Crystalline Splitting of $d$ Orbitals in Regular Optical Lattices

TL;DR

This paper demonstrates how high-order optical potentials in ultracold atom systems can split degenerate $d$ orbitals into specific multiplets, enabling simulation of orbital physics akin to solid-state materials.

Contribution

It reveals the splitting of $d$ orbitals in optical lattices due to high-order potential terms and derives effective models for orbital interactions, bridging ultracold atoms and solid-state physics.

Findings

$d$ orbitals split into singlet and doublet in triangular lattices.

Effective Heisenberg-Compass model describes orbital Mott insulators.

Different multiplet structures arise in square lattices due to symmetry.

Abstract

In solids, crystal field splitting refers to the lifting of atomic orbital degeneracy by the surrounding ions through the static electric field. Similarly, we show that the degenerated $d$ orbitals, which were derived in the harmonic oscillator approximation, are split into a low-lying $d_{x^2+y^2}$ singlet and a $d_{x^2-y^2/xy}$ doublet by the high-order Taylor polynomials of triangular optical potential. The low-energy effective theory of the orbital Mott insulator at $2/3$ filling is generically described by the Heisenberg-Compass model, where the antiferro-orbital exchange interactions of compass type depend on the bond orientation and are geometrically frustrated in the triangular lattice. While, for the square optical lattice, the degenerated $d$ orbitals are split into a different multiplet structure, i.e. a low-lying $d_{x^2\pm y^2}$ doublet and a $d_{xy}$ singlet, which has its physical origin in the $C_{4v}$ point group symmetry of square optical potential. Our results build a novel bridge between ultracold atom systems and solid-state systems for the investigation of $d$-orbital physics.