Symmetry enforced topological phases : topological insulators and topological semimetals

Abstract

Symmetry protected nontrivial band topology has become an area of paramount research interest for unraveling novel dimensions in condensed matter physics (CMP). The time reversal invariant topological insulator (TI) has stimulated intense interest due to their intriguing properties, such as gapless boundary states, unconventional spin texture and so on. The recent years have witnessed a series of theoretical developments which have enabled us to classify the Z2 non-magnetic band insulators. For example, the Z2 even (ordinary) and Z2 odd (topological) states are separated by a topological phase transition, where the bulk gap diminishes during the adiabatic deformation between these two states. In two-dimensional (2D) system, Z2 odd class can be distinguished by the odd number of Kramer’s pairs of counter propagating helical edge states, whereas in three-dimensional (3D) system, it can be characterized by the odd number of Fermi loops on the surface that enclose certain high symmetry points in the Brillouin zones (BZs). Soon after the experimental realization of quantum spin hall effect in 2D HgTe quantum wall, a number of 2D and 3D TIs have been theoretically predicted and experimentally verified. In fact, the search for new TI has been extended to zintl compounds, antiperovskites, and heavy fermion f-electron Kondo type systems. With the conceptual developments in the topological field, research on topological material has been extended from insulators to semimetals and metals. In topological semimetals, symmetry protected band crossing or accidental band touching leads to a nontrivial band topology in 3D momentum space. The topological properties of such semimetals mainly depend on the degeneracy of the bands at the crossing/touching point. A zero dimensional band crossing with two and four fold band degeneracy defines the Dirac and Weyl semimetals, respectively, which are quasi-particle counterparts of Dirac and Weyl fermions in high energy physics. Low energy Dirac fermions in condensed matter are essentially protected by time reversal symmetry (TRS), inversion symmetry (IS) and certain crystal symmetry. Quasi-particle Weyl fermion state can be realized by breaking either space inversion or time reversal of the crys- tal. In case of nodal line semimetal (NLS), the conduction and valance band touches along a line to form a one dimensional close loop. All the topological semimetals show some exotic phenomena, such as quantum magneto-resistance, chiral anomaly and so on. In chapter 1, we have extensively discussed the topological band theory in CMP. We have discussed the general notion of topology in geometrical system and then we extend the discussion into the CMP through Berry phase formalism. We have started with quantum Hall effect in 2D electron gas where we discuss the non-trivial topological quantum Hall physics with external magnetic field. Then we generalize it to the quantum spin Hall (QSH) effect by using Kane-Mele Model in spin-orbit graphene. SQH effect is the first topological phenomena in that does not require an external magnetic field like quantum Hall case. In QSH effect, SOC of the crystal takes similar part as external magnetic field in quantum Hall effect. We have then discussed the 3D topological insulators and computation of Z2 topological index, surface states, adiabatic continuation approach etc. We finally extended our discussion in gapless systems such as Dirac semimetal (DSM), triple point semimetal (TSM), nodal line semimetal (NLS), and Weyl semimetal (WSM). The importance of symmetry (both crystalline and TRS) on the semimetalic systems has been extensively discussed using toy models and symmetry based arguments. For the completeness of the discussions, we have also provided the detail of Berry phase, time reversal symmetry, model analysis for type-II DSM, description of the chiral anomaly for Weyl and Dirac metals and so on. Density Functional Theory(DFT) has been emerged as a versatile tool to study the structural, electronic, optical properties of molecular and material based systems with a crux idea of replacing high dimensional wavefunction with electron density drastically reducing the complexity of many body Schr¨odinger equation. In this chapter 2, we have discussed the computational formalism of DFT which includes many-body Problem, Kohn-Sham equations, exchange-correlation functional etc. We have also provider the tight binding formalism based on localized Wannier orbital. We have used tight binding formalism for the computations of Fermi surfaces, surface states, topological index and so on. In chapter 3, we predict the emergence of non-trivial band topology in the family of XX0 Bi compounds having P62m (# 189) space group. Using first principle calculations within hybrid functional framework, we demonstrate that NaSrBi and NaCaBi are strong topological insulator under controlled band engineering. Here, we propose three different ways to engineer the band topology to get a non-trivial order: (i) hydrostatic pressure, (ii) biaxial strain (due to epitaxial mismatch), and (iii) doping. Non-triviality is confirmed by investigating bulk band inversion, topological Z2 invariant, surface dispersion and spin texture. Interestingly, some of these compounds also show a three dimensional topological nodal line semi-metal (NLS) state in the absence of spin orbit coupling (SOC). In these NLS phases, the closed loop of band degeneracy in the Brillouin zone lie close to the Fermi level. Moreover, a drumhead like flat surface state is observed on projecting the bulk state on the [001] surface. The inclusion of SOC opens up a small band gap making them behave like a topological insulator. In chapter 4, we predict three full Heusler compounds XInPd2 (X = Zr, Hf and Ti) to be potential candidates for type-II Dirac semimetals. The crystal symmetry of these compounds have appropriate chemical environment with a unique interplay of inversion, time reversal and mirror symmetry. These symmetries help to give six pairs of type-II Dirac nodes on the C4 rotation axis, closely located at/near the Fermi level. Using first principle calculations, symmetry arguments and crystal field splitting analysis, we illustrate the occurrence of such Dirac nodes in these compounds. Bulk Fermi surfaces have been studied to understand the Lorentz symmetry breaking and Lifshitz transition (LT) of Fermi surfaces. By analyzing the evolution of arcs with changing chemical potential, we prove the fragile nature and the absence of topological protection of the Dirac arcs. Noble metal surfaces (Au, Ag and Cu etc.) have been extensively studied for the Shockley type surface states (SSs). Very recently, some of these Shockley SSs have been understood from the topological consideration, with the knowledge of global properties of electronic structure. In chapter 5, we show the existence of Dirac like excitations in the elemental noble metal Ru, Re and Os based on symmetry analysis and first principle calculations. The unique SSs driven Fermi arcs have been investigated in details for these metals. Our calculated SSs and Fermi arcs are consistent with the previous transport and photoemission results. We attribute these Dirac excitation mediated Fermi arc topology to be the possible reasons behind several existing transport anomalies, such as large non-saturating magneto resistance, anomalous Nernst electromotive force and its giant oscillations, magnetic breakdown etc. We further show that the Dirac like excitations in these elemental metal can further be tuned to three component Fermionic excitations, using symmetry allowed alloy mechanism. In chapter 6, we show the occurrence of Dirac, Triple point, Weyl semimetal and topological insulating phase in a single ternary compound using specific symmetry preserving perturbations. Based on first principle calculations, k.p model and symmetry analysis, we show that alloying induced precise symmetry breaking in SrAgAs (space group P63/mmc) leads to tune various low energy excitonic phases transforming from Dirac to topological insulating via intermediate triple point and Weyl semimetal phase. We also consider the effect of external magnetic field, causing time reversal symmetry (TRS) breaking, and analyze the effect of TRS towards the realization of Weyl state. Importantly, in this material, the Fermi level lies extremely close to the nodal point with no extra Fermi pockets which further, make this compound as an ideal platform for topological study. The multi fold band degeneracies in these topological phases are analyzed based on point group representation theory. Topological insulating phase is further confirmed by calculating Z2 index. Furthermore, the topologically protected surface states and Fermi arcs are investigated in some detail. Composite quantum compounds (CQCs) have become an important avenue for the investigation of inter-correlation between two apparently distinct phenomenon in physics. Topological superconductors, axion insulators etc. are few such CQCs which have recently drawn tremendous attention in the community. Topological nontriviality and Rashba spin physics are seemingly two incompatible quantum phenomena but can be intertwined within a CQC platform. In chapter 7, we present a general symmetry based mechanism, supported by ab-initio calculations to achieve intertwined giant Rashba splitting and topological non-trivial states simultaneously in a single crystalline system. Such co-existent properties can further be tuned to achieve other rich phenomenon. We have achieved Rashba splitting energy (∆E) and Rashba coefficient (α) values as large as 161 meV and 4.87 eV˚A respectively in conjunction with Weyl semimetal phase in KSnSb0.625Bi0.375. Interestingly, these values are even larger than the values reported for widely studied topologically trivial Rashba semiconductor BiTeI. The advantage of our present analysis is that one can achieve various topological phases without compromising the Rashba parameters, within this CQC platform. In chapter 8, I present the conclusive remarks and the outcome of my research work. I also briefly discuss my future plans here.