Abstract
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 {{{mathcal{T}}}} and glide mirror symmetry {bar{M}}_{y}. Our results indicate that surface degeneracies can be generically protected by symmetries such as {{{mathcal{T}}}}{bar{M}}_{y}, and phonons provide an ideal platform to explore such degeneracies.
Highlights
Degeneracies in the bulk energy bands of crystals were intensively studied in the early days of band theory[1]
When the band crossings are one-dimensional in momentum space, they can form nodal lines, nodal rings or nodal chains, depending on their shape[23,24,25,26,27,28,29,30,31], and these line crossings are protected by symmetries such as mirror or PT32,33
Our results suggest that similar surface degenerate lines/points will be found in other materials when there are extra symmetries to protect these surface degeneracies, and we believe that phonons can be a primary platform to study these degeneracies because of the presence of time-reversal symmetry and the flexibility to study the degeneracies in the entire phonon spectrum
Summary
Degeneracies in the bulk energy bands of crystals were intensively studied in the early days of band theory[1]. When the band crossings are one-dimensional in momentum space, they can form nodal lines, nodal rings or nodal chains, depending on their shape[23,24,25,26,27,28,29,30,31], and these line crossings are protected by symmetries such as mirror or PT (where P is the spatial inversion symmetry and T is the time-reversal symmetry)[32,33] Recent advances, using both group-theoretical analysis and high-throughput calculations, have enabled a comprehensive understanding of bulk band degeneracies in both electrons[34,35,36] and phonons[22]. These topological degeneracies exhibit unique physical properties such as a ‘quantum anomaly’[37,38], enabling various applications from spintronics[39] to topological quantum computation[40]
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