Exact diagonalization of Heisenberg $SU(N)$ chains in the fully symmetric and antisymmetric representations

TL;DR

This paper introduces an exact diagonalization method for $SU(N)$ Heisenberg chains using Young tableaux, enabling analysis of larger systems and revealing critical behaviors in various representations and sizes.

Contribution

The authors develop a symmetry-based exact diagonalization technique for $SU(N)$ chains in symmetric and antisymmetric representations, extending system size reach and analyzing criticality.

Findings

Most systems are gapless at accessible scales.

Results are consistent with $SU(N)$ level $k$ WZW universality class.

A crossover between universality classes is suggested.

Abstract

Motivated by recent experimental progress in the context of ultra-cold multi-color fermionic atoms in optical lattices, we have developed a method to exactly diagonalize the Heisenberg $SU(N)$ Hamiltonian with several particles per site living in a fully symmetric or antisymmetric representation of $SU(N)$. The method, based on the use of standard Young tableaux, takes advantage of the full $SU(N)$ symmetry, allowing one to work directly in each irreducible representations of the global $SU(N)$ group. Since the $SU(N)$ singlet sector is often much smaller than the full Hilbert space, this enables one to reach much larger system sizes than with conventional exact diagonalizations. The method is applied to the study of Heisenberg chains in the symmetric representation with two and three particles per site up to $N=10$ and up to 20 sites. For the length scales accessible to this approach, all systems except the Haldane chain ($SU(2)$ with two particles per site) appear to be gapless, and the central charge and scaling dimensions extracted from the results are consistent with a critical behaviour in the $SU(N)$ level $k$ Wess-Zumino-Witten universality class, where $k$ is the number of particles per site. These results point to the existence of a cross-over between this universality class and the asymptotic low-energy behavior with a gapped spectrum or a critical behavior in the $SU(N)$ level $1$ WZW universality class.