Gauge choices in the Hamiltonian theory of fractionally filled Chern bands

In the Hamiltonian approach to Chern bands one constructs an algebraically exact mapping which expresses the electron density projected to the Chern band ρFCB as a sum of Girvin-MacDonald-Platzman density operators ρGMP. The ρGMP operators, which obey the magnetic translation algebra, are expressed in terms of auxiliary composite fermion (CF) variables, while preserving the algebra. This produces, in a natural way, a unique Hartree-Fock ground state for the CFs, which can be used as a springboard for various computations. In previous presentations we had not realized that this procedure, which works in any gauge, in principle, is greatly optimized by one family of gauges introduced by Y.-L. Wu, N. Regnault, and B. A. Bernevig [Phys. Rev. B 86, 085129 (2012)1098-xs012110.1103/PhysRevB.86.085129; Phys. Rev. Lett. 110, 106802 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.106802] in their exact diagonalization program for Laughlin-like fractions. Here we explain in detail why their gauge choices also enable us to obtain better variational energies in our Hamiltonian approach. We illustrate the ideas with some results on the Haldane model, comparing our results to exact diagonalizations.