Scale Invariance in the Lowest Landau Level

Abstract

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 ${\tilde{g}}_2$.

18 pages, 3 figures

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Cited by in corpus (2)

• Scale and conformal invariance in rotating interacting few-fermion systems
• Two-body contact of a Bose gas near the superfluid--Mott-insulator transition