On Effective Hamiltonians for Adiabatic Perturbations of Magnetic Schrodinger Operators

0.11 MB·English

by  M. Dimassi

On Effective Hamiltonians for Adiabatic Perturbations of
Magnetic Schro¨dinger Operators
Mouez Dimassi, Jean-Claude Guillot and James Ralston
Abstract.We construct almost invariant subspaces and the corresponding effective
Hamiltonian for magnetic Bloch bands. We also discuss the question of the dynam-
ics related to the effective Hamiltonian. We assume that the magnetic and electric
4
potentials are slowly varying perturbations of the potential of a constant magnetic
0
0 field and a periodic lattice potential, respectively.
2
n 1. Introduction
a In [5] we constructed wave packets for adiabatic perturbations of Schr¨odinger
J
operators in periodic media. The recent work of Panati-Spohn-Teufel, [19], led us
8
to consider the relation of those constructions to effective Hamiltonians. In §3 of

1
this article we give a simple derivation of effective Hamiltonians for these problems.
v
7 The main simplification in our method is the omission of the Floquet-Bloch
1 transformation. This transformation has many nice properties. In particular, it is
0
1 unitary, andthis makesinuseful instudying spectral properties ofoperators. Inthe
0 work of Helffer-Sj¨ostrand [9] and G´erard-Martinez-Sj¨ostrand [7] this transformation
4
was used quite effectively in the computation of spectra, both of perturbed and ef-
0
/ fective Hamiltonians. However, if one is simply interested in effective Hamiltonians,
h
p the Floquet-Bloch transformation requires that one transform the Hamiltonian by
-
a Fourier integral unitary operator only to transform it back at the end of the
h
t calculation. In this article we need to assume that eigenspaces of the unperturbed
a
m Hamiltonian depend smoothly on quasi-momentum, and form trivial bundles over
: a fundamental domain for the dual lattice.
v
i There is also the interesting question of how one interprets the lower order terms
X
in the effective Hamiltonian. If one considers the propagation of observables in the
r
a Heisenberg picture, it is natural to think of these terms as lower order corrections
to the dynamics. This point of view is adopted in [19], and it is implicit in [3] and
[4]. However, the highest order contributions of these terms to the wave packets are
in a precession of the phase. Thus in [5] we did not include them in the dynamics,
and did not see how to reconcile our results with those of [3] and [4]. It now appears
that the two points of view complement each other instead of conflicting.
2. Preliminaries
The Hamiltonian for an electron in a crystal lattice Γ in R3 in the presence of a
constant magnetic field ω = (ω ,ω ,ω ) is given by
1 2 3
2
1 ∂ ω ×x
H = −ih +e +V(x), (1)
0
2m (cid:18) ∂x 2 (cid:19)
where V is a smooth, real-valued potential, periodic with respect to Γ. Here m and
e are the mass and charge of the electron. To simplify notation we will use units in
which h = 2m = e = 1.
We will assume that Γ is generated by the basis {e ,e ,e } for R3,
1 2 3
Γ = e Z+e Z+e Z, (2)
1 2 3
2
3
and let E be the fundamental domain { t e ,t ∈ [0,1)}. We will use the dual
j=1 j j j
lattice Γ∗ = e∗Z+e∗Z+e∗Z, where e∗P·e = 2πδ , with the fundamental domain
1 2 3 j k jk
E∗ = { 3 t e∗,t ∈ [0,1)}.
j=1 j j j
To rPealize H as a self-adjoint operator in L2(R3) we define it first on the
0
Schwartz functions S(R3), and then take the Friedrichs extension. The resulting
operator commutes with the magnetic translations introduced by Zak [24],
T f(x) = eihω×x,γi/2f(x−γ) (3)
γ
for γ ∈ Γ. We assume that
hω,Γ×Γi ⊂ 4πZ.
With this assumption G = {T ,γ ∈ Γ} is an abelian group, and we can reduce H
γ 0
by the eigenspaces of G, i.e. setting
D = {u ∈ H2 (R3),T u = e−ik·γu,γ ∈ Γ}, (4)
k loc γ
considered as a subspace of L2(E), H restricted to D is self-adjoint with compact
0 k
resolvant. We denote its spectrum by
E (k) ≤ E (k) ≤ ...
1 2
Then by standard results the spectrum of H as an operator in L2(R3) is equal to
0
∪k∈E∗ ∪∞m=1 Em(k).
Note that, since Dk+γ∗ = Dk for γ∗ ∈ Γ∗, Em(k +γ∗) = Em(k).
Standard perturbation theory shows that the function E (k) is continuous for
m
k ∈ R3 and real analytic in a neighborhood of any k such that
E (k) < E (k) < E (k) (5)
m−1 m m+1
The closed interval Λm = ∪k∈E∗Em(k) is known as the “m-th magnetic Bloch
band” in the spectrum of H .
0
In what follows it will be convenient to replace H acting on D by
0 k
2
∂ ω ×x
H (k) = e−ik·xH eik·x = −i + +k +V(x)
0 0
(cid:18) ∂x 2 (cid:19)
with the domain
D = {u ∈ H2 (R3),T u = u,γ ∈ Γ}.
loc γ
for all k. As with D , we consider D as a subspace of L2(E).
k
Assumption A. For a given m we will assume that E satisfies (5) for all k.
m
Under this assumption we can choose the eigenfunction Ψ(x,k) associated to
E (k) to be a real-analytic function of k with values in D, such that
m
H (k)Ψ(k) = E (k)Ψ(k) for all k |Ψ(x,k)|2dx = 1.
0 m
Z
3
Assumption B. We assume that
∗
Ψ(x,k+γ∗) = eiγ ·xΨ(x,k),γ∗ ∈ Γ∗.
This assumption makes the complex line bundle of the eigenspaces a trivial
bundle over the torus , R3/Γ∗. In general one has
∗ ∗
Ψ(x,k +γ∗) = ei(γ ·x+θ(k,γ ))Ψ(x,k),γ∗ ∈ Γ∗,
where θ(k,γ∗) isreal-valued, anddetermines the structure oftheeigenspace bundle.
Since
θ(k,m e∗ +m e∗ +m e∗) = m θ(k,e∗)+m θ(k,e∗)+m θ(k,e∗) mod 2π,
1 1 2 2 3 3 1 1 2 2 3 3
when θ(k,γ∗) is nonzero, the derivatives of Ψ with k will be unbounded and Ψ
will not belong to the class of symbols B which we introduce below. Thus we need
Assumption B.
Remark 1. The general method of constructing effective Hamiltonians which
we give here will apply under the weaker hypothesis: for a given m there exist p and
q such that
E (k) < E (k) and E (k) < E (k) for all k.
m−p−1 m−p m+q m+q+1
However, in this case the effective Hamiltonian will be a matrix operator acting on
functions with values in Cp+q+1, as in [6],[7], [9] and [19].
3. Main Result
The adiabatically perturbed Hamiltonian is
2
∂ ω ×x
H = −i + +A(ǫx) +V(x)+W(ǫx),
ǫ
(cid:18) ∂x 2 (cid:19)
where W and A = (A ,A ,A ) are smooth, and bounded together with all of their
1 2 3
derivatives. As before, we define H first on S(R3), and then take the Friedrichs
ǫ
extension to get a self-adjoint operator in L2(R3).
The essential step in applying multi-scale techniques is simply to consider y = ǫx
as a new independent variable in H . Let
ǫ
2
∂ ∂ ω ×x
˜
H = −i −iǫ + +A(y) +V(x)+W(y).
ǫ
(cid:18) ∂x ∂y 2 (cid:19)
Then, for u(x,y) we can define w(x) = u(x,ǫx) and conclude that
[H˜ u](x,ǫx) = [H w](x). (6)
ǫ ǫ
The identity (6) enables us to solve the Schr¨odinger equation for H uniformly in
ǫ
ǫ by solving the Schro¨dinger equation for H˜ uniformly in (y,ǫ). The latter might
ǫ
sound more difficult, but it turns out not to be.
4
Let B denote the subspace of C∞(R3 ×R3 ×R3) consisting of functions of the
form
P(x,y,k,ǫ) = P (x,y,k)+ǫP (x,y,k)+···+ǫNP (x,y,k)
0 1 N
such that P(x+γ,y,k,ǫ)= eihω×x,γi/2P(x,y,k,ǫ) and
supk∂yα∂kβPj(·,y,k)kL2(E) < ∞, for all α,β ∈ N3.
y,k
To P ∈ B we associate the ǫ-pseudo-differential operator
P(x,y,ǫD ,ǫ)f(x,y,ǫ)= (2πǫ)−3 eik·(y−z)/ǫP(x,y,k,ǫ)f(z)dzdk,f ∈ S(R3).
y
Z
Note that here we are using the standard quantization – as opposed to the Weyl
quantization. Our main result is the following:
Theorem. For every N ∈ N there exist P = F + ǫF + ··· + ǫNF ∈ B and
N 0 1 N
HN = h +ǫh +···+ǫNh ∈ B (independent on x) such that
eff 0 1 N
H˜ (P (x,y,ǫD ,ǫ)u)−P (x,y,ǫD ,ǫ)HN (y,ǫD )u = O(ǫN+1) (7)
ǫ N y N y eff y
for u ∈ S(R3). Moreover, considered as an operator from L2(R3) into L2(E×R3),
P is approximately isometric, i.e. P∗P = I +O(ǫN+1).
N N N
We interpret HN as the effective Hamiltonian up to order ǫN. The leading term
eff
in its symbol is h (y,k) = E (k +A(y))+W(y). This is the well-known “Peierls
0 m
substitution”, [20]. The symbol of h is also quite interesting, and we discuss it in
1
§4.
Proof. As in §2 it will be convenient to work with
2
∂ ∂ ω ×x
H˜ (k) = e−ik·xH˜ eik·x = −i −iǫ + +A(y)+k +V(x)+W(y),
ǫ ǫ
(cid:18) ∂x ∂y 2 (cid:19)
acting on functions in B, in place of H˜ . Note that
ǫ
H˜ (P (x,y,ǫD ,ǫ)u) = (2πǫ)−3 eik·(y−z)/ǫH˜ (k)P (x,y,k,ǫ)u(z)dzdk.
ǫ N y ǫ N
Z
The Hamiltonian H˜ (k) can be written as H˜ (k) = H˜ (k)+ǫH˜ (k) +ǫ2H˜ (k),
ǫ ǫ 0 1 2
where
ω ×x
H˜ (k) = (−i∂ + +k +A(y))2 +V(x)+W(y) = H (k +A(y))+W(y)
0 x 0
2
ω ×x
H˜ (k) = −2i(−i∂ + +k +A(y))·∂ −i∂ ·A(y), (8)
1 x y y
2
and H˜ (k) = −∆ .
2 y
We will simply construct the symbols of the pairs (h ,F ), (h ,F ),..., succes-
0 0 1 1
sively so that (7) holds to order O(ǫN+1) in the ǫ-pseudo-idfferential calculus. To
5
cancel the order zero terms in (7) we set h (y,k) = E (k + A(y)) + W(y) and
0 m
F (x,y,k) = Ψ(x,k +A(y)). Then, since the symbol of F (x,y,ǫD )h (y,ǫD ) is
0 0 y 0 y
ǫ∂F ∂h
F (x,y,k)h (y,k)+ 0(x,y,k)· 0(y,k)+O(ǫ2),
0 0
i ∂k ∂y
we must have
1∂F ∂h
(H˜ (k)−h )F − 0 · 0 +(H˜ (k)−h )F = 0. (9)
0 0 1 1 1 0
i ∂k ∂y
By the Fredholm Alternative in L2(E), we can solve (9) for F if and only if
1
1∂F ∂h
hF (·,y,k),− 0(·,y,k)· 0 +(H˜ (k)−h (y,k))F (·,y,k)i = 0,
0 1 1 0
i ∂k ∂y
where h·,·i denotes the inner product in L2(E). Hence, we choose
−1∂F ∂h
h (y,k) = hF (·,y,k), 0(·,y,k)· 0(y,k)+H˜ (k)F (·,y,k)i and (10)
1 0 1 0
i ∂k ∂y
1∂F ∂h
F (x,y,k) = (H˜ (k)−h (y,k))−1( 0· 0+h F −H˜ (k)F )+a (y,k)F , (11)
1 0 0 1 0 1 0 1 0
i ∂k ∂y
where (H˜ (k)−h (y,k))−1 denotes the inverse which maps the orthogonal comple-
0 0
ment of F (·,y,k) in L2(E) onto itself. We recall that
0
˜
H (k)−h (y,k) = H (k +A(y))−E (k +A(y)).
0 0 0 m
We determine a (y,k) by the requirement that P∗P = I + O(ǫN+1). Given
1 N N
f,g ∈ S(R3), this implies
dx [(F (x,y,ǫD )+ǫF (x,y,ǫD ))f][(F (x,y,ǫD )+ǫF (x,y,ǫD ))g]dy =
0 y 1 y 0 y 1 y
ZE ZR3
fgdy +O(ǫ2). (12)
ZR3
Since |F (x,y,k)|2dx = 1 for all (k,y), one can calculate Re{a (y,k)} from (12),
E 0 1
and seRe that it is smooth and bounded, by the pseudo-differential calculus. We
choose Im{a (y,k)} = 0.
1
The calculation of (F ,h ) for j > 1 proceeds in the same manner: we calculate
j j
the terms of order ǫj in the symbol of the left hand side of (7) which come from
(F ,h ) for l < j and choose (F ,h ) so that the left hand side of (7) has the desired
l l j j
form up to terms of order ǫj+1. At each stage we use the Fredholm Alternative in
L2(E), and F is only determined modulo a term of the form a (y,k)F (x,y,k).
j j 0
Then we choose the real part of a (y,k) so that P∗P = I +O(ǫj+1), and take the
j j j
imaginary part of a to be zero. Continuing in this way we complete the proof of
j
the Theorem.
Remark 2 If we set Π = P P∗, then Π is approximately a projection:
N N N N
Π2 = Π +O(ǫN+1) and Π = Π∗ . The equation (7) implies that
N N N N
P∗H˜ = H P∗ +O(ǫN+1).
ǫ eff
6
Hence
P P∗H˜ = P H P∗ +O(ǫN+1).
N N ǫ N eff N
If we replace P H in the equality above by the left hand side of (7),we obtain:
N eff
P P∗H˜ = H˜ P P∗ +O(ǫN+1).
N N ǫ ǫ N N
Thus Π is a projection which commutes with H˜ to order O(ǫN+1) as in [19].
N ǫ
In [19], the construction of the almost invariant subspaces is based on the method
of Nenciu-Sordoni [17] and Sordoni[18] (see also [10], [14],[15]). This method is
heavily related to the construction of Moyal projections.
4. Relations with Previous Work
To relate the results here to what has already been done we need to complete
the calculation of the effective Hamiltonian up to terms of order ǫ2, i.e. to compute
the symbol h (y,k) from (10). The explicit computation of
1
hF (·,y,k),H˜ (k)F (·,y,k)i= E (y,k)
0 1 0 def 1
is contained in the computations in [5] (it is also in [8] with a small error – see
Remark 1 in [5]). When one replaces k(y,s) and Ψ(x,y,s) in [5, pp. 7601-3] by
k+A(y) and Ψ(x,k+A(y)) respectively, E (y,k) is “ih” in the notation of [5, (25)]
1
and we have
˜ ˜ ˜
1 ∂ ∂E (k) ∂Ψ(·,k) ∂E (k)
m ˜ m
E = · −L·B −ihΨ(·,k), i· ,
1
2i∂y ∂k ∂y ∂k
˜
where k = k +A(y). Here B(y) = ∇×A(y) and L =
∂Ψ ∂Ψ ∂Ψ ∂Ψ ∂Ψ ∂Ψ
L = Im hM(y,k) , i,hM(y,k) , i hM(y,k) , i .
,
(cid:18) ∂k ∂k ∂k ∂k ∂k ∂k (cid:19)
2 3 3 1 1 2
with M(y,k) = H˜ (k) − h (y,k). The vector L is an angular momentum and
0 0
L·B contributes the “Rammal-Wilkinson” term to the energy, cf. [1]. Adding the
additional term from (10) to E to obtain h , we obtain
1 1
1 ∂ ∂E (k +A(y))
h (y,k) = · m −L·B −ihΨ(·,k +A(y),Ψ˙ (·,k+A(y))i, (13)
1
2i∂y ∂k
where
∂Ψ(x,k+A(y)) ∂Ψ(x,k+A(y))
˙ ˙
Ψ(x,k+A(y)) = ·y˙ + ·k
∂y ∂k
˙
and y˙ and k are defined by the Hamiltonian system
∂(E (k +A(y))+W(y)) ∂(E (k +A(y))+W(y))
m ˙ m
y˙ = k = − .
∂k ∂y
ThusonerecognizesihΨ(·,k+A(y)),Ψ˙ (·,k+A(y))iasthetermgeneratingtheBerry
phase precession, cf. [13], [22]. Comparing (13) with [19, (22)] (in the case l = 1),
one sees that they agree completely when one takes into account the difference in
7
the choice of sign in the magnetic potential, A(y), and the use of Weyl quantization
in [19]. The sign of the Berry phase term in (13) may appear inconsistent with
˙
[5, (29)], but it is not. In [5] (y˙,k) was the vector field from the Hamiltonian
−E (k +A(y)).
m
In [5] and [8] instead of introducing effective Hamiltonians we constructed wave
packets. These packets are nonetheless related to effective Hamiltonians in that
one can compute what the effective Hamiltonian must be – assuming that there is
one – from the packets. To see this one can proceed as follows. The packets have
the form (here s = ǫt and W(y) = 0)
∂φ
u(x,y,s,ǫ) = eiφ(y,s)/ǫ[f(y,s)Ψ(x, +A(y))+O(ǫ)],
∂y
where φ and f are solutions of
∂φ ∂φ ∂f ∂E ∂φ ∂f
= E ( +A(y)) and = m( +A(y))· +(D−iL·B+hΨ,Ψ˙i)f. (14)
m
∂s ∂y ∂s ∂k ∂y ∂y
˜
Here all functions of (k,y) are evaluated at k = k(y,s) = ∂ φ, and D = (1/2)∂ ·
y y
(∂ E (∂ φ+A(y))). Assuming that the evolution of f is governed by an effective
k m y
Hamiltonian H = h (y,ǫD )+ǫh (y,ǫD )+O(ǫ2), we must have (on bounded
eff 0 y 1 y
intervals in s)
[eisHeff/ǫeiφ(·,0)/ǫf(·,0)](y,s)= eiφ(y,s)/ǫf(y,s)+O(ǫ2). (15)
Differentiating (15) with respect to s, one concludes
i i ∂φ ∂f
H (eiφ(y,s)/ǫf(y,s)) = ( f + )eiφ(y,s)/ǫ +O(ǫ). (16)
eff
ǫ ǫ ∂s ∂s
Using the symbol expansion from the pseudo-differential calculus
e−iφ/ǫH (eiφ/ǫf) = h (y,k˜)+
eff 0
1∂h ∂f 1 ∂2h ∂2φ
ǫ[ 0(y,k˜)· (y)+ 0 (y,k˜) (y)+h (y,k˜)]f(y)+O(ǫ2). (17)
1
i ∂k ∂y 2i ∂k ∂k ∂y ∂y
X j l j l
j,l
Substituting (17) into (16) and comparing the result with (14), one recovers the
formulas given earlier for h (y,k) and h (y,k).
0 1
In[8]wewereunabletoreconcileourresultswiththoseofChang andNiu, see[3],
[4] and also [23]). We thought that this might have resulted from different choices
of scales. This is partially true, since Chang and Niu do not distinguish the scale
y = ǫx, but, as Panati, Spohn and Teufel point out in [19], the differences largely
disappear when one considers the Heisenberg formulation of quantum dynamics.
Letting a(y,ǫD ) be an observable, the propagation of a in the Heisenberg picture
y
is given by
da
a(s) = e−isHeff/ǫaeisHeff/ǫ or ǫ = i[a,H ].
eff
ds
If one considers this propagation at the symbol level, then the symbol of a is
propagating along the trajectories of the Hamiltonian system
∂H ∂H
eff ˙ eff
y˙ = k = − .
∂k ∂y
8
Henceonecanconsiderthecontributionofh asanorderǫcorrectiontotheclassical
1
Peierls dynamics arising from h . This is the point of view taken in [19]. However,
0
it is worth noting that the wave packets are propagating along the trajectories from
h with a precession in their phases arising from (the imaginary part of) h .
0 1
Acknowledgements We wish to thank Professors Panati, Spohn and Teufel for
sendingusthepreprintversionof[19]andforseveralhelpfuldiscussionsofthiswork.
We especially thank Stefan Teufel for pointing out the necessity of Assumption B
here.
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M. Dimassi and J.-C. Guillot
D´epartement de Math´ematiques, Universit´e Paris 13, Villetaneuse, France
email: [email protected]
email: [email protected]
J. Ralston
University of California, Los Angeles, CA 90095, USA
email: [email protected]