Bloch Electron in a Strong Magnetic Field - WordPress.com · Outline 1 Introduction An overview on our research project 2 The space adiabatic perturbation theory The philosophy The

Bloch Electron in a Strong Magnetic FieldAdiabatic Derivation of the Harper Model

Giuseppe De Nittis

Mathematical Physics SectorSISSA International School for Advanced Studies, Trieste

IIIrd Mathematical Methods in Quantum MechanicsBressanone, February 16-21, 2009

supervisor:

prof. Gianfausto Dell’Antoniobased on joint work with:

Gianluca Panati & Frédéric Faure

Giuseppe De Nittis (SISSA, Trieste) Bloch Electron in a Strong Magnetic Field IIIrd M3Q 1 / 30

Outline

1 IntroductionAn overview on our research project

2 The space adiabatic perturbation theoryThe philosophyThe ingredients

3 Adiabatic theory for the strong magnetic field regimeThe physical modelSeparation of the scales and adiabatic parameterFormal expansion of the semiclassical symbol: order δ 4 approximationThe statement of the main results

4 Is the periodic magnetic field important?Some commentsThe effective model


The research project consists in the solution of 3 different (at mathematicallevel) but related problems:

Adiabatic problem. Starting from the physical model (free electron in a2D lattice with an orthogonal uniform magnetic field), and using thespace adiabatic perturbation theory of Panati, Spohn and Teufel [PST1],[PST2], one wants to deduce rigorously two effective models: the Harpermodel in the strong limit (B−1 → 0) and the Hofstdater model in theweak limit (B → 0).

Algebraic problem. Using the C∗-algebraic framework one wants toshow that the two models are isomorphic at algebraic level(isospectrality) but not unitarly equivalent (different spectral type).

Geometric duality. As a consequence of the non unitary equivalence andusing the tools of the differential geometry one wants to show that thetwo models are characterized by a different topology. This amounts toprove that the two models have different values of the first Chern classrelated by the TKNN formula MCHof +NCHar = 1 when the adiabaticparameter is rational M/N. This explains the structure of the two colorcoded quantum butterflies [Av].












Color-Coded Quantum Butterflies:

Courtesy of J. Avron [Av]


Outline






When the system shows a separation into slow and fast degrees offreedom certain dynamical degrees of freedom lose their autonomousstatus. The fast modes quickly adapt to the slow modes which in turn aregoverned by a suitable effective Hamiltonian. This mechanism is calledadiabatic decoupling. The paradigm is the Born-Oppenheimerapproximation for the motion of nuclei.

The slow degrees of freedom are “semiclassical” in the sense that the fullHamiltonian can be seen as the Weyl quantization of an operator-valuedsymbol on the phase space of the classical degrees of freedom. Theorigin of decoupling can be traced to a spectral property of the symbol inthe sense that it can be decomposed in a relevant separated part. This canbe associated with an almost invariant subspace (invariant under theevolution up to errors small to any order in ε) of the Hilbert space of thesystem.The a. i. subspace depends on ε and is not easily accessible. In order toobtain a useful description of the effective intraband dynamics (effectiveHamiltonian) we need a unitary map from the a. i. subspace into aneasily accessible and ε-independent reference subspace.


When the system shows a separation into slow and fast degrees offreedom certain dynamical degrees of freedom lose their autonomousstatus. The fast modes quickly adapt to the slow modes which in turn aregoverned by a suitable effective Hamiltonian. This mechanism is calledadiabatic decoupling. The paradigm is the Born-Oppenheimerapproximation for the motion of nuclei.The slow degrees of freedom are “semiclassical” in the sense that the fullHamiltonian can be seen as the Weyl quantization of an operator-valuedsymbol on the phase space of the classical degrees of freedom. Theorigin of decoupling can be traced to a spectral property of the symbol inthe sense that it can be decomposed in a relevant separated part. This canbe associated with an almost invariant subspace (invariant under theevolution up to errors small to any order in ε) of the Hilbert space of thesystem.

The a. i. subspace depends on ε and is not easily accessible. In order toobtain a useful description of the effective intraband dynamics (effectiveHamiltonian) we need a unitary map from the a. i. subspace into aneasily accessible and ε-independent reference subspace.


When the system shows a separation into slow and fast degrees offreedom certain dynamical degrees of freedom lose their autonomousstatus. The fast modes quickly adapt to the slow modes which in turn aregoverned by a suitable effective Hamiltonian. This mechanism is calledadiabatic decoupling. The paradigm is the Born-Oppenheimerapproximation for the motion of nuclei.The slow degrees of freedom are “semiclassical” in the sense that the fullHamiltonian can be seen as the Weyl quantization of an operator-valuedsymbol on the phase space of the classical degrees of freedom. Theorigin of decoupling can be traced to a spectral property of the symbol inthe sense that it can be decomposed in a relevant separated part. This canbe associated with an almost invariant subspace (invariant under theevolution up to errors small to any order in ε) of the Hilbert space of thesystem.The a. i. subspace depends on ε and is not easily accessible. In order toobtain a useful description of the effective intraband dynamics (effectiveHamiltonian) we need a unitary map from the a. i. subspace into aneasily accessible and ε-independent reference subspace.


Outline






I) The physical state space of the system decomposes (up to a unitarytransform W), as

HphyW−→ L2(Rd)⊗Hf

where L2(Rd) =: Hs is the state space of the slow degrees of freedomand Hf an (arbitrary separable) state space of the fast degrees offreedom. The classical phase space of the slow degree of freedom is thusT∗Rd 'R2d with z := (ps,xs) ∈R2d.

II) The physical Hamiltonian H (up to a unitary transform), generating thetime-evolution, is given as the Weyl quantization of a semiclassicalsymbol Hε : T∗Rd →B(D,Hf) where D is a suitable normed domainon which Hε is bounded. “Semiclassical” means that Hε admits a formalexpansion Hε(z) ∑j ε j Hj(z) whose principal symbol is H0(z).

III) Constant Gap Condition (CGC): for any z ∈ T∗Rd the spectrum σ(z) ofH0(z) contains a relevant subset σ∗(z) which is uniformly separated fromthe remainder by a gap, namely

infz∈T∗Rd

dist(σ(z)\σ∗(z),σ∗(z)) = Cg > 0.







infz∈T∗Rd

dist(σ(z)\σ∗(z),σ∗(z)) = Cg > 0.







infz∈T∗Rd

dist(σ(z)\σ∗(z),σ∗(z)) = Cg > 0.


Relevant separated part of the spectrum:


Outline






The Bloch-Landau Hamiltonian

HBL :=1

2m

[−i~∇− q

c~AΓ (~x)− q

c~A(~x)

]2+VΓ (~x)

It acts on the physical Hilbert space Hphy := L2(R2). Γ ⊂R2 is the latticespanned by ~a;~b (non orthogonal in general) such thatΩΓ := (axby−bxay) > 0 is the volume of the fundamental cell MΓ of thelattice. ~AΓ and VΓ are Γ-periodic and smooth.

~A(~x) :=B2~ez∧~x =

B2

(−y,x) (symmetric gauge)

where~ez is the normalized vector orthogonal to Γ.


The Bloch-Landau Hamiltonian

HBL :=1

2m

[−i~∇− q

c~AΓ (~x)− q

c~A(~x)

]2+VΓ (~x)

It acts on the physical Hilbert space Hphy := L2(R2). Γ ⊂R2 is the latticespanned by ~a;~b (non orthogonal in general) such thatΩΓ := (axby−bxay) > 0 is the volume of the fundamental cell MΓ of thelattice. ~AΓ and VΓ are Γ-periodic and smooth.

~A(~x) :=B2~ez∧~x =

B2

(−y,x) (symmetric gauge)

where~ez is the normalized vector orthogonal to Γ.


Experimental setting


Outline






Let (for the moment)~AΓ = 0 and denote Px :=−i∂x and Qx :=multiplicationby x (and similarly for y):

HBL :=1

2m

[(Px +

qB2c

Qy

)2

+(

Py−qB2c

Qx

)2]

+VΓ(Qx,Qy)

Define the new variables ~X(fast) := (K1,K2) (magnetic momenta),~X(slow) := (G1,G2) (centre of the cyclotron orbit)

K1 :=−~b∗ ·~Q2√

ε−σq

√ε

~a ·~P

K2 :=~a∗ ·~Q2√

ε−σq`

√ε

~b ·~P

G1 :=

~b∗ ·~Q2

−σqε

~a ·~P

G2 :=~a∗ ·~Q

2+σq

ε

~b ·~P

σq is the sign of the charge q, ε := c|q|ΩΓB = 1

ZΦ0ΦB

is the semiclassical

parameter, with Z := |q|e , Φ0 the quantum of flux and ΦB the flux of B per unit

cell.~a∗ and~b∗ satisfies~a ·~a∗ = 1 =~b ·~b∗ and~b ·~a∗ = 0 =~a ·~b∗ (dual vectors).


Let (for the moment)~AΓ = 0 and denote Px :=−i∂x and Qx :=multiplicationby x (and similarly for y):

HBL :=1

2m

[(Px +

qB2c

Qy

)2

+(

Py−qB2c

Qx

)2]

+VΓ(Qx,Qy)

Define the new variables ~X(fast) := (K1,K2) (magnetic momenta),~X(slow) := (G1,G2) (centre of the cyclotron orbit)

K1 :=−~b∗ ·~Q2√

ε−σq

√ε

~a ·~P

K2 :=~a∗ ·~Q2√

ε−σq`

√ε

~b ·~P

G1 :=

~b∗ ·~Q2

−σqε

~a ·~P

G2 :=~a∗ ·~Q

2+σq

ε

~b ·~P

σq is the sign of the charge q, ε := c|q|ΩΓB = 1

ZΦ0ΦB

is the semiclassical

parameter, with Z := |q|e , Φ0 the quantum of flux and ΦB the flux of B per unit

cell.~a∗ and~b∗ satisfies~a ·~a∗ = 1 =~b ·~b∗ and~b ·~a∗ = 0 =~a ·~b∗ (dual vectors).


With respect the new variables we have:

H′BL :=

1E0

HBL =1ε

Ξ(K1,K2)+V(G2 +

√ε K2,G1−

√ε K1

).

The new variables make evident the adiabatic separation between slowand fast degrees of freedom, indeed

[K1;K2] = iσq 1, [G1;G2] = iσq ε 1, [Kj;Gk] = 0, j,k = 1,2

They are dimensionless, then we can factorize all the physical constantswith E0 := 2

mΩΓwhich fixes a natural unit of energy. Fixed the energy

scale, the dynamics is governed by the dimensionless Hamiltonian H′BL.

The function V is related with the Γ-periodic potential VΓ by the relationV(~a∗ ·~x,~b∗ ·~x) := 1

E0VΓ(~x); it is dimensionless and bi-periodic with

periods 1, i.e. V(x+1,y) = V(x,y+1) = V(x,y).

Ξ :=1

2ΩΓ

(|~a|2K2

2 + |~b|2K12−~a ·~bK1;K2

)with discrete spectrum

λn := (n+ 12) : n ∈N (Landau levels).



H′BL :=

1E0

HBL =1ε

Ξ(K1,K2)+V(G2 +

√ε K2,G1−

√ε K1

).









Ξ :=1

2ΩΓ

(|~a|2K2

2 + |~b|2K12−~a ·~bK1;K2





H′BL :=

1E0

HBL =1ε

Ξ(K1,K2)+V(G2 +

√ε K2,G1−

√ε K1

).









Ξ :=1

2ΩΓ

(|~a|2K2

2 + |~b|2K12−~a ·~bK1;K2





H′BL :=

1E0

HBL =1ε

Ξ(K1,K2)+V(G2 +

√ε K2,G1−

√ε K1

).









Ξ :=1

2ΩΓ

(|~a|2K2

2 + |~b|2K12−~a ·~bK1;K2





H′BL :=

1E0

HBL =1ε

Ξ(K1,K2)+V(G2 +

√ε K2,G1−

√ε K1

).









Ξ :=1

2ΩΓ

(|~a|2K2

2 + |~b|2K12−~a ·~bK1;K2




Let t be the (slow) microscopic time-scale. The physic of the problem has anatural (fast) ultramicroscopic time-scale fixed by the cyclotron frequencyωc := |q|B

mc by the relation τ := ωct. With respect this new scale thetime-dependent (dimensionless) Schrödinger equation reads

H′BLψ = i

E0

∂

∂ tψ = i

ωc

E0

∂

∂τψ = i

1ε

∂

∂τψ

then the relevant (dimensionless) Hamiltonian from the physical viewpoint inthe limit of a strong magnetic field is

εH′BL = Ξ(K1,K2)+ ε V

(G2 +

√ε K2,G1−

√ε K1

).

The Stone-von Neumann uniqueness Theorem assures that there exists aunitary map W : Hphy −→ L2(R)s⊗L2(R)f := Hs⊗Hf such thatW : (G1,G2)→ (Qs := σq xs,Ps : −iε∂s) on Hs andW : (K1,K2)→ (Qf := σq xf,Pf : −i∂f) on Hf, moreover the relevantHamiltonian reads

H := WεH′BLW−1 = 1s⊗Ξ(Qf,Pf)+ ε V

(Ps +

√ε Pf,Qs−

√ε Qf

)


Let t be the (slow) microscopic time-scale. The physic of the problem has anatural (fast) ultramicroscopic time-scale fixed by the cyclotron frequencyωc := |q|B

mc by the relation τ := ωct. With respect this new scale thetime-dependent (dimensionless) Schrödinger equation reads

H′BLψ = i

E0

∂

∂ tψ = i

ωc

E0

∂

∂τψ = i

1ε

∂

∂τψ

then the relevant (dimensionless) Hamiltonian from the physical viewpoint inthe limit of a strong magnetic field is

εH′BL = Ξ(K1,K2)+ ε V

(G2 +

√ε K2,G1−

√ε K1

).

The Stone-von Neumann uniqueness Theorem assures that there exists aunitary map W : Hphy −→ L2(R)s⊗L2(R)f := Hs⊗Hf such thatW : (G1,G2)→ (Qs := σq xs,Ps : −iε∂s) on Hs andW : (K1,K2)→ (Qf := σq xf,Pf : −i∂f) on Hf, moreover the relevantHamiltonian reads

H := WεH′BLW−1 = 1s⊗Ξ(Qf,Pf)+ ε V

(Ps +

√ε Pf,Qs−

√ε Qf

)Giuseppe De Nittis (SISSA, Trieste) Bloch Electron in a Strong Magnetic Field IIIrd M3Q 16 / 30

Outline






Summarizing:

the von Neumann W provides a decomposition of the physical Hilbertspace Hphy →Hs⊗Hf related to the existence of a separation of scales;

the physically relevant Hamiltonian H, acting on Hs⊗Hf, can be seen asthe Weyl-quantization of the symbol

Hδ (ps,xs) := Ξ(Qf,Pf)+δ2 V (ps +δ Pf,qs−δ Qf)

where δ :=√

ε and the quantization rule is ps → Ps, xs → Qs;

the symbol Hδ is a function on T∗R with values in the unboundedoperators on Hf but can be see as a symbol with values in B(D,Hf)which is the Banach space of bounded operators from the Hilbert spaceD (the domain of Ξ with the graph-norm) into the Hilbert space Hf;

the principal symbol is H0 = Ξ which is “morally” an harmonicoscillator and its spectrum satisfies the CGC.

The last step is to show in what sense Hδ is “semiclassical”.


Summarizing:




where δ :=√






Summarizing:




where δ :=√






Summarizing:




where δ :=√






Summarizing:




where δ :=√






Separated family of Landau levels:


“Formally” one has the following expansion:

δ2V (ps +δ Pf,qs−δ Qf) =

+∞

∑n,m=−∞

δ2vn,m ei2π(nps+mxs)ei2πδ (nPf−mQf)

=+∞

∑n,m=−∞

δ2vn,m ei2π(nps+mxs)

(+∞

∑j=0

(i2πδ )j

j!(nPf−mQf)j

)

=+∞

∑j=0

δj+2

((i2π)j

j!

+∞

∑n,m=−∞

vn,m ei2π(nps+mxs)(nPf−mQf)j

)

=+∞

∑j=0

δj+2Hj+2(ps,xs)

the derivation is rigorous since: (i) V is bi-periodic and smooth (Fourierexpansion), (ii) there exists a dense set F ⊂Hf of analytic vectors(exponential expansion), (iii) the double sum converges in norm on the denseset F. Danger!! when j increases Hj+2 becomes “more unbounded” and thedomains of definition “shrink”. The domain of Hk with k > 4 is too smallcompared to the domain D of Ξ and so we have problems with the definitionof a common domain of selfadjointness.


“Formally” one has the following expansion:

δ2V (ps +δ Pf,qs−δ Qf) =

+∞

∑n,m=−∞

δ2vn,m ei2π(nps+mxs)ei2πδ (nPf−mQf)

=+∞

∑n,m=−∞

δ2vn,m ei2π(nps+mxs)

(+∞

∑j=0

(i2πδ )j

j!(nPf−mQf)j

)

=+∞

∑j=0

δj+2

((i2π)j

j!

+∞

∑n,m=−∞

vn,m ei2π(nps+mxs)(nPf−mQf)j

)

=+∞

∑j=0

δj+2Hj+2(ps,xs)

the derivation is rigorous since: (i) V is bi-periodic and smooth (Fourierexpansion), (ii) there exists a dense set F ⊂Hf of analytic vectors(exponential expansion), (iii) the double sum converges in norm on the denseset F. Danger!! when j increases Hj+2 becomes “more unbounded” and thedomains of definition “shrink”. The domain of Hk with k > 4 is too smallcompared to the domain D of Ξ and so we have problems with the definitionof a common domain of selfadjointness.Giuseppe De Nittis (SISSA, Trieste) Bloch Electron in a Strong Magnetic Field IIIrd M3Q 20 / 30

To solve this problem we define the order δ 4 approximate symbol

Hδ (ps,xs) := Ξ+4

∑j=2

δjHj(ps,xs)

which is selfadjoint on the domain D of Ξ and we define the remainder asRδ (ps,xs) := Hδ (ps,xs)− Hδ (ps,xs).

Proposition (key argument)

The remainder Rδ is a B(D,Hf)-valued symbol of order O(δ 4), namely‖Rδ (ps,xs)‖B(D,Hf) 6 δ 4C for all (ps,xs) ∈R2 and for a suitable constantC > 0. Moreover if πr := ∑

mi=1 |ψki〉〈ψki | is the projection on the subspace

spanned by the finite family ψkimi=1 of eigenfunctions of Ξ, then one has

‖Rδ (ps,xs)πr‖B(Hf) = ‖πrRδ (ps,xs)‖B(Hf) 6 δ 5Ck, for all (ps,xs) ∈R2 andfor a suitable constant Ck > 0, namely Rδ πr, πrRδ and [Rδ ;πr] are aB(Hf)-valued symbols of order O(δ 5).


To solve this problem we define the order δ 4 approximate symbol

Hδ (ps,xs) := Ξ+4

∑j=2

δjHj(ps,xs)

which is selfadjoint on the domain D of Ξ and we define the remainder asRδ (ps,xs) := Hδ (ps,xs)− Hδ (ps,xs).

Proposition (key argument)

The remainder Rδ is a B(D,Hf)-valued symbol of order O(δ 4), namely‖Rδ (ps,xs)‖B(D,Hf) 6 δ 4C for all (ps,xs) ∈R2 and for a suitable constantC > 0. Moreover if πr := ∑

mi=1 |ψki〉〈ψki | is the projection on the subspace

spanned by the finite family ψkimi=1 of eigenfunctions of Ξ, then one has

‖Rδ (ps,xs)πr‖B(Hf) = ‖πrRδ (ps,xs)‖B(Hf) 6 δ 5Ck, for all (ps,xs) ∈R2 andfor a suitable constant Ck > 0, namely Rδ πr, πrRδ and [Rδ ;πr] are aB(Hf)-valued symbols of order O(δ 5).


Outline






Now we can give the statement of the space adiabatic theorem for the strongfield limit:

I) Existence of the almost invariant subspaceThere exists an orthogonal projection Πε ∈B(Hs⊗Hf) such that

[H∼;Πδ ] = O0(δ ∞), [H;Π

δ ] = O0(δ 5) for δ → 0.

Moreover Πδ = π +O0(δ ∞), where π is the Weyl quantization of asemiclassical symbol π(ps,xs) ∑j δ j πj(ps,xs). The principal part π0 of thesymbol π is the spectral projection of Ξ corresponding to the given isolatedfamily of Landau bands λkim

i=1, namely π0 := ∑mi=1 |ψki〉〈ψki | where ψki is

the kith eigenfunction of Ξ.


Now we can give the statement of the space adiabatic theorem for the strongfield limit:

I) Existence of the almost invariant subspaceThere exists an orthogonal projection Πε ∈B(Hs⊗Hf) such that

[H∼;Πδ ] = O0(δ ∞), [H;Π

δ ] = O0(δ 5) for δ → 0.

Moreover Πδ = π +O0(δ ∞), where π is the Weyl quantization of asemiclassical symbol π(ps,xs) ∑j δ j πj(ps,xs). The principal part π0 of thesymbol π is the spectral projection of Ξ corresponding to the given isolatedfamily of Landau bands λkim

i=1, namely π0 := ∑mi=1 |ψki〉〈ψki | where ψki is

the kith eigenfunction of Ξ.


II) Reference subspace and intertwining unitary operator

Let πr := π0(ps,xs) for all (ps,xs) ∈R2, let Πr := 1Hs ⊗πr ∈B(Hs⊗Hf) beits Weyl quantization and K := Ran Πr. Evidently K' L2(R)s⊗Cm since Πris a m-dimensional projection. There exists a unitary operatorUδ ∈B(Hs⊗Hf) such that Πr := Uδ Πδ Uδ−1

and Uδ = u+O0(δ ∞),where u ∑j δ j uj has principal symbol u0 = 1Hf .

For the last part of the theorem we need to define the following second orderdifferential operator

D2(~a,~b) :=

(|~a|2 ∂ 2

∂x2s−2~a ·~b ∂ 2

∂xs∂ps+ |~b|2 ∂ 2

∂p2s

),

when~a ·~b = 0 and |~a|= |~b|= 1 (square lattice) then D2(~a,~b)

=4.


II) Reference subspace and intertwining unitary operator

Let πr := π0(ps,xs) for all (ps,xs) ∈R2, let Πr := 1Hs ⊗πr ∈B(Hs⊗Hf) beits Weyl quantization and K := Ran Πr. Evidently K' L2(R)s⊗Cm since Πris a m-dimensional projection. There exists a unitary operatorUδ ∈B(Hs⊗Hf) such that Πr := Uδ Πδ Uδ−1

and Uδ = u+O0(δ ∞),where u ∑j δ j uj has principal symbol u0 = 1Hf .

For the last part of the theorem we need to define the following second orderdifferential operator

D2(~a,~b) :=

(|~a|2 ∂ 2

∂x2s−2~a ·~b ∂ 2

∂xs∂ps+ |~b|2 ∂ 2

∂p2s

),

when~a ·~b = 0 and |~a|= |~b|= 1 (square lattice) then D2(~a,~b)

=4.


III) The effective Hamiltonian in a single band reference subspaceConsider the case in which the isolated family of Landau bands reduces to asingle Landau band λn, namely πr = |ψn〉〈ψn|. Let h and h be theresummations of the formal symbols u]π]Hδ ]π]u−1 and u]π]Hδ ]π]u−1

respectively and denote by h =: Heffδ

(effective Hamiltonian) and h∼ theirquantization. One has that Heff

δand h∼ are in B(Hs⊗Hf) and

h∼− Heffδ

= O0(δ 5). Moreover [h∼;Πr] = [Heffδ

;Πr] = 0 hence they can beseen as elements of B(K), namely as a bounded operators on L2(R)s. Then

B(Hs⊗Hf) 3Πδ H∼δ

Πδ Uδ

−→ h∼+O0(δ ∞) = Heffδ

+O0(δ 5) ∈B(L2(R)s).Up to the order δ 4 on has that

Heffδ

= λn1Hs + ε V(Ps,Qs)+ ε2 1

ΩΓ

λn

2D2(~a,~b)

(V)(Ps,Qs)+O0

(ε

52

)where δ 2 = ε . V and D2

(~a,~b)(V) are the Weyl quantization of the bi-periodic

symbols V(ps,xs) and D2(~a,~b)

(V)(ps,xs) respectively.


Outline






This result extends (non-orthogonal lattice, periodic magnetic field) andreinforce (local asymptotic equivalence vs. local isospectrality) aprevious result of Helffer and Sjostrand [HS].

The first non-trivial order in the expansion of Heffδ

is δ 2 = ε .

The term of order δ vanishes since H1(ps,xs) = 0 identically and theterm of order δ 3 vanishes for technical reasons which can besummarized in the fact that H3(ps,xs) is an odd polynomial in Pf and Qf.

If the periodic vector potential is switched; i.e. ~AΓ 6= 0, thenH1(ps,xs) 6= 0 but since it is odd in Pf and Qf it gives no contribution toHeff

δin the theory for a single decoupled Landau band.

If we consider a pair of consecutive decoupled Landau band then also theodd terms give contribution to Heff

δ, hence the leading order in the

perturbation is δ =:√

ε if we consider also the effects due to themicroscopical magnetic field caused by the the crystal nuclei.




is δ 2 = ε .











is δ 2 = ε .











is δ 2 = ε .











is δ 2 = ε .









Outline






Now suppose that~AΓ 6= 0 and assume (to simplify the notation) that Γ =Z2

(square lattice). Let gA be the coupling constant of the periodic magnetic fieldand gV the coupling constant of the periodic electric field.

In this case thesemiclassical symbol is given by

Hδ (ps,xs) = Ξ+δgA(f (ps,xs)a+ f (ps,xs)a†)+δ

2gVV(ps,xs)+O(δ 2gA)

where f = Ax− iAy and a, a† are the usual annihilation and creation operatorsof the harmonic oscillator Ξ.Let πr := |ψ0〉〈ψ0|+ |ψ1〉〈ψ1|, Πr its quantization and K' L2(R)s⊗C2 thereference subspace. The effective dynamics is given (up to a constant term) by

Heffδ

=

121 δgA f (Ps,Qs)†

δgA f (Ps,Qs) −121

+O0(δ

2gV).

In general gA gV , then the approximation is meaningful if δ gAgV

. The

Hamiltonian Heffδ

is a Dirac-like operator and

σ(Heffδ

) =±√

14

+σ (f f †)∪±√

14

+σ (f † f )

.



(square lattice). Let gA be the coupling constant of the periodic magnetic fieldand gV the coupling constant of the periodic electric field.In this case thesemiclassical symbol is given by



where f = Ax− iAy and a, a† are the usual annihilation and creation operatorsof the harmonic oscillator Ξ.

Let πr := |ψ0〉〈ψ0|+ |ψ1〉〈ψ1|, Πr its quantization and K' L2(R)s⊗C2 thereference subspace. The effective dynamics is given (up to a constant term) by

Heffδ

=



+O0(δ

2gV).


. The

Hamiltonian Heffδ


σ(Heffδ

) =±√

14

+σ (f f †)∪±√

14

+σ (f † f )

.



(square lattice). Let gA be the coupling constant of the periodic magnetic fieldand gV the coupling constant of the periodic electric field.In this case thesemiclassical symbol is given by



where f = Ax− iAy and a, a† are the usual annihilation and creation operatorsof the harmonic oscillator Ξ.Let πr := |ψ0〉〈ψ0|+ |ψ1〉〈ψ1|, Πr its quantization and K' L2(R)s⊗C2 thereference subspace. The effective dynamics is given (up to a constant term) by

Heffδ

=



+O0(δ

2gV).


. The

Hamiltonian Heffδ


σ(Heffδ

) =±√

14

+σ (f f †)∪±√

14

+σ (f † f )

.Giuseppe De Nittis (SISSA, Trieste) Bloch Electron in a Strong Magnetic Field IIIrd M3Q 29 / 30

References

[Av] J. Avron. Colored Hofstadter butterflies.http://arxiv.org/abs/math-ph/0308030.

[HS] H. Helffer and J. Sjostrand. Équation de Schrödinger avecchamp magnétique et équation de Harper in Lecture Notes inPhysics. Schrödinger operators (Sønderborg, 1988), 345,118–198 (1989). Springer.

[PST1] G. Panati, H. Spohn and S. Teufel. Effective dynamics for Blochelectrons: Peierls substitution and beyond. Comm. Math. Phys.242, 547-578 (2003).

[PST2] G. Panati, H. Spohn and S. Teufel. Space-AdiabaticPerturbation Theory. Adv. Theor. Math. Phys. 7, 145-204(2003).


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