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NANOSYSTEMS:
PHYSICS, CHEMISTRY, MATHEMATICS
Popov I.Y. Nanosystems: Phys. Chem. Math., 2023,14 (4), 418-420.
http://nanojournal.ifmo.ru
Original article
DOI 10.17586/2220-8054-2023-14-4-418-420
A model of charged particle on the flat Möbius strip in a magnetic field
IgorY. Popov
ITMO University, St. Petersburg, Russia [email protected]
PACS 02.30.Tb, 03.65.-w; 02.30.Jr
Abstract The spectral problem for the Schrodinger operator with a magnetic field on the flat Mobius strip is considered. The model construction is described. It is compared with the case of the Laplace operator.
Keywords Landau operator, flat Mobius strip, spectrum
Acknowledgements The author thanks Dr. I. V. Blinova and L. S. Blinov for useful discussions.
For citation Popov I.Y. A model of charged particle on the flat Mobius strip in a magnetic field. Nanosystems: Phys. Chem. Math., 2023, 14 (4), 418-420.
1. Introduction
The quantum Hall effect discovered at the end of the twentieth century (see, e.g., [1-5]) is intensively used in nano-electronics. From the mathematical point of view the problem is related to the investigation of two-dimensional magnetic Schrodinger operator HL [3-5]. The magnetic Schrodinger operator in a strip on the plane was studied in many papers [6-9]. There are papers devoted to the spectral problem for three-dimensional Hamiltonian with a magnetic field (see, e.g., [10,11]).
We use the results of work [12] which studied the magnetic Schrodinger operator in an infinite strip on the plane.
Last time, curved nanostructures attract a special attention. Physicists investigate the properties of nanosystems caused by the nanostructure curvature (see, e.g., [13-18]). We can mention also a model based on quantum mechanics in spaces of constant curvature [19,20]. Hamiltonians on curved manifolds are especially interesting. In the present paper we deal with the Mobius strip. Recently, a work [21] appeared which considers the Dirichlet Laplacian on the Mobius strip. The authors deal with Courant-sharp property for Dirichlet eigenfunctions on the flat Mobius strip. In the present paper we consider the Dirichlet eigenfunctions for the magnetic Schrodinger operator (Landau operator) on the flat Mobius strip.
Let us describe the flat Mobius strip. Usually, it was made by the following way [21]. We start with the infinite strip = (-a, a) x (-to, to) with width 2a, equipped with the flat metric dx2 + dy2 of R2. Given b > 0, define the following isometry of Sinfty:
The group G2 is a subgroup of G, of index 2, generated by The action of G on is smooth, isometric, totally discontinuous, without fixed points. Correspondingly, we can consider the quotient manifolds with boundary
equipped with the flat metric induced from the metric of Sœ. Here Cb is the cylinder and Mb is the flat Mobius strip.
This construction is convenient for the authors of [21] because they deal with the Laplacian which "doesn't feel" a direction, i. e. it is invariant in respect to the map (x, y) ^ (-x, y). We will deal with the magnetic Schrodinger operator (Landau operator) which "feels" the direction. That is why, we will use another construction of the flat Mobius strip related to gluing of rectangles.
In the present short rapid note we present the main theorem only. Detailed proof, description of the model and analysis of the result will be published in the next paper.
ab : (x,y) ^ ( x, y + b).
Define the groups
G = ja6fc|k G Z}, G2 = jak|k G 2Z}.
Cb = STO/G 2, Mb = Sœ/G,
© Popov I.Y., 2023
Flat Möbius strip in a magnetic field
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2. Flat Möbius strip in the magnetic field
For the case of the magnetic Schrödinger operator (Landau operator), the situation is more complicated than for the Laplace operator. Consider four copies Qj, j = 1, 2,3,4, of the rectangle Q = (-a, a) x (-b, b). The initial operator is the orthogonal sum of operators Hf defined in ¿2(Qj ): HM = Hf © H2M © Hf © H4M where
rrM
H 1,2
rrM
H3,4
d 2 d
-- (dy - 2inex)2' d 2 d
-- (dy + 2^x)2 •
(1)
It is the Hamiltonian of free two-dimensional particle with charge e in the homogeneous magnetic field B orthogonal to the plane of the particle confinement. Here the vector potential of the magnetic field is chosen in the Landau gauge. Let = 2nhc/|e| be the magnetic flux quantum playing a role of a unit for the magnetic flux in the system, £ = |B|/$0 is the density of the magnetic flux, i.e. the number of the magnetic flux quanta through the unit area on the plane of the system, x, y are the Cartesian coordinates on the plane. The system of physical units is chosen in such a way that the charge of the particle e, the speed of light c and the Planck constant h equal 1, the mass of the particle is one half.
j=4
We include in the domain of HM functions (u^ u2, u3, u4) e ^ ©W22(Qj), W22(Qj) is the Sobolev space in Qj,
j=i
satisfying the following conditions:
D(HM ) :
Uj (—a, y)= Uj (a, y) = 0, j = 1, 2, 3, 4 Ui(x,b) = «2(—x, b), dui(x,b) = — dU2(—x,6),
dyi dy2
«2(x, — b) = «3(x, —b), —2 (x, —b) = — t«3 (x, —b), (2)
dy2 dy3 v 7
U3(x,b)= U4(—x,b), dU3(x, b) = — dU4(—x,b), dy3 dy4
U4(x, —b) = ui(x, —b), dU4 (x, —b) = — dui (x, —b).
dy4 dyi
Remark. Each rectangle Qj presents, actually, one side of the rectangular sheet. The magnetic flux is related to the side of the surface. Correspondingly, if the sheet is turned over, then the sign of the flux (£) changes. That is why, there
are different signs in expressions for H^M and H3M in (1). There is no change of the sign between EM1 and HM (HM
M
M
M
M
and ) because when one glues Qi to Q2 (Q3 to Q4) in accordance with (2), there is, simultaneously, a replacement
x ^ —x.
Solving equations HM ^ = E^ at each rectangle and satisfying the gluing and boundary conditions (2), one obtains the spectral equation and the eigenfunctions. The spectral equation is as follows:
$i,n(-a) $2,n(-a) $i,n(a) $2,n(a)
Here
$i,n(x;0 = e-n|«|(x-w)2$( -
E 1 1 ( n ,
$2,n(x; £) = e-n|«|:
(x - -
E 3 3 ( n \2 , A
S^M + 4' 2^x - T^J ,
(3)
(4)
(5)
where T
8b, $(a, z) is the Kummer function:
$(a,2;z) = i + £>)fc z
k = 1
(1/2)fc k!
(6)
(a)k = a(a+1)...(a + k — 1), (a)o = 1.
Roots En m of equation (3) gives us the eigenvalues of the operator. It is known [12] that the roots of equation (3) can be ordered increasingly. Correspondingly, En m is the m—th root of n—th equation (3). The main result is the following theorem.
0
(7)
420 I. Y. Popov
Theorem 2.1. The eigenvalues Enm of the operator HM are the roots of equation (3) with T = 8b. The corresponding eigenfunctions have the following form:
*ntm = An,mei^^n,m(x; C), (x,y) G Ol = inAn,me-i^^n,m(x; C), (x, y) G O2, ^L = (-1)nAn,mei^^n,m(x; C), (x,y) G O3 ^21 = (-i)nAn,me-inny^n,m(x; C), (x,y) G O4, where An,m is some constant, ^n,m(x; C) is given by (8).
(x; C) = $2 (a; C)$
(x; C) + $ (a; C)$2 (x; C), (8)
where $j,n,m(x; C) is $j,n(x; C) for E = Enm.
Functions $1jn(x; C) and $2,n(x; C) are two linearly independent solutions of the following equation
V>"(x) - ^(2nC)2 (x - T^) - ^ ^(x) = 0. (9)
One can note that function ^n m(x; C) satisfies the following property:
^n,m(-x; C) = ^n,m(x; -C) = ^-n,m(x; C). (10)
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Submitted 02 June 2023; revised 05 August 2023; accepted 08 August 2023
Information about the author:
IgorY. Popov - Center of Mathematics, ITMO University, Kroverkskiy, 49, St. Petersburg, 197101, Russia; ORCID 0000-0002-5251-5327; [email protected]
