#### 2.1 Effective Hamiltonian for HgTe QWs

We will discuss the QSHIs in the context of their experimental realization in HgTe/CdTe quantum wells (QWs) 5–7. HgTe is a zinc-blende-type semiconductor. Due to large relativistic corrections, it has an inverted band structure where a metallic s-type electron band (usually acting as the conduction band) has a lower energy than p-type hole bands 191. Consequently, in HgTe QWs with large thicknesses *d* the energy subbands are also inverted (see Fig. 3a). In Fig. 3, subbands *H*1, *H*2, … originate from the heavy-hole band, whereas the electron-like subbands are denoted by *E*1, *E*2, …. With the decreasing QW thickness *d*, the energies of the *E* subbands increase as a result of quantum confinement, whereas those of the *H* subbands decrease (see Fig. 3a). This results in the normal band sequence in narrow QWs. The different *d*-dependences of *E*1 and *H*1 subbands imply a critical thickness, *d*_{c}, at which the band gap is closed. For *d* ≈ *d*_{c} and near the *Γ* point, an effective four-band model involving double (Kramers) degenerate *E*1 and *H*1 subbands can be derived from the eight-band Kane model 4, 26. Introducing basis states , , , and (where ± denotes Kramers partners), one can write the effective four-band Hamiltonian for the system as follows 4, 26:

- ((1))

- ((2))

- ((3))

The two diagonal blocks of *H*_{HgTe} (1) describe pairs of states related to each other by time reversal symmetry (Kramers partners). Each of the blocks has a matrix 2 × 2 structure with Pauli matrices *σ*_{x,y,z} and unit matrix *σ*_{0} representing the two lowest-energy subbands *E*1 and *H*1. The linear terms in Eq. (2) (proportional to constant and in-plane wave-vectors *k*_{x,y}) describe the *E*1–*H*1 hybridization, while yields the band gap at the *Γ* () point of the Brillouin zone. The positive quadratic terms and take into account the details of the band curvature in HgTe QWs 4. The Hamiltonian (1) can be extended to include the spin–orbit coupling between the Kramers partners 26, 6.

#### 2.2 Simple analytic model of the QSH insulator

The QSHI state is realized when a QW with an inverted gap is sided by ordinary band insulators with (see also Fig. 1b). The system boundaries play a role of topological defects – “mass” domain walls – that bind electronic states near the edge so that they decay on both sides of the boundary. In this subsection we discuss specific properties of such edge states:

In order to illustrate these properties we will make two simplifications. First, we will omit all the terms in Hamiltonian (4). This is justified since the edge states occur in the vicinity of the point. Hamiltonian (4) takes, then, the form which in position representation corresponds to the following equation for the four-component wave function :

- ((5))

Second, we will assume that our system is confined by a normal band insulator with the infinite mass, . It is known 192 that such infinite mass confinement can be modeled by an effective local boundary condition which in our geometry (e.g., at *y* = 0) reads

- ((6))

This boundary condition is specific to Dirac fermions with linear spectrum. It ensures vanishing of the normal component of the particle current without putting to zero at the boundary. The use of the infinite mass confinement (6) is complementary to the tight-binding calculations (see e.g., Refs. 6, 18) and other continuum models of the helical edge states, which include the quadratic () terms and employ the hard wall boundary conditions (see e.g., Refs. 11, 13).

The helicity [see Eq. (14)] protects the edge states from local perturbations that does not break the time-reversal symmetry. Concretely, let us consider a perturbation *V* which is (up to a phase) invariant under time reversal:

- ((16))

Using Eqs. (15) and (16) we can transform the matrix element as follows

- ((17))

- ((18))

i.e., for a hermitian its matrix element is zero:

- ((19))

In particular, spin-independent potential disorder cannot cause scattering between the helical edge states and their localization. The physics of localization in the QSHIs has been studied by using both field-theoretical 20, 24 and numerical (see e.g., 18, 35, 37, 52, 51) techniques. In particular, Ref. 18 has found that the edge backscattering and magnetoresistance can occur as a combined effect of bulk-inversion asymmetry, sufficiently strong potential disorder and an external magnetic field.

Another question regarding the role of the time-reversal symmetry is what happens to the QSHIs in a strong quantizing magnetic field? In this case the helical edge states appear within the gap between the highest hole () and lowest particle () Landau levels 19. The edge-state dispersion is still nodal, but nonlinear, i.e., the two crossing spectral branches have now different group velocities. The latter circumstance makes the helical edge states prone to get coupled by spin-flip scattering, which also generates the edge magnetoresistance 19, 50.

We should emphasize that Eqs. (17)–(19) do not generally hold for interacting helical channels where the electron backscattering may occur provided that the axial spin rotation symmetry of Hamiltonian (1) is broken, e.g. by the Bychkov–Rashba spin–orbit coupling 20, 47, 60. The spatially random Bychkov–Rashba coupling has been predicted to cause localization of the edge states in the presence of weakly screened electron–electron interaction 20. Ref. 60 has shown that weak electron–electron interaction in the absence of the axial spin rotation symmetry allows for inelastic backscattering of a single electron, accompanied by forward scattering of another. The inelastic phonon-induced backscattering in the helical liquid has been considered in Ref. 47.

#### 2.3 Helical versus chiral edge transport. Nonlocal detection of the QSHI state

The helical edge transport in a QSHI differs markedly from the chiral edge transport in QH systems. In order to illustrate this we follow Ref. 7 and consider a four-terminal device shown in Fig. 4. Using the ballistic Landauer–Bütikker approach, we express the current, *I*_{i}, injected through contact *i* in terms of voltages *V*_{j} induced on all contacts as

- ((20))

where is the transmission probability from contact i to contact *j*. For a chiral QH edge channel, connects the neighboring contacts only in one propagation direction (see also Fig. 4a), such that

- ((21))

where *N* is the number of the terminals (e.g., *N* = 4 in Fig. 4), with the convention that describes the transmission from terminal *N* to terminal 1. Assuming, for concreteness, that the current flows from terminal 1 to terminal 4, while leads 2 and 3 are used as voltage probes, we have

- ((22))

This yields a finite two-terminal resistance and zero four-terminal (nonlocal) resistances , , and .

#### 2.4 Helical carriers and weak antilocalization in n-type HgTe quantum wells

The role of the geometrical phases in transport quantum-interference effects in HgTe quantum wells has recently been studied numerically in Ref. 62. In Ref. 63 analytical results for the WAL in HgTe quantum wells have been obtained with the account of both bulk-inversion asymmetry and Rashba SO coupling. Experimentally, the WAL in HgTe has been observed in 2D quantum wells 67, 75 and in strained 3D layers 112. Also, experiments have revealed cyclotron resonance phenomena for the helical carriers in HgTe quantum wells 70, 72.