Symmetry and topology of nodal semimetals
Agterberg, Daniel F.
The subject matter of this thesis is the appearance of various nodal semimetals in crystalline solids. The valence and the conduction bands of such materials touch at points or along lines in momentum space, and their chemical potential is adjusted to the energy of this touching. As a consequence, these materials have a vanishing or a very small density of states at the Fermi level, while not being characterized by an energy gap in the bulk excitation spectrum. This places nodal semimetals at the fine borderline between metals and insulators. In this work, we investigate the role of space group symmetries in enabling such band touchings, the so-called nodes, and the way they are protected by the topology of the electron wave functions, their so-called topological invariants. The most famous example of a nodal semimetal is certainly graphene. In this two-dimensional material, a pair of bands disperses linearly around a touching point. As a consequence, electrons in graphene can be described by the massless Dirac equation borrowed from high-energy physics. This analogy is manifested in various transport signatures, for example in the absence of back-scattering on a certain type of barrier, known as Klein tunnelling. More recently, three-dimensional cousins of graphene called Weyl semimetals have been theoretically proposed in 2011 and experimentally observed in TaAs crystals in 2015. These materials allow for a condensed matter realization of another effect originally considered in high-energy physics, the chiral anomaly, which is responsible for an increase of conductivity if parallel magnetic field is applied to a sample. Weyl semimetals also exhibit unusual surface states, which have the form of open-ended Fermi arcs connecting the projections of bulk Weyl points inside the surface Brillouin zone. Soon after the proposal of Weyl semimetals, a plethora of other three-dimensional nodal semimetals with more intricately structured nodes have been identified and studied. This thesis is a collection of three projects concerning nodal semimetals and related gapless materials. In the first project, we focus on Weyl semimetals. Our strategy to realize this phase is to start with a three-dimensional band structure which contains Dirac points at time-reversal invariant momenta on the Brillouin zone boundary. Such a situation arises for example in certain pyrochlore iridates due to the non-symmorphic character of their space group. We consider a tight-binding description of such materials which incorporates the leading coupling of the electrons to an elastic degree of freedom, and show that such a model may exhibit a lattice instability at low temperatures. We present detailed group-theoretical arguments to explain why such a lattice distortion alters the band structure from Dirac semimetal through Weyl semimetal to an insulator. We numerically investigate the phase diagram of this model and find that for a certain range of parameters it exhibits an exotic reentrant behaviour when a symmetry-broken phase can be reached from a symmetric phase at both higher and lower temperatures. We further observe that the connectivity of the surface Fermi arcs can be changed by tuning the model parameters. We call this phenomenon a Weyl-Lifshitz transition, and we suggest a way for its detection using quantum oscillation experiments. The second project is concerned with nodal line semimetals. Such materials are associated with nearly-flat surface bands, which were predicted to trigger various magnetic or superconducting surface instabilities. After briefly reviewing the previously known realizations of nodal lines, we show that space group symmetry imposes their existence in spin-orbit coupled materials with a glide plane and no centre of inversion. Furthermore, this species of nodal lines occurs automatically at the Fermi level provided that a few simple criteria are met. We study the relation of such nodal lines to the usual massless Dirac spectrum, and use this to explain their peculiar Landau levels which exhibit a direction-selective chiral anomaly. We further show that similar systems with a pair of glide planes lead to a band touching along two connected loops, which in the extended momentum space look like an infinite chain. We thereby call such materials nodal-chain metals. We show that such a phase might be realized in an existing compound IrF4. We use group theory and Wilson operators to explain the complex nature of its surface states. Finally, in the last project we attempt at more generality. We consider band structures of centrosymmetric systems, and classify them according to their global symmetries (i.e. time-reversal symmetry, particle-hole symmetry and chiral symmetry) into ten groups which we call centrosymmetric extensions of the Atland-Zirnbauer classes. We use homotopy theory to show that some of these classes exhibit nodes characterized by a pair of topological charges. As a consequence, such nodes reach the very high degree of stability similar to that of Weyl points: A continuous evolution of the underlying Hamiltonian can freely move such nodes throughout the Brillouin zone, but the only way to get rid of them is a pairwise annihilation. We remark that no further crystalline symmetries beyond the spatial inversion are necessary for the appearance of such nodes. We describe this property as robustness. We find that robust nodes appear in four out of the ten centrosymmetric extensions, which are relevant for both semimetallic and superconducting systems. For example, we predict a qualitatively new species of nodal lines of singlet superconducting gap in certain topologically non-trivial Fermi surfaces. As another example, exotic nodal surfaces were very recently identified in certain time-reversal breaking multiorbital superconductors. We develop a simple tight-binding model and provide a geometric interpretation of the pair of charges for all four species of doubly charged nodes possible in three spatial dimensions. We indicate how these concepts generalize to higher spatial dimensions. The thesis contains an additional chapter in the beginning, which provides an introduction into topological aspects of band structures. Furthermore, two mathematical appendices are included at the very end, one concerning irreducible representations of non-symmorphic space groups, and the other containing the derivations of the classifying spaces of the individual Atland-Zirnbauer classes. We conclude with some closing remarks, containing both comments on certain unresolved issues touched upon in the text as well as possible future directions.
Symmetry and topology of nodal semimetals
In Copyright - Non-Commercial Use Permitted
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02010 - Dep. Physik / Dep. of Physics::02511 - Institut für Theoretische Physik / Institute for Theoretical Physics::03571 - Sigrist, Manfred / Sigrist, Manfred
Files in this item
Doctoral Thesis