Exact Diagonalization of $\mathrm{SU}(N)$ Fermi-Hubbard Models

TL;DR

This paper introduces a method for exact diagonalization of $ ext{SU}(N)$ Fermi-Hubbard models using Young tableaux, enabling analysis of complex quantum phases on small lattices.

Contribution

It develops a basis based on semistandard Young tableaux for efficient diagonalization of $ ext{SU}(N)$ models, and applies it to study phase robustness and emergent order.

Findings

Long-range color order appears at intermediate U for N=4 on triangular lattice.

Method allows analysis of models with up to 12 sites and N up to 6.

Identifies stability of $ ext{SU}(N)$ phases under varying interaction strengths.

Abstract

We show how to perform exact diagonalizations of $\mathrm{SU}(N)$ Fermi-Hubbard models on $L$-site clusters separately in each irreducible representation ({irrep}) of $\mathrm{SU}(N)$. Using the representation theory of the unitary group $\mathrm{U}(L)$, we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of {\it semistandard Young tableaux} (or, equivalently the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some $\mathrm{SU}(N)$ phases predicted in the Heisenberg limit upon decreasing the on-site interaction $U$ on various lattices of size $L \leq 12$ and for $2 \leq N \leq 6$. In particular, we show that a long-range color ordered phase emerges for intermediate $U$ for $N=4$ at filling $1/4$ on the triangular lattice.