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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-06-01
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| Series: | Comptes Rendus. Physique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.137/ |
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| Summary: | 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 $\tilde{g}_2$. |
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| ISSN: | 1878-1535 |