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Two Topologically Distinct Dirac-Line Semimetal Phases and Topological Phase Transitions in Rhombohedrally Stacked Honeycomb Lattices

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Abstract

Three-dimensional topological semimetals can support band crossings along one-dimensional curves in the momentum space (nodal lines or Dirac lines) protected by structural symmetries and topology. We consider rhombohedrally (ABC) stacked honeycomb lattices supporting Dirac lines protected by time-reversal, inversion and spin rotation symmetries. For typical band structure parameters there exists a pair of nodal lines in the momentum space extending through the whole Brillouin zone in the stacking direction. We show that these Dirac lines are topologically distinct from the usual Dirac lines which form closed loops inside the Brillouin zone. In particular, an energy gap can be opened only by first merging the Dirac lines going through the Brillouin zone in a pairwise manner so that they turn into closed loops inside the Brillouin zone, and then by shrinking these loops into points. We show that this kind of topological phase transition can occur in rhombohedrally stacked honeycomb lattices by tuning the ratio of the tunneling amplitudes in the directions perpendicular and parallel to the layers. We also discuss the properties of the surface states in the different phases of the model.

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Notes

  1. Berry phase \(\phi \) depends on the convention used for the overall phase of the wavefunctions. Therefore, it is uniquely defined only up to \(n 2 \pi \) (\(n \in {\mathbb {Z}}\)). Here, for simplicity we fix the convention for the overall phase in such a way that \(0 \le \phi < 2 \pi \). This automatically fixes also a specific convention for the Berry connection \({\varvec{\mathcal{A}}}({\mathbf {k}})\).

  2. We can generalize the argument also to the case where the system has a chiral symmetry. In this case, the Hamiltonian can be always block-off-diagonalized and the Berry phase \(\phi /\pi \) in Eq. (5) can be replaced with the winding number of the determinant of the off-diagonal block of the Hamiltonian. The difference is that this new winding number does not have the same ambiguity as the Berry phase related to the shifts of \(n 2 \pi \) (\(n \in {\mathbb {Z}}\)) and therefore \(Q_M\) becomes a \({\mathbb {Z}}\) topological invariant. Because in these symmetry classes \(|Q_M|=\nu \) only the Dirac lines with \(\nu =0\) can be created and annihilated individually. In the special case of 2\(\times 2\) Hamiltonian with time-reversal and inversion symmetries, the Hamiltonian automatically has a chiral symmetry up to terms proportional to \(\sigma _0\). Because the terms proportional to \(\sigma _0\) do not influence the existence of the Dirac lines the \({\mathbb {Z}}\) topological invariant can be defined also in this case.

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Acknowledgements

We thank G. E. Volovik, T. Bzdusek and A. Bouhon for fruitful discussions and comments. This work was supported by the Academy of Finland Centre of Excellence and Key Funding programs (Project Nos. 284594 and 305256).

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  1. Department of Physics and Nanoscience Center, University of Jyväskylä, P.O. Box 35 (YFL), 40014, Jyväskylä, Finland

    T. Hyart, R. Ojajärvi & T. T. Heikkilä

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Hyart, T., Ojajärvi, R. & Heikkilä, T.T. Two Topologically Distinct Dirac-Line Semimetal Phases and Topological Phase Transitions in Rhombohedrally Stacked Honeycomb Lattices. J Low Temp Phys 191, 35–48 (2018). https://doi.org/10.1007/s10909-017-1846-3

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