Decomposing momentum scales in the Hubbard Model: From Hatsugai-Kohmoto to Aubry-Andr\'e

TL;DR

This paper introduces a momentum-space clustering scheme for the Hubbard model, generalizing Hatsugai-Kohmoto models, enabling efficient low-energy approximations especially in systems with dominant scattering channels.

Contribution

The authors develop a generalized clustering scheme that connects HK models to finite-site Hubbard models, improving low-energy modeling of complex lattice systems.

Findings

Clusters of two sites can accurately recover ground state energy in Aubry-Andre9-Hubbard model.

The scheme generalizes HK models and relates to twist-averaged boundary conditions.

Physically motivated momentum-space truncations enable accurate low-energy descriptions.

Abstract

The all-to-all momentum coupling of the Hubbard interaction makes interacting lattice models generically unsolvable. In many settings, however, from Peierls instabilities to Moir\'e superlattice physics, the low-energy behavior is dominated by scattering at a few characteristic wavevectors. We exploit this by constructing a momentum-space clustering scheme that retains only a chosen subset of interaction channels. Our scheme can be considered a generalization of twist-averaged boundary conditions. In proving this, we also prove that our scheme can be considered as a generalization of Hatsugai-Kohmoto (HK) models, and all versions of the HK model previously considered in the literature arise as special cases. This shows that the surprising phenomenological success of HK models arises from their correspondence to the finite-site Hubbard model. In particular, the recently introduced "Momentum-Mixing HK" model corresponds to a specific choice of clustering limit, which is equal to the original finite-site Hubbard model with twist-averaged boundary conditions. Our scheme becomes particularly powerful when a spatially varying potential selects the dominant momentum channels. We demonstrate this on the one-dimensional analogue of interacting moir\'e systems: the Aubry-Andr\'e-Hubbard model. We show that for sufficiently strong onsite potential, clusters as small as two sites can recover the ground state energy to below 1% error relative to DMRG benchmarks. This establishes that physically motivated momentum-space truncations can yield accurate low-energy descriptions at feasible computational cost, opening a path toward tractable interacting models of Moir\'e systems in two dimensions. Code for reproducing all numerical results is available at https://github.com/chainik1125/decomposing-hubbard.