# Spin-valley density wave in moir\'e materials

## TL;DR

This paper introduces a two-orbital Hubbard model on a triangular lattice to study spin-valley density waves in moiré materials, capturing key features of graphene heterostructures and TMDs, and analyzes their stability and responses.

## Contribution

It develops a minimal Hubbard model incorporating valley-contrasting flux and derives a spin-orbital exchange Hamiltonian revealing a spin-valley density wave ground state.

## Key findings

- A semiclassical spin-valley density wave is stabilized in the strong-coupling regime.
- Small second-neighbor exchange interactions can stabilize the ordered state.
- The effects of Zeeman fields and thermal fluctuations on the order are characterized.

## Abstract

We introduce and study a minimum two-orbital Hubbard model on a triangular lattice, which captures the key features of both the trilayer ABC-stacked graphene-boron nitride heterostructure and twisted transition metal dichalcogenides in a broad parameter range. Our model comprises first- and second-nearest neighbor hoppings with valley-contrasting flux that accounts for trigonal warping in the band structure. For the strong-coupling regime with one electron per site, we derive a spin-orbital exchange Hamiltonian and find the semiclassical ground state to be a spin-valley density wave. We show that a relatively small second-neighbor exchange interaction is sufficient to stabilize the ordered state against quantum fluctuations. Effects of spin- and valley Zeeman fields as well as thermal fluctuations are also examined.