Precise ground state of multi-orbital Mott systems via the variational discrete action theory

TL;DR

This paper introduces an advanced variational method called VDAT for accurately determining the ground state of multi-orbital Hubbard models in the infinite-dimensional limit, overcoming previous computational limitations.

Contribution

The authors develop a decoupled minimization algorithm for VDAT applied to multi-orbital Hubbard models, demonstrating its effectiveness at various levels of the variational ansatz and capturing complex physical phenomena.

Findings

VDAT at  recovers the multi-orbital Gutzwiller approximation.

At , VDAT accurately captures the interplay of U, J, and  in the two-orbital model.

Interactions influence orbital polarization effects even at small U/t.

Abstract

Determining the ground state of multi-orbital Hubbard models is critical for understanding strongly correlated electron materials, yet existing methods struggle to simultaneously reach zero temperature and infinite system size. The \textit{de facto} standard is to approximate a finite dimension multi-orbital Hubbard model with a $d=\infty$ version, which can then be formally solved via the dynamical mean-field theory (DMFT), though the DMFT solution is limited by the state of unbiased impurity solvers for zero temperature and multiple orbitals. The recently developed variational discrete action theory (VDAT) offers a new approach to solve the $d=\infty$ Hubbard model, with a variational ansatz that is controlled by an integer $\mathcal{N}$, and monotonically approaches the exact solution at an increasing computational cost. Here we propose a decoupled minimization algorithm to implement VDAT for the multi-orbital Hubbard model in $d=\infty$ and study $\mathcal{N}=2-4$ . At $\mathcal{N}=2$, VDAT rigorously recovers the multi-orbital Gutzwiller approximation, reproducing known results. At $\mathcal{N}=3$, VDAT precisely captures the competition between the Hubbard $U$, Hund $J$, and crystal field $\Delta$ in the two orbital Hubbard model over all parameter space, with a negligible computational cost. For sufficiently large $U/t$ and $J/U$, we show that $\Delta$ drives a first-order transition within the Mott insulating regime. In the large orbital polarization limit with finite $J/U$, we find that interactions have a nontrivial effect even for small $U/t$. VDAT will have far ranging implications for understanding multi-orbital model Hamiltonians and strongly correlated electron materials.