Edge currents in the absence of edges

Article (PDF Available)inPhysics Letters A 264:124-130 · September 1999with 23 Reads 
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DOI: 10.1016/S0375-9601(99)00804-X · Source: arXiv
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Abstract
We investigate a charged two-dimensional particle in a homogeneous magnetic field interacting with a periodic array of point obstacles. We show that while Landau levels remain to be infinitely degenerate eigenvalues, between them the system has bands of absolutely continuous spectrum and exhibits thus a transport along the array. We also compute the band functions and the corresponding probability current. The fact that the presence of boundaries can induce a transport in a system with a homogeneous magnetic field is known for long [1, 2] and has numerous consequences in solid state physics. The mentioned pioneering papers were followed by tremendous number of studies in which the magnetic transport was analyzed numerically in various models as well as experimentally. The obstacles at which the particle “bounces ” need not be hard walls but also objects with openings such as various antidot lattices; for a sample of literature
Edge currents in the absence of edges
P. Exnera,b, A. Joyec, and H. Kovaˇr´ıka,d,
a) Department of Theoretical Physics, Nuclear Physics Institute, Academy
of Sciences, 25068 ˇ
Reˇz near Prague
b) Doppler Institute, Czech Technical University, Bˇrehov´a 7, 11519 Prague,
Czech Republic
c) Institut Fourier, Universit´e de Grenoble 1, 38402 Saint-Martin d’Heres,
France
d) Faculty of Mathematics and Physics, Charles University,
V Holeˇsoviˇck´ach 2, 18000 Prague
exner@ujf.cas.cz, joye@ujf-grenoble.fr, kovarik@ujf.cas.cz
Abstract
We investigate a charged two-dimensional particle in a homoge-
neous magnetic field interacting with a periodic array of point ob-
stacles. We show that while Landau levels remain to be infinitely
degenerate eigenvalues, between them the system has bands of abso-
lutely continuous spectrum and exhibits thus a transport along the
array. We also compute the band functions and the corresponding
probability current.
The fact that the presence of boundaries can induce a transport in a system
with a homogeneous magnetic field is known for long [1, 2] and has numerous
consequences in solid state physics. The mentioned pioneering papers were
followed by tremendous number of studies in which the magnetic transport
was analyzed numerically in various models as well as experimentally. The
obstacles at which the particle “bounces” need not be hard walls but also
objects with openings such as various antidot lattices; for a sample of liter-
ature see, e.g., Refs. [3, 4], the following papers by the same authors, and
references therein. The closest to the subject of this letter is a very recent
1
paper [5] where the array of obstacles is strictly one-dimensional and there
are no boundaries to help the transport.
One can say therefore that at the heuristic level most aspects of the
two-dimensional magnetic transport are understood. With this fact and the
extensive literature in mind it is a bit surprising how little attention was paid
during a decade and a half to a strict derivation of transport properties from
the first principles – which is after all the raison d’ˆetre of theoretical physics.
The situation concerning this aspect of the problem changed recently where
several rigorous studies appeared. It was shown, e.g., that the edge currents
in a halfplane survive a mild disorder so that away of the Landau levels the
spectrum remains absolutely continuous [6, 7], and that the result extends
to planar domains containing an open wedge [8].
Recall that the wall producing the edge states need not be of a potential
type. It is known, e.g., that a step of the magnetic field or another variation
exhibiting a translational symmetry will smear again the Landau levels into
a continuous spectrum [9, 10]. Similarly to the usual edge states, this type
of propagation allows for a classical explanation in terms of the cyclotronic
radius changing with the magnetic field – see Ref. [11], Sec. 6.5.
The aim of the present letter is to contribute to this series of rigorous stud-
ies with a simple solvable model in which a charged quantum particle in the
plane exposed to a homogeneous magnetic field of intensity Bperpendicular
to the plane interacts with a periodic array of point obstacles situated at the
xaxis and described by δpotentials. We shall show that the model exhibits
a magnetic transport of which we can with a license say that it is a purely
quantum effect in the sense that a quantum particle propagates despite the
fact that its classical counterpart moves on localized circular trajectories –
apart of a zero-measure family of the initial conditions. The delta potentials
are certainly an idealization; a more realistic model with potential-type ob-
stacles will be discussed in a forthcoming paper. Let us mention, however,
that point interactions play a distinguished role being the only obstacles that
can preserve the Landau levels in the spectrum.
Using the Landau gauge, we can write the Hamiltonian formally as
Hα,` = (i∂x+By)22
y+X
j
˜αδ(xx0j`),(1)
where ` > 0 is the array spacing. Since we are interested mainly in the essence
of the effect, we use everywhere rationalized units ~=c=e= 2m= 1.
The interaction term, in particular the formal coupling constant ˜α, needs an
2
explanation, since the two-dimensional δpotential is an involved object. We
adopt here the conventional definition from Ref. [12] which determines the
latter by means of the boundary conditions
L1(ψ, ~aj)+2παL0(ψ,~aj) = 0 , j = 0,±1,±2, . . . (2)
with ~aj:= (x0+j`, 0), where Lkare the generalized boundary values
L0(ψ, ~a) := lim
|~x~a|→0
ψ(~x)
ln |~x~a|, L1(ψ, ~a) := lim
|~x~a|→0hψ(~x)L0(ψ, ~a) ln |~x~a|i,
(3)
and αis the (rescaled) coupling constant; the free (Landau) Hamiltonian
corresponds to α=. Recall that since the magnetic field amounts locally
to a regular potential in the s-wave subspace, the non-magnetic boundary
conditions of Ref. [12] need not be modified – see, e.g., Ref. [13].
The difference between αand ˜αreflects the nontrivial way in which the
two-dimensional point interaction arises in the limit of scaled potentials.
Due to the coupling constant renormalization, a caution is required when
interpreting spectral properties of such a Hamiltonian. On the other hand,
the two-dimensional point interaction yields a generically correct description
of low-energy scattering which can be tested in experiments – see, e.g., a
fresh example in [14]. To understand the above remark about the purely
quantum nature of the transport here, recall the well-known expression for
the scattering length which shows that the obstacles have a “point” character
if e2πα `, i.e., if the point interaction is strong enough. Below we shall
show that the transport exists here for any finite value of α.
Using the periodicity, we can perform the Bloch decomposition in the x
direction writing
Hα,` =`
2πZ
|θ`|≤π
Hα,`(θ)dθ , (4)
where the fiber operator Hα,` (θ) is of the form (1) on the strip 0 x`
with the boundary conditions
i
xψ(`, y) = eiθ` i
xψ(0+, y), i = 0,1.(5)
The Green’s function of the operator Hα,`(θ) is given by means of the Krein
formula [12, App.A],
(Hα,`(θ)z)1(~x, ~x0) = G0(~x, ~x0;θ, z)
+(αξ(~a0;θ, z))1G0(~x,~a0;θ, z )G0(~a0, ~x0;θ, z),(6)
3
where G0is the free Green’s function and
ξ(~a;θ, z) := lim
|~x~a|→0G0(~a, ~x;θ, z)1
2πln |~x~a|(7)
is its regularized value at the point ~a. The Bloch conditions (5) determine
eigenvalues and eigenfunctions of the transverse part of the free operator,
µm(θ) = 2πm
`+θ2
, ηθ
m(x) = 1
`ei(2πm+θ`)x/` ,(8)
where mruns through integers. Then we have
G0(~x, ~x0;θ, z) =
X
m=−∞
uθ
m(y<)vθ
m(y>)
W(uθ
m, vθ
m)ηθ
m(x)ηθ
m(x0),(9)
where y<, y>is the smaller and larger value, respectively, of y, y0, and uθ
m, vθ
m
are solutions to the equation
u00(y) + By +2πm
`+θ2
u(y) = zu(y) (10)
such that uθ
mis L2at −∞ and vθ
mis L2at +; in the denominator we have
their Wronskian. By the argument shift we get
uθ
m(y) = uy+2πm +θ`
B` (11)
and a similar relation for vθ
m, where u, v are the corresponding oscillator
solutions. Of course, we have W(uθ
m, vθ
m) = W(u, v). The functions u, v
express in terms of the confluent hypergeometric functions [15, Chap. 13]:
v(y) = eBy2/2UBz
4B,1
2;By2(12)
away from zero, and uis obtained by analytical continuation in the y2vari-
able; together we have
u
v(y) = πeBy2/2MBz
4B,1
2;By2
Γ3Bz
4B±2By M3Bz
4B,3
2;By2
ΓBz
4B.
(13)
4
From here and Ref. [15, Chap. 6] we compute the Wronskian; in combination
with (8) we get
G0(~x, ~x0;θ, z) = 2(z /2B)(3/2)
πB` ΓBz
2Be(xx0)
×
X
m=−∞
uy<+2πm +θ`
B` vy>+2πm +θ`
B` e2πim(xx0)/` .(14)
As expected the function has singularities which are independent of θand
coincide with the Landau levels, i.e., zn=B(2n+1), n = 0,1,2, . . .. Let us
observe first that each znremains to be infinitely degenerate eigenvalue of the
“full” fiber operator Hα,` (θ). To this end, one has to adapt the argument of
Refs. [16, 17] to the set of functions wksin πw
`eB|w|2/4, k = 0,1, . . ., with
w:= x+iy which vanish at the points of the array so the conditions (2) are
satisfied for them automatically.
On the other hand, Hα,`(θ) has also eigenvalues away of znwhich we
denote as n(θ)(α,`)
n(θ). In view of (6) they are given by the implicit
equation
α=ξ(~a0;θ, ) (15)
and the corresponding eigenfunctions are
ψ(α,`)
n(~x;θ) = G0(~x, ~a0;θ, n(θ)) .(16)
In order to evaluate them, we have to assess the convergence of the series in
(14). Using the asymptotic behavior
u
v(y) = e∓{±}By2/2By
zB
2B1 + O(|y|2)(17)
for y→ ∓∞, we find that the product
sm:= uy<+2πm +θ`
B` vy>+2πm +θ`
B`
is for y6=y0governed by the exponential term,
smexp B
2y2
<y2
>+θ2π|m|
`(y>y<)|m|1+O(|m|2)
(18)
5
as |m| → ∞, while for y=y0we have
sm=1
4π|m|1+O(|m|2),
so the series (14) is not absolutely convergent. Summing now the contribu-
tions from ±mwe see that in the limit x0xit diverges at the same rate
as the Taylor series of (1/2π) ln ζdoes for ζ0+. Hence we get
ξ(~x;θ, z) =
X
m=−∞ 1δm,0
4π|m|22ζ1
πB` Γ(2ζ) (uv)y+2πm +θ`
B`  ,(19)
where ζ:= Bz
4B. The expression is independent of x, because the regularized
resolvent does not change if the array is shifted in the xdirection. We can
write it by means of the first hypergeometric function alone, since
(uv)(ξ/B) = πeξ2M(ζ, 1
2;ξ2)2
Γ(ζ+1
2)24ξ2M(ζ+1
2,3
2;ξ2)2
Γ(ζ)2,(20)
where ξ:= By+2πm+θ`
B` .
Spectral bands of our model are given by the ranges of the functions
n(·). Solutions of the condition (15) do not cross the Landau levels, because
ξ(~a0;θ, ·) is increasing in the intervals (−∞, B) and (B(2n1), B(2n+ 1))
and diverges at the endpoints; this is a general feature [18]. The spectrum
will be continuous away of znif the latter are nowhere constant. In view
of the spectral condition (15) one has to check that ξ(~x;θ, z) is nowhere
constant as a function of θ. Notice that each term in (19) is real-analytic
for real zand the series has a convergent majorant independent of θ; hence
ξ(~x;·, z) is real-analytic as well and one has to check that it is non-constant
in the whole Brillouin zone [π/`, π/`).
Suppose that the opposite is true. Then the Fourier coefficients
ck:= Zπ/`
π/`
ξ(~x;θ, z)eik`θ (21)
should vanish for any non-zero integer k. Since the summand in (19) be-
haves as O(|m|2) as |m|→∞, we may interchange the summation and
integration. A simple change of variables then gives
ck=22ζ1
πB` Γ(2ζ) lim
M→∞ Zπ(2M+1)
π(2M+1)
(uv)y+ϑ
B` eikϑ dϑ , (22)
6
so
ˆ
Fy(k) := Z
−∞
Fy(ϑ)eikϑ = 0 ,(23)
where Fy(ϑ) := (uv)y+ϑ
B` . The same reasoning applies to any finitely
periodic extension of ξ(~x;θ, z), hence (23) is valid for each non-zero rational
k. However, the function decays O(|ϑ|1) and the integral makes sense only
as the principal value. We shall use the above mentioned asymptotic behavior
which implies, in particular,
Fy(ϑ) = 1
4π1 + ϑ2+fy(ϑ),(24)
where fy(ϑ) = O(|ϑ|2) uniformly in y[0, `]. Thus
ˆ
Fy(k) = 1
2πK0(k) + ˆ
fy(k),(25)
see [19, 3.754.2]. Since fyL1, the second term at the r.h.s. is continuous
w.r.t. kand the same is then true for ˆ
Fy; this means that the relation (23)
is valid for any nonzero k. Furthermore, ˆ
fyis bounded and K0diverges
logarithmically at k= 0, hence RN
NFy(ϑ)eikϑ can be bounded by an
integrable function independent of N. Then
Z
−∞
ˆ
Fy(k)φ(k)dk =Z
−∞
dk φ(k) lim
N→∞ ZN
N
Fy(ϑ)eikϑ =Z
−∞
Fy(ϑ)ˆ
φ(ϑ)
(26)
holds for any φ∈ S(R), i.e., ˆ
Fy(k) is the Fourier transform of Fy(ϑ) in the
sense of tempered distributions. Since this is a one-to-one correspondence
[20, Thm.IX.2], we arrive at the absurd conclusion that Fy= 0. We get thus
the following result:
Theorem. For any real αthe spectrum of Hα,` consists of the Landau levels
B(2n+1), n = 0,1,2, . . ., and absolutely continuous spectral bands situated
between adjacent Landau levels and below B.
Let us remark that during the final stage of the work we learned about a
similar result for a chain of point scatterers in a three-dimensional space
with a homogeneous magnetic field [21 ]. Due to the higher dimensionality,
the spectrum is purely a.c. in that case and has at most finitely many gaps.
The above theorem says a little about the character of the transport. To
get a better idea we solve the spectral condition (15) numerically for several
7
INSERT FIG. 1 (spect2.eps)
Figure 1: The eigenvalues of Hα,l(θ) in the units of B. (Top) n= 20, B= 4
(full line), B= 6 (full dotted line) and B= 8 (dotted line). (Bottom) n= 1,
B= 10 (full line) and B= 15 (dotted line). The thick lines represent the
Landau levels.
INSERT FIG. 2 (tokn20.eps)
Figure 2: Probability current for n= 20, B= 4, α= 0.5 and two different
values of θcorresponding to the extremal points of 20(·): θ= 0 (top), θ= 2.2
(bottom), see the arrows in Fig. 1. The star marks the point perturbation
position.
values of the parameters. The results are plotted in Fig. 1 for the second and
21st spectral band. We see that the bands move downwards with decreasing
αand their profile becomes more complicated with the band index n; a higher
Btends to smear the structure.
The Bloch functions (16) are in general complex-valued and yield thus
a nontrivial probability current, ~n(~x;θ) = 2 Im ¯
ψ(α,`)
n(~
∇ − i~
A)ψ(α,`)
n(~x;θ).
The current pattern changes with θoscillating between a symmetric “two-
way” picture and the situations where one direction clearly prevails, these
extremal behaviors occurring at the extrema of the corresponding band func-
tion. This is illustrated in Fig. 2. In addition, while the pattern has pre-
dominantly “laminar” character, in some parts current vortices may form,
mainly in low spectral bands as it is illustrated in Fig. 3. Similar effects have
also been observed in the numerical analysis of related models mentioned in
the introduction.
To sum up the above discussion, we have analyzed the behavior of a
quantum particle in the plane exposed to a homogeneous magnetic field and
interacting with a periodic array of point perturbations. We have shown
that while the Landau levels survive, the spectrum develops an absolutely
continuous part, i.e. a sequence of spectral bands. Depending on the quasi-
INSERT FIG. 2 (tokn1b.eps)
Figure 3: Probability current for n= 1, B= 20, α=1 and θ= 0. In the
bottom graph the inset shows a vortex between the point perturbations.
8
momentum, the particle is transported along the array with zero or nonzero
mean longitudinal momentum, and the probability current pattern may ex-
hibit vortices in some regions.
Acknowledgments
A.J. wishes to thank his hosts at the Nuclear Physics Institute in ˇ
Reˇz, where
this work was initiated. The research has been partially supported by the
GAAS grant 1048801. We are grateful to V. Geyler for making Ref. 21
available to us prior to publication.
References
[1] B.I. Halperin, Phys. Rev. B25, 2185 (1982).
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(1992).
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[5] T. Ueta, Phys. Rev. B60, 8213 (1999).
[6] S. De Bi`evre, J.V. Pul´e, Math. Phys. El. J. 5(1999), No. 3.
[7] J. Fohlich, G.M. Graf, J. Walcher: On the extended nature of edge
states of quantum Hall Hamiltonians, math-ph/9903014.
[8] J. Fohlich, G.M. Graf, J. Walcher: Extended quantum Hall edge states:
general domains, mp arc 99-327.
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tors with Applications to Quantum Mechanics and Global Geometry,
Springer, Berlin 1987.
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els in Quantum Mechanics, Springer, Berlin 1988.
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Seba, Phys. Lett. A228, 146 (1997).
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9
[16] Y. Avishai, R.M. Redheffer, Y.M. Band, J. Phys. A25, 3883 (1992).
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[21 ] S. Albeverio, V.A. Geyler, O.G. Kostov, Rep. Math. Phys. 44 (1999),
to appear.
Figure captions
Figure 1. The eigenvalues of Hα,l(θ) in the units of B. (Top) n= 20,
B= 4 (full line), B= 6 (full dotted line) and B= 8 (dotted line).
(Bottom) n= 1, B= 10 (full line) and B= 15 (dotted line). The
thick lines represent the Landau levels.
Figure 2. Probability current for n= 20, B= 4, α= 0.5 and two different
values of θcorresponding to the extremal points of 20(·): θ= 0 (top),
θ= 2.2 (bottom), see the arrows in Fig. 1. The star marks the point
perturbation position.
Figure 3. Probability current for n= 1, B= 20, α=1 and θ= 0. In the
bottom graph the inset shows a vortex between the point perturbations.
10
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