The consequence of such a relation in 3D is discussed. The organization of this chapter is as follows. We introduce the cubic lattice Dirac model and a transformation on the low energy Dirac fermions to bring it into a ‘normal’ form in Sec. 3.2.2, from which one can easily read off the orders that lead to an energy gap. The chiral topological insulator is identified, and a microscopic model with hopping along the body diagonals of the cube, is shown to lead to this phase. An intuitive picture in terms of a quasi 2D starting point is developed in Sec. 3.3.2, which directly demonstrates the existence of surface Dirac states. Here, we also show how a Z2 topological insulator protected by TRS with bulk Dirac nodes at TRIM can be understood within this picture. In Sec. 3.4 the magnetoelectric coefficient ‘θ’ of such an insulator is argued to be quantized , and is calculated to be θ = π for the spinless fermion model we discuss. Introducing spin, and studying gapped superconducting states, we show in Sec. 3.6 that only a pair of singlet superconductors is allowed, which, in addition to the regular onsite s-wave paired state, bucket flower includes a singlet topological superconductor, with pairing along the body diagonals. Sec. 3.7 describes an attempt to move towards a more physical realization of the cTI phase, utilizing a layered honeycomb lattice structure.
Finally, in Sec. 3.8, we explore some topological properties of the 3D Dirac fermion system by studying its physics in the presence of point topological defects in its order parameters, and deriving the Berry’s phase terms that determine quantum interference of different orders.In many ways, a chiral topological insulator can be viewed as a close cousin of the known topological states in 3D, such as a Z2 topological insulator: A hallmark of both of these states is an appearance of non-trivial surface modes when topological bulk states are terminated by a boundary. However, for a chiral topological insulator, an arbitrary number of flavors of Dirac fermions can appear at the surface and be stable. We discussed a physically transparent picture of the chiral topological insulator, which explains the appearance of surface Dirac fermion states, and their stability in the presence of chiral symmetry. A similar picture also explains the stability of TRS Z2 topological insulators whose bulk Dirac nodes are centred at TRIM. It is shown that the θ = π axion electrodynamics can also be realized in chiral topological insulators, in addition to the known realization in Z2 topological insulators. We should also stress that chiral symmetry, which is realized in the models discussed here as sublattice symmetry, is likely broken in any realistic systems. Nevertheless, as far as breaking of chiral symmetry is sufficiently weak, θ is expected to be close to π, and can It is also worth while mentioning that chiral symmetry need not to be realized only as sublattice symmetry: class AIII symmetry can be realized in the BdG Hamiltonians for Szconserving superconductors. Thus, chiral symmetry when realized in this way is much more robust than sublattice symmetry.
Furthermore, in the 3D π-flux lattice model with inclusion of spin degree of freedom, we found a spin-chiral topological insulator , and also a singlet topological superconductor . The latter is stable as long as the physical symmetries of SU spin rotation and time reversal are present. Finally, utilizing the proximity to a Dirac state, we derived an interesting correlation, or “duality”, between the singlet topological superconductor and Neel order. These order parameters are dual in the sense that a topological defect in either one of these phases carry complementary quantum numbers: e.g.. a defect in the Neel vector can carry electric charge. We also find many such 6-tuplets of order parameters, including a six component vector consists of three Neel order and three VBS order parameters. These dualities are a natural extension of those discussed in 1D and 2D quantum spin models, the latter in the context of deconfined criticality. While in this chapter we have studied the properties of these topological defects at single particle level, and hence the topological defects are static objects, we cannot resist contemplating more interesting situation where they are dynamical entities. In particular, it is interesting to ask if there is a counterpart of the non Landau-Ginzburg transition, realized in two dimensions, can exist, possibly in the presence of strong electron correlations in three dimensions. This is left for future study. P.H. would like to thank Shinsei Ryu and Ashvin Vishwanath for the collaboration in this work. We wish to thank Ying Ran and Tarun Grover for insightful discussions, and the Center for Condensed Matter Theory at University of California, Berkeley and NSFDMR-0645691 for support. This work is published in Physical Review B, 81, 045120 .In the last chapter, we saw interesting physics emerge when appropriate sets of phases were combined together to form three component order parameters, and topological defects were created in them. A natural question to ask is whether there are other topologically non-trivial ways of combining the phases which give rise to unconventional phenomena. The strong topological insulator is a good candidate for being involved in some novel physics, since rich phenomena have already been predicted to arise when this phase is combined with conventional orders such as magnetism, crystalline order and superconductivity.
The last is particularly interesting. Superconductivity induced on the surface of a TI was predicted to have vortices harboring Majorana zero modes. These are of interest to quantum information processing, since they are intrinsically robust against errors. Recently, superconductivity was discovered in a doped TI, which could be used to induce surface superconductivity. Below we discuss a new theoretical approach to studying this remarkable superconducting phase, which provides different insights and directions for experiments. In this case, the low energy description of the system is entirely in terms of bosonic coordinates , much as the Landau Ginzburg order parameter theory describes superconductors at energies below the gap. Can fermions ever emerge is such a theory? While it is easy to imagine obtaining bosons from a fermionic theory, the reverse is harder to imagine. However, it has been shown in principle that bosonic theories that contain additional Berry’s phase terms, can accomplish this transmutation of statistics. We show that this indeed occurs in the superconductor-TI system; the order parameter theory contains a Berry phase term which implies that a particular configuration of fields – the Hopf soliton – carries fermionic statistics. While such statistics transmutation is common in one dimension, it is a rare phenomenon in higher dimensions. In the condensed matter context, an physically realizable example exists in two dimensions: solitons of quantum Hall ferromagnets are fermionic and charged, and have been observed. However, the superconductor-TI system is, to our knowledge, the first explicit condensed matter realization of this phenomenon in three dimensions. The organization of this chapter is as follows. First, we introduce our simplified model of a topological insulator with surface superconductivity, cut flower bucket and review properties of the Hopftexture. We then discuss evidence from numerical calculations on a lattice model, that demonstrate the Hopfions are fermions. We also discuss the connection between fermionic Hopfions and 3D non-abelian statistics of [128]. A simplified two dimensional example, where skyrmions are fermionic, is also discussed. Next, we provide a field theoretical derivation of the same result in the continuum, and introduce the necessary theoretical tools to compute a topological term, that leads to the fermionic statistics. Finally, we mention physical consequences, for tunneling experiments as well as for Josephson junctions.In the previous chapters, the focus has largely been on bulk topological phases and especially on the bulk of the topological insulator. Although the Majorana zero mode discussed in the previous chapter was localized on the surface, its existence was governed by the bulk topological phase. In contrast, in this chapter we focus only on the surface states of the strong topological insulator . In their simplest incarnation, the SSs of TIs correspond to the dispersion of a single Dirac particle, which cannot be realized in a purely two dimensional band structure with time reversal invariance. This dispersion is endowed with the property of spin-momentum locking, i.e., for each momentum there is a unique spin direction of the electron.
Since several materials were theoretically predicted to be in this phase, most of the experimental focus on TIs so far has been towards trying to directly observe these exotic SSs in real or momentum space, in tunneling and photo emission experiments, respectively, and establish their special topological nature. However, there has so far been a dearth of experiments which study the response of these materials to external perturbations, such as an external electromagnetic field. In order to fill this gap, we calculate here the response of TI surfaces to circularly polarized light. Since photons in CP light have a well-defined angular momentum, CP light can couple to the spin of the surface electrons. Then, because of the spin-momentum-locking feature of the SSs, this coupling can result in dc transport which is sensitive to the helicity of the incident light. This phenomenon is known as the circular photogalvanic effect . In this work, we derive general expressions for the direct current on a TI surface as a result of the CPGE at normal incidence within a two-band model and estimate its size for the surface of Bi2Se3, an established TI, and find it to be well within measurable limits. Since bulk Bi2Se3 has inversion symmetry and the CPGE, which is a second-order non-linear effect, is forbidden for inversion symmetric systems, this current can only come from the surface.We find, remarkably, that the dominant contribution to the current is controlled by the Berry curvature of the electron bands and grows linearly with time. In practice this growth is cut-off by a scattering event which resets the current to zero. At the microscopic level, this part of the current involves the absorption of a photon to promote an electron from the valence to the conduction band. The total current contains two other terms – both time-independent – one again involving an interband transition and the other resulting from intraband dynamics of electrons. However, for clean samples at low temperatures, the scattering or relaxation time is expected to be large, and these contributions will be eclipsed by the linear-in-time one. Hence, this experiment can also be used to measure the relaxation time for TI SSs. Historically, the Berry curvature has been associated with fascinating phenomena such as the anomalous Hall effect and the integer quantum Hall effect and therefore, it is exciting that it appears in the response here. Its main implication here is that is gives us a simple rule, in addition to the requirement of the right symmetries, for identifying the perturbations that can give a linear-in-time CPGE at normal incidence: we look for perturbations that result in a non-zero Berry curvature. Put another way, we can identify perturbations that have the right symmetries but still do not give this current because the Berry curvature vanishes for these perturbations. Importantly, for TI SSs, the requirement of a non-zero Berry curvature amounts to the simple physical condition that the spin-direction of the electrons have all three components non-zero. In other words, if the electron spin in the SSs is completely in-plane, the Berry curvature is zero and no linear-in-time CPGE is expected. The spins must somehow be tipped slightly out of the plane, as shown in Figure 6.1.1a, in order to get such a response. Thus, a pure Dirac dispersion, for which the spins are planar, cannot give this response; deviations from linearity, such as the hexagonal warping on the surface of Bi2Te3 , are essential for tilting the spins out of the plane. CPGE has been observed in the past in GaAs, SiGe and HgTe/CdHgTe quantum wells – all systems with strong spin-orbit coupling. The effect in these systems can be understood within a four-band model consisting of two spin-orbit split valence bands and two spin-degenerate conduction bands. In contrast, TI SSs can be faithfully treated within a two-band model. The simplicity of the latter system makes it more convenient for studying theoretically compared to semiconductor quantum wells, and hence, enables us to determine a connection between the CPGE and the Berry curvature.