On the Atomic Orbital Magnetism: A Rigorous Derivation of the Larmor and Van Vleck Contributions

Savoie, Baptiste

doi:10.1007/s00023-014-0313-9

Cited by 3 publications

(5 citation statements)

References 38 publications

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“…The proof leans on an approximation of the finite-volume semigroup operator via a geometric perturbation theory. Although this method had been originally developed for the resolvent operators, see [9] and also [10,17], below we extend the method to the semigroup operators.…”

Section: Proof Of Proposition 24mentioning

confidence: 99%

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Gaussian decay for a difference of traces of the Schrödinger semigroup associated with the isotropic harmonic oscillator

Beau¹,

Savoie²

2014

Journal of Mathematical Analysis and Applications

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This paper deals with the derivation of a sharp estimate on the difference of traces of the one-parameter Schrödinger semigroup associated to the quantum isotropic harmonic oscillator. Denoting by H∞,κ the self-adjoint realization ine −tH L,κ , t > 0 has for L sufficiently large a Gaussian decay in L. Furthermore, the estimate that we derive is sharp in the two following senses: its behavior when t ↓ 0 is similar to the one given by Tr L 2 (R d ) e −tH∞,κ = (2 sinh( κ 2 t)) −d and the exponential decay in t arising from Tr L 2 (R d ) e −tH∞,κ when t ↑ ∞ is preserved. For illustrative purposes, we give a simple application within the framework of quantum statistical mechanics.MSC-2010 number: 35J10, 47D08, 81Q10, 81Q15, 82B10.

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Section: Proof Of Proposition 24mentioning

confidence: 99%

“…Its proof relies on a geometric perturbation theory applied to bounded operators on separable Hilbert spaces. Such a method has been originally developed to treat approximations of resolvent operators in [9], see also [10,17] for further applications. Our paper extends the method to treat approximations of semigroups.…”

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confidence: 99%

Gaussian decay for a difference of traces of the Schrödinger semigroup associated with the isotropic harmonic oscillator

Beau¹,

Savoie²

2014

Journal of Mathematical Analysis and Applications

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“…In 2012, Briet et al gave for the first time in [5] a rigorous justification of the Landau-Peierls approximation for the bulk zero-field orbital susceptibility of Bloch electron gases. Recently, the present author revisited the atomic orbital magnetism in [48] and gave a rigorous derivation of the diamagnetic Larmor contribution and the 'complete' orbital Van Vleck paramagnetic contribution in the tight-binding approximation.…”

Section: The Motivations Of the Papermentioning

confidence: 99%

“…Let us turn to the geometric perturbation theory, for further applications see [9,10,48]. For any h > 0, let Ω h := (− 1 2h , 1 2h ) 3 be the dilated unit cell centered at the origin of coordinates.…”

Section: Concluding Remarks and A Few Open Problemsmentioning

confidence: 99%

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A rigorous proof of the Bohr-van Leeuwen theorem in the semiclassical limit

Savoie

2014

Preprint

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The original formulation of the Bohr-van Leeuwen (BvL) theorem states that, in a uniform magnetic field and in thermal equilibrium, the magnetization of an electron gas in the classical Drude-Lorentz model vanishes identically. This stems from classical statistics which assign the canonical momenta all values ranging from −∞ to ∞ what makes the free energy density magnetic-field-independent. When considering a classical (Maxwell-Boltzmann) interacting electron gas, it is usually admitted that the BvL theorem holds upon condition that the potentials modeling the interactions are particle-velocities-independent and do not cause the system to rotate after turning on the magnetic field. From a rigorous viewpoint, when treating large macroscopic systems one expects the BvL theorem to hold provided the thermodynamic limit of the free energy density exists (and the equivalence of ensemble holds). This requires suitable assumptions on the many-body interactions potential and on the possible external potentials to prevent the system from collapsing or flying apart. Starting from quantum statistical mechanics, the purpose of this article is to give, within the linear-response theory, a proof of the BvL theorem in the semiclassical limit when considering a dilute electron gas in the canonical conditions subjected to a class of translational invariant external potentials.

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Spectral edge regularity of magnetic Hamiltonians

Cornean

Purice

2015

J. London Math. Soc.

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We analyse the spectral edge regularity of a large class of magnetic Hamiltonians when the perturbation is generated by a globally bounded magnetic field. We can prove Lipschitz regularity of spectral edges if the magnetic field perturbation is either constant or slowly variable. We also recover an older result by G. Nenciu who proved Lipschitz regularity up to a logarithmic factor for general globally bounded magnetic field perturbations.

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On the Atomic Orbital Magnetism: A Rigorous Derivation of the Larmor and Van Vleck Contributions

Cited by 3 publications

References 38 publications

Gaussian decay for a difference of traces of the Schrödinger semigroup associated with the isotropic harmonic oscillator

Gaussian decay for a difference of traces of the Schrödinger semigroup associated with the isotropic harmonic oscillator

A rigorous proof of the Bohr-van Leeuwen theorem in the semiclassical limit

Spectral edge regularity of magnetic Hamiltonians

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