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arXiv:cond-mat/9910121

Orthogonality Catastrophe Revisited

Nic Shannon

Published Oct 7, 1999• 0 citations
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Abstract

We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, $α=2 \sum_l(2l+1) (δ_l(ε_f)/π)^2 $ %$α= 2\sum_l [2l+1] (δ_l(ε_f)/π)^2$, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.

DOI
10.48550/arxiv.cond-mat/9910121

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We prove a simple theorem on the overlap of the wavefunctions of a manybody system with and without a single impurity and show how, and under which conditions, this leads to the ``Orthogonality Catastrophe'' (OC) described by Anderson. A compact new derivation of the known result for the OC exponent in a free electron gas, α=2∑l(2l+1)(δl(εf)/π)2α=2 \sum_l(2l+1) (δ_l(ε_f)/π)^2 α=2∑l​(2l+1)(δl​(εf​)/π)2 %α=2∑l[2l+1](δl(εf)/π)2α= 2\sum_l [2l+1] (δ_l(ε_f)/π)^2α=2∑l​[2l+1](δl​(εf​)/π)2, is given. We also introduce a simple way of calculating core level photoemission (XPS) lineshapes from the finite size scaling of groundstate energies, reproducing and extending results previously derived using boundary conformal field theory. Different conditions under which the OC fails are discussed, and simple physical predictions for XPS lineshapes made in each case.
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 renormalization-group algorithm. Keeping up to 200020002000 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 noninteracting 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. Neverthless we find indirect evidence that the backward scattering contribution to the exponent scales to 1/161/161/16 based on predictions of boundary conformal field theory.
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.
The sudden introduction of a local impurity in a Fermi sea leads to an anomalous disturbance of its quantum state that represents a local quench, leaving the system out of equilibrium and giving rise to the Anderson orthogonality catastrophe. The statistics of the work done describe the energy fluctuations produced by the quench, providing an accurate and detailed insight into the fundamental physics of the process. We present here a numerical approach to the non-equilibrium work distribution, supported by applications to phenomena occurring at very diverse energy ranges. One of them is the valence electron shake-up induced by photo-ionization of a core state in a fullerene molecule. The other is the response of an ultra-cold gas of trapped fermions to an embedded two-level atom excited by a fast pulse. Working at low thermal energies, we detect the primary role played by many-particle states of the perturbed system with one or two excited fermions. We validate our approach through the comparison with some photoemission data on fullerene films and previous analytical calculations on harmonically trapped Fermi gases.
The behavior of a single impurity in a one-dimensional Luttinger liquid is numerically investigated by means of the density-matrix renormalization group. By analyzing the finite-size scaling behavior of the low-energy spectrum, we confirm the theoretical prediction of Kane and Fisher [Phys. Rev. Lett. 68, 1220 (1992)] both for attractive and repulsive interactions. Moreover, we calculate the exponent of the orthogonality catastrophe, which gives a further support to the above theoretical prediction.