Spectral scrambling in Coulomb - blockade quantum dots

We estimate the fluctuation width of an energy level as a function of the number of electrons added to a Coulomb-blockade quantum dot. A microscopic calculation in the limit of Koopmans’ theorem predicts that the standard deviation of these fluctuations is linear in the number of added electrons, in agreement with a parametric random-matrix approach. We estimate the number of electrons it takes to scramble the spectrum completely in terms of the interaction strength, the dimensionless Thouless conductance, and the symmetry class. PACS numbers: 73.23.Hk, 05.45+b, 73.21.La, 73.23.-b Typeset using REVTEX 1 The simplest model for describing a quantum dot in the Coulomb-blockade regime [1] is the constant interaction (CI) model, in which the electrons occupy single-particle levels and the Coulomb interaction is taken as an average electrostatic energy that depends only on the number of electrons. When the single-particle dynamics in the dot are chaotic (or diffusive), one can apply random matrix theory (RMT) to describe the statistical properties of the single-particle wave functions and energies within an energy window whose width is the Thouless energy. RMT was successful in describing the conductance peak height distributions and their sensitivity to time-reversal symmetry [2,3]. However, other measured statistics, most notably the peak spacing distribution [4], indicated that it is necessary to include interaction effects beyond the simple CI model. The best way to take into account interactions while retaining a single-particle framework is the Hartree-Fock (HF) approximation, which has been used to explain some of the observed features of the peak spacing statistics [5–7]. The HF single-particle wave functions and energies are calculated self-consistently and therefore can change as electrons are added to the dot. In the statistical regime (i.e., in a chaotic or diffusive dot), this phenomenon is called scrambling. Scrambling was observed in the decay of correlations between the m-th excited state in the dot and the ground-state of a dot with an additional m electrons [8], and indirectly through the saturation of the peakto-peak correlations as a function of temperature [9,10]. Numerical evidence of scrambling was found in the HF calculations (see, e.g., in Ref. [6]). A phenomenological way to describe scrambling is to consider a discrete set of Hamiltonians (corresponding to the different number of electrons in the dot) that are random but have the correct symmetries. Such a set is known as a discrete Gaussian process (GP) [11], and can be embedded in a continuous GP, i.e., random matrices that depend on a continuous parameter. This approach leads to a nearly Gaussian peak spacing distribution [12], and explains the saturation of the number of correlated peaks versus temperature [10]. While the parametric approach is appealing in its simplicity, it is not clear how well it describes features obtained in a microscopic approach of adding electrons into the dot. For example, in the parametric approach, the addition of m 2 electrons into the dot results in a fluctuation width of a typical level that is proportional to m. One can also evaluate the fluctuation width of a typical energy level in the microscopic approach through the fluctuations of the diagonal interaction matrix elements (in the limit where the single-particle wave functions do not change with the addition of electrons). If the addition of electrons is a random process, we expect the fluctuation width of a level to behave like √ m. We shall show, however, that since the correlations among the diagonal interaction matrix elements are comparable to their variance, the standard deviation of the fluctuations is linear in m (for 1 ≪ m ≪ g, where g is the Thouless conductance), in overall agreement with parametric RMT. The main results of the microscopic approach are summarized in Eqs. (10), (14) and (15). Extrapolating our results to larger values of m, we also estimate the dependence of the number of added electrons required for complete scrambling on g and the interaction strength (see Eq. (16)). Scrambling implies variation of both eigenstates and energy levels. However, in this work we are concerned with the limit where the wave functions do not change and only the energy levels scramble. In the parametric approach this is valid when the change in the parameter upon the addition of m electrons is small compared to the mean (parametric) distance between avoided crossings. In the microscopic approach the appropriate limit is known as Koopmans’ theorem [13], where the variation of an energy level upon the addition of an electron is just a corresponding diagonal interaction matrix element. We first discuss the parametric approach. The variation of the single-particle energies and eigenfunctions (e.g., in a mean-field approximation) with the addition of electrons into the dot is described by a parametric variation of the Hamiltonian [14]. We denote by H(xN ) the effective single-particle Hamiltonian of the dot with N electrons. We assume that the dot is either diffusive or ballistic with chaotic dynamics, and that the statistical properties of H(xN ) are not modified by the interactions. Restricting ourselves to the universal regime, i.e., to ∼ g levels in the vicinity of the Fermi energy [15], we assume that H(xN ) belongs to one of the Gaussian ensemble of random matrices whose symmetry class β is independent of xN . The sequence of Hamiltonians H(xN ) forms a discrete GP that can be embedded in 3 a continuous GP H(x) [19]. A simple GP is given by [11,20] H(x) = cosxH1 + sin xH2 , (1) where H1 and H2 are N × N uncorrelated random matrices chosen from the Gaussian ensemble of symmetry class β: P (H) ∝ e β 2a2 2. We choose a = (2/N)ζ so that the average level density (a semicircle) has a constant band width of 2a √ 2N = 4ζ , and the mean level spacing in the middle of the spectrum is ∆ = πa/ √ 2N = πζ/N . The average distance between avoided crossings is given by the inverse of the rms level velocity δxc = ∆[(∂ǫα/∂x)] −1/2 = π(β/2N), where ǫα is an energy level. This distance is larger for the GUE by a factor of √ 2 compared with the GOE. Two-point parametric correlators become universal when the parameter x is measured in units of δxc; i.e., as a function of a scaled parameter x̄ ≡ x/δxc. The energy levels scramble as the parameter x̄ changes. A change of δx̄ leads to a corresponding change in the energy of a single-particle level α: δǫα = ǫα(x̄ + δx̄) − ǫα(x̄). The variance of this parametric fluctuation of a level can be estimated in the limit δx̄ ≪ 1 using first order perturbation theory in δx̄, i.e., ignoring the change of the single-particle wave function as x̄ → x̄+ δx̄: σ(δǫα) = ∆ (δx̄) . (2) In the parametric approach it is assumed that x̄ changes by δx̄1 upon the addition of one electron into the dot (independently of the number of electrons N ). Thus for the addition of m electrons δx̄m = mδx̄1, and as long as δx̄m ≪ 1, we can use (2) to estimate the variance of the change in the energy level α when m electrons are added σ(δǫ α ) = ∆mδx̄1 = mσ(δǫ (1) α ) , (3) where δǫ α ≡ ǫ α − ǫ ) α (ǫ α is the energy of level α in a dot with N +m electrons). To relate the parametric approach to a microscopic mean-field approach, we describe the effect that adding one or a few electrons has on a particular single-particle HF level. The Hamiltonian of the dot is 4