Real-space entanglement spectra of lowest Landau level projected fractional quantum Hall states using Monte Carlo methods

Real-space entanglement spectrum (RSES) of a quantum Hall (QH) wave function gives a natural route to infer the nature of its edge excitations. Computation of RSES becomes expensive with increase in number of particles and number of included Landau levels (LL). RSES can be efficiently computed using Monte Carlo (MC) methods for trial states that can be written as products of determinants such as the composite fermion (CF) and parton states. This computational efficiency applies also to the RSES of lowest Landau level (LLL) projected CF and parton states, however, LLL projection to be used here requires approximations that generalize the Jain Kamilla (JK) projection. This work is a careful study of how this approximation should be done. We identify the approximation that is closest in spirit to JK projection, and perform tests of the approximations involved in the projection by comparing the MC results with the RSES obtained from computationally expensive but exact methods. We present the techniques and use them to calculate exact RSES of the exact LLL projected bosonic Jain 2/3 state in bipartition of systems of sizes up to N = 24 on the sphere. For the lowest few angular momentum sectors of the RSES, we present evidence to show that MC results closely match the exact spectra. We also compare other plausible projection schemes. We also calculate the exact RSES of the unprojected fermionic Jain 2/5 state obtained from exact diagonalization of the Trugman-Kivelson Hamiltonian in the two lowest LLs on the sphere. By comparing with the RSES of the unprojected 2/5 state from Monte Carlo methods, we show that the latter is practically exact.