Scaling at the Chaos Threshold for Interacting Electrons in a Quantum Dot

Abstract

The chaotic mixing by random two-body interactions of many-electron Fock states in a confined geometry is investigated numerically and compared with analytical predictions. Two distinct regimes are found in the dependence of the inverse participation ratio in Fock space I on the dimensionless conductance of the quantum dot g and the excitation energy ε. In both regimes I ≫ 1, but only the small-g regime is described by the golden rule. The crossover region is characterized by a maximum in a scaling function that becomes more pronounced with increasing excitation energy. The scaling parameter that governs the transition is (ε/g) ln g.PACS numbers: 05.45.+b, 71.10.-w, 73.23.-b The highly excited atomic nucleus was the first example of a quantum chaotic system, although the interpretation of Wigner's distribution of level spacings [1] as a signature of quantum chaos came many years later, from the study of electron billiards [2]. While the spectral statistics of the nucleus and the billiard is basically the same, the origin of the chaotic behavior is entirely different [3]: In the billiard chaos appears in the single-particle spectrum as a result of boundary scattering, while in the nucleus chaos appears in the many-particle spectrum as a result of interactions.The study of the interaction-induced transition to chaos entered condensed matter physics with the realization that a semiconductor quantum dot could be seen as an artificial atom or compound nucleus [4]. A particularly influential paper by Altshuler, Gefen, Kamenev, and Levitov [5] studied the interaction-induced decay of a quasiparticle in a quantum dot and interpreted the broadening of the peaks in the single-particle density of states as a delocalization transition in Fock space. Different scenario's leading to a smooth rather than an abrupt transition from localized to extended states were considered later [6][7][8]. Recent computer simulations [9,10] also confirm the smooth crossover from localized to delocalized regime for quasiparticle decay.As emphasized by Altshuler et al. [5], the delocalized regime in the quasiparticle decay problem is not yet chaotic because the states do not extend uniformly over the Fock space. One may study the transition to chaos in the single-particle density of states, but theoretically it is easier to consider instead the mixing by interactions of arbitrary many-particle states. This was the approach taken in Refs. [6,8,[11][12][13][14], focusing on two quantities: The distribution of the energy level spacings and the inverse participation ratio (IPR) of the wave functions in Fock space. Both quantities can serve as a signature for chaotic behavior, the spacing distribution by comparing with Wigner's distribution [1] and the IPR by comparing with the golden rule (according to which the IPR is the mean spacing δ of the many-particle states divided by the mean decay rate Γ of a non-interacting manyparticle state [12]). Two fundamental questions in these investigations are: 1) What is the scaling parameter that governs th...

Multifractality of wave functions on a Cayley tree: From root to leaves

“…The t dependence of the solution in this range is determined by the factor e −e t 1 − e t in the integrand of Eq. (14). This yields, in view of 1 = 1,…”

“…3 where solutions of the recursion relation (14) for ψ r (t) are shown (blue curves) for the Cayley tree with a connectivity m = 2 and radius R = 25. To solve numerically the recursion relation, we discretized the integral recursions with mesh points chosen to be equally spaced in t, with 1000 points covering the range t = [ −40, 40].…”

“…4 suggested an approximate mapping between the problem of Fock-space localization in a quantum dot and a single-particle localization on a Bethe lattice. (See also subsequent works [5][6][7][8][9][10][11][12][13][14][15][16] .) Later, these ideas were extended to explore the manybody-localization (MBL) in spatially extended systems with localized single-particle states and with short-range interaction 17,18 .…”

Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices for Bosons

“…Eq. (27) applies only in the unitary case (β = 2) and implies that in the dense limit, the average spectrum has Gaussian shape. We have been unable to extend this result to the orthogonal case (β = 1).…”

“…This holds also true in the dense limit m → ∞ with k and l fixed as given by Eq. (27). For the two-point function we drop terms that vanish as 1/N for N → ∞.…”