Bloch Electron in a Strong Magnetic Field. Adiabatic Derivation of the Harper Model

Bloch Electron in a Strong Magnetic Field

Giuseppe De Nittis

Mathematical Physics Sector

SISSA International School for Advanced Studies, Trieste

IIIrdMathematicalMethods inQuantumMechanics

Bressanone, February 16-21, 2009

supervisor:

prof.Gianfausto Dell’Antonio based on joint work with:

Gianluca Panati & Frédéric Faure

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

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

Adiabatic problem. Starting from the physical model (free electron in a 2D lattice with an orthogonal uniform magnetic field), and using the

space adiabatic perturbation theoryof Panati, Spohn and Teufel[PST1],

[PST2], one wants to deduce rigorously two effective models: theHarper

modelin the strong limit (B−1→0) and theHofstdater modelin the

weak limit (B→0).

Algebraic problem.Using theC∗-algebraic framework one wants to

show that the two models areisomorphicat algebraic level

(isospectrality) butnot unitarly equivalent(different spectral type).

Geometric duality.As a consequence of the non unitary equivalence and using the tools of the differential geometry one wants to show that the two models are characterized by a different topology. This amounts to

prove that the two models have different values of thefirst Chern class

related by theTKNN formulaMC_Hof+NC_Har=1 when the adiabatic

parameter is rationalM/N. This explains the structure of the twocolor

coded quantum butterflies [Av].

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

Adiabatic problem. Starting from the physical model (free electron in a 2D lattice with an orthogonal uniform magnetic field), and using the

space adiabatic perturbation theoryof Panati, Spohn and Teufel[PST1],

[PST2], one wants to deduce rigorously two effective models: theHarper

modelin the strong limit (B−1→0) and theHofstdater modelin the

weak limit (B→0).

Algebraic problem.Using theC∗-algebraic framework one wants to

show that the two models areisomorphicat algebraic level

(isospectrality) butnot unitarly equivalent(different spectral type).

Geometric duality.As a consequence of the non unitary equivalence and using the tools of the differential geometry one wants to show that the two models are characterized by a different topology. This amounts to

prove that the two models have different values of thefirst Chern class

related by theTKNN formulaMC_Hof+NC_Har=1 when the adiabatic

parameter is rationalM/N. This explains the structure of the twocolor

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

Adiabatic problem. Starting from the physical model (free electron in a 2D lattice with an orthogonal uniform magnetic field), and using the

space adiabatic perturbation theoryof Panati, Spohn and Teufel[PST1],

[PST2], one wants to deduce rigorously two effective models: theHarper

modelin the strong limit (B−1→0) and theHofstdater modelin the

weak limit (B→0).

Algebraic problem.Using theC∗-algebraic framework one wants to

show that the two models areisomorphicat algebraic level

(isospectrality) butnot unitarly equivalent(different spectral type).

Geometric duality.As a consequence of the non unitary equivalence and using the tools of the differential geometry one wants to show that the two models are characterized by a different topology. This amounts to

prove that the two models have different values of thefirst Chern class

related by theTKNN formulaMC_Hof+NC_Har=1 when the adiabatic

parameter is rationalM/N. This explains the structure of the twocolor

coded quantum butterflies [Av].

Color-Coded Quantum Butterflies:

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory}

The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Some comments The effective model

When the system shows aseparationintoslowandfast degrees of freedomcertain dynamical degrees of freedom lose their autonomous status. The fast modes quickly adapt to the slow modes which in turn are

governed by a suitableeffective Hamiltonian. This mechanism is called

adiabatic decoupling. The paradigm is the Born-Oppenheimer approximation for the motion of nuclei.

The slow degrees of freedom are “semiclassical” in the sense that the full

Hamiltonian can be seen as theWeyl quantizationof an operator-valued

symbol on the phase space of the classical degrees of freedom. The origin of decoupling can be traced to a spectral property of the symbol in

the sense that it can be decomposed in arelevant separated part. This can

be associated with analmost invariant subspace(invariant under the

evolution up to errors small to any order inε) of the Hilbert space of the

system.

The a. i. subspace depends onε and is not easily accessible. In order to

obtain a useful description of the effective intraband dynamics (effective

Hamiltonian) we need aunitarymap from the a. i. subspace into an

When the system shows aseparationintoslowandfast degrees of freedomcertain dynamical degrees of freedom lose their autonomous status. The fast modes quickly adapt to the slow modes which in turn are

governed by a suitableeffective Hamiltonian. This mechanism is called

adiabatic decoupling. The paradigm is the Born-Oppenheimer approximation for the motion of nuclei.

The slow degrees of freedom are “semiclassical” in the sense that the full

Hamiltonian can be seen as theWeyl quantizationof an operator-valued

symbol on the phase space of the classical degrees of freedom. The origin of decoupling can be traced to a spectral property of the symbol in

the sense that it can be decomposed in arelevant separated part. This can

be associated with analmost invariant subspace(invariant under the

evolution up to errors small to any order inε) of the Hilbert space of the

system.

The a. i. subspace depends onε and is not easily accessible. In order to

obtain a useful description of the effective intraband dynamics (effective

Hamiltonian) we need aunitarymap from the a. i. subspace into an

easily accessible andε-independentreference subspace.

When the system shows aseparationintoslowandfast degrees of freedomcertain dynamical degrees of freedom lose their autonomous status. The fast modes quickly adapt to the slow modes which in turn are

governed by a suitableeffective Hamiltonian. This mechanism is called

adiabatic decoupling. The paradigm is the Born-Oppenheimer approximation for the motion of nuclei.

The slow degrees of freedom are “semiclassical” in the sense that the full

Hamiltonian can be seen as theWeyl quantizationof an operator-valued

symbol on the phase space of the classical degrees of freedom. The origin of decoupling can be traced to a spectral property of the symbol in

the sense that it can be decomposed in arelevant separated part. This can

be associated with analmost invariant subspace(invariant under the

evolution up to errors small to any order inε) of the Hilbert space of the

system.

The a. i. subspace depends onε and is not easily accessible. In order to

obtain a useful description of the effective intraband dynamics (effective

Hamiltonian) we need aunitarymap from the a. i. subspace into an

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory}

The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Some comments The effective model

I) Thephysicalstate space of the system decomposes (up to a unitary

transformW), as

Hphy−→W L2(Rd)⊗Hf

whereL2(Rd) =:Hsis the state space of theslow degrees of freedom

andHfan (arbitrary separable) state space of thefast degrees of

freedom. Theclassical phase spaceof the slow degree of freedom is thus

T∗Rd_'R2d_{withz:= (p}

s,xs)∈R2d.

II) Thephysical HamiltonianHb (up to a unitary transform), generating the

time-evolution, is given as the Weyl quantization of asemiclassical

symbolH_ε :T∗_Rd→B(D,Hf)whereDis a suitablenormeddomain

on whichHε is bounded. “Semiclassical” means thatHε admits a formal

expansionHε(z)∑jεjHj(z)whoseprincipal symbolisH0(z). III) Constant Gap Condition (CGC):for anyz∈T∗Rd_{the spectrum}

σ(z)of

H0(z)contains arelevant subsetσ∗(z)which isuniformlyseparated from

the remainder by a gap, namely

inf

z∈T∗_Rd dist(σ(z)

I) Thephysicalstate space of the system decomposes (up to a unitary

transformW), as

Hphy−→W L2(Rd)⊗Hf

whereL2(Rd) =:Hsis the state space of theslow degrees of freedom

andHfan (arbitrary separable) state space of thefast degrees of

freedom. Theclassical phase spaceof the slow degree of freedom is thus

T∗Rd_'R2d_{withz:= (p}

s,xs)∈R2d.

II) Thephysical HamiltonianHb (up to a unitary transform), generating the

time-evolution, is given as the Weyl quantization of asemiclassical

symbolH_ε :T∗_Rd→B(D,Hf)whereDis a suitablenormeddomain

on whichHε is bounded. “Semiclassical” means thatHε admits a formal

expansionHε(z)∑jεjHj(z)whoseprincipal symbolisH0(z).

III) Constant Gap Condition (CGC):for anyz∈T∗Rd_{the spectrum}

σ(z)of

H0(z)contains arelevant subsetσ∗(z)which isuniformlyseparated from

the remainder by a gap, namely

inf

z∈T∗_Rd dist(σ(z)

\σ∗(z),σ∗(z)) =Cg>0.

I) Thephysicalstate space of the system decomposes (up to a unitary

transformW), as

Hphy−→W L2(Rd)⊗Hf

whereL2(Rd) =:Hsis the state space of theslow degrees of freedom

andHfan (arbitrary separable) state space of thefast degrees of

freedom. Theclassical phase spaceof the slow degree of freedom is thus

T∗Rd_'R2d_{withz:= (p}

s,xs)∈R2d.

II) Thephysical HamiltonianHb (up to a unitary transform), generating the

time-evolution, is given as the Weyl quantization of asemiclassical

symbolH_ε :T∗_Rd→B(D,Hf)whereDis a suitablenormeddomain

on whichHε is bounded. “Semiclassical” means thatHε admits a formal

expansionHε(z)∑jεjHj(z)whoseprincipal symbolisH0(z). III) Constant Gap Condition (CGC):for anyz∈T∗Rd_{the spectrum}

σ(z)of

H0(z)contains arelevant subsetσ∗(z)which isuniformlyseparated from

the remainder by a gap, namely

inf

Relevant separated part of the spectrum:

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

TheBloch-Landau Hamiltonian

HBL:=

1

2m

h

−i_}~∇−q

c~AΓ(~x)− q c~A(~x)

i2

+V_Γ(~x)

It acts on thephysicalHilbert spaceHphy:=L2(R2).Γ⊂R2is the lattice

spanned by{~a;~b}(non orthogonal in general) such that

ΩΓ:= (axby−bxay)>0 is the volume of thefundamental cellMΓof the

lattice.~AΓandVΓareΓ-periodicandsmooth.

~_A(~x):=B

2~ez∧~x=

B

2(−y,x) (symmetric gauge)

where~ezis the normalized vector orthogonal toΓ.

TheBloch-Landau Hamiltonian

HBL:=

1

2m

h

−i_}~∇−q

c~AΓ(~x)− q c~A(~x)

i2

+V_Γ(~x)

It acts on thephysicalHilbert spaceHphy:=L2(R2).Γ⊂R2is the lattice

spanned by{~a;~b}(non orthogonal in general) such that

ΩΓ:= (axby−bxay)>0 is the volume of thefundamental cellMΓof the

lattice.~AΓandVΓareΓ-periodicandsmooth.

~_A(~x):=B

2~ez∧~x=

B

2(−y,x) (symmetric gauge)

Experimental setting

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Let (for the moment)~A_Γ=0and denotePx:=−i}∂xandQx:=multiplication

byx(and similarly fory):

HBL:=

1

2m

"

Px+

qB 2cQy

2

+

Py−

qB 2cQx

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∗_·~Q

2√ε −σq

√ ε

} ~a·

~_P

K2:= ~a∗·~Q

2√ε −σq`

√ ε

}

~_b·~_P

      

G1:=

~_b∗_·~Q

2 −σq

ε

}~a·

~_P

G2:=

~a∗·~Q

2 +σq

ε

}

~_b·~_P

σqis thesign of the chargeq,ε:=_|q|Ωc} ΓB =

1

Z

Φ0

ΦB is thesemiclassical

parameter, withZ:=|q_e|,Φ0thequantum of fluxandΦBtheflux ofBper unit

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

Let (for the moment)~A_Γ=0and denotePx:=−i}∂xandQx:=multiplication

byx(and similarly fory):

HBL:=

1

2m

"

Px+

qB 2cQy

2

+

Py−

qB 2cQx

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∗_·~Q

2√ε −σq

√ ε

} ~

a·~P

K2:= ~a∗·~Q

2√ε −σq`

√ ε

}

~_b·~_P

      

G1:=

~_b∗_·~Q

2 −σq

ε

}~a·

~_P

G2:=

~a∗·~Q

2 +σq

ε

}

~_b·~_P

σqis thesign of the chargeq,ε:=_|q|Ωc} ΓB =

1

Z

Φ0

ΦB is thesemiclassical

parameter, withZ:=|q_e|,Φ0thequantum of fluxandΦBtheflux ofBper unit

With respect the new variables we have:

H_BL0 := 1

E0

HBL=

1

εΞ(K1,K2) +V G2+ √

ε K2,G1−

√ ε K1

.

The new variables make evident theadiabatic separationbetween slow

and fast degrees of freedom, indeed

[K1;K2] =iσq1, [G1;G2] =iσqε1, [Kj;Gk] =0, j,k=1,2

They are dimensionless, then we can factorize all the physical constants

withE0:= }

2

mΩΓ which fixes anatural unit of energy. Fixed the energy

scale, the dynamics is governed by thedimensionlessHamiltonianH0_BL.

The functionV is related with theΓ-periodic potentialVΓby the relation

V(~a∗·~x,~b∗·~x):=_E1

0VΓ(~x); it is dimensionless andbi-periodicwith

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

Ξ:= 1

2ΩΓ

|~a|2_K

22+|~b|2K12−~a·~b{K1;K2}

with discrete spectrum

{λn:= (n+1₂) : n∈N}(Landau levels).

With respect the new variables we have:

H_BL0 := 1

E0

HBL=

1

εΞ(K1,K2) +V G2+ √

ε K2,G1−

√ ε K1

.

The new variables make evident theadiabatic separationbetween slow

and fast degrees of freedom, indeed

[K1;K2] =iσq1, [G1;G2] =iσqε1, [Kj;Gk] =0, j,k=1,2

They are dimensionless, then we can factorize all the physical constants

withE0:= }

2

mΩΓ which fixes anatural unit of energy. Fixed the energy

scale, the dynamics is governed by thedimensionlessHamiltonianH0_BL.

The functionV is related with theΓ-periodic potentialVΓby the relation

V(~a∗·~x,~b∗·~x):=_E1

0VΓ(~x); it is dimensionless andbi-periodicwith

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

Ξ:= 1

2ΩΓ

|~a|2_K

22+|~b|2K12−~a·~b{K1;K2}

with discrete spectrum

With respect the new variables we have:

H_BL0 := 1

E0

HBL=

1

εΞ(K1,K2) +V G2+ √

ε K2,G1−

√ ε K1

.

The new variables make evident theadiabatic separationbetween slow

and fast degrees of freedom, indeed

[K1;K2] =iσq1, [G1;G2] =iσqε1, [Kj;Gk] =0, j,k=1,2

They are dimensionless, then we can factorize all the physical constants

withE0:= }

2

mΩΓ which fixes anatural unit of energy. Fixed the energy

scale, the dynamics is governed by thedimensionlessHamiltonianH_BL0 .

The functionV is related with theΓ-periodic potentialVΓby the relation

V(~a∗·~x,~b∗·~x):=_E1

0VΓ(~x); it is dimensionless andbi-periodicwith

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

Ξ:= 1

2ΩΓ

|~a|2_K

22+|~b|2K12−~a·~b{K1;K2}

with discrete spectrum

{λn:= (n+1₂) : n∈N}(Landau levels).

With respect the new variables we have:

H_BL0 := 1

E0

HBL=

1

εΞ(K1,K2) +V G2+ √

ε K2,G1−

√ ε K1

.

The new variables make evident theadiabatic separationbetween slow

and fast degrees of freedom, indeed

[K1;K2] =iσq1, [G1;G2] =iσqε1, [Kj;Gk] =0, j,k=1,2

They are dimensionless, then we can factorize all the physical constants

withE0:= }

2

mΩΓ which fixes anatural unit of energy. Fixed the energy

scale, the dynamics is governed by thedimensionlessHamiltonianH_BL0 .

The functionV is related with theΓ-periodic potentialVΓby the relation

V(~a∗·~x,~b∗·~x):=_E1

0VΓ(~x); it is dimensionless andbi-periodicwith

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

Ξ:= 1

2ΩΓ

|~a|2_K

22+|~b|2K12−~a·~b{K1;K2}

with discrete spectrum

With respect the new variables we have:

H_BL0 := 1

E0

HBL=

1

εΞ(K1,K2) +V G2+ √

ε K2,G1−

√ ε K1

.

The new variables make evident theadiabatic separationbetween slow

and fast degrees of freedom, indeed

[K1;K2] =iσq1, [G1;G2] =iσqε1, [Kj;Gk] =0, j,k=1,2

They are dimensionless, then we can factorize all the physical constants

withE0:= }

2

mΩΓ which fixes anatural unit of energy. Fixed the energy

scale, the dynamics is governed by thedimensionlessHamiltonianH_BL0 .

The functionV is related with theΓ-periodic potentialVΓby the relation

V(~a∗·~x,~b∗·~x):=_E1

0VΓ(~x); it is dimensionless andbi-periodicwith

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

Ξ:= 1

2ΩΓ

|~a|2_K

22+|~b|2K12−~a·~b{K1;K2}

with discrete spectrum

{λn:= (n+1₂) : n∈N}(Landau levels).

Lettbe the (slow)microscopic time-scale. The physic of the problem has a

natural (fast)ultramicroscopic time-scalefixed by thecyclotron frequency

ωc:=|mcq|B by the relationτ :=ωct. With respect this new scale the

time-dependent (dimensionless) Schrödinger equation reads

H_BL0 ψ=i}

E0

∂

∂tψ=i

}ωc

E0

∂

∂ τψ=i

1

ε ∂

∂ τψ

then therelevant(dimensionless)Hamiltonianfrom the physical viewpoint in

the limit of a strong magnetic field is

εH_BL0 =Ξ(K1,K2) +ε V G2+

√

εK2,G1−

√ εK1

.

TheStone-von Neumann uniqueness Theoremassures that there exists a unitary mapW:Hphy−→L2(R)s⊗L2(R)f:=Hs⊗Hfsuch that W:(G1,G2)→(Qs:=σqxs,Ps:−iε ∂s)onHsand

W:(K1,K2)→(Qf:=σqxf,Pf:−i∂f)onHf, moreover the relevant

Hamiltonian reads

b

H:=WεH_BL0 W−1=1s⊗Ξ(Qf,Pf) +ε V Ps+

√

ε Pf,Qs−

√ εQf

Lettbe the (slow)microscopic time-scale. The physic of the problem has a

natural (fast)ultramicroscopic time-scalefixed by thecyclotron frequency

ωc:=|mcq|B by the relationτ :=ωct. With respect this new scale the

time-dependent (dimensionless) Schrödinger equation reads

H_BL0 ψ=i}

E0

∂

∂tψ=i

}ωc

E0

∂

∂ τψ=i

1

ε ∂

∂ τψ

then therelevant(dimensionless)Hamiltonianfrom the physical viewpoint in

the limit of a strong magnetic field is

εH_BL0 =Ξ(K1,K2) +ε V G2+

√

εK2,G1−

√ εK1

.

TheStone-von Neumann uniqueness Theoremassures that there exists a unitary mapW:Hphy−→L2(R)s⊗L2(R)f:=Hs⊗Hfsuch that W:(G1,G2)→(Qs:=σqxs,Ps:−iε ∂s)onHsand

W:(K1,K2)→(Qf:=σqxf,Pf:−i∂f)onHf, moreover the relevant

Hamiltonian reads

b

H:=WεH_BL0 W−1=1s⊗Ξ(Qf,Pf) +ε V Ps+

√

ε Pf,Qs−

√ εQf

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Summarizing:

the von NeumannWprovides a decomposition of the physical Hilbert

spaceHphy→Hs⊗Hfrelated to the existence of a separation of scales;

the physically relevant HamiltonianHb, acting onHs⊗Hf, can be seen as

theWeyl-quantizationof the symbol

H_δ(ps,xs):=Ξ(Qf,Pf) +δ2V(ps+δ Pf,qs−δ Qf)

whereδ :=

√

ε and the quantization rule isps→Ps,xs→Qs;

the symbolH_δ is a function onT∗_Rwith values in theunbounded

operators onHfbut can be see as a symbol with values inB(D,Hf)

which is the Banach space ofbounded operatorsfrom the Hilbert space

D(the domain ofΞwith thegraph-norm) into the Hilbert spaceHf;

the principal symbol isH0=Ξwhich is “morally” an harmonic

oscillator and its spectrum satisfies theCGC.

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

Summarizing:

the von NeumannWprovides a decomposition of the physical Hilbert

spaceHphy→Hs⊗Hfrelated to the existence of a separation of scales;

the physically relevant HamiltonianHb, acting onHs⊗Hf, can be seen as

theWeyl-quantizationof the symbol

H_δ(ps,xs):=Ξ(Qf,Pf) +δ2V(ps+δ Pf,qs−δ Qf)

whereδ :=

√

ε and the quantization rule isps→Ps,xs→Qs;

the symbolH_δ is a function onT∗_Rwith values in theunbounded

operators onHfbut can be see as a symbol with values inB(D,Hf)

which is the Banach space ofbounded operatorsfrom the Hilbert space

D(the domain ofΞwith thegraph-norm) into the Hilbert spaceHf;

the principal symbol isH0=Ξwhich is “morally” an harmonic

oscillator and its spectrum satisfies theCGC.

Summarizing:

the von NeumannWprovides a decomposition of the physical Hilbert

spaceHphy→Hs⊗Hfrelated to the existence of a separation of scales;

the physically relevant HamiltonianHb, acting onHs⊗Hf, can be seen as

theWeyl-quantizationof the symbol

H_δ(ps,xs):=Ξ(Qf,Pf) +δ2V(ps+δ Pf,qs−δ Qf)

whereδ :=

√

ε and the quantization rule isps→Ps,xs→Qs;

the symbolH_δ is a function onT∗_Rwith values in theunbounded

operators onHfbut can be see as a symbol with values inB(D,Hf)

which is the Banach space ofbounded operatorsfrom the Hilbert space

D(the domain ofΞwith thegraph-norm) into the Hilbert spaceHf;

the principal symbol isH0=Ξwhich is “morally” an harmonic

oscillator and its spectrum satisfies theCGC.

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

Summarizing:

the von NeumannWprovides a decomposition of the physical Hilbert

spaceHphy→Hs⊗Hfrelated to the existence of a separation of scales;

the physically relevant HamiltonianHb, acting onHs⊗Hf, can be seen as

theWeyl-quantizationof the symbol

H_δ(ps,xs):=Ξ(Qf,Pf) +δ2V(ps+δ Pf,qs−δ Qf)

whereδ :=

√

ε and the quantization rule isps→Ps,xs→Qs;

the symbolH_δ is a function onT∗_Rwith values in theunbounded

operators onHfbut can be see as a symbol with values inB(D,Hf)

which is the Banach space ofbounded operatorsfrom the Hilbert space

D(the domain ofΞwith thegraph-norm) into the Hilbert spaceHf;

the principal symbol isH0=Ξwhich is “morally” an harmonic

oscillator and its spectrum satisfies theCGC.

Summarizing:

the von NeumannWprovides a decomposition of the physical Hilbert

spaceHphy→Hs⊗Hfrelated to the existence of a separation of scales;

the physically relevant HamiltonianHb, acting onHs⊗Hf, can be seen as

theWeyl-quantizationof the symbol

H_δ(ps,xs):=Ξ(Qf,Pf) +δ2V(ps+δ Pf,qs−δ Qf)

whereδ :=

√

ε and the quantization rule isps→Ps,xs→Qs;

the symbolH_δ is a function onT∗_Rwith values in theunbounded

operators onHfbut can be see as a symbol with values inB(D,Hf)

which is the Banach space ofbounded operatorsfrom the Hilbert space

D(the domain ofΞwith thegraph-norm) into the Hilbert spaceHf;

the principal symbol isH0=Ξwhich is “morally” an harmonic

oscillator and its spectrum satisfies theCGC.

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

“Formally” one has the following expansion:

δ2V(ps+δ Pf,qs−δ Qf) =

+∞

∑

n,m=−∞

δ2vn,mei2π(nps+mxs)ei2π δ(nPf−mQf)

= +∞

∑

n,m=−∞

δ2vn,mei2π(nps+mxs)

+∞

∑

j=0

(i2π δ)j

j! (nPf−mQf)

j ! = +∞

∑

δj+2 (i2π)

j

j!

+∞

∑

vn,mei2π(nps+mxs)(nPf−mQf)j

!

= +∞

∑

j=0

δj+2Hj+2(ps,xs)

the derivation is rigorous since:(i)Vis bi-periodic and smooth (Fourier

expansion),(ii)there exists a dense setF⊂Hfofanalytic vectors

(exponential expansion),(iii)the double sum converges in norm on the dense

setF.Danger!!whenjincreasesHj+2becomes “more unbounded” and the

domains of definition “shrink”. The domain ofHk withk>4 is too small

compared to the domainDofΞand so we have problems with the definition

of a common domain of selfadjointness.

“Formally” one has the following expansion:

δ2V(ps+δ Pf,qs−δ Qf) =

+∞

∑

n,m=−∞

δ2vn,mei2π(nps+mxs)ei2π δ(nPf−mQf)

= +∞

∑

n,m=−∞

δ2vn,mei2π(nps+mxs)

+∞

∑

j=0

(i2π δ)j

j! (nPf−mQf)

j ! = +∞

∑

δj+2 (i2π)

j

j!

+∞

∑

vn,mei2π(nps+mxs)(nPf−mQf)j

!

= +∞

∑

j=0

δj+2Hj+2(ps,xs)

the derivation is rigorous since:(i)V is bi-periodic and smooth (Fourier

expansion),(ii)there exists a dense setF⊂Hfofanalytic vectors

(exponential expansion),(iii)the double sum converges in norm on the dense

setF.Danger!!whenjincreasesHj+2becomes “more unbounded” and the

domains of definition “shrink”. The domain ofHk withk>4 is too small

compared to the domainDofΞand so we have problems with the definition

To solve this problem we define theorderδ4approximate symbol

e

H_δ(ps,xs):=Ξ+ 4

∑

j=2

δjHj(ps,xs)

which is selfadjoint on the domainDofΞand we define theremainderas

R_δ(ps,xs):=Hδ(ps,xs)−He_δ(p_s,x_s).

Proposition(key argument)

The remainderR_δ is aB(D,Hf)-valued symbol of orderO(δ4), namely

kR_δ(p_s,x_s)k_B(D,Hf)6δ4Cfor all(ps,xs)∈R2and for a suitable constant

C>0. Moreover ifπr:=∑mi=1|ψkiihψki|is the projection on the subspace spanned by the finite family{ψk_i}m

i=1of eigenfunctions ofΞ, then one has

kR_δ(p_s,x_s)πrkB(Hf)=kπrRδ(ps,xs)kB(Hf)6δ 5_C

k,for all(ps,xs)∈R2and

for a suitable constantCk>0, namelyRδπr,πrRδ and[Rδ;πr]are a

B(Hf)-valued symbols of orderO(δ5).

To solve this problem we define theorderδ4approximate symbol

e

H_δ(ps,xs):=Ξ+ 4

∑

j=2

δjHj(ps,xs)

which is selfadjoint on the domainDofΞand we define theremainderas

R_δ(ps,xs):=Hδ(ps,xs)−He_δ(p_s,x_s).

Proposition(key argument)

The remainderR_δ is aB(D,Hf)-valued symbol of orderO(δ4), namely

kR_δ(p_s,x_s)k_B(D,Hf)6δ4Cfor all(ps,xs)∈R2and for a suitable constant

C>0. Moreover ifπr:=∑mi=1|ψkiihψki|is the projection on the subspace spanned by the finite family{ψk_i}m

i=1of eigenfunctions ofΞ, then one has

kR_δ(p_s,x_s)πrkB(Hf)=kπrRδ(ps,xs)kB(Hf)6δ 5_C

k,for all(ps,xs)∈R2and

for a suitable constantCk>0, namelyRδπr,πrRδ and[Rδ;πr]are a

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation The statement of the main results

4 Is the periodic magnetic field important?

Some comments The effective model

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

I) Existence of the almost invariant subspace

There exists an orthogonal projectionΠε ∈B(Hs⊗Hf)such that

[Hb

∼

;Πδ] =O0(δ∞), [Hb;Πδ] =O0(δ5) for δ→0.

MoreoverΠδ =πb+O0(δ

∞_{), where}

b

πis the Weyl quantization of a

semiclassical symbolπ(ps,xs)∑jδj πj(ps,xs). The principal partπ0of the

symbolπis the spectral projection ofΞcorresponding to the given isolated

family of Landau bands{λk_i}m

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

I) Existence of the almost invariant subspace

There exists an orthogonal projectionΠε ∈B(Hs⊗Hf)such that

[Hb

∼

;Πδ] =O0(δ∞), [Hb;Πδ] =O0(δ5) for δ →0.

MoreoverΠδ =πb+O0(δ

∞_{), where}

b

πis the Weyl quantization of a

semiclassical symbolπ(ps,xs)∑jδjπj(ps,xs). The principal partπ0of the

symbolπis the spectral projection ofΞcorresponding to the given isolated

family of Landau bands{λk_i}m

i=1, namelyπ0:=∑mi=1|ψkiihψki|whereψki is thekitheigenfunction ofΞ.

II) Reference subspace and intertwining unitary operator

Letπr:=π0(ps,xs)for all(ps,xs)∈R2, letΠr:=1_Hs⊗πr∈B(Hs⊗Hf)be

its Weyl quantization andK:=RanΠr. EvidentlyK'L2(R)s⊗CmsinceΠr

is am-dimensional projection. There exists a unitary operator

Uδ _∈B₍Hs_⊗Hf_{)such that}

Πr:=Uδ Πδ Uδ −1

andUδ ₌

b

u+O0(δ∞),

whereu∑jδjujhas principal symbolu0=1Hf.

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

D2

(~a,~b):=

|~a|2 ∂

2

∂x2_s −2~a·

~_b ∂2

∂xs∂ps

+|~b|2 ∂

2

∂p2_s

,

when~a·~b=0 and|~a|=|~b|=1 (square lattice) thenD2

II) Reference subspace and intertwining unitary operator

Letπr:=π0(ps,xs)for all(ps,xs)∈R2, letΠr:=1_Hs⊗πr∈B(Hs⊗Hf)be

its Weyl quantization andK:=RanΠr. EvidentlyK'L2(R)s⊗CmsinceΠr

is am-dimensional projection. There exists a unitary operator

Uδ _∈B₍Hs_⊗Hf_{)such that}

Πr:=Uδ Πδ Uδ −1

andUδ ₌

b

u+O0(δ∞),

whereu∑jδjujhas principal symbolu0=1Hf.

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

D2

(~a,~b):=

|~a|2 ∂

2

∂x2_s −2~a·

~_b ∂2

∂xs∂ps

+|~b|2 ∂

2

∂p2_s

,

when~a·~b=0 and|~a|=|~b|=1 (square lattice) thenD2

(~a,~b)=4.

III) The effective Hamiltonian in a single band reference subspace

Consider the case in which the isolated family of Landau bands reduces to a

single Landau bandλn, namelyπr=|ψnihψn|. Lethandehbe the

resummations of the formal symbolsu]π]Hδ]π]u

−1_andu]

π]He_δ]π]u−1

respectively and denote bybh=:Hbeff

δ (effective Hamiltonian) andbh

∼_their

quantization. One has thatHb_δeffandbh∼are inB(Hs⊗Hf)and

b

h∼−Hbeff

δ =O0(δ

5_{). Moreover[}

b

h∼;Πr] = [Hbeff

δ ;Πr] =0 hence they can be

seen as elements ofB(K), namely as a bounded operators onL2(R)s. Then

B(Hs⊗Hf)3ΠδHb∼_δΠδ Uδ

−→ bh∼+O0(δ∞) =Hbeff

δ +O0(δ

5_)∈B(L2_(R)s).

Up to the orderδ4on has that

b

H_δeff=λn1_Hs+ε Vb(Ps,Qs) +ε2

1 ΩΓ λn 2 \ D2

(~a,~b)(V)(Ps,Qs) +O0

ε

5 2

whereδ2=ε.VbandD\2

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

symbolsV(ps,xs)andD2

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Some comments

The effective model

This result extends (non-orthogonal lattice,periodic magnetic field) and reinforce (local asymptotic equivalence vs. local isospectrality) a

previous result of Helffer and Sj˝ostrand[HS].

The first non-trivial order in the expansion ofHbeff

δ isδ 2₌

ε.

The term of orderδ vanishes sinceH1(ps,xs) =0identically and the

term of orderδ3vanishes for technical reasons which can be

summarized in the fact thatH₃(ps,xs)is anodd polynomialinPfandQf.

If the periodic vector potential is switched; i.e.~A_Γ6=0, then

H1(ps,xs)6=0but since it is odd inPfandQfit gives no contribution to

b

H_δeffin the theory for asingle decoupled Landau band.

If we consider apair of consecutive decoupled Landau bandthen also the

odd terms give contribution toHbeff

δ , hence the leading order in the

perturbation isδ =: √

ε if we consider also the effects due to the

This result extends (non-orthogonal lattice,periodic magnetic field) and reinforce (local asymptotic equivalence vs. local isospectrality) a

previous result of Helffer and Sj˝ostrand[HS].

The first non-trivial order in the expansion ofHbeff

δ isδ 2₌

ε.

The term of orderδ vanishes sinceH1(ps,xs) =0identically and the

term of orderδ3vanishes for technical reasons which can be

summarized in the fact thatH₃(ps,xs)is anodd polynomialinPfandQf.

If the periodic vector potential is switched; i.e.~A_Γ6=0, then

H1(ps,xs)6=0but since it is odd inPfandQfit gives no contribution to

b

H_δeffin the theory for asingle decoupled Landau band.

If we consider apair of consecutive decoupled Landau bandthen also the

odd terms give contribution toHbeff

δ , hence the leading order in the

perturbation isδ =: √

ε if we consider also the effects due to the

microscopical magnetic field caused by the the crystal nuclei.

This result extends (non-orthogonal lattice,periodic magnetic field) and reinforce (local asymptotic equivalence vs. local isospectrality) a

previous result of Helffer and Sj˝ostrand[HS].

The first non-trivial order in the expansion ofHbeff

δ isδ 2₌

ε.

The term of orderδ vanishes sinceH1(ps,xs) =0identically and the

term of orderδ3vanishes for technical reasons which can be

summarized in the fact thatH₃(ps,xs)is anodd polynomialinPfandQf.

If the periodic vector potential is switched; i.e.~A_Γ6=0, then

H1(ps,xs)6=0but since it is odd inPfandQfit gives no contribution to

b

H_δeffin the theory for asingle decoupled Landau band.

If we consider apair of consecutive decoupled Landau bandthen also the

odd terms give contribution toHbeff

δ , hence the leading order in the

perturbation isδ =: √

ε if we consider also the effects due to the

This result extends (non-orthogonal lattice,periodic magnetic field) and reinforce (local asymptotic equivalence vs. local isospectrality) a

previous result of Helffer and Sj˝ostrand[HS].

The first non-trivial order in the expansion ofHbeff

δ isδ 2₌

ε.

The term of orderδ vanishes sinceH1(ps,xs) =0identically and the

term of orderδ3vanishes for technical reasons which can be

summarized in the fact thatH₃(ps,xs)is anodd polynomialinPfandQf.

If the periodic vector potential is switched; i.e.~A_Γ6=0, then

H1(ps,xs)6=0but since it is odd inPfandQfit gives no contribution to

b

Heff

δ in the theory for asingle decoupled Landau band.

If we consider apair of consecutive decoupled Landau bandthen also the

odd terms give contribution toHbeff

δ , hence the leading order in the

perturbation isδ =: √

ε if we consider also the effects due to the

microscopical magnetic field caused by the the crystal nuclei.

This result extends (non-orthogonal lattice,periodic magnetic field) and reinforce (local asymptotic equivalence vs. local isospectrality) a

previous result of Helffer and Sj˝ostrand[HS].

The first non-trivial order in the expansion ofHbeff

δ isδ 2₌

ε.

The term of orderδ vanishes sinceH1(ps,xs) =0identically and the

term of orderδ3vanishes for technical reasons which can be

summarized in the fact thatH₃(ps,xs)is anodd polynomialinPfandQf.

If the periodic vector potential is switched; i.e.~A_Γ6=0, then

H1(ps,xs)6=0but since it is odd inPfandQfit gives no contribution to

b

Heff

δ in the theory for asingle decoupled Landau band.

If we consider apair of consecutive decoupled Landau bandthen also the

odd terms give contribution toHbeff

δ , hence the leading order in the

perturbation isδ =: √

ε if we consider also the effects due to the

Outline

1 _Introduction

An overview on our research project

2 _{The space adiabatic perturbation theory} The philosophy

The ingredients

3 Adiabatic theory for the strong magnetic field regime

The physical model

Separation of the scales and adiabatic parameter

Formal expansion of the semiclassical symbol: orderδ4approximation

The statement of the main results

4 Is the periodic magnetic field important?

Some comments

The effective model

Now suppose that~A_Γ6=0 and assume (to simplify the notation) thatΓ=Z2

(square lattice). LetgAbe thecoupling constantof the periodic magnetic field

andgV thecoupling constantof the periodic electric field.

In this case the semiclassical symbol is given by

H_δ(ps,xs) =Ξ+δgA f(ps,xs)ba+f(ps,xs)ba

†

+δ2gVV(ps,xs) +O(δ2gA)

wheref =Ax−iAyandba,ba

†_{are the usualannihilationandcreation operators}

of the harmonic oscillatorΞ.

Letπr:=|ψ0ihψ0|+|ψ1ihψ1|,Πrits quantization andK'L2(R)s⊗C2the

reference subspace. The effective dynamics is given (up to a constant term) by

b

Heff_δ =





1

21 δgAbf(Ps,Qs) †

δgAbf(P_s,Q_s) −1₂1



+O₀ δ2gV

.

In generalgAgV, then the approximation is meaningful ifδ g_gA

V. The

HamiltonianHbeff

δ is aDirac-likeoperator and

σ(Hb_δeff) =± r

1

4+σ(bfbf

†_{)∪ ±}

r

1 4+σ(bf

†

bf)

Now suppose that~A_Γ6=0 and assume (to simplify the notation) thatΓ=Z2

(square lattice). LetgAbe thecoupling constantof the periodic magnetic field

andgV thecoupling constantof the periodic electric field.In this case the semiclassical symbol is given by

H_δ(ps,xs) =Ξ+δgA f(ps,xs)ba+f(ps,xs)ba

†

+δ2gVV(ps,xs) +O(δ2gA)

wheref =Ax−iAyandba,ba

†_{are the usualannihilationandcreation operators}

of the harmonic oscillatorΞ.

Letπr:=|ψ0ihψ0|+|ψ1ihψ1|,Πrits quantization andK'L2(R)s⊗C2the

reference subspace. The effective dynamics is given (up to a constant term) by

b

Heff_δ =





1

21 δgAbf(Ps,Qs) †

δgAbf(P_s,Q_s) −1₂1



+O₀ δ2gV

.

In generalgAgV, then the approximation is meaningful ifδ g_gA

V. The

HamiltonianHbeff

δ is aDirac-likeoperator and

σ(Hb_δeff) =± r

1

4+σ(bfbf

†_{)∪ ±}

r

1 4+σ(bf

†

bf)

.

Now suppose that~A_Γ6=0 and assume (to simplify the notation) thatΓ=Z2

(square lattice). LetgAbe thecoupling constantof the periodic magnetic field

andgV thecoupling constantof the periodic electric field.In this case the semiclassical symbol is given by

H_δ(ps,xs) =Ξ+δgA f(ps,xs)ba+f(ps,xs)ba

†

+δ2gVV(ps,xs) +O(δ2gA)

wheref =Ax−iAyandba,ba

†_{are the usualannihilationandcreation operators}

of the harmonic oscillatorΞ.

Letπr:=|ψ0ihψ0|+|ψ1ihψ1|,Πrits quantization andK'L2(R)s⊗C2the

reference subspace. The effective dynamics is given (up to a constant term) by

b

Heff_δ =





1

21 δgAbf(P_s,Q_s)†

δgAbf(P_s,Q_s) −1₂1



+O₀ δ2gV

.

In generalgAgV, then the approximation is meaningful ifδ g_gA

V. The

HamiltonianHbeff

δ is aDirac-likeoperator and

σ(Hb_δeff) =± r

1

4+σ(bfbf

†_{)∪ ±}

r

1 4+σ(bf

†_bf)

References

[Av] J. Avron.Colored Hofstadter butterflies.

http://arxiv.org/abs/math-ph/0308030.

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

champ magnétique et équation de HarperinLecture Notes in

Physics. Schrödinger operators (Sønderborg, 1988),345,

118–198 (1989). Springer.

[PST1] G. Panati, H. Spohn and S. Teufel.Effective dynamics for Bloch

electrons: Peierls substitution and beyond. Comm. Math. Phys.

242, 547-578 (2003).

[PST2] G. Panati, H. Spohn and S. Teufel.Space-Adiabatic

Perturbation Theory. Adv. Theor. Math. Phys. 7, 145-204

(2003).