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arXiv:2104.05492 · Physical Review A

Ground-state-energy universality of noninteracting fermionic systems

Douglas F. C. A. Silva, Massimo Ostilli, Carlo Presilla A

Published Aug 11, 2021• 0 citations
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Abstract

When noninteracting fermions are confined in a $D$-dimensional region of volume $O({L}^{D})$ and subjected to a continuous (or piecewise-continuous) potential $V$ which decays sufficiently fast with distance, in the thermodynamic limit, the ground-state energy of the system does not depend on $V$. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in $D$ dimensions. We find that this universality property holds under a quite mild condition on $V$, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.

DOI
10.1103/physreva.104.023309

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We quantify the asymptotic vanishing of the ground-state overlap of two non-interacting Fermi gases in d -dimensional Euclidean space in the thermodynamic limit. Given two one-particle Schrödinger operators in finite-volume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding non-interacting N -Fermion systems. We interpret the decay exponent \gamma in terms of scattering theory and find \gamma = \pi^{-2}\|\mathrm {arcsin}|T_E/2|\|_{\mathrm {HS}}^2 , where T_E is the transition matrix at the Fermi energy E . This exponent reduces to the one predicted by Anderson [Phys. Rev. 164, 352–359 (1967)] for the exact asymptotics in the special case of a repulsive point-like perturbation.
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