Download r interaction * Michael R. Geller

PHYSICAL REVIEW B
VOLUME 53, NUMBER 11
15 MARCH 1996-I
Quantum breathing mode for electrons with 1/r 2 interaction
Michael R. Geller* and Giovanni Vignale
Department of Physics, University of Missouri Columbia, Missouri 65211
~Received 18 August 1995!
We show that the collective excitation spectrum of electrons with 1/r 2 interaction in a parabolic quantum dot
of frequency v 0 contains a ‘‘breathing’’ mode of frequency 2V, where V 2 [ v 20 1 41 v 2c and v c is the cyclotron
frequency, a result first obtained by Johnson and Quiroga @Phys. Rev. Lett. 74, 4277 ~1995!#.
In a recent paper,1 Johnson and Quiroga have obtained
some exact results for electrons with 1/r 2 interaction in a
two-dimensional quantum dot. A parabolic confining potential of the form 21 m v 20 r 2 is assumed, and the system is subjected to a uniform perpendicular magnetic field. In particular, they have shown that there exists a collective
‘‘breathing’’ mode excitation with frequency
v 52V,
~2!
and where v c is the cyclotron frequency. The exact spectrum
of the interacting electron system therefore contains an infinite ladder of energy levels at integer multiples of 2\V.
However, these authors do not explain why the unphysical
1/r 2 interaction is special, apart from the mathematical fact
that it permits a separation of the many-particle Schrödinger
equation in the hyperradial and hyperangular coordinates
employed. Furthermore, the physical nature of the breathing
mode is not fully explained.
Given the simplicity of the result ~1!, it is natural to ask
whether there is a more direct way of obtaining it. The purpose of this paper is to point out that the quantum breathing
mode excitation spectrum can also be obtained directly from
the behavior of the Hamiltonian under a scale transformation, and in a manner that makes evident the special property
of the inverse-square interaction for the quantum breathing
mode. We shall work in the symmetric gauge and write the
Hamiltonian as
H5T1 \ v cL z 1V1U,
1
2
T[
p 2n
(n 2m
(n ~ rn 3pn ! •ez
(
n,n
g
a
u
r
2r
n 8u
8 n
~7!
is any power-law electron-electron interaction.
We first note the properties of H under a scale transformation O→e ilS/\ Oe 2ilS/\ generated by
S[
1
2
(n ~ rn •pn 1pn •rn ! .
~8!
This transformation performs a radial displacement of each
coordinate by an amount proportional to its distance from the
origin; that is, it generates a ‘‘breathing’’ motion. Under this
transformation,
T→T22lT,
L z →L z ,
V→V12lV,
and
U→U2 a lU,
~9!
to first order in l. Therefore,
H→H1
~4!
~5!
is the z component of the canonical angular momentum,
0163-1829/96/53~11!/6979~2!/$10.00
~6!
i
l @ S,H # 5H22lT12lV2 a lU.
\
~10!
Equation ~10!, however, may be regarded as an equation of
motion for S. In fact, noting that
is the canonical kinetic energy, with pn the canonical momentum,
L z[
U[
~3!
where
1
(n 2 mV 2 r 2n
is the effective field-dependent parabolic confining potential,
and
~1!
where
V 2 [ v 20 1 41 v 2c ,
V[
53
dV
5V 2 S,
dt
~11!
we obtain the operator equation of motion
d 2V
1 ~ 21 a ! V 2 V5 a V 2 ~ H2 21 \ v cL z ! 1 ~ 22 a ! V 2 T,
dt 2
~12!
6979
© 1996 The American Physical Society
BRIEF REPORTS
6980
which is the same as one would obtain classically.
The breathing mode of the corresponding classical system
of point charges may be obtained from ~12! by considering
small oscillations about an equilibrium configuration, where
the velocities are zero. Because H and L z are constants of the
motion, whereas the physical kinetic energy
T1
\ v c L z v 2c V
1
2
4V 2
~13!
is zero to first order in the displacements, the classical
breathing mode frequency is generally
v 5 A~ 21 a ! v 20 1 v 2c .
~14!
For example, the classical breathing frequency of electrons
with Coulomb interactions ( a 51) in a parabolic dot with no
magnetic field is A3 v 0 , a result recently discovered by
Peeters, Schweigert, and Bedanov.2
Quantum zero-point motion, however, generally modifies
the breathing mode and the other classical normal modes, by
*Present address: Department of Physics, Simon Fraser University,
Burnaby B.C., Canada V5A 1S6.
1
N. F. Johnson and L. Quiroga, Phys. Rev. Lett. 74, 4277 ~1995!.
53
shifting their frequencies and by giving them a finite width
of the order of a B /R, with a B denoting the Bohr radius and
R the radius of the droplet of charge in the dot.
An exception occurs when a 52: In this case V becomes
an exact quantum collective coordinate with frequency ~1!,
independent of g and N, where N is the number of electrons.
The collective coordinate V may also be separated into a
center-of-mass and relative-coordinate part, V5V c.m.1V rel .
For a 52, it can be shown that each component separately
satisfies a harmonic oscillator equation of motion of the form
~12! with frequency 2V. V rel is the collective coordinate
corresponding to the breathing mode discussed in Ref. 1.
This work was supported by NSF Grants Nos. DMR9403908 and DMR-9416906. We gratefully acknowledge the
hospitality of the Condensed Matter Theory Group at Indiana
University, where this work was initiated, and we thank Allan MacDonald for stimulating discussions, and Francois
Peeters for first drawing our attention to the breathing mode
in classical systems.
2
F. M. Peeters, V. A. Schweigert, and V. M. Bedanov, Physica B
212, 710 ~1995!. See also V. A. Schweigert and F. M. Peeters,
Phys. Rev. B 51, 7700 ~1995!.