Vortex lattices and the Bogoliubov-de Gennes equations

Chenn, Ilias; Sigal, Israel Michael

doi:10.1016/j.aim.2020.107546

Cited by 6 publications

(8 citation statements)

References 22 publications

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“…Open problem: Show that for superconductors of Type II and for n = 1, the vortex lattice solution is stable. (For discussion of Type I and II superconductors and the notion of stability in the context of the BdG, see [74]. These notions as well as the notion of self-duality still need to be elucidated in the BdG theory.…”

Section: Bogolubov-de Gennes Systemmentioning

confidence: 99%

“…Sigal ([74]). In particular, Theorem 6.2 was proven in [74]. An asymptotic behaviour of critical temperature in weak magnetic fields was established in [131,102].…”

mentioning

confidence: 95%

“…For the HFB system, one inserts the delta-function potentials and in the BdG case one sets a = 0. The full, time-dependent systems in the general form as they appear in this paper were written out and formally derived in [33] and [74,52], respectively.…”

mentioning

confidence: 99%

“…Clearly, the HFB and BdG systems generalize the Hartree and Hartree-Fock equations. For the relation between the Hartree and Hartree-Fock approximations, on one hand, and quasi-free (Wick) states, on the other, see [51] and Appendix E of [74].…”

mentioning

confidence: 99%

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Differential Equations of Quantum Mechanics

Sigal

2022

Preprint

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We review very briefly the main mathematical structures and results in some important areas of Quantum Mechanics involving PDEs and formulate open problems. Contents 1. Preface 1 Acknowledgments 2 2. Schrödinger equation 2 3. Including photons (Nonrelativistic QED) 4 4. Effective Equations 6 5. Hartree-Fock-Bogolubov system 8 6. Bogolubov-de Gennes system 9 7. Ginzburg-Landau equations 12 8. Summary 14 9. Remarks on literature 15 Appendix A. The NR QED Hamiltonian 17 Appendix B. Hartree-Fock-Bogolubov equations 18 Appendix C. Bogolubov-de Gennes Equations: Discussion 19 References 20 1 ONEPAS and MCQM stand for Online Northeast PDE and Analysis Seminar and Mathematical Challenges in Quantum Mechanics.

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Section: Bogolubov-de Gennes Systemmentioning

confidence: 99%

“…Sigal ([74]). In particular, Theorem 6.2 was proven in [74]. An asymptotic behaviour of critical temperature in weak magnetic fields was established in [131,102].…”

mentioning

confidence: 95%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Differential Equations of Quantum Mechanics

Sigal

2022

Preprint

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“…We refer to [28,16,31,32,22,4,19,13] for works that investigate the translation-invariant BCS functional with a local pair interaction. BCS theory in the presence of external fields has been studied in [33,5,20,12,6].…”

Section: Introductionmentioning

confidence: 99%

Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field

Deuchert¹,

Hainzl²,

Maier³

2021

Preprint

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Starting from the Bardeen-Cooper-Schrieffer (BCS) free energy functional, we derive the Ginzburg-Landau functional for the case of a weak homogeneous magnetic field. We also provide an asymptotic formula for the BCS critical temperature as a function of the magnetic field. This extends the previous works [17,18] of Frank, Hainzl, Seiringer and Solovej to the case of external magnetic fields with non-vanishing magnetic flux through the unit cell.

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The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

et al. 2022

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We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.

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Vortex lattices and the Bogoliubov-de Gennes equations

Cited by 6 publications

References 22 publications

Differential Equations of Quantum Mechanics

Differential Equations of Quantum Mechanics

Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field

The time-dependent Hartree–Fock–Bogoliubov equations for Bosons

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