Precursor of the Laughlin state of hard-core bosons on a two-leg ladder

Abstract

We study hard core bosons on a two leg ladder lattice under the orbital effect of a uniform magnetic field. At densities which are incommensurate with flux, the ground state is a Meissner state, or a vortex state, depending on the strength of the flux. When the density is commensurate with the flux, analytical arguments predict the existence of a ground state of central charge $c = 1$, which displays signatures compatible with the expected Laughlin state at $\nu=1/2$. This differs from the coupled wire construction of the Laughlin state in that there exists a nonzero backscattering term in the edge Hamiltonian. We construct a phase diagram versus density and flux in order to delimit the region where this precursor to the Laughlin state is the ground state, by using a combination of bosonization and numerics based on the density matrix renormalization group (DMRG) and exact diagonalization. We obtain the phase diagram from local observables and central charge. We use bipartite charge fluctuations to deduce the Luttinger parameter for the edge Luttinger liquid corresponding to the Laughlin state. The properties studied with local observables are confirmed by the long distance behavior of correlation functions. Our findings are consistent with a calculation of the many body ground state transverse conductivity in a thin torus geometry for parameters corresponding to the Laughlin state. The model considered is simple enough such that the precursor to the Laughlin state could be realized in current ultracold atom, Josephson junction array, and quantum circuit experiments.