Hourglass nodal loop phonons in two dimensions jointly supported by fourfold rotation and glide mirror symmetries

Hourglass nodal loops have garnered significant attention due to their unique band dispersion; yet their existence in two-dimensional (2D) solid materials has been scarcely studied, particularly in spinless phonon systems. While recent solutions have been proposed for spin systems, none have been proposed for spinless 2D phonon systems. In this paper, we perform a thorough search of all 80 layer groups (LGs) and identify only three LGs capable of hosting hourglass nodal loop phonons. Using symmetry analysis, we discover the minimum symmetry requirement for hourglass nodal loop phonons. In addition to a horizontal glide mirror and the time-reversal symmetry, which are necessary for spinful systems, the existence of hourglass nodal loops in 2D spinless systems requires a fourfold rotation symmetry, which differs from that in 2D spinful systems. With the presence of the fourfold rotation symmetry, 2D hourglass nodal loop phonons always appear with a quadratic Weyl point located at the center of this loop. Using first-principles calculations, we identify a realistic material that hosts the hourglass nodal loop to demonstrate our symmetry analysis. Furthermore, we demonstrate that the nodal loop can transform into a half-hourglass-like dispersion while still persisting when uniaxial strain is applied to break the fourfold rotation symmetry. Simultaneously, the quadratic nodal point splits into two linear Weyl points with opposite chirality. Our work highlights the differences between identifying hourglass nodal loops in spinless and spinful symmetry requirements and offers an effective approach to studying materials with hourglass nodal loops in spinless systems.