On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $\lambda$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue $z$ and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we provide the 'edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach \cite{BourgadeDubach}.Comment: published versio

Scaling for Orthogonality

In updating algorthms where orthogonal transformations are accumulated, it is important to preserve the orthogonality of the product in the presence of rounding error. Moonen, Van Dooren, and Vandewalle have pointed out that simply normalizing the columns of the product tends to preserve orthogonality\,---\,though not, as DeGroat points out, to working precision. In this note we give an analysis of the phenomenon. (Also cross-referenced as UMIACS-TR-92-43

Orthogonality catastrophe in a one-dimensional system of correlated electrons

We present a detailed numerical study of the orthogonality catastrophe exponent for a one-dimensional lattice model of spinless fermions with nearest neighbor interaction using the density matrix remormalization group algorithm. Keeping up to 1200 states per block we achieve a very great accuracy for the overlap which is needed to extract the orthogonality exponent reliably. We discuss the behavior of the exponent for three different kinds of a localized impurity. For comparison we also discuss the non-interacting case. In the weak impurity limit our results for the overlap confirm scaling behavior expected from perturbation theory and renormalization group calculations. In particular we find that a weak backward scattering component of the orthogonality exponent scales to zero for attractive interaction. In the strong impurity limit and for repulsive interaction we demonstrate that the orthogonality exponent cannot be extracted from the overlap for systems with up to 100 sites, due to finite size effects. This is in contradiction to an earlier interpretation given by Qin et al. based on numerical data for much smaller system sizes. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/16 based on predictions of boundary conformal field theory.Comment: 16 pages, Latex, 8 eps figures, submitted to Phys. Rev.

Entropy, fidelity, and double orthogonality for resonance states in two-electron quantum dots

Resonance states of a two-electron quantum dot are studied using a variational expansion with both real basis-set functions and complex scaling methods. The two-electron entanglement (linear entropy) is calculated as a function of the electron repulsion at both sides of the critical value, where the ground (bound) state becomes a resonance (unbound) state. The linear entropy and fidelity and double orthogonality functions are compared as methods for the determination of the real part of the energy of a resonance. The complex linear entropy of a resonance state is introduced using complex scaling formalism