Mathematical Models and Biological Meaning: Taking Trees Seriously

We compare three basic kinds of discrete mathematical models used to portray phylogenetic relationships among species and higher taxa: phylogenetic trees, Hennig trees and Nelson cladograms. All three models are trees, as that term is commonly used in mathematics; the difference between them lies in the biological interpretation of their vertices and edges. Phylogenetic trees and Hennig trees carry exactly the same information, and translation between these two kinds of trees can be accomplished by a simple algorithm. On the other hand, evolutionary concepts such as monophyly are represented as different mathematical substructures are represented differently in the two models. For each phylogenetic or Hennig tree, there is a Nelson cladogram carrying the same information, but the requirement that all taxa be represented by leaves necessarily makes the representation less efficient. Moreover, we claim that it is necessary to give some interpretation to the edges and internal vertices of a Nelson cladogram in order to make it useful as a biological model. One possibility is to interpret internal vertices as sets of characters and the edges as statements of inclusion; however, this interpretation carries little more than incomplete phenetic information. We assert that from the standpoint of phylogenetics, one is forced to regard each internal vertex of a Nelson cladogram as an actual (albeit unsampled) species simply to justify the use of synapomorphies rather than symplesiomorphies.Comment: 15 pages including 6 figures [5 pdf, 1 jpg]. Converted from original MS Word manuscript to PDFLaTe

Magnetic field effects in few-level quantum dots: theory, and application to experiment

We examine several effects of an applied magnetic field on Anderson-type models for both single- and two-level quantum dots, and make direct comparison between numerical renormalization group (NRG) calculations and recent conductance measurements. On the theoretical side the focus is on magnetization, single-particle dynamics and zero-bias conductance, with emphasis on the universality arising in strongly correlated regimes; including a method to obtain the scaling behavior of field-induced Kondo resonance shifts over a very wide field range. NRG is also used to interpret recent experiments on spin-1/2 and spin-1 quantum dots in a magnetic field, which we argue do not wholly probe universal regimes of behavior; and the calculations are shown to yield good qualitative agreement with essentially all features seen in experiment. The results capture in particular the observed field-dependence of the Kondo conductance peak in a spin-1/2 dot, with quantitative deviations from experiment occurring at fields in excess of $\sim$ 5 T, indicating the eventual inadequacy of using the equilibrium single-particle spectrum to calculate the conductance at finite bias.Comment: 15 pages, 12 figures. Version as published in PR

A local moment approach to the degenerate Anderson impurity model

The local moment approach is extended to the orbitally-degenerate [SU(2N)] Anderson impurity model (AIM). Single-particle dynamics are obtained over the full range of energy scales, focussing here on particle-hole symmetry in the strongly correlated regime where the onsite Coulomb interaction leads to many-body Kondo physics with entangled spin and orbital degrees of freedom. The approach captures many-body broadening of the Hubbard satellites, recovers the correct exponential vanishing of the Kondo scale for all N, and its universal scaling spectra are found to be in very good agreement with numerical renormalization group (NRG) results. In particular the high-frequency logarithmic decays of the scaling spectra, obtained here in closed form for arbitrary N, coincide essentially perfectly with available numerics from the NRG. A particular case of an anisotropic Coulomb interaction, in which the model represents a system of N `capacitively-coupled' SU(2) AIMs, is also discussed. Here the model is generally characterised by two low-energy scales, the crossover between which is seen directly in its dynamics.Comment: 23 pages, 7 figure

Correlated electron physics in multilevel quantum dots: phase transitions, transport, and experiment

We study correlated two-level quantum dots, coupled in effective 1-channel fashion to metallic leads; with electron interactions including on-level and inter-level Coulomb repulsions, as well as the inter-orbital Hund's rule exchange favoring the spin-1 state in the relevant sector of the free dot. For arbitrary dot occupancy, the underlying phases, quantum phase transitions (QPTs), thermodynamics, single-particle dynamics and electronic transport properties are considered; and direct comparison is made to conductance experiments on lateral quantum dots. Two distinct phases arise generically, one characterised by a normal Fermi liquid fixed point (FP), the other by an underscreened (USC) spin-1 FP. Associated QPTs, which occur in general in a mixed valent regime of non-integral dot charge, are found to consist of continuous lines of Kosterlitz-Thouless transitions, separated by first order level-crossing transitions at high symmetry points. A `Friedel-Luttinger sum rule' is derived and, together with a deduced generalization of Luttinger's theorem to the USC phase (a singular Fermi liquid), is used to obtain a general result for the T=0 zero-bias conductance, expressed solely in terms of the dot occupancy and applicable to both phases. Relatedly, dynamical signatures of the QPT show two broad classes of behavior, corresponding to the collapse of either a Kondo resonance, or antiresonance, as the transition is approached from the Fermi liquid phase; the latter behavior being apparent in experimental differential conductance maps. The problem is studied using the numerical renormalization group method, combined with analytical arguments.Comment: 22 pages, 18 figures, submitted for publicatio

Quantum phase transition in capacitively coupled double quantum dots

We investigate two equivalent, capacitively coupled semiconducting quantum dots, each coupled to its own lead, in a regime where there are two electrons on the double dot. With increasing interdot coupling a rich range of behavior is uncovered: first a crossover from spin- to charge-Kondo physics, via an intermediate SU(4) state with entangled spin and charge degrees of freedom; followed by a quantum phase transition of Kosterlitz-Thouless type to a non-Fermi liquid `charge-ordered' phase with finite residual entropy and anomalous transport properties. Physical arguments and numerical renormalization group methods are employed to obtain a detailed understanding of the problem.Comment: 4 pages, 3 figure

Two-channel Kondo physics in tunnel-coupled double quantum dots

We investigate theoretically the possibility of observing two-channel Kondo (2CK) physics in tunnel-coupled double quantum dots (TCDQDs), at both zero and finite magnetic fields; taking the two-impurity Anderson model (2AIM) as the basic TCDQD model, together with effective low-energy models arising from it by Schrieffer-Wolff transformations to second and third order in the tunnel couplings. The models are studied primarily using Wilson's numerical renormalization group. At zero-field our basic conclusion is that while 2CK physics arises in principle provided the system is sufficiently strongly-correlated, the temperature window over which it could be observed is much lower than is experimentally feasible. This finding disagrees with recent work on the problem, and we explain why. At finite field, we show that the quantum phase transition known to arise at zero-field in the two-impurity Kondo model (2IKM), with an essentially 2CK quantum critical point, persists at finite fields. This raises the prospect of access to 2CK physics by tuning a magnetic field, although preliminary investigation suggests this to be even less feasible than at zero field.Comment: 10 pages, 8 figures. Version as published in PR