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)2−∂2

y+X

j

˜αδ(x−x0−j`),(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)

+(α−ξ(~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) = e−By2/2UB−z

4B,1

2;By2(12)

away from zero, and uis obtained by analytical continuation in the y2vari-

able; together we have

u

v(y) = √πe−By2/2MB−z

4B,1

2;By2

Γ3B−z

4B±2√By M3B−z

4B,3

2;By2

ΓB−z

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` ΓB−z

2Beiθ(x−x0)

×

∞

X

m=−∞

uy<+2πm +θ`

B` vy>+2πm +θ`

B` e2πim(x−x0)/` .(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

`e−B|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∓{±}By2/2∓√By

z−B

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,

sm∼exp B

2y2

<−y2

>+θ−2π|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 x0→xit diverges at the same rate

as the Taylor series of −(1/2π) ln ζdoes for ζ→0+. Hence we get

ξ(~x;θ, z) =

∞

X

m=−∞ 1−δm,0

4π|m|−2−2ζ−1

√πB` Γ(2ζ) (uv)y+2πm +θ`

B` ,(19)

where ζ:= B−z

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) = πe−ξ2M(ζ, 1

2;ξ2)2

Γ(ζ+1

2)2−4ξ2M(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(2n−1), 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 dθ (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=−2−2ζ−1

√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 dϑ = 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

4π√1 + 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 fy∈L1, 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 dϑ 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 dϑ =Z∞

−∞

Fy(ϑ)ˆ

φ(ϑ)dϑ

(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.

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9

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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