Degenerate topological line surface phonons in quasi-1D double helix crystal SnIP

Abstract

<jats:title>Abstract</jats:title><jats:p>Degenerate points/lines in the band structures of crystals have become a staple of the growing number of topological materials. The bulk-boundary correspondence provides a relation between bulk topology and surface states. While line degeneracies of bulk excitations have been extensively characterised, line degeneracies of surface states are not well understood. We show that SnIP, a quasi-one-dimensional van der Waals material with a double helix crystal structure, exhibits topological nodal rings/lines in both the bulk phonon modes and their corresponding surface states. Using a combination of first-principles calculations, symmetry-based indicator theories and Zak phase analysis, we find that two neighbouring bulk nodal rings form doubly degenerate lines in their drumhead-like surface states, which are protected by the combination of time-reversal symmetry <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\mathcal{T}}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> and glide mirror symmetry <jats:inline-formula><jats:alternatives><jats:tex-math>$${\bar{M}}_{y}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>. Our results indicate that surface degeneracies can be generically protected by symmetries such as <jats:inline-formula><jats:alternatives><jats:tex-math>$${{{\mathcal{T}}}}{\bar{M}}_{y}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:msub> <mml:mrow> <mml:mover> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, and phonons provide an ideal platform to explore such degeneracies.</jats:p>