Je fliyaiqut; Nuclйaire, 'J1106 Orsay Cedex, France the classical ari

42 Pages·2009·0.36 MB·English

OH THF LANDAU LEVELS ON THE
HYPERBOLIC PLANE
Alain Coratet
Division de Physique Théorique , Institut
(Je fliyaiqut; Nucléaire, 'J1106 Orsay Cedex, France
the classical arid quantum mechanics of A charged particle
moving on the hyperbolic plane in a constant magnetic field is discussed.
The underlying SL(2,H) symmetry leads to a general description of various
possible trajectories. In contrast with the flat case. It is shown that
closed orbits only arise for sufficiently strong fields. Ar the quantum
level a group theoretical approach including both hound and continuum
states is presented. It is shown that the semiclassical approximation
leaJs to the e*act hound state spectrum. The resolvent and its flat space
limit are constructed in closed form.
IPNO/TH 06-31 April litib
* Laboratoire Associe au CNRS
Introduction
In this work we study the classical anû quantum dynamics of a
charged particle moving on tne hyperbolic plane under the action of a constant
magnetic Field. A preliminary study of this problem was started in an earlier
paper in à different context ; namely we considered the hyperbolic plane as
the Wick rotated form of a two dimensional space-time manifold over which a
constant external field was acting. Constructing the effective action for
scalar or spinorial fields defined on this curved manifold requires a know-
ledçfr of the spectrum of various elliptic Operators. Both the discrete and
continuous spectra were determined exactly. As a result, closed expressions
for the zeta function, and the corresponding functional determinants wure
oPtdined. The physics of the problem is essentially contained in an explicit
expression for the pair creation rate.
The approach developed in this wort is quite different. Mere
we deal with the classical and quantum mechanics of a charged particle moving
in a constant electromagnetic field in a two dimensional universe with
constant negative curvature. Part of our motivation is to present an elemen
tary generalization of the Landau levels to the case of a curved manifold. A
possible physical application would be for instance to probe the role of the
curvature on tne statistical mechanics of a gas of electrons.
However the main interest of this work lies elsewhere, fiecent
theoretical and numerical analyses'" have led to a nev. understanding of the
correspondance between the classical behavior of a system and its energy
level fluctuations. In particular, level repulsion has been conjectured on
2
a numerical basis for chaotic systems with two degrees of freedom, fl
different type of behavior has been observed for time reversal breaking
systems such as billiards in magnetic fields . Vie believe that It would be
extremely desirable to obtain an analytical treatment of these problems.
A suitable framework to carry out such an investigation is the geometry of
two dimensional spaces of constant negative curvature. On the one h?nd, the
thuory of geodesies over such manifolds has been developed quite extensively,
on theotner, semiclassical methods have found in this case there most appro
priate field of action through the famous Selberg trace formula . As is well
known, the latter re la lei the partition function for the free laplscian
inside a fundamental domain of the hyperbolic plane to a purely classical
expression, namely a certain sum over closed geodesies. Though the basic
problem of extracting the spectrum from the geodesies remfins essentially
untouched (see the recent review of Balazs and Voros ), we believe it is of
some interest to explore more general situations in which time reversal
breaking effects are expected to play a role. As emphasized by Berry and
Gutïwiller , the most simple set up involves pure gauge fields with aon
trivia! f-ux. In this case the geodesic motion is unperturbed and the
spoctrum is thus entirely controlled by topology.
We here consider a different situation, namely a constant
magnetic field acting on the open hyperbolic plane realised as the Poincaré
plane h. Though the non trivial dynamics is induced by its tesselation by
discrete subgroups of SL(2,R), the knowledge of the spectrum and of the
Green's functions in the infinite case is clearly needed as a preliminary
step before carrying out a more detailed investigation.
Sincp 5L(2,R) is an is one try group of H, one expects on
physical grounds that at least part of this symmetry will be maintained in
the presence of a constant magnetic field {the system remains homogeneous
and isotropic). In the second section we Indeed show that Sl(2,R) acts as
a dynamical symmetry «hen one considers the whole Poincare plane. At the
classical level, this leads to a general description of trajectories. The
quantum mechanics on the half plane is discussed in section III and a
classification of the spectrum according to suitable representations of
SL[?,R) is presented. In the fourth section we analyse the spectrum from a
seraiclassical viewpoint. The Ust section is devoted to the construction of
the resolvent.
In the course of this work we have realized that some of our
formulae have appeared in the mathematical literature in the context of
automorphic forms. The most relevant references are the works of S.J.
Patterson and J. Elstrodt . Since the physical interpretation i-- new and
interesting, we believe it is of some interest to draw attention on these
works. Material presented in the last section is based on Patterson's work.
Let us now briefly establish our notations. We denote by H
tne upper halfplane of the complex variable H = X,f ivi l\l'2 ^)
endowed with the metric
(ia}*=-2L[(<UjV (^/] (i.D
The factor a is here to set the scale. The scalar curvature is a negative
constant P _*
a'
As shown in our previous work . a constant magnetic field on H is obtained
by considering the one form
A = A»u.«.;d.K= _ l2i'«l«. (I-?:
and its corresponding field strength
iA = B^ <A^oU (8 SO)
A |l3
We sometimes use the rescaled field
II) Classical mechanics of a charged particle in a constant field
We first study the classical trajectories of a charged
particle in the external field (1.3). The Euler Lagrange equations are
derived from the lagrangian
£ = ji(*V <i'j _ Bo.' A. (2-»
t "i
where the dot denotes derivative with respect to time.
For convenience the mass of the particle is set equal to 2.
The equations, of motion are second order coupled differential equations
which can be integrated by quadrature. However the geometric description
of the paths only relies on the underlying dynamical symmetry of the system.
Ihe lagrangian [2.1) is invariant with respect to the
following infinitesimal trans formations :
1) dilatation |x, = X X Sy = > H j
2) translation along x $ •*. - \ S-u. = 0 ,
9 J (2.2)
3) non linear rotation hx - ^ ( ***- X.1)
{in this case « changes by a total time derivative).
The respective constants of motion obtained through Noether
theorem are
L= (^'-x^h.»- 2x4-fvJ + iBa'i
3
1 •>* r ? ;
are the canonical nomenta.
It is well known that when the magnetic field vanishes the
three parameter group Or of Teal unimodular 2x2 matrices
acting on H as a group of fractional linear '-ansformation
l-î + S !
maps H into itself and leaves otj invariant. Far a non-zero magnetic
field the action of G (2.4) induces a gaugo transformation on the gauge
potentials
A —» A + ^A
where A = Z Bo.1 <vuj, (a '_ *-J .
This implies that 3o is changed by a total derivative, hence SL(2,R) is
still a symmetry of the problem.
The SL(2,R) Lie algebra is generated from the canonical Poisson brackets
{L.,L.j=L
v
The 3-generators correspond to the following one parameter subgroups of
SL(2,R)
The hsmiltonlan obtained through the canonical procedure is a quadratic
operator in the L;,
Aa1 L
Eliminating p^ and h-, from eq.(2.3), the equation of the classical paths
describes a circle in the euclidean plane.
It is in fact more appropriate to describe the trajectory in an intrinsic
manner.
Starting from the hyperbolic distance r between (x,y) and (x',y')
1,.^ *,a^J, \A * ^-«-W^-ïO'j
(29)
we derive the equation of an H circle of radius ft = CX-iS , centered at
x,,v. (r.>o;.
(x-XoJ1^ (<j,- Yo <^f j = i^c 4-^.y . (2.10)
Comparing with eq. (i\H) we conclude that for L <" O and R ^> 2 v H
a Ci, | tne
trajectory is a circle of center
X„ = -il Y. - V1 S'a.4- 4 «.'H
and radius R = q-a^a £& 2. V"H (2.1I)
As *e shall discuss later on, this corresponds to a periodic motion. Me check
that the previous formula goes to the proper flat space limit * * H when
,— B
a -> co . The condition 6 > ^ » C*. implies that the magnetic field
has. to be strong enough to capture the particle on closed orbits. If this
condition is not fulfilled the motion will be unbounded and tne particle
(•scapes to infinity (see fig.l).
The equations of motion are obtained from the hamiltDniiin (2.7)
by the canonical procedure :
y = Vf1-? , (213)
" la' ;
h = ° /
It Is convenient to parametrize the constants L,, L-, L-, in the following
form
Li = 2.%o Ka1
U = 2. Ka2
(2.14)
L}=(i£._ 2xtKj a*.
1 K '
i for C > O K > O H = IC +• £.' J a.1
K 8 +-I/B'+^C JixJTj (215)
K B-H/BVÏC' J(T/C~J
The trajectory is the arc of the euclidean circle
(X.-X.) H- (y-<fr.J = BN-4C
4K'
which lies in H. It obviously has infinite length (see fig.l).
2) For K < O
Ha'(£_,c,J
one obtains =
x = x„ _ /iiL./KViïï- s^ïw_
K 8 + /B'-4/c/wr/ic7'216'
aid *
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