Abstract
The overlap, \({\mathcal{D}_N}\), between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\). We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\). The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\), and \({\tilde{\gamma}}\). In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.
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Communicated by Jean Bellissard.
This work was supported by the research network SFB TR 12—‘Symmetries and Universality in Mesoscopic Systems’ of the German Research Foundation (DFG).
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Küttler, H., Otte, P. & Spitzer, W. Anderson’s Orthogonality Catastrophe for One-Dimensional Systems. Ann. Henri Poincaré 15, 1655–1696 (2014). https://doi.org/10.1007/s00023-013-0287-z
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DOI: https://doi.org/10.1007/s00023-013-0287-z