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The spin response function for electrons confined in a quantum dot is studied within the time-dependent local spin density approximation (TDLSDA) of density functional theory. In the long-wavelength regime we predict the existence of a low-energy collective dipole (ℓ = 1) spin mode. The evolution with electron number of the spin response is studied and compared with that of the density response. Predictions for the static dipole polarizability are given.

We present ground and excited state energies obtained from Diffusion Monte Carlo (DMC) calculations, using accurate multiconfiguration wave functions, for N electrons (N ≤ 13) confined to a circular quantum dot. We compare the density and correlation energies to the predictions of local spin density approximation theory (LSDA), and Hartree-Fock theory (HF), and analyze the electron-electron pair-correlation functions. The DMC estimated change in electrochemical potential as a function of the number of electrons in the dot is compared to that from LSDA and HF calculations. Hund's first rule is found to be satisfied for all dots except N = 4 for which there is a near degeneracy. 85.30.Vw,

This book is printed on acid-free paper. PrefaceThis book is the fruit of my lectures on the Theory of Many Body Systems which I have been teaching for many years in the degree course in Physics at the University of Trento. As often happens, the outline of the book was my students' notes; in particular, the notes of the students of the academic year 1999-2000 which were extremely useful for me. Chapter 6 on the Monte Carlo methods is the work of Francesco Pederiva, a research assistant in our Department. During the course, Francesco, apart from illustrating the method, teaches the students all the computer programmes, continually referred to in the book, by means of practical exercises in our computational laboratory. In particular, he teaches the Hartree-Fock, Brueckner-Hartree-Fock, Kohn-Sham, Diffusion Monte Carlo programmes for the static properties, RPA, and Time-Dependent HF and LSDA for Boson and Fermion finite systems. These programmes are available to anyone who is interested in using them.The book is directed towards students who have had a conventional course in quantum mechanics and have some basic understanding of condensed matter phenomena. I have often gone into extensive mathematical details, trying to be as clear as possible and I hope that the reader will be able to rederive many of the formulas presented without too much difficulty.In the book, even though a lot of space is devoted to the description of the homogeneous systems, such as electron gas in different dimensions, quantum wells in intense magnetic field, liquid helium and nuclear matter, the most relevant part is dedicated to the study of finite systems. Particular attention is paid to those systems realized recently in the laboratories throughout the world: metal clusters, quantum dots and the condensates of cold and dilute atoms in magnetic traps. However, some space is also devoted to the more traditional finite systems, like the helium drops and the nuclei. I have tried to treat all these systems in the most unifying way possible, attempting to bring all the analogies to light. My intention was to narrow the gap between the usual undergraduate lecture course and the literature on these systems presented in scientific journals. VI PrefaceIt is important to remark that the book takes a "Quantum chemist's" approach to many-body theories. It focuses on methods of getting good numerical approximations to energies and linear response based on approximations to first-principle Hamiltonians. There is another approach to many-body physics that focuses on symmetries and symmetry breaking, quantum field theory and renormalization groups, and aims to extracting the emergent features of the many body systems. This works with "effective" model theories, and does not attempt to do "06 initio computations". These two ways of dealing with many-body systems complement each other, and find common ground in the study of atomic gases, metal clusters, quantum dots and quantum Hall effect systems that are the main application of the book.

A magnetic state of orbital nature is predicted in deformed metal clusters. The restoring force of this state originates from a quantum effect associated with the kinetic Fermi motion. For Na clusters the frequency is predicted to be COM\ =54.67V ~1 /3 eV, where S is the deformation of the cluster.PACS numbers: 71.45.-d, 36.40.+d Collective electric dipole excitations have been recently observed in alkali-metal clusters. 1 These excitations correspond to the classical Mie 2 surface-plasmon oscillations and have revealed the importance of deformation effects. 3 In this Letter we suggest the existence of a new magnetic collective state in deformed clusters occurring at energies much smaller than the plasma frequency. Such an excitation has no classical counterpart and emerges from the quantum effect associated with the Fermi motion of valence electrons. A collective excitation of similar nature has been recently observed in deformed atomic nuclei via inelastic electron scattering as well as (y,/) reactions. 4 A macroscopic illustration of the new magnetic state is suggested by the following form for the displacement field relative to the electronic motion: 5In Eq. (1) QI is the unit vector in the x direction (the cluster is assumed to be axially deformed along the z direction) and 5= f (R 2 -R 2 )/(R 2 + 2R 2 ) is the deformation of the electron density profile which is assumed to be of spheroidal shape: p e =p e (x 2 /R 2 +y 2 /R 2 +z 2 / R 2 ) (R z and R x = R y are the radii parallel and perpendicular to the symmetry axis, respectively).The term o>xr of Eq. (1) corresponds to a rigid rotation of the electrons with respect to the jellium background [scissor mode; 6 see Fig. 1(a)]. If only this term were included in the electronic motion, the electrons would experience a restoring force originating from the Coulomb interaction with the jellium, similarly to what happens in the dipole Mie oscillation. The cost in the Coulomb energy is minimized by including the quadrupole term in V(yz) in the displacement field. This is well understood by noting that the change in the electron density p e ,is zero with choice (l) since w\p e =0 in the spheroidal model. The resulting motion is illustrated in Fig. 1(b) where we have taken a sharp density profile. In this case one gets a rotation within a spheroid with a rigid surface.The relevant restoring force taking place during the motion goes not originate from the Coulomb interaction. Rather it is produced by the quadrupole component V(yz) of the velocity field which gives rise to a distortion of the Fermi sphere of the valence electrons. This effect of quantum nature has been extensively studied in the case of atomic nuclei and reveals an elastic behavior exhibited by Fermi systems. 7The frequency of the resulting mode can be estimated through the expression co M \ = (tf/6» 1/2 , where K^lEin)is fixed by the energy change E(u) associated with the displacement field (1), and j 0=* jmpofdVu 2 is the collective mass parameter. In the limit of small deformations we find (ft =c ...

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Several of the energies in Tables I and II are incorrect. Most of the errors are due to incorrectly inputting the symmetry of some of the states, others are due to incorrectly converting or transcribing the energies. In particular, for N = 4 we had a near degeneracy and a violation of Hund's first rule. The corrected result has |L = 0, S = 1 as the ground state, as predicted by Hund's rule. Hund's first rule is satisfied for all N for the confining potential used, but there is a near degeneracy for N = 10. We present corrected versions of Tables I and II. All quantum Monte Carlo results have been recomputed with an improved Jastrow factor. All LSDA values in Table II have also been recalculated. The rms fluctuations of the local energy in VMC range from 0.008 H * for N = 2 to 0.24 H * for the N = 13, |L = 1, S = 1/2 state. The number of determinants, N det , in the wave functions depends on whether real or complex orbitals are employed. The values shown in Table I are for real orbitals. Not only the DMC energies, but also the VMC energies computed with LSDA and LDA orbitals agree within 1 mH * in all cases tested. All LSDA values in Table II have also been recalculated. The effective Bohr radius a * 0 should be 97.9373Å, rather than 97.93 A.