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 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.

41 pages, 21 figures, references added, figure 2(j) modified, abstract and introduction revised

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• Efficient conversion from fermionic Gaussian states to matrix product states
• Symmetric projected entangled-pair states analysis of a phase transition in coupled spin-1/2 ladders
• Simulating chiral spin liquids with projected entangled-pair states
• A Promising Method for Strongly Correlated Electrons in Two Dimensions: Gutzwiller-Guided Density Matrix Renormalization Group
• Bridging conformal field theory and parton approaches to SU(n)_k chiral spin liquids
• Unit-density SU(3) Fermi-Hubbard Model with Spin Flavor Imbalance
• Entanglement Spectrum as a diagnostic of chirality of Topological Spin Liquids: Analysis of an $\mathrm{SU}(3)$ PEPS
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Detuning the Floquet anomalous chiral spin liquid