Exact Diagonalization of Heisenberg SU(N) models

TL;DR

This paper introduces a method for exact diagonalization of Heisenberg SU(N) models using standard Young tableaux, enabling analysis of larger N and revealing diverse quantum phases on a square lattice.

Contribution

It provides an explicit basis construction for SU(N) models that simplifies calculations and extends the feasible size of exact diagonalizations beyond previous limits.

Findings

Long-range color order observed for SU(5)

Spontaneous dimerization detected for SU(8)

Evidence of quantum liquid phase for SU(10)

Abstract

Building on advanced results on permutations, we show that it is possible to construct, for each irreducible representation of SU(N), an orthonormal basis labelled by the set of {\it standard Young tableaux} in which the matrix of the Heisenberg SU(N) model (the quantum permutation of N-color objects) takes an explicit and extremely simple form. Since the relative dimension of the full Hilbert space to that of the singlet space on $n$ sites increases very fast with N, this formulation allows to extend exact diagonalizations of finite clusters to much larger values of N than accessible so far. Using this method, we show that, on the square lattice, there is long-range color order for SU(5), spontaneous dimerization for SU(8), and evidence in favor of a quantum liquid for SU(10).