Many-body theory of electronic transport in single-molecule heterojunctions

A many-body theory of molecular junction transport based on nonequilibrium Green's functions is developed, which treats coherent quantum effects and Coulomb interactions on an equal footing. The central quantity of the many-body theory is the Coulomb self-energy matrix $\Sigma_{\rm C}$ of the junction. $\Sigma_{\rm C}$ is evaluated exactly in the sequential tunneling limit, and the correction due to finite tunneling width is evaluated self-consistently using a conserving approximation based on diagrammatic perturbation theory on the Keldysh contour. Our approach reproduces the key features of both the Coulomb blockade and coherent transport regimes simultaneously in a single unified transport theory. As a first application of our theory, we have calculated the thermoelectric power and differential conductance spectrum of a benzenedithiol-gold junction using a semi-empirical $\pi$-electron Hamiltonian that accurately describes the full spectrum of electronic excitations of the molecule up to 8--10eV.Comment: 13 pages, 7 figure

Noncommutative Blowups of Elliptic Algebras

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal structure of T(d) is quite rigid. Our results generalise those of the first author. In the companion paper "Classifying Orders in the Sklyanin Algebra", we apply our results to classify orders in (a Veronese subalgebra of) a generic cubic or quadratic Sklyanin algebra.Comment: 39 pages. Minor changes from previous version. The final publication is available from Springer via http://dx.doi.org/10.1007/s10468-014-9506-

The Auslander-Gorenstein property for Z-algebras

We provide a framework for part of the homological theory of Z-algebras and their generalizations, directed towards analogues of the Auslander-Gorenstein condition and the associated double Ext spectral sequence that are useful for enveloping algebras of Lie algebras and related rings. As an application, we prove the equidimensionality of the characteristic variety of an irreducible representation of the Z-algebra, and for related representations over quantum symplectic resolutions. In the special case of Cherednik algebras of type A, this answers a question raised by the authors.Comment: 31 page

Noncommutative curves and noncommutative surfaces

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing up and down, has been developed.Comment: Suggestions by many people (in particular Haynes Miller and Dennis Keeler) have been incorporated. The formulation of some results has been improve