Orbital order in a bosonic $p$-band triangular lattice

TL;DR

This paper investigates orbital order in a bosonic p-band triangular lattice, revealing two superfluid phases with distinct orbital orders, analyzing their excitations, and connecting these to the Mott insulator phase through an orbital exchange model.

Contribution

It provides a detailed analysis of orbital order evolution across the superfluid-Mott insulator transition in a p-band triangular lattice, including new insights into the phases and their excitations.

Findings

Two superfluid phases with distinct orbital orders identified.

Emergent Dirac points in the excitation spectrum.

In-plane ferro-orbital order likely evolves from the superfluid phase.

Abstract

We present a detailed study of the Bose-Hubbard model in a $p$-band triangular lattice by focusing on the evolution of orbital order across the superfluid-Mott insulator transition. Two distinct phases are found in the superfluid regime. One of these phases adiabatically connects the weak interacting limit. This phase is characterized by the intertwining of axial $p_\pm=p_x \pm ip_y$ and in-plane $p_\theta=\cos\theta p_x+\sin\theta p_y$ orbital orders, which break the time-reversal symmetry and lattice symmetries simultaneously. In addition, the calculated Bogoliubov excitation spectrum gaps the original Dirac points in the single-particle spectrum but exhibits emergent Dirac points. The other superfluid phase in close proximity to the Mott insulator with unit boson filling shows a detwined in-plane ferro-orbital order. Finally, an orbital exchange model is constructed for the Mott insulator phase. Its classical ground state has an emergent SO$(2)$ rotational symmetry in the in-plane orbital space and therefore enjoys an infinite degeneracy, which is ultimately lifted by the orbital fluctuation via the order by disorder mechanism. Our systematic analysis suggests that the in-plane ferro-orbital order in the Mott insulator phase agrees with and likely evolves from the latter superfluid phase.