Finite-size Energy of Non-interacting Fermi Gases

Gebert, Martin

doi:10.1007/s11040-015-9198-1

Cited by 6 publications

(15 citation statements)

References 17 publications

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“…( 1) is positive. Except for a restricted technical discussion within experts in the spectral theory of Hilbert spaces [1,2], as well as in the alternative derivations and the mathematical foundations of Anderson's orthogonality catastrophe (AOC) [3] discussed in [4][5][6], where Eq. ( 3) -in principle -might be recovered as a byproduct, we are not aware of any explicit statement of such a general result, or its direct derivation for general dimensions D, or even its connection with the condition (1) and the scaling (4).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Ground-state-energy universality of noninteracting fermionic systems

Silva,

Ostilli,

Presilla

2021

Preprint

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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 , but only on the imposed boundary conditions. Here, we review this almost unknown result from several perspectives and derive a proof for radially symmetric potentials valid in D = 2 and D = 3 dimensions, generalizing an existing approach to the case D = 1. We find that this universality property holds under a quite mild condition on V and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Ground-state-energy universality of noninteracting fermionic systems

Silva,

Ostilli,

Presilla

2021

Preprint

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“…We want to emphasize that the asymptotic terms may depend, in general, how we perform the thermodynamic limit, cf. [7,8] for the case with an (electric) potential.…”

Section: N Non-interacting Fermions and Their Ground-state Energymentioning

confidence: 99%

“…Up to the constant 2 π 2 , the negative exponent δ 2 of the exact asymptotics in Eq. (7) and the negative exponent sin 2 (δ) of the upper bound in Ineq. (8) satisfy "Anderson's rule".…”

Section: Anderson's Orthogonality Catastrophementioning

confidence: 99%

“…The papers [4,5,6] focus on AOC (caused by potentials and not magnetic fields) and prove upper bounds on the overlap which are all in agreement with Anderson's bounds. In [7] and [8], the finite-size energy and its relation to the coefficient γ in AOC is studied in a one-dimensional Fermi system. Yet, we do not know whether Anderson's prediction about the decay rate γ of the overlap is sharp.…”

Section: Introductionmentioning

confidence: 99%

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Anderson’s orthogonality catastrophe in one dimension induced by a magnetic field

Knörr¹,

Otte²,

Spitzer³

2015

J. Phys. A: Math. Theor.

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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.

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“…The main work of [GKM14] consists in deriving a lower bound of the form tr(I − A) γ 1 ln L for the Anderson integral with γ 1 given by (1.4). There are only few other mathematically rigorous works on Anderson's orthogonality catastrophe [KüOS14,G15a,KnOS15,G15b]. It is shown in [KüOS14] that (1.4) in fact provides the exact coefficient in the asymptotics tr(I − A) ∼ γ 1 ln L of the Anderson integral in the thermodynamic limit for one-dimensional systems.…”

Section: Introductionmentioning

confidence: 99%

The exponent in the orthogonality catastrophe for Fermi gases

Gebert¹,

Küttler²,

Müller³

et al. 2016

J. Spectr. Theory

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Dedicated to Hajo Leschke on the occasion of his 70 th birthday ABSTRACT. 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 finitevolume which differ by a compactly supported bounded potential, we prove a power-law upper bound on the ground-state overlap of the corresponding noninteracting N -particle systems. We interpret the decay exponent γ in terms of scattering theory and find γ = π −2 arcsin |T E /2| 2 HS , 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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Finite-size Energy of Non-interacting Fermi Gases

Cited by 6 publications

References 17 publications

Ground-state-energy universality of noninteracting fermionic systems

Ground-state-energy universality of noninteracting fermionic systems

Anderson’s orthogonality catastrophe in one dimension induced by a magnetic field

The exponent in the orthogonality catastrophe for Fermi gases

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