Scale Invariance in the Lowest Landau Level

Johannes Hofmann; Wilhelm Zwerger

doi:10.5802/crphys.137

Scale Invariance in the Lowest Landau Level
[Invariance d’échelle dans le niveau de Landau fondamental]

Johannes Hofmann ¹ ; Wilhelm Zwerger ²

¹ Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
² Technische Universität München, Physik Department, James-Franck-Strasse, 85748 Garching, Germany

Comptes Rendus. Physique, Online first (2023), pp. 1-18.

Résumé
Abstract

Nous montrons que l’ensemble discret des amplitudes de paires $A_{m}$ introduit par Haldane est une généralisation résolue en moment cinétique du coefficient de contact à deux corps de Tan, qui paramétrise les corrélations universelles à courte distance dans les gaz quantiques atomiques. Les amplitudes de paires fournissent une description complète des états invariants par translation et par rotation dans le niveau de Landau fondamental (LLL), qu’ils soient compressibles ou incompressibles. Au premier ordre non nul au-delà de la limite de haute température sans interaction, elles sont déterminées analytiquement en fonction des paramètres $V_{m}$ du pseudopotentiel de Haldane, ce qui fournit une description qualitative du passage vers des états fondamentaux incompressibles pour différents taux de remplissage. De plus, nous montrons que pour les interactions de contact $g_{2} δ^{(2)} (x)$ , qui sont invariantes d’échelle au niveau classique, la non-commutation des coordonnées du centre de giration donne naissance à une anomalie quantique dans le commutateur $i [{\hat{H}}_{LLL}, {\hat{D}}_{R}] = (2 + ℓ \partial_{ℓ}) {\hat{H}}_{LLL}$ de l’hamiltonien avec le générateur des dilatations ${\hat{D}}_{R}$ dans le LLL, qui remplace l’anomalie de Weyl sur la trace en l’absence de champ magnétique. La brisure de l’invariance d’échelle induite par l’interaction conduit à un déplacement de fréquence du mode de respiration dans un piège harmonique, qui reflète des transitions entre différents niveaux de Landau et dont nous estimons la valeur en termes de la constante de couplage sans dimension pertinente ${\tilde{g}}_{2}$ .

We show that the discrete set of pair amplitudes $A_{m}$ introduced by Haldane are an angular-momentum resolved generalization of the Tan two-body contact, which parametrizes universal short-range correlations in atomic quantum gases. The pair amplitudes provide a complete description of translation-invariant and rotation-invariant states in the lowest Landau level (LLL), both compressible and incompressible. To leading nontrivial order beyond the non-interacting high-temperature limit, they are determined analytically in terms of the Haldane pseudopotential parameters $V_{m}$ , which provides a qualitative description of the crossover towards incompressible ground states for different filling factors. Moreover, we show that for contact interactions $\sim g_{2} δ^{(2)} (x)$ , which are scale invariant at the classical level, the non-commutativity of the guiding center coordinates gives rise to a quantum anomaly in the commutator $i [{\hat{H}}_{LLL}, {\hat{D}}_{R}] = (2 + ℓ \partial_{ℓ}) {\hat{H}}_{LLL}$ with the dilatation operator ${\hat{D}}_{R}$ in the LLL, which replaces the trace anomaly in the absence of a magnetic field. The interaction-induced breaking of scale invariance gives rise to a finite frequency shift of the breathing mode in a harmonic trap, which describes transitions between different Landau levels, the strength of which is estimated in terms of the relevant dimensionless coupling constant ${\tilde{g}}_{2}$ .

Reçu le : 2022-10-16
Révisé le : 2022-12-13
Accepté le : 2022-12-15
Première publication : 2023-06-13

DOI : 10.5802/crphys.137

Keywords: Bose–Einstein condensates under rapid rotation, Lowest Landau Level, Virial Expansion, Scale invariance and quantum scale anomaly, Universal/exact (Tan) relations
Mot clés : Condensats de Bose–Einstien en rotation rapide, niveau de Landau fondamental, développement du viriel, invariance d’échelle et anomalie d’échelle quantique, relations universelles/exactes de Tan

Affiliations des auteurs :

Johannes Hofmann ¹ ; Wilhelm Zwerger ²

¹ Department of Physics, Gothenburg University, 41296 Gothenburg, Sweden
² Technische Universität München, Physik Department, James-Franck-Strasse, 85748 Garching, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Johannes Hofmann and Wilhelm Zwerger},
     title = {Scale {Invariance} in the {Lowest} {Landau} {Level}},
     journal = {Comptes Rendus. Physique},
     publisher = {Acad\'emie des sciences, Paris},
     year = {2023},
     doi = {10.5802/crphys.137},
     language = {en},
     note = {Online first},
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Johannes Hofmann; Wilhelm Zwerger. Scale Invariance in the Lowest Landau Level. Comptes Rendus. Physique, Online first (2023), pp. 1-18. doi : 10.5802/crphys.137.

Bibliographie
Cité par

[1] D. C. Tsui; H. L. Störmer; A. C. Gossard Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Lett., Volume 48 (1982) no. 22, pp. 1559-1562 | DOI

[2] R. B. Laughlin Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Lett., Volume 50 (1983) no. 18, pp. 1395-1398 | DOI

[3] J. Fröhlich; T. Kerler Universality in quantum Hall systems, Nucl. Phys., B, Volume 354 (1991) no. 2, pp. 369-417 | DOI | MR

[4] X. G. Wen; A. Zee Classification of Abelian quantum Hall states and matrix formulation of topological fluids, Phys. Rev. B, Volume 46 (1992) no. 4, pp. 2290-2301 | DOI

[5] Carlos Hoyos; Dam Thanh Son Hall Viscosity and Electromagnetic Response, Phys. Rev. Lett., Volume 108 (2012) no. 6, 066805, 5 pages | DOI

[6] Dam Thanh Son Newton-Cartan Geometry and the Quantum Hall Effect (2013) (https://arxiv.org/abs/1306.0638)

[7] Dam Thanh Son Is the Composite Fermion a Dirac Particle?, Phys. Rev. X, Volume 5 (2015) no. 3, 031027, 14 pages | DOI

[8] Yvan Castin; Zoran Hadzibabic; Sabine Stock; Jean Dalibard; Sandro Stringari Quantized Vortices in the Ideal Bose Gas: A Physical Realization of Random Polynomials, Phys. Rev. Lett., Volume 96 (2006) no. 4, 040405, 4 pages | DOI

[9] Immanuel Bloch; Jean Dalibard; Wilhelm Zwerger Many-body physics with ultracold gases, Rev. Mod. Phys., Volume 80 (2008) no. 3, pp. 885-964 | DOI

[10] N. R. Cooper Rapidly rotating atomic gases, Adv. Phys., Volume 57 (2008) no. 6, pp. 539-616 | DOI

[11] N. R. Cooper; E. H. Rezayi; S. H. Simon Vortex Lattices in Rotating Atomic Bose Gases with Dipolar Interactions, Phys. Rev. Lett., Volume 95 (2005) no. 20, 200402, 4 pages | DOI

[12] N. R. Cooper; N. K. Wilkin; J. M. F. Gunn Quantum Phases of Vortices in Rotating Bose-Einstein Condensates, Phys. Rev. Lett., Volume 87 (2001) no. 12, 120405, 4 pages | DOI

[13] N. R. Cooper Optical Flux Lattices for Ultracold Atomic Gases, Phys. Rev. Lett., Volume 106 (2011) no. 17, 175301, 4 pages | DOI

[14] N. R. Cooper; Jean Dalibard Optical flux lattices for two-photon dressed states, Eur. Phys. Lett., Volume 95 (2011) no. 6, p. 66004 | DOI

[15] N. Goldman; Jean Dalibard Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields, Phys. Rev. X, Volume 4 (2014) no. 3, 031027, 29 pages | DOI

[16] André Eckardt Atomic quantum gases in periodically driven optical lattices, Rev. Mod. Phys., Volume 89 (2017) no. 1, 011004, 30 pages | DOI | MR

[17] Julian Léonard; Sooshin Kim; Joyce Kwan; Perrin Segura; Fabian Grusdt; Cécile Repellin; Nathan Goldman; Markus Greiner Realization of a fractional quantum Hall state with ultracold atoms (2022) (https://arxiv.org/abs/2210.10919)

[18] Logan W. Clark; Nathan Schine; Claire Baum; Ningyuan Jia; Jonathan Simon Observation of Laughlin states made of light, Nature, Volume 582 (2020), pp. 41-45 | DOI

[19] Richard F. Fletcher; Airlia Shaffer; Cedric C. Wilson; Parth B. Patel; Zhenjie Yan; Valentin Crépel; Biswaroop Mukherjee; Martin W. Zwierlein Geometric squeezing into the lowest Landau level, Science, Volume 372 (2021) no. 6548, pp. 1318-1322 | DOI | MR | Zbl

[20] Biswaroop Mukherjee; Airlia Shaffer; Parth B. Patel; Zhenjie Yan; Cedric C. Wilson; Valentin Crépel; Richard F. Fletcher; Martin Zwierlein Crystallization of bosonic quantum Hall states in a rotating quantum gas, Nature, Volume 601 (2022) no. 7891, pp. 58-62 | DOI

[21] F. Duncan; F. D. M. Haldane The Hierarchy of Fractional States and Numerical Studies, The Quantum Hall Effect (R. E. Prange; S. M. Girvin, eds.) (Graduate Texts in Contemporary Physics), Springer, 1987, pp. 303-352 | DOI

[22] F. D. M. Haldane Fractional Quantization of the Hall Effect: A Hierarchy of Incompressible Quantum Fluid States, Phys. Rev. Lett., Volume 51 (1983) no. 7, pp. 605-608 | DOI | MR

[23] Shina Tan Energetics of a strongly correlated Fermi gas, Ann. Phys., Volume 323 (2008) no. 12, pp. 2952-2970 | DOI | MR | Zbl

[24] Shina Tan Large momentum part of a strongly correlated Fermi gas, Ann. Phys., Volume 323 (2008) no. 12, pp. 2971-2986 | DOI | MR | Zbl

[25] Shina Tan Generalized virial theorem and pressure relation for a strongly correlated Fermi gas, Ann. Phys., Volume 323 (2008) no. 12, pp. 2987-2990 | DOI | MR | Zbl

[26] Shizhong Zhang; Anthony J. Leggett Universal properties of the ultracold Fermi gas, Phys. Rev. A, Volume 79 (2009) no. 2, 02360, 12 pages | DOI

[27] Eric Braaten Universal Relations for Fermions with Large Scattering Length, The BCS–BEC Crossover and the Unitary Fermi Gas (W. Zwerger, ed.) (Lecture Notes in Physics), Volume 836, Springer, 2012, pp. 193-231 | DOI

[28] S. M. Girvin Anomalous quantum Hall effect and two-dimensional classical plasmas: Analytic approximations for correlation functions and ground-state energies, Phys. Rev. B, Volume 30 (1984) no. 2, pp. 558-560 | DOI

[29] F. D. M. Haldane Geometrical Description of the Fractional Quantum Hall Effect, Phys. Rev. Lett., Volume 107 (2011) no. 11, 116801, 5 pages | DOI

[30] S. M. Girvin; A. H. MacDonald; P. M. Platzman Magneto-roton theory of collective excitations in the fractional quantum Hall effect, Phys. Rev. B, Volume 33 (1986) no. 4, pp. 2481-2494 | DOI

[31] Dung Xuan Nguyen; Tankut Can; Andrey Gromov Particle-Hole Duality in the Lowest Landau Level, Phys. Rev. Lett., Volume 118 (2017) no. 20, 206602, 6 pages | DOI

[32] Martin R. Zirnbauer Particle–hole symmetries in condensed matter, J. Math. Phys., Volume 62 (2021) no. 2, 021101 | DOI | MR | Zbl

[33] Bhilahari Jeevanesan; Sergej Moroz Thermodynamics of two-dimensional bosons in the lowest Landau level, Phys. Rev. Res., Volume 2 (2020) no. 3, 033323 | DOI

[34] S. A. Trugman; S. Kivelson Exact results for the fractional quantum Hall effect with general interactions, Phys. Rev. B, Volume 31 (1985) no. 8, p. 5280 | DOI

[35] Maxim Olshanii; Hélène Perrin; Vincent Lorent Example of a Quantum Anomaly in the Physics of Ultracold Gases, Phys. Rev. Lett., Volume 105 (2010) no. 9, 095302, 4 pages | DOI

[36] Johannes Hofmann Quantum Anomaly, Universal Relations, and Breathing Mode of a Two-Dimensional Fermi Gas, Phys. Rev. Lett., Volume 108 (2012) no. 18, 185303, 5 pages | DOI

[37] L. P. Pitaevskii; A. Rosch Breathing modes and hidden symmetry of trapped atoms in two dimensions, Phys. Rev. A, Volume 55 (1997) no. 2, p. R853-R856 | DOI

[38] Yvan Castin Exact scaling transform for a unitary quantum gas in a time dependent harmonic potential, C. R. Physique, Volume 5 (2004) no. 3, pp. 407-410 | DOI

[39] Félix Werner; Yvan Castin Unitary gas in an isotropic harmonic trap: Symmetry properties and applications, Phys. Rev. A, Volume 74 (2006) no. 5, 053604, 10 pages | DOI

[40] Yusuke Nishida; Dam Thanh Son Nonrelativistic conformal field theories, Phys. Rev. D, Volume 76 (2007) no. 8, 086004, 14 pages | DOI | MR

[41] Wilhelm Zwerger Basic Concepts and some current Directions in Ultracold Gases (2021) (Lectures on many-body phenomena in ultracold gases, Collège de France, https://pro.college-de-france.fr/jean.dalibard/CdF/2021/Zwerger/notes_Lecture1.pdf)

[42] T. Peppler; P. Dyke; M. Zamorano; I. Herrera; S. Hoinka; C. J. Vale Quantum Anomaly and 2D-3D Crossover in Strongly Interacting Fermi Gases, Phys. Rev. Lett., Volume 121 (2018) no. 12, 120402, 5 pages | DOI

[43] M. Holten; L. Bayha; A. C. Klein; P. A. Murthy; P. M. Preiss; S. Jochim Anomalous Breaking of Scale Invariance in a Two-Dimensional Fermi Gas, Phys. Rev. Lett., Volume 121 (2018) no. 12, 120401, 6 pages | DOI

[44] Gentaro Watanabe Breathing mode of rapidly rotating Bose-Einstein condensates, Phys. Rev. A, Volume 73 (2006) no. 1, 013616, 8 pages | DOI

[45] Daniel S. Fisher; P. C. Hohenberg Dilute Bose gas in two dimensions, Phys. Rev. B, Volume 37 (1988) no. 10, pp. 4936-4943 | DOI

[46] Sabrine Stock; V. Bretin; F. Chevy; Jean Dalibard Shape oscillation of a rotating Bose-Einstein condensate, Europhysics Letters, Volume 65 (2004) no. 5, p. 594 | DOI

[47] Mauro Antezza; Marco Cozzini; Sandro Stringari Breathing modes of a fast rotating Fermi gas, Phys. Rev. A, Volume 75 (2007) no. 5, 053609, 5 pages | DOI

[48] Dung Xuan Nguyen; Dam Thanh Son; Chaolun Wu owest Landau Level Stress Tensor and Structure Factor of Trial Quantum Hall Wave Functions (2014) (https://arxiv.org/abs/1411.3316)

[49] Siavash Golkar; Dung Xuan Nguyen; Dam Thanh Son Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect, J. High Energy Phys., Volume 2016 (2016) no. 1, p. 21 | DOI

[50] Thomas Busch; Berthold-Georg Englert; Kazimierz Rzażewski; Martin Wilkens Two Cold Atoms in a Harmonic Trap, Found. Phys., Volume 28 (1998) no. 4, pp. 549-599 | DOI

[51] Viktor Bekassy; Johannes Hofmann Nonrelativistic Conformal Invariance in Mesoscopic Two-Dimensional Fermi Gases, Phys. Rev. Lett., Volume 128 (2022) no. 19, 193401, 7 pages | DOI | MR

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