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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!.