Download Dynamics of a classical Hall system driven by a time-dependent

JOURNAL OF MATHEMATICAL PHYSICS 48, 052901 共2007兲
Dynamics of a classical Hall system driven
by a time-dependent Aharonov-Bohm flux
J. Ascha兲
CPT-CNRS, Luminy Case 907, F-13288 Marseille Cedex 9, France
P. Šťovíček
Department of Mathematics, Faculty of Nuclear Science, Czech Technical University,
Trojanova 13, 120 00 Prague, Czech Republic
共Received 4 September 2006; accepted 16 March 2007; published online 10 May 2007兲
We study the dynamics of a classical particle moving in a punctured plane under
the influence of a homogeneous magnetic field, an electric background, and driven
by a time-dependent singular flux tube through the hole. We exhibit a striking
共de兲localization effect: when the electric background is absent we prove that a
linearly time-dependent flux tube opposite to the homogeneous flux eventually
leads to the usual classical Hall motion: the particle follows a cycloid whose center
is drifting orthogonal to the electric field; if the fluxes are additive, the drifting
center eventually gets pinned by the flux tube whereas the kinetic energy is growing with the additional flux. © 2007 American Institute of Physics.
关DOI: 10.1063/1.2723550兴
I. INTRODUCTION
The motivation to study the dynamics of this classical system is to sharpen our intuition on its
quantum counterpart which is, following Laughlin’s13 and Halperin’s11 proposals, widely used for
an explanation of the integer quantum Hall effect. Of special interest is how the topology influences on the dynamics. In the mathematical physics literature Bellissard et al.5 and Avron et al.3,4
used an adiabatic limit of the model to introduce indices. The indices explain the quantization of
charge transport observed in the experiments.12 See Refs. 6, 9, 7, 8, and 10 for recent developments. We discussed aspects of the adiabatics of the quantum system in Ref. 2, its quantum and
semiclassical dynamics will be treated elsewhere. The dynamics of the classical system without
magnetic field were discussed in Ref. 1. We state the model and our main results.
Consider a classical point particle of mass m ⬎ 0 and charge e ⬎ 0 moving in the punctured
plane R2 \ 共0兲. Suppose that a magnetic flux line with time varying strength ⌽ pierces the origin
and further the presence of a homogeneous magnetic field of strength B orthogonal to the plane
and an interior electric field with smooth bounded potential V.
Let
⌽:R → R and V:R ⫻ R2 → R be smooth functions.
Denote q⬜ ª 共−q2 , q1兲. The force on the particle at q 苸 R2 \ 共0兲 with velocity q̇ is
冉
− e Bq̇⬜ −
冊
⳵ t⌽ q ⬜
+ ⳵qV = e共q̇ ∧ rot共A兲 − ⳵tA兲,
2␲ 兩q兩2
with
a兲
Electronic mail: [email protected]
0022-2488/2007/48共5兲/052901/14/$23.00
48, 052901-1
© 2007 American Institute of Physics
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052901-2
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
A共⌽共t兲,q兲 ª
冉
冊
B
⌽共t兲
−
q⬜ + t⳵qV共q兲.
2 2␲兩q兩2
Remark that the electromotive force induced by the flux line has circulation e⳵t⌽, constant
torque e⳵t⌽ / 2␲, vanishing rotation, and is long range with a 1 / r singularity at the origin, we call
it the circular part. V is smooth on the entire plane so the circulation of the corresponding field is
zero. The total magnetic flux through a circle of radius R is ␲BR2 − ⌽ if it encircles the flux line,
else ␲BR2. So the two fluxes are “opposite” if B and ⌽ have the same sign.
The equations of motions are Hamiltonian. For a point 共q , p兲 = 共共q1 , q2兲 , 共p1 , p2兲兲 in phase
space
P = R2 \ 共0兲 ⫻ R2 ,
the time-dependent Hamiltonian is
H共t,B;q,p兲 =
1
共p − eA共⌽共t兲,q兲兲2 .
2m
Suppose
⌽共− t兲 = − ⌽共t兲,
then there is the time reversal symmetry
H共− t,− B;q,− p兲 = H共t,B;q,p兲.
So in order to fix the ideas we convene that growing time means growing flux opposite to the
homogeneous flux and suppose furthermore
B ⬎ 0,
⳵t⌽共t兲 艌 ⌽0 ⬎ 0.
Recall that when only the constant magnetic field is present, the particle follows the Landau
orbits: circles around a fixed center with the cyclotron frequency
␻=
eB
m
and radius R such that the magnetic flux through the Landau circle satisfies e␻ / 2␲共␲BR2兲 = H.
With the above convention this motion is clockwise.
If V = 0 then intuitively the physics is the following: the additional field makes the center q
− 共q̇⬜ / ␻兲 drift orthogonal to the electric field. The constant torque exerced by the circular part
accelerates orbits which encircle the origin, the orbit shrinks with growing flux and grows with
decreasing flux; orbits not encircling the flux line should have constant radius.
We have the following results for the case ⌽ = ⌽0t, V = 0 共see Corollary 7.2兲.
a.
The above intuition is asymptotically correct. Furthermore
H共t,B兲⬃⌽共t兲→−⬁
e␻
兩⌽共t兲兩
2␲
H共t,B兲→⌽共t兲→+⬁const.
b.
In the accelerating regime the center c is eventually trapped by the flux line, the particle is
spiraling outward
c共t兲→⌽共t兲→−⬁const,
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052901-3
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
FIG. 1. Typical trajectory of the Hamiltonian 1 / 2共p − 共共1 / 2兲q⬜ − s共q⬜ / q2兲兲兲2.
冑
c.
␲B
q共t兲⬃⌽共t兲→−⬁共cos共− ␻t兲,sin共− ␻t兲兲.
兩⌽共t兲兩
In the decelerating regime the orbit ends up drifting diffusively orthogonal to the field
冑
␲B
q共t兲→⌽共t兲→⬁const.
⌽共t兲
In addition we expand the solution up to an error O共1 / t3/2兲 关see Theorem 共5.1兲兴. We further
compute the adiabatic limit 共i.e., the solution of the equations of motions averaged over the
Landau orbits兲 in the perspective to obtain information on the transition between the two dynamics. We find 关see Theorem 共6.2兲兴 that in the adiabatic limit the transition between the two dynamics is sharp and that the center gets stuck after a finite time if there is no electric background; it is
a challenging problem to study if the adiabatic limit provides an approximation of the true dynamics.
For a general increasing flux and a background field whose torque is controlled by the constant torque of the circular part we show 关see Corrollary 共7.1兲兴
2␲ H共t,B兲
⬎ 0,
⌽共t兲→−⬁ e␻ 兩⌽共t兲兩
lim inf
␲Bc2共t兲
⬎ 0.
⌽共t兲→⬁ ⌽共t兲
lim inf
We may state our observation as follows: States if submitted to an accelerating flux line will
eventually become energy conducting; if no electric background is present they get trapped by the
flux tube.
We give two numerical illustrations in Figs. 1 and 2.
II. DYNAMICS OF THE FROZEN SYSTEM
Upon scaling q, p, t to dimensionless coordinates, which we call q, p, s, we work with the
Hamilitonian, H共␧s兲 where
1
H共s;q,p兲 = 共p − a共s,q兲兲2 ,
2
with
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052901-4
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J. Asch and P. Šťovíček
FIG. 2. Typical trajectory of the Hamiltonian 1 / 2共p − 共共1 / 2兲q⬜ − s共q⬜ / q2兲 + s⳵qV兲兲2 with the background potential V chosen
to be V共x , y兲 = 1 / 10共sin x + sin y兲 on a region 关−10, 10兴2. The background shows the potential lines of V共x , y兲 − arg共x , y兲.
and N, Ṽ smooth functions,
N共− s兲 = − N共s兲,
⳵sN 艌 1,
and ␧ a dimensionless parameter. We discuss the scaling in Sec. VII.
The function aE is smooth on R2 \ 共0兲 with rot共aE兲 = 0. Define E共s兲 : R2 \ 共0兲 → R2 by
冉
E共s兲 ª − ⳵saE共␧s兲 = ␧ 共⳵sN兲共␧s兲
冊
q⬜
− ⳵qṼ共q兲 .
q2
共1兲
We discuss first the solution of the equation of motions for a frozen time ␴ 苸 R. As
⳵saE共␴ ; q兲 = 0, the solution of the frozen equations generated by the Hamiltonian H共␴兲 goes along
the lines of the classical Landau problem 关which means the case ⌽ = 0, V共q兲 = 0兴.
For ␴ 苸 R define
1.
2.
3.
the velocity field: v共␴兲 : P → R2, v共␴ ; q , p兲 ª p − a共␴ ; q兲;
the center: c共␴兲 : P → R2, c共␴ ; q , p兲 ª q − v⬜共␴ ; q , p兲;
the angular momentum: L : P → R, L共q , p兲 ª q ∧ p.
Denote the Poisson bracket: 兵f , g其 = ⳵q f ⳵ pg − ⳵ p f ⳵qg.
We list some useful formulas.
Proposition 2.1: The following identities hold as functions on phase space P for all ␴ 苸 R:
1.
2.
3.
兵v1 , v2其 = 1, 兵c1 , c2其 = −1, 兵c , c2 / 2其 = c⬜, 兵ci , v j其 = 0;
H = 21 v2, 兵v , H其 = −v⬜, 兵c , H其 = 0;
1 2
c 共␴兲 = H共␴兲 + L − q ∧ aE共␴兲 = H + L + 共N共␴兲 − ␴q ∧ ⳵qṼ兲;
2
4.
共2兲
the frozen flow 共q共␴ ; s兲 , p共␴ ; s兲兲 defined by ⳵sq共␴ ; s兲 = ⳵ pH共␴兲, ⳵s p共␴ ; s兲 = −⳵qH共␴兲,
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052901-5
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
共q共␴ ; 0兲 , p共␴ ; 0兲兲 = 共q , p兲 is
q共␴ ;s兲 = c共␴兲 + cos共s兲v⬜共␴兲 + sin共s兲v共␴兲
1
p共␴ ;s兲 = 共c⬜共␴兲 + cos共s兲v共␴兲 − sin共s兲v⬜共␴兲兲 + aE共␴ ;q共␴ ;s兲兲.
2
Proof:
a.
b.
1, 2, 3: 兵v1 , v2其 = 兵p1 − a1共␴ , q兲 , p2 − a2共␴ , q兲其 = rot共a共␴兲兲 = 1, 兵qi , v j其 = ␦ij. H = 21 v2 so 兵q , H其
= v , 兵v , H其 = −v⬜. c2 = q2 + v2 + 2q ∧ v; on the other hand, L = q ∧ v + 21 q2 + q ∧ aE共␴ ; q兲.
4: The force is −q̇⬜ independently of ␴; Newton’s equation q̈ = −q̇⬜ is readily verified. On
the other hand, p = v + a = a + c⬜ − q⬜ = c⬜ − 21 q⬜ + aE共␴ ; q兲. So p共s兲 follows from q共s兲.
䊐
Remark 2.1:
1.
Since the energy H共␴兲 = 21 v共␴兲2 is conserved under the frozen flow, the projections of the
trajectories to q space are circles around c共␴兲 with radius 冑2H共␴兲. An orbit encircles the
origin (has nontrivial homotopy) in R2 \ 共0兲 if and only if
c2 ⬍ 2H ⇔ L − q ∧ aE共␴ ;q兲 ⬍ 0;
2.
1 2 1 ⬜ 2
2 c = 2 共c 兲
is the Hamiltonian for the reversed magnetic field.
III. GENERAL FEATURES
Set ␧ = 1 and denote by abuse of notation O共s兲 ª O共s ; q共s兲 , p共s兲兲 for an observable O.
We have the following general qualitative behavior.
Proposition 3.1: Suppose that there exists a 苸 关0 , 1兲 such that for all s , q:
兩q ∧ ⳵qṼ共q兲兩 艋 ⳵sN共s兲a,
then for any initial condition there exists a unique “hitting” time s0 such that
±c2共s兲 艌 ± 2H共s兲,
± s ⬎ ± s0 .
Furthermore,
2H共s兲
艌 共1 − a兲 ⬎ 0,
N共s兲→−⬁ 兩N共s兲兩
lim inf
c2共s兲
艌 共1 − a兲 ⬎ 0.
N共s兲→⬁ 2N共s兲
lim inf
For radially symmetric potentials it holds
兩q ∧ ⳵qṼ共q兲兩 = 0 ⇒
c2共s兲
− H共s兲 = s − s0 .
2
共3兲
Proof: By Eq. 共2兲 and as 兵c2 , H其 = 0,
冉 冊 冉 冊
c2
d c2
− H = ⳵s
− H = 共⳵sN − q ∧ ⳵qṼ兲 艌 ⳵sN共1 − a兲 艌 共1 − a兲,
2
ds 2
this gives the first claim. The second follows from positivity,
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052901-6
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
冉 冊
冉 冊
c2共s兲
c2
c2
艌 H共s兲 +
− H 共0兲 + 共1 − a兲N共s兲 艌
− H 共0兲 + 共1 − a兲N共s兲,
2
2
2
which implies
lim inf
s→⬁
c2共s兲
c2共s兲
= lim inf
艌 共1 − a兲.
2N共s兲 N共s兲→⬁ 2N共s兲
Analogously,
lim inf
s→−⬁
H共s兲
H共s兲
= lim inf
艌 共1 − a兲.
− N共s兲 N共s兲→−⬁ 兩N共s兲兩
IV. ACTION ANGLE COORDINATES
In order to discuss the dynamics we introduce action angle coordinates. The frozen dynamics
as discussed in Proposition 2.1 suggests to take as coordinates the angles and absolute values of c
and v⬜, i.e., with the notation,
e共␪兲 ª 共cos ␪,sin ␪兲,
q = c + v⬜ = 兩c兩
c
v⬜
+ 兩v兩
¬ 兩c兩e共␸1兲 + 兩v兩e共− ␸2兲,
兩c兩
兩v兩
1
1
p = 共c⬜ + v兲 + aE共␴ ;q兲 = 共兩c兩e⬜共␸1兲 − 兩v兩e⬜共− ␸2兲兲 + aE共␴ ;q兲.
2
2
Motivated by this we define for ␴ 苸 R,
q共␴ ; ␸,I兲 ª 冑2I1e共␸1兲 + 冑2I2e共− ␸2兲 ¬ q共␸,I兲,
1
p共␴ ; ␸,I兲 ª 共冑2I1e⬜共␸1兲 − 冑2I2e⬜共− ␸2兲兲 + aE共␴ ;q共␴ ; ␸,I兲兲,
2
and denote by C the nullset 兵共␸ , I兲 ; ␸1 + ␸2 = ␲ , I1 = I2其 where q共␴ ; ␸ , I兲 = 0, by D the nullset
兵共q , p兲 ; v2 = 0 or c2 = 0其. Thus for each frozen time ␴ 苸 R the transformation to action angle coordinates T共␴兲 is defined by
T共␴兲:S1 ⫻ S1 ⫻ 兵共I1,I2兲;I1 艌 0,I2 艌 0其 \ C → P \ D,
T共␴ ; ␸,I兲 = T共␴ ; ␸1, ␸2,I1,I2兲 ª 共q共␴ ; ␸,I兲,p共␴ ; ␸,I兲兲.
We have the following.
Lemma 4.1:
1.
2.
T共␴兲 is a canonical diffeomorphism.
T−1共␴兲 is determined by
I 1共 ␴ 兲 =
冉 冉
冊冊
冉 冉
冊冊
1
c 2共 ␴ 兲 1
= p − − q⬜ + aE共␴ ;q兲
2
2
2
I2共␴兲 = H共␴兲 =
1
1
p − q⬜ + aE共␴ ;q兲
2
2
2
,
2
,
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052901-7
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
e共␸1共␴兲兲 =
共1/2兲q − p⬜ + aE⬜共␴ ;q兲
c
共␴兲 =
冑2共H共␴兲 + L − q ∧ aE共␴ ;q兲兲 ,
兩c兩
e共− ␸2共␴兲兲 =
共1/2兲q + p⬜ − aE⬜共␴ ;q兲
v⬜
.
共␴兲 =
冑2H共␴兲
兩v兩
Proof: These identities follow immediately from Proposition 2.1:
兵I1,I2其 = 0,
兵e共␸1兲,e共␸2兲其 = 0,
兵e共␸1兲,I1其 =
兵I1,e共␸2兲其 = 0 = 兵I2,e共␸1兲其,
再 冎
1
c2
c⬜
c,
=
= e⬜共␸1兲.
2
兩c兩
兩c兩
On the other hand, 兵e共␸1兲 , I1其 = e⬜共␸1兲兵␸1 , I1其, so 兵␸1 , I1其 = 1. Similarly, 兵␸2 , I2其 = 1.
䊐
We now write the equations of motion for time-dependent flux in these action angle coordinates. As rot共E兲 = 0 there exists a 共possibly multivalued兲 function which we denote by m
= m共s ; q兲 such that
⳵qm共s兲 = E共s兲 = − ⳵saE共␧s兲.
Then T共s兲 is generated by m:
⳵sT共s; ␸,I兲 = 共0, ⳵saE共␧s,q共␸,I兲兲兲 = 共⳵ pm,− ⳵qm兲 ⴰ T共s; ␸,I兲.
Denote by U共s兲 : P → P the Hamiltonian flow of H共⑀s兲 defined by U共s兲 ª 共q共s兲 , p共s兲兲,
q̇共s兲 = ⳵ pH,ṗ共s兲 = − ⳵qH,
共q共0兲,p共0兲兲 = 共q,p兲,
then for the flow Û共s兲 = 共␸共s兲 , I共s兲兲 in action angle coordinates defined by
T共s兲 ⴰ Û共s兲 = U共s兲 ⴰ T共s = 0兲,
it holds
␸˙ 共s兲 = ⳵IK ⴰ Û共s兲,
İ共s兲 = − ⳵␸K ⴰ Û共s兲,
共␸共0兲,I共0兲兲 = 共␸,I兲,
where the Hamiltonian in action angle coordinates, K = H ⴰ T − m ⴰ T, is
K共s; ␸,I兲 = I2 − m共s;q共␸,I兲兲
and the equations of motion are 共with the notation 具·,·典 for the scalar product兲
␸˙ 共s兲 = ⳵IK =
冉冊
0
1
− 具E共s,q共␸,I兲兲, ⳵Iq典,
İ共s兲 = − ⳵␸K = 具E共s;q共␸,I兲兲, ⳵␸q典.
共4兲
共5兲
Remark 4.1: Another way to derive these equations is to start from Newton’s equation,
q̈ = − q̇⬜ + E共s;q兲.
From the very definition of c and v one gets
ċ = − E⬜共c + v⬜兲,
v̇ = − v⬜ + E共c + v⬜兲,
which in action angle coordinates gives Eqs. (4) and (5).
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052901-8
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
V. LARGE TIME ASYMPTOTICS, POTENTIAL FREE CASE
For the case N共s兲 = s, V = 0 we can precise the large time asymptotics and develop the solution
up to order O共1 / s3/2兲 We have
E共s兲 =
q⬜
.
q2
So m共q兲 = arg共q兲. Observe that
K = K共␸,I兲 = I2 − arg共冑2I1e共␸1兲 + 冑2I2e共− ␸2兲兲
is an integral of motion.
Theorem 5.1: Let V = 0, N共s兲 = s. Denote by I = 共I1 , I2兲, ␸ = 共␸1 , ␸2兲 the solution of the equations of motion (4) and (5),
␸˙ 共s兲 = ⳵IK共␸共s兲,I共s兲兲,
I共0兲 = 共I01,I02兲,
İ共s兲 = − ⳵␸K共␸共s兲,I共s兲兲,
␸共0兲 = 共␸01, ␸02兲,
then the following asymptotic behaviors hold.
a.
In the future, s → ⬁.
The following limits exist and define the constants a0 ⬎ 0, b0:
lim I2共s兲 ¬
s→⬁
a20
,
4
lim 共␸1共s兲 + ␸2共s兲 − s兲 ¬ b0,
s→⬁
lim 共I2共s兲 − ␸1共s兲兲 = K.
s→⬁
The asymptotics are
I2共s兲 =
冉
冊 冉 冊
a20 a0
a2
1 1
1
1
−
sin共s + b0兲 + 1 + 0 sin共2共s + b0兲兲 + O 3/2 ,
冑s 4
4
2
2
s
s
I1共s兲 = I2共s兲 + 共s − s0兲,
␸1共s兲 =
␸2共s兲 = s + b0 −
+O
b.
冉
冊
a20
1 1
4
1
1
cos共s + b0兲 + − 1 + 2 cos共2共s + b0兲兲 − 2 sin共2共s + b0兲兲
+K−
冑s 8
4
a0
s
a0
冉 冊
1
s
冉 冊
a20
1
1
−K−
+ O 3/2 ,
4
4s
s
3/2
,
with s0 defined as in Eq. (3).
In the past, s → −⬁.
The following limits exist and define the constants ã0 ⬎ 0, b̃0:
lim I1共s兲 ¬
s→−⬁
ã20
,
4
lim 共␸1共s兲 + ␸2共s兲 − s兲 ¬ b̃0,
s→−⬁
lim 共I2共s兲 + ␸2共s兲兲 = K.
s→−⬁
The asymptotics are
I1共s兲 =
冉
冊 冉 冊
ã2
ã20 ã0
1
1
1
1
− 1 − 0 sin共2共s + b̃0兲兲 + O
,
+
sin共s + b̃0兲
冑
4
2
2
s
兩s兩3/2
兩s兩 4
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052901-9
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
I2共s兲 = I1共s兲 − 共s − s0兲,
␸1共s兲 = s0 + b̃0 +
−
ã20
1
1
cos共s + b̃0兲
−K+
冑兩s兩 ,
4
ã0
冉
冊 冉 冊
1
4
1
1
1 − 2 cos共2共s + b̃0兲兲 − 2 sin共2共s + b̃0兲兲 + O
,
8
s
兩s兩3/2
ã0
␸2共s兲 = s − s0 −
冉 冊
ã20
1
1
.
+K−
+O
4
4s
兩s兩3/2
Proof: We give an outline of the main steps of the proof for the case t → ⬁. Some particular
computations in the proof turned out to be quite tedious and thus computer algebra systems were
employed to facilitate them.
Suppose t ⬎ 0.
Step 1. From Eq. 共2兲 we know I1共s兲 − I2共s兲 = 共s − s0兲. So the equations of motion only involve
J ª I1 + I2 and ␺ ª ␸1 + ␸2 and transform to
␺˙ = 1 +
s sin ␺
冑J2 − s2共J + 冑J2 − s2 cos ␺兲 ,
J̇ =
s
,
J + 冑J2 − s2 cos ␺
Step 2. Do a second transformation,
x1 ª 冑J2 − s2 cos ␺,
x2 ª 1 + 冑J2 − s2 sin ␺ ,
the J, ␺ equations transform to
ẋ1 −
x1
+ x2 = F共s,x1,x2兲,
s
ẋ2 − x1 = 0,
with
F共s,x1,x2兲 ª 1 −
x1
s
.
−
s 冑x21 + 共x2 − 1兲2 + s2 + x1
The corresponding homogeneous system is equivalent to
ẍ1 −
冉 冊
1
ẋ1
+ 1 + 2 x1 = 0
s
s
or sÿ + ẏ + sy = 0,
with y defined by x1 = sy. The latter is Bessel’s equation of order 0 so one has two independent
solutions of the homogeneous system:
冉 冊冉 冊 冉 冊冉 冊
x1共s兲
sJ0共s兲
=
x2共s兲
sJ1共s兲
and
x1共s兲
sY 0共s兲
=
x2共s兲
sY 1共s兲
with the Bessel functions Jm 共Y m兲 of the first 共second兲 kind.
Step 3. Transform the x-differential equation to the integral equation,
x1共s兲 = c1sJ0共s兲 + c2sY 0共s兲 −
␲s
2
冕
⬁
共Y 0共s兲J1共␶兲 − J0共s兲Y 1共␶兲兲F共␶,x1共␶兲,x2共␶兲兲d␶ ,
s
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052901-10
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
x2共s兲 = c1sJ1共s兲 + c2sY 1共s兲 −
␲s
2
冕
⬁
共Y 1共s兲J1共␶兲 − J1共s兲Y 1共␶兲兲F共␶,x1共␶兲,x2共␶兲兲d␶ ,
s
where the numbers c1, c2 involve the initial conditions.
The equation is of the form x = K共x兲; the solution is constructed as the limit of the sequence
xn+1 = K共xn兲 starting from x0 = 0. To verify the convergence one can apply yet another substitution
x共s兲 = y共s兲 / 冑s, G共s , y兲 = s−1/2F共s , s−1/2y兲. Consequently, the integral equation takes the form
y共s兲 = y 0共s兲 −
冕
⬁
F共s, ␶兲G共␶,y 1共␶兲,y 2共␶兲兲d␶ ,
s
where
y 0j共s兲 = c1冑sJ j−1共s兲 + c2冑sY j−1共s兲,
F j共s, ␶兲 =
j = 1,2,
␲冑
s␶共Y j−1共s兲J1共␶兲 − J j−1共s兲Y 1共␶兲兲,
2
j = 1,2.
Considering the new integral equation in the Banach space L⬁共关s* , ⬁关兲 丢 R2, one can show that the
iteration process is indeed contracting provided s* 艌 1 is sufficiently large. It is then straightforward to derive from the integral equation the asymptotic expansion of the solution x共s兲. One finds
that
x共s兲 = a0e共s + b0兲冑s +
冉
冊
冉冊
a30
5
1
1
+O
.
e共s + b0兲 − a0e⬜共t + b0兲
冑
8
8
s
s
Step 4. Transforming back first to the J, ␺ then to I1, I2, ␸1, ␸2 variables gives the claimed
asymptotic expansion.
䊐
VI. AVERAGED DYNAMICS
In the perspective to investigate the behavior at the transition point between the two dynamics
in the case of small epsilon we analyze the equation averaged over the fast angle ␸2. This might
provide a good approximation for the actions for small ␧ for times of order 1 / ␧.14 We consider
again the case N共s兲 = s. We set up the averaged equations and show that in the adiabatic limit the
energy grows if and only I1 ⬍ I2.
We then solve the equations explicitly for the case V = 0 and show that in the adiabatic limit
the transition between the pinned and the “Hall dynamics” happens for a unique value of the
driving flux and that the center moves if and only if the Landau orbit does not encircle the origin.
We apply averaging with respect to the fast angle ␸2 to the system 共4兲 and 共5兲.
E共s兲 = ␧
冉
冊
q⬜
− ⳵qṼ共q兲 .
q2
Further, choose m and thus K,
m共q兲 = ␧共arg共q兲 − Ṽ兲,
K共␸,I兲 = I2 − m共冑2I1e共␸1兲 + 冑2I2e共− ␸2兲兲.
Denote the average of a function f on the phase space by
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052901-11
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
f av共␸1,I兲 ª
1
2␲
冕
2␲
f共␸1, ␸2,I兲d␸2 .
0
In particular, for a function f defined on the plane thus depending only on the variable q we denote
f av共␸1,I兲 =
1
2␲
Making use of the identities
冓 冔
冕
2␲
f共冑2I1e共␸1兲 + 冑2I2e共− ␸2兲兲d␸2 .
0
冊 冓 冔冉
冉
sin共␸1 + ␸2兲 冑I2/I1
q⬜
,
2 , ⳵ Iq =
q
q2
− 冑I1/I2
the system 共4兲 and 共5兲 reads
冉冊
0
␸˙ 共s兲 =
1
İ共s兲 = ␧
−␧
冊
关共I1 − I2兲/q2兴 + 1/2
q⬜
,
⳵
q
=
,
␸
关共I1 − I2兲/q2兴 − 1/2
q2
sin共␸1 + ␸2兲
2共I1 + I2 + 2冑I1I2 cos共␸1 + ␸2兲兲
I1 − I2
2共I1 + I2 + 2冑I1I2 cos共␸1 + ␸2兲兲
冉 冑冑 冊
I2/I1
− I1/I2
+ ␧⳵IṼ共q共I, ␸兲兲,
冉冊 冉 冊
1
1
+
␧ 1
− ␧⳵␸Ṽ共q共I, ␸兲兲.
2 −1
The averaged quantities are readily calculated: using
冉冊
1
q2
=
av
1
,
2兩I1 − I2兩
冉
sin共␸1 + ␸2兲
q2
冊
= 0,
av
one finds for the averaged vector field
共⳵IK兲av共␸1,I兲 =
冉冊
0
1
+ ␧⳵IṼav共␸1,I兲 − 共⳵␸K兲av共␸1,I兲 = ␧
冉
冊 冉
冊
␹共I1 ⬎ I2兲
⳵␸1Ṽav共␸1,I兲
−␧
,
− ␹共I1 ⬍ I2兲
0
共6兲
where we used the binary function ␹: ␹共True兲 ª 1, ␹共False兲 ª 0.
Remark 6.1: Remark that the averaged vector field is the Hamiltonian vector field derived
from the from the “averaged” Hamiltonian Kav. Indeed, using the splitting of arg共q兲, which is a
multivalued function defined on the covering space of R2 \ 共0兲, into a linear and oscillating part,
arg共q共␸,I兲兲 =
冦
冉
冑
冉 冑
␸1 + arg 共1,0兲 +
− ␸2 + arg 共1,0兲 +
I2
e共− ␸1 − ␸2兲
I1
I1
e共␸1 + ␸2兲
I2
冊
冊
if I1 ⬎ I2
if I2 ⬎ I1 ,
冧
and the equality
冕
2␲
arg共共1,0兲 + ae共s兲兲ds = 0
for 0 艋 a ⬍ 1,
0
one finds that for
Kav共␸,I兲 ª I2 − ␧共共␸1␹共I1 ⬎ I2兲 − ␸2␹共I1 ⬍ I2兲兲 − Ṽav共␸1,I兲兲,
it holds ⳵␸Kav = 共⳵␸K兲av, ⳵IKav = 共⳵IK兲av.
The result on the averaged dynamics now is as follows.
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052901-12
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
Theorem 6.1: Let N共s兲 = s. Denote by J = 共J1 , J2兲, ␺ = 共␺1 , ␺2兲 the solution of the averaged
equations (6)
␺˙ 共s兲 = ⳵IKav共␺共s兲,J共s兲兲,
J̇共s兲 = − ⳵␸Kav共␺共s兲,J共s兲兲,
J共0兲 = 共J01,J02兲,
␺共0兲 = 共␺01, ␺02兲,
then it holds the following.
1.
Let V = 0, denote ⌬J = J02 − J01 then
冉
冉 冊
J共s兲 = min兵J01,J02其 + 共␧s − ⌬J兲
␺共s兲 =
2.
For any V and any s1 , s2 苸 R,
兩J2共s2兲 − J2共s1兲兩 = ␧
␺01
␺02 + s
冏冕
s2
冊
␹共␧s ⬎ ⌬J兲
,
− ␹共␧s ⬍ ⌬J兲
.
冏
␹共J1共u兲 ⬍ J2共u兲兲du .
s1
Proof: Using that for V = 0 it holds J1共s兲 − J2共s兲 − ␧s = ⌬J the first assertion follows by inspection. The second assertion follows from integration of Eq. 共6兲.
䊐
Remark 6.1:
1.
Loosely speaking the second assertion of the theorem means that, on the average, one has
兩energy change兩 = 兩flux change through the orbit during stay time兩,
where the stay time means the time where the “orbit surrounds the origin.” This should be
like this as the change in energy equals the work of the electric field along the orbit:
H共s;q共s兲兲 − H共s0 ;q共s0兲兲 =
冕
s
具aE共s兲,ds典.
s0
2.
3.
4.
In situations where the averaging approximation is valid one gets estimates of the type
兩I共s兲 − J共s兲兩 = O共␧兲 for 兩s兩 艋 O共1 / ␧兲. Because of the singular behavior of the averaged equation it is a challenging problem to investigate if such an estimate is true or not and how the
error would depend on the initial conditions.
In the case when a smooth potential is present in view of the second assertion of the above
theorem, it would be interesting to investigate if the kinetic energy only fluctuates by small
amounts as soon as the particle is in the region c2 / 2 ⬎ H.
Remark that a finite time adiabatic analysis would apply also to the case where ⌽ is a
switching function which is linear for some time.
VII. SCALING
Let T, L, B, 关⌽兴 be units of time, length, magnetic field, and flux. Define dimensionless
parameters T␻¬␧−1; 关⌽兴 / 共2␲BL2兲¬␩, denote N共s兲 ª ⌽共sT兲 / 关⌽兴 and choose L such that ␩ = 1
then
H共t;q,p兲 =
␻e
关⌽兴H共t/T,q/L,p/共eLB兲兲,
2␲
where
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052901-13
J. Math. Phys. 48, 052901 共2007兲
Hall effect driven by an Aharonov-Bohm flux
1
H共s;q,p兲 = 共p − a共s,q兲兲2 ,
2
with
and
2␲T
V共Lq兲.
e关⌽兴
Ṽ共q兲 ª
The scaled function 共qsc共s兲 , psc共s兲兲 ª 共q / L共s / ␻兲 , p / 共eBL兲共s / ␻兲兲 then solves the Hamilton equations for the Hamiltonian H共␧s兲.
Corollary 7.1: (To Proposition 7.1). Suppose that the torque of the background field is smaller
than the circular one, i.e., that there exists a 苸 关0 , 1兲 such that for all t, q:
e兩q ∧ ⳵qV共q兲兩 艋 e
⳵t⌽共t兲
a,
2␲
then for any initial condition there exists a unique hitting time t0 such that
± ␲Bc2共t兲 艌 ±
2␲
H共t,B兲,
e␻
± t ⬎ ± t0 .
Furthermore for any initial condition we have
2␲ H共t,B兲
艌 共1 − a兲 ⬎ 0,
e␻ 兩⌽共t兲兩
lim inf
⌽共t兲→−⬁
lim inf
⌽共t兲→⬁
␲Bc2共t兲
艌 共1 − a兲 ⬎ 0.
⌽共t兲
䊐
We have
H共t,B兲 =
e␻
关⌽兴I2共␧␻t兲,
2␲
q共t兲 = Lqsc共␻t兲,
H共t,− B兲 =
e␻
关⌽兴I1共␧␻t兲,
2␲
qsc = 冑2I1e共␸1兲 + 冑2I2e共− ␸2兲.
Corollary 7.2: (To Theorem 5.1). Let ␾共t兲 = 关⌽兴t / T, V = 0. The following limits are valid for
any fixed initial condition:
␲Bq2共t兲
→兩⌽共t兲兩→⬁1,
兩⌽共t兲兩
H共t,B兲
兩⌽共t兲兩
→⌽共t兲→−⬁
H共t,− B兲
e␻
,
←⌽共t兲→⬁
2␲
⌽共t兲
H共t,B兲→⌽共t兲→⬁
␻a20
,
4
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052901-14
J. Math. Phys. 48, 052901 共2007兲
J. Asch and P. Šťovíček
H共t,− B兲→⌽共t兲→−⬁
␲B
冑兩⌽共t兲兩
冑
q共t兲→t→⬁e
␻ã20
,
4
冉 冊
a20
−K ,
4
␲B
q共t兲⬃t→−⬁e共− ␻t兲.
兩⌽共t兲兩
Remark also that for the rescaled center it holds
H共t,− B兲 =
1 2
c
2m␻2
so
␲Bc2共t兲⬃t→⬁⌽共t兲.
ACKNOWLEDGMENTS
One of the authors 共P. Š.兲 wishes to acknowledge gratefully partial support from the Grant
Nos. 201/05/0857 of the Grant Agency of the Czech Republic and LC06002 of the Ministry of
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