Analogous to exchanging charges between Gaussian surfaces, changing the topological invariant necessarily requires two sets of Brillouin zone wave functions or equivalently, two bands to touch and exchange quanta of the invariant. Thus, the system becomes gapless as it is tuned across a topological phase transition. Table 1.1 contains a list of analogies between Gauss’s law in electrostatics and topological phenomena in non-interacting band structures. An important consequence of the gaplessness of band structures as they are tuned across a topological phase transition is the existence of protected gapless states on the surface of a topological insulator or superconductor. If a topological insulator or superconductor is placed next to a topologically trivial one, including vacuum, the phases on the two sides of the interface are topologically distinct. Thus, gapless states exist at the interface and survive in the presence of weak perturbations as long as certain basic symmetries of the system are preserved by the perturbations. The number of such protected surface states is intimately tied to the value of the bulk topological invariant. These surface states are very important from a practical point of view as well as they are more easily accessible to many experimental probes than the bulk states and hence, help in identifying the topology of the bulk bands without requiring the bulk bands to be probed directly. Moreover, the surface states fully determine the low energy physics in the presence of a bulk gap and hence, are the only ones relevant for transport. And finally, black flower buckets the surface states of many topological phases display various unusual phenomena that ordinary quasiparticle states do not.
As mentioned earlier, the precise form of the topological invariant in terms of the Bloch functions depends on the symmetry properties of the system of interest. Table 1.2 lists the classes of non-interacting Hamiltonians classified using their symmetry properties, and the sets of topologically distinct gapped phases each class of Hamiltonians may have in one, two and three spatial dimensions. The only physical symmetries used in the classification process are time-reversal symmetry and particle-hole symmetry , since these are the only ones generic to random, disordered systems. Thus, crystal symmetries, which are in general absent in disordered systems, are not considered. The sublattice symmetry in Table 1.2 is defined as the product of TRS and PHS. The ten classes are obtained as follows: both time-reversal and particle-hole conjugation are anti-unitary operations and hence, can either be absent, or be present and square to either +1 or −1. This gives 9 classes of Hamiltonians. The tenth class, AIII is one where both TRS and PHS are broken but their product, the SLS, is preserved.The second way in which topology appears in band Hamiltonians is in the form of topological defects in momentum space. These defects occur at points where the band structure becomes gapless and are analogous to real space topological defects such as domain walls, vortices and hedgehogs, where an order parameter has a non-trivial winding around the defect and vanishes at the defect. Like real space topological defects, topological defects in momentum space can be combined by moving them in momentum space while keeping their total topological quantum number conserved. On a finite system, band structure singularities result in peculiar surface states. Whereas gapped topological phases carry dispersing surface states, gapless topological phases host dispersionless or flat bands on their surface. The flatness of these bands is protected by the topological nature of the bulk band structure and cannot be removed as long as the topological objects in the bulk band structure survive.
One crucial manner in which the topological nature of gapless band structures is different from that of gapped ones is that the former is not immune arbitrary disorder. Since the topological entities occur in momentum space, it is vital that momentum be a good quantum number. Thus, perturbations that break translational symmetry can potentially gap out the topological defects. We now briefly review two common instances of band structure singularities and their associated flat surface bands.The Fermi surface of a Weyl semimetal on a slab consists of unusual states known as Fermi arcs. The Fermi arcs are like a two dimensional Fermi surface, except that the two dimensional Fermi surface is broken into two parts and one part is localized on the top surface while the other is localized on the bottom surface. On each surface, the Fermi arc connects the projection of the bulk Weyl nodes onto the surface, as shown in Figure 1.2.2. A simple way to understand the presence of Fermi arcs is by recalling that the Weyl points are sources of Chern flux. Each two dimensional slice in momentum space perpendicular to the line joining the Weyl nodes can be thought of as a Chern insulator, and the Chern numbers of two slices on either side of a Weyl node differ by one. Thus, if the slices in the region far away from the Weyl nodes have Chern number 0, the Chern number of the slices between the Weyl nodes must be 1 and so on. The Fermi arcs, then, are simply the edge states of the Chern insulators. In summary, there are two broad ways in which topology manifests itself in the spectra of non-interacting band Hamiltonians. For gapped Hamiltonians, a topological phase exists when the Bloch wave functions of the occupied bands form non-trivial topological textures across the Brillouin zone. The exact class of textures that is stable against perturbations depends on the symmetries and the dimensionality of the system. Gapless Hamiltonians, on the other hand, can be characterized by topological invariants by focusing only on the states near the gapless region. The latter manifestation of topology in band structures is, in some sense, less stable than the former because it requires translational invariance and is thus notoblivious to disorder. Both classes of topological phases are associated with unconventional states on the surface of a finite system – dispersing Dirac, Weyl or Majorana modes for gapped phases and flat bands for gapless phases.
In the next chapter, the physics of a famous gapped topological phase – the strong topological insulator – will be briefly introduced and reviewed. Some novel phenomena associated with this phase as well as some material realizations will be mentioned.In the last half of a decade, the buzzword that has perhaps stolen the greatest amount of limelight in condensed matter physics is topological insulators. Two examples of insulators with non-trivial band topology, hence topological insulators according to the definition, were touched upon in the previous chapter. This section does a brief review of the phase that is colloquially called the topological insulator and is the one that has actually captured most of the attention. This is the three-dimensional insulator with time-reversal symmetry – class AII according to Table 1.2 – which has a non-trivial Z2 topological invariant. Henceforth, unless otherwise mentioned, the term ‘topological insulator’ will be used to refer to this particular realization of an insulating topological phase. It is closely related to the quantum spin Hall state; thus, the latter is often also termed the ‘two-dimensional topological insulator’.Having reviewed the main ideas in band structure topology and the basics of topological insulators, we now investigate other possible gapped phases of band Hamiltonians in three dimensions. A convenient starting point is a three Dirac dispersion, french flower bucket since it is proximate to a variety of orders, which when established, lead to an energy gap. In the context of graphene, charge density wave and valence bond solid order as well as antiferromagnetism are known to induce a gap, and lead to an insulating state. Several years back Haldane pointed out that the integer quantum Hall state could be realized starting from the graphene semimetal, in the absence of external magnetic fields. A valuable outcome of this Dirac proximity approach, was the discovery of an entirely new phase of matter, the Z2 Quantum Spin Hall insulator, obtained in theory by perturbing the graphene Dirac dispersion. By analogy, here we study three dimensional Dirac fermions, and their proximate gapped phases, on a cubic lattice. In three dimensions, Dirac points naturally occur in some heavy materials like bismuth and antimony, with strong spin orbit interactions. A three dimensional version of the quantum spin Hall state – the Z2 topological insulator , can be realized by appropriately perturbing such a state, as demonstrated in [38], in a toy model on the diamond lattice. According to recent experiments, this phase is believed to be realized by several Bibased materials including Bi0.9Sb0.1, Bi2Se3 and Bi2Te3. Both the Z2 quantum spin Hall and the Z2 topological insulator phases require time-reversal symmetry to be preserved. The Z2 index represents the fact that only an odd number of edge or surface Dirac nodes are stable in these phases. In contrast, in this chapter we study a toy model on the cubic lattice, with π flux through the faces, which realizes three dimensional Dirac fermions, and identify the proximate states.
To begin with, we consider insulating phases of spin polarized electrons. In addition to conventional insulators, e.g., with charge or bond order, we also find an additional novel topological insulator phase within this model, the chiral Topological Insulator . This provides a concrete realization of this phase, which was recently predicted on the basis of a general topological classification of three dimensional insulators in different symmetry classes. This phase is distinct from the spin-orbit Z2 topological insulators in two main respects. First, it is realized in the absence of time-reversal symmetry. Instead, it relies on another discrete symmetry called the chiral symmetry. Second, these insulators also host protected Dirac nodes at their surface, but any integer number of Dirac nodes is stable on its surface. Thus, it has a Z rather than Z2 character. In an insulator, chiral symmetry restricts us to Hamiltonians with only hopping terms between opposite sublattices. Clearly, this is not a physical symmetry, and hence such insulators are less robust than topological insulators protected by time reversal symmetry. However our results will be relevant if such symmetry breaking terms are weak, or in engineered band structures in lattice cold atom systems . With spin, an interesting gapped state that can be reached from the Dirac limit is the singlet topological superconductor , also first discussed in [119]. This state also possesses protected Dirac surface states. The stability of these states is guaranteed, as long as time reversal symmetry and SU spin symmetry, both physical symmetries, are preserved. The Dirac limit allows for an easy calculation of the charge response of the cTI , and provides an intuitive picture of these phases. For example, the cTI can be understood as arising from a quasi 2D limit of layered Dirac semi-metals, with a particular pattern of node pairings, leading to a bulk gap, but protected surface states. It is hoped that this intuition will help in the search for realistic examples of these phases. Additionally, this picture helps in understanding Z2 topological insulators protected by TRS whose bulk Dirac nodes are at time-reversal-invariant momenta , such as Bi2Se3, Bi2Te3 and Sb2Se3. Finally, we utilize the Dirac starting point to derive relations between different gapped phases. We show that there is a duality between Neel and VBS phases: point defects of the Neel order are found to carry quantum numbers of the VBS state and vice versa. This is done by studying the midgap states induced by these defects, and the results agree with spin model calculations that are appropriate deep in the insulating limit. Thus, the Dirac approach is a convenient way to capture universal properties of the gapped phases in its vicinity. These results are also derived following a technique applied to the one and two dimensional cases, by integrating out the Dirac fermions and deriving an effective action for a set of orders. In particular we focus on the Berry’s phase term which, when present, implies non-trivial quantum interference between them. Such sets of ‘quantum competing’ orders can be readily identified within this formalism. We show that in addition to Neel and VBS orders, interestingly, Neel order and the singlet topological superconductor also share such a relation.