[Sur la stabilité du réseau d’Abrikosov dans le plus bas niveau de Landau]
We study the lowest Landau level equation set on simply and doubly-periodic domains (in other words, rectangles and strips with appropriate boundary conditions). To begin with, we study well-posedness and establish the existence of stationary solutions. Then we investigate the linear stability of the lattice solution and prove it is stable for the (hexagonal) Abrikosov lattice, but unstable for rectangular lattices.
Nous étudions l’équation du plus bas niveau de Landau sur des domaines simplement et doublement périodiques (en d’autres termes, sur des rectangles et des bandes avec des conditions aux limites appropriées). Tout d’abord, nous montrons que l’équation d’évolution est bien posée et établissons l’existence de solutions stationnaires. Nous étudions ensuite la stabilité linéaire de la solution sur un réseau et prouvons qu’elle est stable pour le réseau d’Abrikosov (hexagonal), mais instable pour les réseaux rectangulaires.
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Keywords: Nonlinear Schrödinger equation, lowest Landau level, stationary solutions, periodical conditions, theta functions
Mots-clés : Équation de Schrödinger, plus bas niveau de Landau, solutions stationnaires, conditions périodiques, fonctions theta
Pierre Germain 1 ; Valentin Schwinte 2 ; Laurent Thomann 2
CC-BY 4.0
@article{JEP_2025__12__585_0,
author = {Pierre Germain and Valentin Schwinte and Laurent Thomann},
title = {On the stability of the {Abrikosov} lattice in the lowest {Landau} level},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {585--640},
publisher = {\'Ecole polytechnique},
volume = {12},
year = {2025},
doi = {10.5802/jep.298},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.298/}
}
TY - JOUR AU - Pierre Germain AU - Valentin Schwinte AU - Laurent Thomann TI - On the stability of the Abrikosov lattice in the lowest Landau level JO - Journal de l’École polytechnique — Mathématiques PY - 2025 SP - 585 EP - 640 VL - 12 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.298/ DO - 10.5802/jep.298 LA - en ID - JEP_2025__12__585_0 ER -
%0 Journal Article %A Pierre Germain %A Valentin Schwinte %A Laurent Thomann %T On the stability of the Abrikosov lattice in the lowest Landau level %J Journal de l’École polytechnique — Mathématiques %D 2025 %P 585-640 %V 12 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.298/ %R 10.5802/jep.298 %G en %F JEP_2025__12__585_0
Pierre Germain; Valentin Schwinte; Laurent Thomann. On the stability of the Abrikosov lattice in the lowest Landau level. Journal de l’École polytechnique — Mathématiques, Tome 12 (2025), pp. 585-640. doi : 10.5802/jep.298. https://jep.centre-mersenne.org/articles/10.5802/jep.298/
[1] - “Observation of vortex lattices in Bose-Einstein condensates”, Science 292 (2001) no. 5516, p. 476-479 | DOI
[2] - “On the magnetic properties of superconductors of the second group”, Soviet Physics-JETP 5 (1957), p. 1174-1182
[3] - Vortices in Bose-Einstein condensates, Progr. in Nonlinear Differential Equations and their Applications, vol. 67, Birkhäuser Boston, Inc., Boston, MA, 2006 | DOI | MR
[4] - “Vortex lattices in rotating Bose-Einstein condensates”, SIAM J. Math. Anal. 38 (2006) no. 3, p. 874-893 | DOI | MR | Zbl
[5] - “Vortex patterns in a fast rotating Bose-Einstein condensate”, Phys. Rev. A 71 (2005) no. 2, article ID 023611, 11 pages | DOI
[6] - “Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates”, J. Funct. Anal. 241 (2006) no. 2, p. 661-702 | DOI | MR | Zbl
[7] - “Lowest Landau level approach in superconductivity for the Abrikosov lattice close to ”, Selecta Math. (N.S.) 13 (2007) no. 2, p. 183-202 | DOI | MR | Zbl
[8] - “Tkachenko waves in rapidly rotating Bose-Einstein condensates”, Phys. Rev. Lett. 92 (2004) no. 16, article ID 160405, 4 pages | DOI
[9] - “Tkachenko modes of vortex lattices in rapidly rotating Bose-Einstein condensates”, Phys. Rev. Lett. 91 (2003) no. 11, article ID 110402, 4 pages | DOI
[10] - “Exact lowest-Landau-level solutions for vortex precession in Bose-Einstein condensates”, Phys. Rev. A 96 (2017), article ID 053615, 6 pages | DOI
[11] - “Two infinite families of resonant solutions for the Gross-Pitaevskii equation”, Phys. Rev. E 98 (2018) no. 3, article ID 032222, 12 pages | DOI | MR
[12] - “Solvable cubic resonant systems”, Comm. Math. Phys. 369 (2019) no. 2, p. 433-456 | DOI | MR | Zbl
[13] - “Lowest-Landau-level vortex structure of a Bose-Einstein condensate rotating in a harmonic plus quartic trap”, Phys. Rev. A 77 (2008) no. 5, article ID 053615, 8 pages | DOI
[14] - Elliptic functions, Grundlehren Math. Wissen., vol. 281, Springer-Verlag, Berlin, 1985 | DOI | MR
[15] - “The transition to a giant vortex phase in a fast rotating Bose-Einstein condensate”, Comm. Math. Phys. 303 (2011) no. 2, p. 451-508 | DOI | MR | Zbl
[16] - “Time-periodic quantum states of weakly interacting bosons in a harmonic trap”, Phys. Rev. A 384 (2020) no. 36, article ID 126930, 11 pages | DOI | Zbl
[17] - “The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation”, J. Amer. Math. Soc. 29 (2016) no. 4, p. 915-982 | DOI | MR | Zbl
[18] - “Rotating trapped Bose-Einstein condensates”, Rev. Modern Phys. 81 (2009) no. 2, p. 647-691 | DOI
[19] - “Vortex states of rapidly rotating dilute Bose-Einstein condensates”, Phys. Rev. Lett. 90 (2003) no. 14, article ID 140402, 4 pages | DOI
[20] - “Averaging of nonlinear Schrödinger equations with strong magnetic confinement”, Commun. Math. Sci. 15 (2017) no. 7, p. 1933-1945 | DOI | MR | Zbl
[21] - “On the cubic lowest Landau level equation”, Arch. Rational Mech. Anal. 231 (2019) no. 2, p. 1073-1128 | DOI | MR | Zbl
[22] - “On the continuous resonant equation for NLS, II: Statistical study”, Anal. PDE 8 (2015) no. 7, p. 1733-1756 | DOI | MR | Zbl
[23] - “On the continuous resonant equation for NLS. I. Deterministic analysis”, J. Math. Pures Appl. (9) 105 (2016) no. 1, p. 131-163 | DOI | MR | Zbl
[24] - “On the high frequency limit of the LLL equation”, Quart. Appl. Math. 74 (2016) no. 4, p. 633-641 | DOI | MR | Zbl
[25] - “Formalism for the quantum Hall effect: Hilbert space of analytic functions”, Phys. Rev. B 29 (1984) no. 10, p. 5617-5625 | DOI | MR
[26] - Analysis. IV, Universitext, Springer, Cham, 2015 | MR
[27] - “Asymptotic behavior of the nonlinear Schrödinger equation with harmonic trapping”, Comm. Pure Appl. Math. 69 (2016) no. 9, p. 1727-1776 | DOI | MR | Zbl
[28] - “Bose-Einstein condensates with large number of vortices”, Phys. Rev. Lett. 87 (2001) no. 6, article ID 060403, 4 pages | DOI
[29] - “Vortex formation in a stirred Bose-Einstein condensate”, Phys. Rev. Lett. 84 (2000) no. 5, p. 806-809 | DOI
[30] - “Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement”, Discrete Contin. Dynam. Systems. Ser. A 36 (2016) no. 9, p. 5097-5118 | DOI | MR | Zbl
[31] - Tata lectures on theta. I, Progress in Math., vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983 | DOI | MR
[32] - “Thomas-Fermi profile of a fast rotating Bose-Einstein condensate”, Pure Appl. Anal. 4 (2022) no. 3, p. 535-569, Erratum-Addendum: Ibid. 7 (2025), no. 1, p. 243–249 | DOI | MR | Zbl
[33] - “À propos des fonctions thêta et des réseaux d’Abrikosov”, in Séminaire équations aux dériées partielles 2006-2007, École Polytechnique, Palaiseau, 2007, Exp. no. XII, 27 p. | Numdam | MR | Zbl
[34] - “Bose-Einstein condensates in the lowest Landau level: Hamiltonian dynamics”, Rev. Math. Phys. 19 (2007) no. 1, p. 101-130 | DOI | MR | Zbl
[35] - Semi-classical limits of 2D fermions under high magnetic fields, Ph. D. Thesis, UMPA-ENSL, 2023, tel-04320293
[36] - “Multiple Landau level filling for a mean field limit of 2D fermions”, J. Math. Phys. 65 (2024) no. 2, article ID 021902, 46 pages | DOI | MR | Zbl
[37] - “Annular Bose-Einstein condensates in the lowest Landau level”, Appl. Math. Res. Express. AMRX (2011) no. 1, p. 95-121 | DOI | MR | Zbl
[38] - “Vortex rings in fast rotating Bose-Einstein condensates”, Arch. Rational Mech. Anal. 203 (2012) no. 1, p. 69-135 | DOI | MR | Zbl
[39] - “Quantum Hall states of bosons in rotating anharmonic traps”, Phys. Rev. A 87 (2013) no. 2, article ID 023618, 9 pages | DOI
[40] - “Rapidly rotating Bose-Einstein condensates in and near the lowest Landau level”, Phys. Rev. Lett. 92 (2004) no. 4, article ID 040404, 4 pages | DOI
[41] - “An optimal minimization problem in the lowest Landau level and related questions”, Comm. Math. Phys. 405 (2024) no. 4, article ID 98, 36 pages | DOI | MR | Zbl
[42] - “Growth of Sobolev norms for coupled lowest Landau level equations”, Pure Appl. Anal. 3 (2021) no. 1, p. 189-222 | DOI | MR | Zbl
[43] - “Abrikosov vortex lattices at weak magnetic fields”, J. Funct. Anal. 263 (2012) no. 3, p. 675-702 | DOI | MR | Zbl
[44] - “Stability of Abrikosov lattices under gauge-periodic perturbations”, Nonlinearity 25 (2012) no. 4, p. 1187-1210 | DOI | MR | Zbl
[45] - “On stability of Abrikosov vortex lattices”, Adv. Math. 326 (2018), p. 108-199 | DOI | MR | Zbl
[46] - “Ground state and Tkachenko modes of a rapidly rotating Bose-Einstein condensate in the lowest-Landau-level state”, Phys. Rev. A 72 (2005) no. 2, article ID 021606, 4 pages | DOI
[47] - Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Series, vol. 43, Princeton University Press, Princeton, NJ, 1993 | MR
[48] - Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Series, vol. 32, Princeton University Press, Princeton, NJ, 1971 | MR
[49] - “On multi-solitons for coupled lowest Landau level equations”, Discrete Contin. Dynam. Systems. Ser. A 42 (2022) no. 10, p. 4937-4964 | DOI | MR | Zbl
[50] - “On vortex lattices”, Sov. Phys. JETP 22 (1996) no. 6, p. 1282-1286
[51] - “Landau levels and the Thomas-Fermi structure of rapidly rotating Bose-Einstein condensates”, Phys. Rev. Lett. 93 (2004) no. 19, article ID 190401, 4 pages | DOI
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