Scale Invariance in the Lowest Landau Level
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-i...
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Académie des sciences
2023-06-01
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| Series: | Comptes Rendus. Physique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.137/ |
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| author | Hofmann, Johannes Zwerger, Wilhelm |
| author_facet | Hofmann, Johannes Zwerger, Wilhelm |
| author_sort | Hofmann, Johannes |
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| description | 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 $\sim g_2 \delta ^{(2)}(\mathbf{x})$, 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 $i [\hat{H}_{\mathrm{LLL}}, \hat{D}_R] = (2 + \ell \partial _\ell ) \hat{H}_{\mathrm{LLL}}$ 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 $\tilde{g}_2$. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2023-06-01 |
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| spelling | doaj-art-48e6afaed0e2420aa2f2e33e554387a72025-02-07T13:53:11ZengAcadémie des sciencesComptes Rendus. Physique1878-15352023-06-0124S3698610.5802/crphys.13710.5802/crphys.137Scale Invariance in the Lowest Landau LevelHofmann, Johannes0Zwerger, Wilhelm1Department of Physics, Gothenburg University, 41296 Gothenburg, SwedenTechnische Universität München, Physik Department, James-Franck-Strasse, 85748 Garching, GermanyWe 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 $\sim g_2 \delta ^{(2)}(\mathbf{x})$, 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 $i [\hat{H}_{\mathrm{LLL}}, \hat{D}_R] = (2 + \ell \partial _\ell ) \hat{H}_{\mathrm{LLL}}$ 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 $\tilde{g}_2$.https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.137/Bose–Einstein condensates under rapid rotationLowest Landau LevelVirial ExpansionScale invariance and quantum scale anomalyUniversal/exact (Tan) relations |
| spellingShingle | Hofmann, Johannes Zwerger, Wilhelm Scale Invariance in the Lowest Landau Level Comptes Rendus. Physique Bose–Einstein condensates under rapid rotation Lowest Landau Level Virial Expansion Scale invariance and quantum scale anomaly Universal/exact (Tan) relations |
| title | Scale Invariance in the Lowest Landau Level |
| title_full | Scale Invariance in the Lowest Landau Level |
| title_fullStr | Scale Invariance in the Lowest Landau Level |
| title_full_unstemmed | Scale Invariance in the Lowest Landau Level |
| title_short | Scale Invariance in the Lowest Landau Level |
| title_sort | scale invariance in the lowest landau level |
| topic | Bose–Einstein condensates under rapid rotation Lowest Landau Level Virial Expansion Scale invariance and quantum scale anomaly Universal/exact (Tan) relations |
| url | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.137/ |
| work_keys_str_mv | AT hofmannjohannes scaleinvarianceinthelowestlandaulevel AT zwergerwilhelm scaleinvarianceinthelowestlandaulevel |