Exact diagonalization of $\mathrm{SU}(N)$ Heisenberg and AKLT chains using the full $\mathrm{SU}(N)$ symmetry

Abstract

We present a method for the exact diagonalization of the $\mathrm{SU}(N)$ Heisenberg interaction Hamiltonian, using Young tableaux to work directly in each irreducible representation of the global $\mathrm{SU}(N)$ group. This generalized scheme is applicable to chains consisting of several particles per site, with any $\mathrm{SU}(N)$ symmetry at each site. Extending some of the key results of substitutional analysis, we demonstrate how basis states can be efficiently constructed for the relevant $\mathrm{SU}(N)$ subsector, which, especially with increasing values of $N$ or numbers of sites, has a much smaller dimension than the full Hilbert space. This allows us to analyze systems of larger sizes than can be handled by existing techniques. We apply this method to investigate the presence of edge states in $\mathrm{SU}(N)$ Heisenberg and AKLT Hamiltonians.

arXiv comment14 pages, 5 figures

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• Combined use of translational and spin-rotational invariance for spin systems
• Dimension-free entanglement detection in multipartite Werner states
• Haldane Gap of the Three-Box Symmetric $\mathrm{SU}(3)$ Chain
• "Haldane" phases with ultracold fermionic atoms in double-well optical lattices
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• Ground State Phase Diagram of $\text{SU}(3)$ $t$-$J$ Chain
• 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
• Edge states and universality class of the critical two-box symmetric SU(3) chain