slow):= (G1,G2)(centre of the cyclotron orbit).

Commutation relations:

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In the new variables:

HBL= 1

δ2Ξ(K1,K2)+V Γ,A₍_G

1+δ K1

| {z }

Qx

,G2−δ K2

| {z }

Qy

)+1

δW(K1,K2,G1,G2).

Ξ:= 1

2

K12+K22

W :=AΓ_x(·,·)K2+AΓy(·,·)K1.

•σ(Ξ) ={λn:= (n+1₂) : n∈N}(Landau levels).

•t=(slow)microscopic time-scale. Thecyclotron frequency

ωc:= |q|B_mc = 1 δ2

1

} fixes the naturalultramicroscopic time-scale

τ:=ωctfor the fast motion.

HBLψ=i}∂

∂tψ=i

1

δ2

∂

∂τψ.

Relevant Hamiltonianfor the Harper regime:

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In the new variables:

HBL= 1

δ2Ξ(K1,K2)+V Γ,A₍_G

1+δ K1

| {z }

Qx

,G2−δ K2

| {z }

Qy

)+1

δW(K1,K2,G1,G2).

Ξ:= 1

2

K12+K22

W :=AΓ_x(·,·)K2+AΓy(·,·)K1.

•σ(Ξ) ={λn:= (n+1₂) : n∈N}(Landau levels).

•t=(slow)microscopic time-scale. Thecyclotron frequency

ωc:= |q|B_mc = 1 δ2

1

} fixes the naturalultramicroscopic time-scale

τ:=ωctfor the fast motion.

HBLψ=i}∂

∂tψ=i

1

δ2

∂

∂τψ.

Relevant Hamiltonianfor the Harper regime:

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TheStone-von Neumann theoremassures that there exists a unitary mapW (2ndingredient).

Hphy=L2(R2,dx dy)−→W L2(R,dxs)

| {z }

=:Hs

⊗L2(R,dxf)

| {z }

=:Hf

such that

(G1,G2)W ...W−1

7−→ (Qs:=xs,Ps:−iδ2∂s)

(K1,K2)W ...W−1

7−→ (Qf:=xf,Pf:−i∂f).

The relevant Hamiltonian becomes

δ2HBLW ...W−1

7−→ HW =1s⊗Ξ+δ W+δ2VΓ,A

Ξ:= 1

2

Pf2+Qf2

(quadratic, harmonic oscillator)

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Outline

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•HW coincides with theWeyl-quantizationof the symbol

H_δ(ps,qs) := Ξ +δ W(·,·) +δ2VΓ,A(qs+δ Qf,ps−δ Pf)

∞ : (ps,qs)7→(Ps,Qs).

•H_δ(ps,qs)is anunboundedoperators onHf. The principal symbolHδ=0= Ξ(harmonic oscillator) has a basis of

eigenvectors (Hermite functions). LetF be the closure of the linear span of the basis with respect thegraph normkΞ · k.F is an Hilbert space andΞ∈B(F,Hf).

LetAΓ,VΓ∈C∞

b(R2). Then

(R×_R')T∗R3(ps,qs) H_δ

7−→H_δ(ps,qs)∈Bs.a.(F,Hf).

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∞ : (ps,qs)7→(Ps,Qs).

LetAΓ,VΓ∈C∞

b(R2). Then

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∞ : (ps,qs)7→(Ps,Qs).

LetAΓ,VΓ∈C∞

b(R2). Then

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Outline

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LetF={VΓ,A_,_AΓ

x,AΓy}.F issmoothandZ2-periodicthen

F(Qx,Qy) :=

∑

n,m∈Z

fn,mei2πnQxei2πmQy

slow-fast variables+W

7→

_∑

n,m∈Z

fn,m ei2πn(Qs+δQf)ei2πm(Ps−δPf)

commutation relations

=

_∑

n,m∈Z

fn,m ei2π(nQs+mPs)ei2π δ(nQf−mPf)

“de-quantization”,Op_ε

∞

−1

7→

_∑

n,m∈Z

fn,m ei2π(nqs+mps)ei2π δ(nQf−mPf)

=F(qs+δ Qf,ps−δ Pf)=:Fδ(ps,qs) whereOp_ε

∞:e

i2π(nqs+mps)_7→_ei2π(nQs+mPs)_⊗₁

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δ-power “formal” expansion of symbols.

LetIn,m:= (nQf−mPf).

δkF_δ(ps,qs) =

∑

n,m∈Z

δkfn,m ei2π(nqs+mps)ei2π δIn,m

on a dense set ofanalytic (Hermite) vectors

=

_∑

n,m∈Z

δkfn,m ei2π(nqs+mps)

+∞

∑

j=0

(i2π δ)j j! In,m

j

!

uniform convergence on the set of analytic vectors

=

+∞

∑

j=0

δj+k

"

(i2π)j

j! _n,m

∑

_∈_Zfn,m e

i2π(nqs+mps)_I

n,mj #

=:

+∞

∑

j=0

δj+k Fj+k(ps,qs)

Danger!!whenjincreasesFj+k becomes “more unbounded”, (jthdeg. inQfandPf). Whenj>2the domain of definition of

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δ-power “formal” expansion of symbols.

LetIn,m:= (nQf−mPf).

δkF_δ(ps,qs) =

∑

n,m∈Z

δkfn,m ei2π(nqs+mps)ei2π δIn,m

on a dense set ofanalytic (Hermite) vectors

=

_∑

n,m∈Z

δkfn,m ei2π(nqs+mps)

+∞

∑

j=0

(i2π δ)j j! In,m

j

!

uniform convergence on the set of analytic vectors

=

+∞

∑

j=0

δj+k

"

(i2π)j

j! _n,m

∑

_∈_Zfn,m e

i2π(nqs+mps)_I

n,mj #

=:

+∞

∑

j=0

δj+k Fj+k(ps,qs)

Danger!!whenjincreasesFj+k becomes “more unbounded”, (jthdeg. inQfandPf). Whenj>2the domain of definition of

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H_δ(ps,qs) := Ξ +

∞

∑

j=1

δj Hj(ps,qs)

| {z }

:=Wj(ps,qs)+V_jΓ,A(ps,qs)

V_j+2Γ,A(ps,qs) :=

(i2π)j

j! _n,m∈Z

∑

vn,me

i2π(nqs+mps)_I

n,mj, (V₁Γ,A:=0)

Wj+1(ps,qs) :=

(i2π)j

j! _n,m∈

∑

_Ze

i2π(nqs+mps)_I

n,mj axn,mPf+a y n,mQf

Remark!!SinceΞis quadratic (inP_f andQ_f) we can control only quadratic perturbation.degV_kΓ,A=k−2,degWk =k.

e Hω

δ (ps,qs) := Ξ + 2ω+2

∑

j=1

δjHj(ps,qs)

(

ω=0 if AΓ6=0 ω=1 if AΓ=0 e

Hω

δ is theorderδ

(2ω+2)_{approximate symbol}_{(quadratic part).}

Rω

δ(ps,qs):=Hδ(ps,qs)−Heω

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H_δ(ps,qs) := Ξ +

∞

∑

j=1

δj Hj(ps,qs)

| {z }

:=Wj(ps,qs)+V_jΓ,A(ps,qs)

V_j+2Γ,A(ps,qs) :=

(i2π)j

j! _n,m∈Z

∑

vn,me

i2π(nqs+mps)_I

n,mj, (V₁Γ,A:=0)

Wj+1(ps,qs) :=

(i2π)j

j! _n,m∈

∑

_Ze

i2π(nqs+mps)_I

n,mj axn,mPf+a y n,mQf

Remark!!SinceΞis quadratic (inPf andQf) we can control only quadratic perturbation.degV_kΓ,A=k−2,degWk =k.

e Hω

δ (ps,qs) := Ξ + 2ω+2

∑

j=1

δjHj(ps,qs)

(

ω=0 if AΓ6=0 ω=1 if AΓ=0 e

Hω

δ is theorderδ

(2ω+2)_{approximate symbol}_{(quadratic part).}

Rω

δ(ps,qs):=Hδ(ps,qs)−He ω

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Control of the order of the remainder

PROPOSITION([DFP1])

i)Heω

δ is self-adjoint with domainF (Kato-Rellich,δ<δ0); ii)Rω

δ is aB(F,Hf)-valued symbol and

kRω

δkB(F,Hf)6δ

(2ω+2)_C_.

Letπr:=∑mi=1|ψkiihψki|be the projection on the subspace

spanned by a finite family{ψki}i=1,...,m of eigenvectors (Hermite

functions) ofΞ(relevant part of the spectrum).

iii)Rω

δπrandπrR ω

δ areB(Hf)-valued symbols; iv)kR_δπrk_B(Hf)=kπrRδkB(Hf)6δ

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Outline

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THEOREM (Invariant and reference subspace[DFP1])

Step 1.There exists an projectionΠε∞:=Op

ε∞(π)onHs⊗Hf

[Op_ε

∞(He

ω δ);Π

ε∞_{] =}O₍_δ∞₎_, _[_HW_;_Πε∞_{] =}O₍_δ(2ω+3)₎_.

π(ps,qs)

_∑

j

δj πj(ps,qs), πj∈S1(B(Hf))∩S1(B(Hf,F)).

Principal symbolπ0=πr:=∑mi=1|ψkiihψki|.

Step 2.There exists a unitaryUε∞ :=Op

ε∞(u)onHs⊗Hf

Uε∞_Πε∞_Uε∞−1₌_Π

r:=Opε∞(πr) =1s⊗πr (constant symbol)

u(ps,qs)

∑

j

δj uj(ps,qs), uj∈S1(B(Hf)).

Principal symbolu0=1f.

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Outline

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LetAΓ₌_{0 (}

ω =1). ThenVΓ,A=VΓ,W =0.

e

H_δ1:= Ξ +δ2H2+δ3H3+δ4H4

annihilatora:= √1

2(Qf+iPf),creatora †_:= _√1

2(Qf−iPf). H2(ps,qs) :=VΓ(qs,ps)1f

H3(ps,qs) :=

h

∂1VΓ(qs,ps)

i(a+a†)

√

2 −

h

∂2VΓ(qs,ps)

i(a−a†)

i√2

H4(ps,qs) := [∆VΓ]

Ξ

2+

h

(∂₁2−∂₂2)VΓ

i(a2+a†2)

4 +

h

∂₁₂2VΓ

i(a2−a†2)

i

Letπr:=|ψ∗ihψ∗|the eigenprojection on the levelλ∗:= (n∗+1₂)

πrH2πr=VΓπr, πrH3πr=0, πrH4πr= λ∗

2[∆V

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THEOREM (Effective dynamics in a single band[DFP1])

Let AΓ₌₀₍

ω =1) andπr:=|ψ∗ihψ∗|(single band). h:=u]π]H_δ1]π]u−1, he:=u]π]He1

δ]π]u −1_.

Hε∞

eff:=Opε∞(h)(effective Hamiltonian).

Hε∞

eff−Opε∞(eh) =O(δ

5₎_, _[_Hε∞

eff; Πr] = [Opε∞(eh); Πr] =0 thenHε∞

eff, Opε∞(eh)∈B(Href)withHref'Hs=L

2₍_R_,_dx s).

Πε∞_Op

ε∞(He

1 δ)Π

ε∞ U ε∞

7−→ Op_ε

∞(eh) +O(δ

∞_{) =}_Hε∞

eff+O(δ 5₎

Up to the orderδ4(δ2= ε∞

2π)

Hε∞

eff=λ∗1s+δ 2_Op

ε∞(V

Γ_{) +}

δ4 λ∗

2 Opε∞(∆V

Γ_{) +}_O

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Reference Hilbert space:Hs:=L2(R,dx).

Effective Hamiltonian:Hε∞

eff:=VΓ(Q,P)where

Q:=multiplication byx, P:=−iδ2 ∂

∂x =−i

ε∞

2π

∂

∂x.

Harper unitaries:

U∞:=e

i2πP_, _V

∞:=e

i2πQ_, _U

∞V∞:=e

i2π ε∞_V

∞U∞.

Weyl quantization:ei2π(nq+mp)7→ei2π(nQ+mP)=e−iπnmε∞V

∞nU∞m.

VΓ₍_q_,_p_{) =}

∑vn,mei2π(nq+mp)(Z2-periodic), then

VΓ(q,p)7−→

_∑

vn,m e−iπnmε∞V∞nU∞m=:H

ε∞

eff(U∞,V∞)

Harper model:LetVΓ(q,p) =2[cos(2πq) +cos(2πp)], then

HHar(ε∞) :=U∞+U∞

−1₊_V

∞+V∞

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Outline

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•J. Bellissard in[Bel88]first and H. Helffer and J. Sj ˝ostrand in

[HS89]then, derived the effective Hofstadter and Harper models. However our derivation in[DEF1], based on SAPT, proves the(asymptotical) unitary equivalenceof the effective models with the physical model (relevant to compute

conductance).

•The first non trivial order ofHε∞

eff isδ 2

∝ε∞.

•The term of orderδ vanishes sinceH1=0and the term of orderδ3vanishes sinceH3is anodd polynomialinaanda†.

•IfAΓ6=0, thenH16=0but since it is odd inaanda†it gives no contribution toHε∞

eff in the theory for asingle decoupled band.

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Outline

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LetAΓ6=0 (ω =0),cAthecoupling constantof the periodic magnetic field andcV thecoupling constantof the periodic electric field.

e

H_δ0(ps,qs) = Ξ +δ√cA

2 g a+g a

†

+δ2cVVΓ+O(δ2cA)

whereg(qs,ps) :=AΓy(qs,ps)−iAΓx(qs,ps).

πr:=|ψ∗ihψ∗|+|ψ∗+1ihψ∗+1|,Href'Hs⊗C2. The effective dynamics (up to a constant term) is given by

Hε∞

eff=





1

21s δcA Opε∞(g) δcAOpε∞(g)

† ₋1

21s



+O

δ2cV

.

In generalcAcV, then the approximation is meaningful if

δc_cA

V. H

ε∞

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(67)

References

[Bel88]J. BellissardC∗-algebras in solid state Physics. 2D Electrons in a

uniform magnetic field. Operator algebras and applications II. University Press 1988.

[DFP1]G. De Nittis, F. Faure and G. Panati.Models for conductance in

magnetic field: derivation of Harper and the Hofstadter models. To appear as preprint, January 2010.

[Hal79]E. H. Hall.On a New Action of the Magnet on Electric Current. Am. J.

Math.2, 1879.

[HS89]H. Helffer and J. Sj ˝ostrand.Équation de Schrödinger avec champ

magnétique et équation de Harper, Lecture Notes in Physics.345, 1989.

[KDP80]K. von Klitzing, G. Dorda and M. Pepper.New Method for

High-Accuracy Determination of the Fine-Structure Constant Based on

Quantized Hall Resistance. Phys. Rev. Lett.45, 1980.

[Lau81]R. B. Laughlin.Quantized Hall conductivity in two dimensions. Phys.

Rev. B23, 1981.

[PST03a]G. Panati, H. Spohn and S. Teufel.Space-Adiabatic Perturbation

Theory. Adv. Theor. Math. Phys.7, 2003.

[PST03b]G. Panati, H. Spohn and S. Teufel.Effective dynamics for Bloch