[Paper Review] Dirac and topological phonons with spin-orbital entangled orders

Yanqi Wang, Xiong-Jun Liu|arXiv (Cornell University)|Oct 5, 2017

Topological Materials and Phenomena3 citations

TL;DR

This paper proposes a 2D spin-orbit-coupled p-orbital bosonic optical lattice system where spin-orbital entangled orders emerge due to strong interactions. Using a self-consistent Bogoliubov approach, the authors predict Dirac and topological phonons with a bulk gap significantly enhanced—up to several times larger than the single-particle p-band gap—highlighting interaction-driven topological effects in ultracold quantum systems.

ABSTRACT

We propose to study novel quantum phases and excitations for a 2D spin-orbit (SO) coupled bosonic $p$-orbital optical lattice based on the recent experiments. The orbital and spin degrees of freedom with SO coupling compete and bring about nontrivial interacting quantum effects. We develop a self-consistent method for bosons and predict a spin-orbital entangled order for the ground phase, in sharp contrast to spinless high-orbital systems. Furthermore, we investigate the Bogoliubov excitations, showing that the Dirac and topological phonons are obtained corresponding to the predicted different spin-orbital orders. In particular, the topological phonons exhibit a bulk gap which can be several times larger than the single-particle gap of $p$-bands, reflecting the enhancement of topological effect by interaction. Our results highlight the rich physics predicted in SO coupled high-orbital systems and shall attract experimental efforts in the future.

Motivation & Objective

• To explore novel quantum phases in a 2D spin-orbit-coupled p-orbital optical lattice using ultracold bosons.
• To investigate how competing spin and orbital degrees of freedom under spin-orbit coupling lead to new many-body effects.
• To identify and characterize topological phonon excitations arising from spin-orbital entangled orders.
• To demonstrate that interaction can significantly enhance the topological gap in phonon spectra beyond single-particle values.

Proposed method

• Formulate a tight-binding Hamiltonian including spin-conserved hopping, Zeeman splitting, and spin-orbit coupling via Raman-induced tunneling.
• Apply a self-consistent mean-field approach to solve for the ground state with spin-orbital entangled order parameters.
• Derive the Bogoliubov Hamiltonian for small-amplitude excitations around the Bose-Einstein condensate (BEC) ground state.
• Calculate the phonon excitation spectrum using the Bogoliubov-de Gennes formalism, including interaction corrections to the effective Zeeman term.
• Estimate the topological gap in the phonon spectrum by analyzing the effective mass term modified by spin-orbital order parameters.
• Use imaginary time evolution to numerically determine the chemical potential and ground state energy in the BEC phase.

Experimental results

Research questions

• RQ1 Can spin-orbital entangled orders emerge in a 2D p-orbital bosonic system with spin-orbit coupling?
• RQ2 What types of collective excitations (phonons) arise in the presence of such spin-orbital order?
• RQ3 How does many-body interaction affect the topological gap in the phonon spectrum compared to the single-particle band gap?
• RQ4 Can the system host Dirac and topological phonons simultaneously, and under what conditions?
• RQ5 What is the role of the relative phase between spin and orbital components in determining the topological character of the phonons?

Key findings

• A novel spin-orbital entangled order is predicted in the ground state of the p-orbital bosonic system with spin-orbit coupling, distinct from spinless high-orbital systems.
• The system supports Dirac phonons when the spin-orbital order is such that the effective Zeeman term vanishes, preserving gapless Dirac cones.
• Topological phonons emerge when the spin-orbital order induces a non-zero effective Zeeman term, leading to a bulk gap in the phonon spectrum.
• The topological phonon gap is estimated to be several times larger than the single-particle p-band gap, indicating strong enhancement by many-body interactions.
• The effective gap is modified by the interaction-induced term $ M_f( heta, heta) = g(|\phi_\uparrow|^2 - |\phi_\downarrow|^2 + \cos^2\varphi|\phi_\uparrow|^2 - \cos^2\theta|\phi_\downarrow|^2) $, which can either increase or decrease the gap depending on the order parameter signs.
• The BEC ground state is a superposition of $ p_x + e^{i\varphi}p_y $ and $ p_x + e^{i\theta}p_y $ states with spin components, with the phase $ \xi = \beta $ determining the topological character.