Effect of electron-electron interactions on the average polarizability of a mesoscopic system.

A comparison between an analytical calculation of the polarizability of a mesoscopic interacting system in the random phase approximation and numerical exact diagonalization results is presented. While for weak interactions the analytical calculation fits the numerical results rather well, deviations appear for stronger interactions. This is the result of the appearance of intermediate range correlations in the electron density, which suppresses the polarizability below its classical value. The relevance to quantum dot systems is discussed. PACS numbers: 71.55.Jv,73.20.Dx,71.27.+a Typeset using REVTEX 1 The study of the polarizability of small mesoscopic systems has a long history. Already in 1965 Gorkov and Eliashberg [1] have attempted to calculate the average polarizability of a small metallic grain by applying the concept of energy level repulsion from the random matrix theory (RMT) [2,3]. Their calculation, which can be backed by a non-linear sigma model calculation [4], shows that the average polarizability of a mesoscopic system 〈α〉 ∝ (κL)L (here L is the length of the system and the inverse screening length κ = Sde N(0), where N(0) is the level density at the Fermi energy, Sd = 2π, (4π) for two (three) dimensional systems and 〈. . .〉 denotes an ensemble average). This result is much larger than the classical value of the polarizability α ∝ L. The reason for this apparent enhancement of the polarizability is, as pointed out by Strassler et. al. [5], the neglect of electron-electron (e-e) interactions which reduce the quantum corrections dramatically. Once κL ≫ 1, which for metallic systems corresponds to L ≫ 1Å, the polarizability is proportional to the volume [6]. The effect of e-e interactions in the random phase approximation (RPA) has been recently incorporated into the RMT formulation of the polarizability by Efetov [7]. The fluctuations in the grand-canonical ensemble were shown to be strongly suppressed by the e-e interactions [8,9], while the suppression is less significant in the canonical ensemble [10]. In recent experiments on disordered quantum-dots [11] it turned out that the groundstate energies of those dots show large fluctuations [12,13]. These fluctuations are larger than expected from RMT theory in which the e-e interactions are incorporated by the RPA [13]. This deviation from RMT theory is the result of the appearance of intermediate range correlations in the electron density as a result of strong e-e interactions. In this paper we shall show that the average polarizability is also affected by strong e-e interactions. While in the weak interaction regime the average polarizability is proportional to L, for strong interactions the polarizability tends to zero. The polarizability may be written as α = e ∂ ∂E ∫ d~r(z − L/2)n(r)Φ(r), (1) where n(r) is the electronic density, Φ(r) is the local electrostatic potential as a result of 2 applying an external electric field Eẑ and the polarizability is calculated in the ẑ direction. We shall begin by considering the weak interaction limit. In this limit one can treat both the disorder and the e-e interactions perturbatively. Taking into account the e-e interactions in the RPA one may rewrite Eq. (1) for the average polarizability as [9] 〈α〉 = eN(0) ∫ d~r d~r′(z − L/2)χ(~r, ~r′)(z′ − L/2), (2) which after assuming a rectangular geometry and performing a Fourier transform results in 〈α〉 = eN(0) ( L 2 )2d ∑ ~ q [z(~q)]χ(~q), (3) where ~q = (nxπ/L)x̂ + (nyπ/L)ŷ + (nzπ/L)ẑ with nx, ny, nz = 0, 1, 2 . . . (of course for 2D systems ~q = (nxπ/L)x̂+ (nzπ/L)ẑ), z(~q) = 2 L δqx=0δqy=0 (−1)(nz+1) qz , (4) and χ(~q) = ( 2 L )d 1 1 +N(0)VU , (5) for short range interactions represented by an interaction potential U(~r, ~r′) = VUδ(~r − ~r′) (V = a where a is the range of the interaction and d is the systems dimensionality). For the Coulomb interaction U(~r, ~r′) = Ua/|~r − ~r′|, where U is the strength of the Coulomb interaction between two particles at distance a, and χ(~q) = ( 2 L )d q q(d−1) + SdN(0)aU . (6) Inserting all the definitions in Eqs. (4-6) and performing the summation in Eq. (3) results, for short range interactions, in 〈α〉 = 2 d Sd90 L(κL) ( 1 + VU Ld∆ )−1 , (7) where N(0) = (L∆), and ∆ is the single electron level spacing. For long range interactions 〈α〉 = 2 d Sd90 L(κL) ∑