Skeletons and Variation

Well known from the sixties, the pressure of e.g. massless phi-four theory may be written as a series of 2PI-diagrams (skeletons) with the lines fully dressed. Varying the self-energy Pi in this expression, it turns into a functional U[Y] having a maximum in function space at Y=Pi. There is also the Feynman-Jensen thermal variational principle V[S], a potentially non-perturbative tool. Here actions S are varied. It is shown, through a few formal but exact steps, that the functional U is covered by V. The corresponding special subset of trial actions is made explicit.Comment: 4 pages RevTeX; talk at the 5th Thermal Field Theory Workshop 1998 at Regensburg, Germany, August 10-14; text around (9), (10) improve

Hamiltonian YM 2+1: note on point splitting regularization

The Hamiltonian of 2+1 dimensional Yang Mills theory was derived by Karabali, Kim and Nair by using point splitting regularization. But in calculating e.g. the vacuum wave functional this scheme was left in favour of arguments. Here we follow up a conjecture of Leigh, Minic and Yelnikov of how this gap might be filled by including all positive powers of the regularization parameter (\ep \to +0). Admittedly, though we concentrate on the ground state in the large $N$ limit, only two such powers could be included due to the increasing complexity of the task.Comment: 25 page

Geometry of Weyl theory for Jacobi matrices with matrix entries

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. The Weyl surface describing the dependence of Green's matrix on the boundary conditions is interpreted as the set of maximally isotropic subspace of a quadratic from given by the Wronskian. Analysis of the possibly degenerate limit quadratic form leads to the limit point/limit surface theory of maximal symmetric extensions for semi-infinite Jacobi matrices with matrix entries with arbitrary deficiency indices. The resolvent of the extensions is explicitly calculated

Perturbation theory for Lyapunov exponents of an Anderson model on a strip

It is proven that the inverse localization length of an Anderson model on a strip of width $L$ is bounded above by $L/\lambda^2$ for small values of the coupling constant $\lambda$ of the disordered potential. For this purpose, a formalism is developed in order to calculate the bottom Lyapunov exponent associated with random products of large symplectic matrices perturbatively in the coupling constant of the randomness.Comment: to appear in GAF

The density of surface states as the total time delay

For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction (REI) representation, as well as an index theorem for adequate associated operator algebras.Comment: Suggestions of referees incorporated and errors corrected. To appear in Lett. Math. Phy

Lifshitz tails for the 1D Bernoulli-Anderson model

By using the adequate modified Pr\"ufer variables, precise upper and lower bounds on the density of states in the (internal) Lifshitz tails are proven for a 1D Anderson model with bounded potential

Topological insulators from the perspective of non-commutative geometry and index theory

Topological insulators are solid state systems of independent electrons for which the Fermi level lies in a mobility gap, but the Fermi projection is nevertheless topologically non-trivial, namely it cannot be deformed into that of a normal insulator. This non-trivial topology is encoded in adequately defined invariants and implies the existence of surface states that are not susceptible to Anderson localization. This non-technical review reports on recent progress in the understanding of the underlying mathematical structures, with a particular focus on index theory.Comment: to appear in Jahresberichte DM

Rotation numbers for Jacobi matrices with matrix entries

A selfadjoined block tridiagonal matrix with positive definite blocks on the off-diagonals is by definition a Jacobi matrix with matrix entries. Transfer matrix techniques are extended in order to develop a rotation number calculation for its eigenvalues. This is a matricial generalization of the oscillation theorem for the discrete analogues of Sturm-Liouville operators. The three universality classes of time reversal invariance are dealt with by implementing the corresponding symmetries. For Jacobi matrices with random matrix entries, this leads to a formula for the integrated density of states which can be calculated perturbatively in the coupling constant of the randomness with an optimal control on the error terms

Persistence of spin edge currents in disordered quantum spin Hall systems

For a disordered two-dimensional model of a topological insulator (such as a Kane-Mele model with disordered potential) with small coupling of spin invariance breaking term (such as the Rashba coupling), it is proved that the spin edge currents persist provided there is a spectral gap and the spin Chern numbers are well-defined and non-trivial. These conditions on being in the quantum spin Hall phase do not require time-reversal symmetry. The result materializes the general philosophy that topological insulators are non-trivial bulk systems with edge currents that are topologically protected against Anderson localization.Comment: Final version to appear in Commun. Math. Phy

Resummation of phi^4 free energy up to an arbitrary order

The consistency condition, which guarantees a well organized small-coupling asymptotic expansion for the thermodynamics of massless $\phi^4$-theory, is generalized to any desired order of the perturbative treatment. Based on a strong conjecture about forbidden two-particle reducible diagrams, this condition is derived in terms of functions of four-momentum in place of the common toy mass in previous treatments. It has the form of a set of gap equations and marks the position in the space of these functions at which the free energy is extremal.Comment: 15 pages, latex, 3 figures using latex, more discussion below (4.14), 3 Ref's added, slight changes in notatio