In the physics of the fractional quantum Hall (FQH) effect, a zoo of Abelian topological phases can be obtained by varying the magnetic field. Aiming to reach the same phenomenology in spin like systems, we propose a family of
SU(N)-symmetric models in the fundamental representation, on the square lattice with short-range interactions restricted to triangular units, a natural generalization for arbitrary
N of an SU(3) model studied previously where time-reversal symmetry is broken explicitly. Guided by the recent discovery of
SU(2)1 and
SU(3)1 chiral spin liquids (CSL) on similar models we search for topological
SU(N)1 CSL in some range of the Hamiltonian parameters via a combination of complementary numerical methods such as exact diagonalizations (ED), infinite density matrix renormalization group (iDMRG) and infinite Projected Entangled Pair State (iPEPS). Extensive ED on small (periodic and open) clusters up to
N=10 and an innovative
SU(N)-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the
SU(N)1 character of the chiral liquids. An SU(4)-symmetric chiral PEPS, constructed in a manner similar to its SU(2) and SU(3) analogs, is shown to give a good variational ansatz of the
N=4 ground state, with chiral edge modes originating from the PEPS holographic bulk-edge correspondence. Finally, we discuss the possible observation of such Abelian CSL in ultracold atom setups where the possibility of varying
N provides a tuning parameter similar to the magnetic field in the physics of the FQH effect.
We study fractional quantum Hall states in the cylinder geometry with open boundaries. We focus on principal fermionic $\ensuremath{\nu}=1/3$ and bosonic $\ensuremath{\nu}=1/2$ fractions in the case of hard-core interactions. The gap behavior as a function of the cylinder radius is analyzed. By adding enough orbitals to allow for edge modes, we show that it is possible to measure the Luttinger parameter of the nonchiral liquid formed by the combination of the two counterpropagating edges when we add a small confining potential. Although we measure a Luttinger exponent consistent with the chiral Luttinger theory prediction for the full hard-core interaction, the exponent remains nontrivial in the Tao-Thouless limit as well as for simple truncated states that can be constructed on the cylinder. If the radius of the cylinder is taken to infinity, the problem becomes a Tonks-Girardeau one-dimensional interacting gas in Fermi and Bose cases. Finally, we show that the Tao-Thouless and truncated states have an edge electron propagator, which decays spatially with a Fermi-liquid exponent, even if the energy spectrum can still be described by a nontrivial Luttinger parameter.