A Convergent Method for Calculating the Properties of Many Interacting Electrons

A method is presented for calculating binding energies and other properties of extended interacting systems using the projected density of transitions (PDoT) which is the probability distribution for transitions of different energies induced by a given localized operator, the operator on which the transitions are projected. It is shown that the transition contributing to the PDoT at each energy is the one which disturbs the system least, and so, by projecting on appropriate operators, the binding energies of equilibrium electronic states and the energies of their elementary excitations can be calculated. The PDoT may be expanded as a continued fraction by the recursion method, and as in other cases the continued fraction converges exponentially with the number of arithmetic operations, independent of the size of the system, in contrast to other numerical methods for which the number of operations increases with system size to maintain a given accuracy. These properties are illustrated with a calculation of the binding energies and zone-boundary spin- wave energies for an infinite spin-1/2 Heisenberg chain, which is compared with analytic results for this system and extrapolations from finite rings of spins.Comment: 30 pages, 4 figures, corrected pd

Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling

The Anderson model for independent electrons in a disordered potential is transformed analytically and exactly to a basis of random extended states leading to a variant of augmented space. In addition to the widely-accepted phase diagrams in all physical dimensions, a plethora of additional, weaker Anderson transitions are found, characterized by the long-distance behavior of states. Critical disorders are found for Anderson transitions at which the asymptotically dominant sector of augmented space changes for all states at the same disorder. At fixed disorder, critical energies are also found at which the localization properties of states are singular. Under the approximation of single-parameter scaling, this phase diagram reduces to the widely-accepted one in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson transition at infinitesimal disorder, there is a transition between two localized states, characterized by a change in the nature of wave function decay.Comment: 51 pages including 4 figures, revised 30 November 200

Local density of states of a d-wave superconductor with inhomogeneous antiferromagnetic correlations

The tunneling spectrum of an inhomogeneously doped extended Hubbard model is calculated at the mean field level. Self-consistent solutions admit both superconducting and antiferromagnetic order, which coexist inhomogeneously because of spatial randomness in the doping. The calculations find that, as a function of doping, there is a continuous cross over from a disordered ``pinned smectic'' state to a relatively homogeneous d-wave state with pockets of antiferromagnetic order. The density of states has a robust d-wave gap, and increasing antiferromagnetic correlations lead to a suppression of the coherence peaks. The spectra of isolated nanoscale antiferromagnetic domains are studied in detail, and are found to be very different from those of macroscopic antiferromagnets. Although no single set of model parameters reproduces all details of the experimental spectrum in BSCCO, many features, notably the collapse of the coherence peaks and the occurence of a low-energy shoulder in the local spectrum, occur naturally in these calculations.Comment: 9 pages, 5 figure

Classical Localization of an Unbound Particle in a Two-Dimensional Periodic Potential and Surface Diffusion

In periodic, two-dimensional potentials a classical particle might be expected to escape from any finite region if it has enough energy to escape from a single cell. However, for a class of sinusoidal potentials in which the barriers between neighboring cells can be varied, numerical tridiagonalization of Liouville's equation for the evolution of functions on phase space reveals a transition from localized to delocalized motion at a total energy significantly above that needed to escape from a single cell. It is argued that this purely elastic phenomenon increases the effective barrier for diffusion of atoms on crystalline surfaces and changes its temperature dependence at low temperatures when inelastic events are rare.Comment: Revised version accepted for publication in Physical Review

Roger Haydock, Phineas Pemberton, July 1, 1682

Letter dated July 1, 1682 (June 21, 1682 Old Style) from Roger Haydock to Phineas Pemberton

Roger Haydock and Elinor Haydock, Phineas Pemberton, June 17, 1684; June 26, 1684

Two letters from Roger and Elinor Haydock to Phineas Pemberton. The first letter, sent from Warrington, is dated June 17, 1684 (June 7, 1684 Old Style). The second, sent from Liverpool, is dated June 26, 1684 (June 16, 1684 Old Style)