Abelian SU(N)1 chiral spin liquids on the square lattice

Abstract

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 $\mathrm{SU}(N)$-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 ${\mathrm{SU}(2)}_{1}$ and ${\mathrm{SU}(3)}_{1}$ chiral spin liquids (CSL) on similar models we search for topological $\mathrm{SU}({N)}_{1}$ 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 $\mathrm{SU}(N)$-symmetric version of iDMRG to compute entanglement spectra on (infinitely long) cylinders in all topological sectors provide unambiguous signatures of the $\mathrm{SU}({N)}_{1}$ 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.