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Progress of Theoretical Physics

Orthogonality Catastrophe due to Local Electron Correlation

Hiroyuki Kaga, Kei Yosida

Published Jan 1, 1978• 7 citations
Publisher

Abstract

We discuss the orthogonality catastrophe between the ground states of the unperturbed and perturbed Fermi systems due to local electron correlation U in the Anderson model. We show that the symmetric Anderson model presents a special case in which no catastrophe appears and that in the asymmetric case the catastrophe results from the self-energy shift.

DOI
10.1143/ptp.59.34

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We discuss the orthogonality catastrophe between the ground states of the unperturbed and perturbed Fermi systems due to local electron correlation U in the Anderson model. We show that the symmetric Anderson model presents a special case in which no catastrophe appears and that in the asymmetric case the catastrophe results from the self-energy shift.
It is shown that the overlap integral between two ground-state wave functions for the Hamiltonian of the interacting electron system with different local potentials can be written generally in terms of the S-matrix describing potential scattering. Here, the local potential is considered to be a general inseparable one and the Green functions and self-energies are treated as operaors with off-diagonal elements. This orthogonality theorem expressed in a general form is applied to some special cases.
We address the phenomenon of statistical orthogonality catastrophe in insulating disordered systems. In more detail, we analyse the response of a system of non-interacting fermions to a local perturbation induced by an impurity. By inspecting the overlap between the pre- and post-quench many-body ground states we fully characterise the emergent statistics of orthogonality events as a function of both the impurity position and the coupling strength. We consider two well-known one-dimensional models, namely the Anderson and Aubry–André insulators, highlighting the arising differences. Particularly, in the Aubry–André model the highly correlated nature of the quasi-periodic potential produces unexpected features in how the orthogonality catastrophe occurs. We provide a quantitative explanation of such features via a simple, effective model. We further discuss the incommensurate ratio approximation and suggest a viable experimental verification in terms of charge transfer statistics and interferometric experiments using quantum probes.
According to Anderson's orthogonality catastrophe, the overlap of the N -particle ground states of a free Fermi gas with and without an (electric) potential decays in the thermodynamic limit. For the finite one-dimensional system various boundary conditions are employed. Unlike the usual setup the perturbation is introduced by a magnetic (vector) potential. Although such a magnetic field can be gauged away in one spatial dimension there is a significant and interesting effect on the overlap caused by the phases. We study the leading asymptotics of the overlap of the two ground states and the two-term asymptotics of the difference of the ground-state energies. In the case of periodic boundary conditions our main result on the overlap is based upon a well-known asymptotic expansion by Fisher and Hartwig on Toeplitz determinants with a discontinuous symbol. In the case of Dirichlet boundary conditions no such result is known to us and we only provide an upper bound on the overlap, presumably of the right asymptotic order.
Advances in experimental tools allow to study quantum fidelity in unprecedented controlled settings. While the fidelity in metals, when adding a local impurity, is well known to show the Anderson orthogonality catastrophe (AOC), there remain outstanding questions in other quantum phases and settings. Here, we aim to tackle these by exploring systematically the ground-state fidelity of the Fermi liquid in the (extended) Aubry-Andr\'e (AA) model, which allows to explore the AOC in both localized extended and critical phases. We discover that the AOC is typically exponential in the critical regime of the AA model and at the mobility edge of the extended AA model for an extended impurity, while it decays in the AA model with a power law for a weak single-site impurity. We explain this in terms of critical correlations and multipoint correlations. The OC is found to be exponential in the insulating regime, due to a fundamentally different, statistical mechanism, which is explained in detail. Furthermore, we consider a parametric perturbation to the AA model, and find an exponential OC numerically, in agreement with an analytical derivation which we provide here.