The Statistical Theory of Quantum Dots

Abstract

A quantum dot is a sub-micron-scale conducting device containing up to several thousand electrons. Transport through a quantum dot at low temperatures is a quantum-coherent process. This review focuses on dots in which the electron's dynamics are chaotic or diffusive, giving rise to statistical properties that reflect the interplay between one-body chaos, quantum interference, and electron-electron interactions. The conductance through such dots displays mesoscopic fluctuations as a function of gate voltage, magnetic field, and shape deformation. The techniques used to describe these fluctuations include semiclassical methods, random-matrix theory, and the supersymmetric nonlinear $\sigma$ model. In open dots, the approximation of noninteracting quasiparticles is justified, and electron-electron interactions contribute indirectly through their effect on the dephasing time at finite temperature. In almost-closed dots, where conductance occurs by tunneling, the charge on the dot is quantized, and electron-electron interactions play an important role. Transport is dominated by Coulomb blockade, leading to peaks in the conductance that at low temperatures provide information on the dot's ground-state properties. Several statistical signatures of electron-electron interactions have been identified, most notably in the dot's addition spectrum. The dot's spin, determined partly by exchange interactions, can also influence the fluctuation properties of the conductance. Other mesoscopic phenomena in quantum dots that are affected by the charging energy include the fluctuations of the cotunneling conductance and mesoscopic Coulomb blockade.

77 pages including 31 figures, manuscript is also available at the RMP web site, see http://rmp.aps.org

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• The strong-coupling limit of a Kondo spin coupled to a mesoscopic quantum dot: effective Hamiltonian in the presence of exchange correlations
• Unfolding a composed ensemble of energy spectra using singular value decomposition
• Decoding the conductance of disordered nanostructures: a quantum inverse problem
• A generation-based particle-hole density-matrix renormalization group study of interacting quantum dots
• Electron-hole generations: A numerical approach to interacting fermion systems
• Significance of the nature of disorder on the universal features of the spatio-temporal correlations of two-dimensional Coulomb-clusters
• Edge channels in a graphene Fabry-Perot interferometer
• Distribution of local density of states in superstatistical random matrix theory
• Path-integral Monte Carlo study of electronic states in quantum dots in an external magnetic field
• Developments in Random Matrix Theory
• Transmission Resonances Anomaly in 1D Disordered Quantum Systems
• Correlation between peak-height modulation and phase-lapses in transport through quantum dots
• Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models
• Manifestation of interdot spin tunneling effects in ordered arrays of quantum dots: An anomalous magnetization of In_{1-x}Mn_xAs
• Ballistic dynamics of a convex smooth-wall billiard with finite escape rate along the boundary
• Tunneling density of states in quantum dots with anisotropic exchange
• The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs
• Detecting quantum chaos via pseudo-entropy
• Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles
• Random Matrices and Quantum Hamilton-Jacobi Method
• Interaction Matrix Element Fluctuations in Ballistic Quantum Dots: Dynamical Effects
• Quantum probes for quantum wells
• Chaotic behavior of the Compound Nucleus, open Quantum Dots and other nanostructures
• Spin-orbit scattering in superconducting nanoparticles
• Electronic transport in three-terminal chaotic systems with a tunnel barrier
• Winding Number Statistics for Chiral Random Matrices: Universal Correlations and Statistical Moments in the Unitary Case
• Geometry effects in the magnetoconductance of normal and Andreev Sinai billiards
• Universal Frequency Correlations and Recurrence Statistics of Complex Impedance Matrices
• Mesoscopic interplay of superconductivity and ferromagnetism in ultra-small metallic grains
• Discrete charging of metallic grains: Statistics of addition spectra
• Quantum Dots: Coulomb Blockade, Mesoscopic Fluctuations, and Qubit Decoherence
• Universality Emerging in a Universality: Derivation of the Ericson Transition in Stochastic Quantum Scattering and Experimental Validation
• Transition-state dynamics in complex quantum systems
• Two gate-voltage periods in a metallic-nanoparticle based single-electron transistor
• The renormalization group for interacting fermions: from Fermi liquids to quantum dots