As opposed to ordinary metals, whose Fermi surfaces are two dimensional, topological (semi-)metals can exhibit protected one-dimensional Fermi lines or zero-dimensional Fermi points, which arise due to an intricate interplay between symmetry and topology of the electronic wavefunctions. Here, we study how reflection symmetry, time-reversal symmetry, SU(2) spin-rotation symmetry, and inversion symmetry lead to the topological protection of line nodes in three-dimensional semi-metals. We obtain the crystalline invariants that guarantee the stability of the line nodes in the bulk and show that a quantized Berry phase leads to the appearance of protected surfaces states, which take the shape of a drumhead. By deriving a relation between the crystalline invariants and the Berry phase, we establish a direct connection between the stability of the line nodes and the drumhead surface states. Furthermore, we show that the dispersion minimum of the drumhead state leads to a van Hove singularity in the surface density of states, which can serve as an experimental fingerprint of the topological surface state. As a representative example of a topological semi-metal, we consider Ca3P2, which has a line of Dirac nodes near the Fermi energy. The topological properties of Ca3P2 are discussed in terms of a low-energy effective theory and a tight-binding model, derived from ab initio DFT calculations. Our microscopic model for Ca3P2 shows that the drumhead surface states have a rather weak dispersion, which implies that correlation effects are enhanced at the surface of Ca3P2.

A single molecular layer of titanium diselenide (TiSe2) is a promising material for advanced electronics beyond graphene—a strong focus of current research. Such molecular layers are at the quantum limit of device miniaturization and can show enhanced electronic effects not realizable in thick films. We show that single-layer TiSe2 exhibits a charge density wave (CDW) transition at critical temperature TC=232±5 K, which is higher than the bulk TC=200±5 K. Angle-resolved photoemission spectroscopy measurements reveal a small absolute bandgap at room temperature, which grows wider with decreasing temperature T below TC in conjunction with the emergence of (2 × 2) ordering. The results are rationalized in terms of first-principles calculations, symmetry breaking and phonon entropy effects. The observed Bardeen-Cooper-Schrieffer (BCS) behaviour of the gap implies a mean-field CDW order in the single layer and an anisotropic CDW order in the bulk.

Two-dimensional materials constitute a promising platform for developing nanoscale devices and systems. Their physical properties can be very different from those of the corresponding three-dimensional materials because of extreme quantum confinement and dimensional reduction. Here we report a study of TiTe2 from the single-layer to the bulk limit. Using angle-resolved photoemission spectroscopy and scanning tunneling microscopy and spectroscopy, we observed the emergence of a (2 × 2) charge density wave order in single-layer TiTe2 with a transition temperature of 92 ± 3 K. Also observed was a pseudogap of about 28 meV at the Fermi level at 4.2 K. Surprisingly, no charge density wave transitions were observed in two-layer and multi-layer TiTe2, despite the quasi-two-dimensional nature of the material in the bulk. The unique charge density wave phenomenon in the single layer raises intriguing questions that challenge the prevailing thinking about the mechanisms of charge density wave formation.

Electronic and optical properties of materials are affected by atomic motion through the electron–phonon interaction: not only band gaps change with temperature, but even at absolute zero temperature, zero-point motion causes band-gap renormalization. We present a large-scale first-principles evaluation of the zero-point renormalization of band edges beyond the adiabatic approximation. For materials with light elements, the band gap renormalization is often larger than 0.3 eV, and up to 0.7 eV. This effect cannot be ignored if accurate band gaps are sought. For infrared-active materials, global agreement with available experimental data is obtained only when non-adiabatic effects are taken into account. They even dominate zero-point renormalization for many materials, as shown by a generalized Fröhlich model that includes multiple phonon branches, anisotropic and degenerate electronic extrema, whose range of validity is established by comparison with first-principles results.