When is a ``wavefunction'' not a wavefunction?: a quantum-geometric reinterpretation of the Laughlin state

The Laughlin state is the fundamental model for the description of fractional quantum Hall (FQH) fluids and was presented as a ``lowest Landau-level (LLL) Schr\"odinger wavefunction'', i.e., of the form $f(z_1,\ldots ,z_N)\exp -\sum_i z_i^*z_i/2$, where $z_i$ = $(x_i + iy_i)/\surd 2\ell_B$, and $|z_i-z_0|^2$ = constant is the shape of a Landau orbit. Its characterization as a LLL wavefunction was generally accepted without question, and leads to ``explanations'' of its success in terms of judicious placement of its zeroes. However, the Laughlin state also occurs in the n=2 LL, and now has been found in Chern-insulator lattice systems. Numerical studies confirm that (without direct reference to which LL is partially-occupied) its success can be explained solely in terms of the short-range repulsion between the non-commuting guiding centers of Landau orbits. These (as a ``quantum geometry'') do not by themselves have a Schr\"odinger (as opposed to Heisenberg) description. A reexamination shows that the variable ``z'' describes the shape of an emergent geometry of the FQH fluid derived from the Coulomb interaction, not the Landau-orbit shape, and that the holomorphic function is a coherent state representation of a Heisenberg state, not a Schr\"odinger wavefunction.