A Derivation of Cooperon equations (25)

# Weak antilocalization in HgTe quantum wells and topological surface states: Massive versus massless Dirac fermions

## Abstract

HgTe quantum wells and surfaces of three-dimensional topological insulators support Dirac fermions with a single-valley band dispersion. In the presence of disorder they experience weak antilocalization, which has been observed in recent transport experiments. In this work we conduct a comparative theoretical study of the weak antilocalization in HgTe quantum wells and topological surface states. The difference between these two single-valley systems comes from a finite band gap (effective Dirac mass) in HgTe quantum wells in contrast to gapless (massless) surface states in topological insulators. The finite effective Dirac mass implies a broken internal symmetry, leading to suppression of the weak antilocalization in HgTe quantum wells at times larger than certain , inversely proportional to the Dirac mass. This corresponds to the opening of a relaxation gap in the Cooperon diffusion mode which we obtain from the Bethe-Salpeter equation including relevant spin degrees of freedom. We demonstrate that the relaxation gap exhibits an interesting nonmonotonic dependence on both carrier density and band gap, vanishing at a certain combination of these parameters. The weak-antilocalization conductivity reflects this nonmonotonic behavior which is unique to HgTe QWs and absent for topological surface states. On the other hand, the topological surface states exhibit specific weak-antilocalization magnetoconductivity in a parallel magnetic field due to their exponential decay in the bulk.

## I Introduction

Recently discovered materials – graphene, (1); (2) two-dimensional (2D) (3); (4); (5); (6) and three-dimensional (3D) (7); (8); (9); (10); (11); (12) topological insulators (TIs) (13); (14) – exhibit a Dirac-like band dispersion which is responsible for their unusual electronic and optical properties. In graphene the low-energy electron spectrum can be approximated by two spin-degenerate Dirac cones at the corners of the Brillouin zone. (15) The 2D TIs have been realized in HgTe quantum wells (QWs) (4); (5) which have a single double-degenerate Dirac valley, as predicted by band-structure calculations and inferred from transport measurements. (16) The double degeneracy of the HgTe QW bands allows for an energy gap at the Dirac point, without breaking time-reversal invariance, which paves the way to study Dirac fermions with a finite (positive and negative) effective mass and related mass disorder. (17); (18); (19) In comparison with HgTe QWs, the ideal 3D TI exhibits a single non-degenerate gapless Dirac cone on the surface of the material, whereas its bulk is insulating. (20) In this case, the opening of the gap in the surface Dirac spectrum requires time-reversal symmetry breaking and has been predicted to cause the surface quantum Hall effect (21); (22); (23); (24) and rich magneto-electric phenomena (21); (22); (23); (24); (25); (26); (27) related to axion electrodynamics. (28)

The number of the Dirac valleys is an essential factor in understanding quantum electron transport in disordered samples. The transport studies of graphene have reported weak localization (29) and more complex quantum-interference patterns (30) instead of the antilocalization effect expected for the symplectic universality class (e.g. Refs. (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46); (47)). Such a situation can occur if the two graphene’s valleys are coupled by scattering off atomically sharp defects. (38); (41); (43); (46); (47) In contrast, in single-valley Dirac systems such scattering processes are forbidden, and recent experiments on HgTe QWs (48); (49) and 3D TIs (50); (51); (52) have found a positive (antilocalization) quantum-interference conductivity.

In this work we conduct a comparative study of the weak antilocalization (WAL) in HgTe QWs and on surfaces of 3D TIs. The goal is to elucidate the difference between these two systems which comes from the finite effective Dirac mass in HgTe QWs in contrast to the massless surface states in 3D TIs. Like in the conventional 2D electron systems (2DESs) with Bychkov-Rashba or Dresselhaus spin-orbit interactions, (32); (34) the WAL in HgTe QWs and on surfaces of 3D TIs reflects the broken rotation symmetry in relevant spin space. However, in addition to the lack of this continuous symmetry, the effective Dirac mass in HgTe QWs implies a broken discrete symmetry which for a single-cone system would play the role of time reversal. In this sense, there is a formal analogy between the effective Dirac mass and an out-of-plane Zeeman field in a 2DES. (44); (53) Therefore, by analogy with weak ferromagnets (44); (53) we find that the WAL in HgTe QWs is suppressed at times larger than certain , inversely proportional to the effective Dirac mass. Such suppression is however absent for the massless surface states in 3D TIs, which can be used to experimentally differentiate between the two systems.

Before going to the calculation details given in Sec. III and IV, in the next section we briefly announce some of our results for HgTe QWs and TIs.

## Ii Overview of the results

To calculate the quantum-interference (Cooperon) conductivity correction, , we adopt the effective Dirac model of HgTe QWs (4) and obtain the following expression for :

 δσxx(n,M)=−α2e2πhlnτ−1τ−1M+τ−1φ, (1) τ−1M=2τ(M+Bk2F)2A2k2F+(M+Bk2F)2,kF=√2πn. (2)

Here the symmetry-breaking-induced relaxation gap is proportional to the total mass term, , in the effective Dirac model of HgTe QWs. (4) is the band gap at the Dirac point, is the positive quadratic correction accounting for the curvature of the filled conduction band ( is the Fermi momentum determined by the 2D carrier concentration ), whereas constant determines the linear (Dirac) part of the spectrum ( the elastic life-time). In Eq. (1) the factor of 2 accounts for the double degeneracy of the Dirac valley in HgTe QWs, constant approaches for (as discussed in detail in Sec. III) and is the dephasing time.

The unique feature of the HgTe QWs is that the band gap can take both positive and negative values depending on the QW thickness. (5); (16) Therefore, the relaxation gap (2) exhibits an interesting nonmonotonic behavior as a function of both and carrier density [see Fig. 1], vanishing when these parameters satisfy the condition,

 M+2πnB=0. (3)

It represents a line in parameter space on which conductivity (1) reaches the maximum [see also Fig. 2]. Such a nonmonotonic behavior of can be used to experimentally identify the symmetry breaking and the resulting relaxation gap . In particular, the predicted carrier-density dependence should hold for the QWs where the potential impurity scattering is much stronger than scattering off random gap fluctuations. This regime can be identified from the carrier-density dependence of the QW mobility. (17) As to the dependence on the gap , it can be extracted from sample-to-sample measurements. The band structure of MBE grown HgTe QWs is controllable to a great extent. (5); (6); (16) For the experiment we suggest here one should select several QWs with distinctly different gaps and comparable dephasing times (inferred from the temperature dependence of the conductivity) and other band structure parameters (inferred from the Hall and Shubnikov-de Haas measurements). Alternatively, the presence of the relaxation gap can be established from the dependence of on the dephasing rate , which is directly related to the temperature dependence (e.g. the dephasing rate due to electron-electron interactions is linearly proportional to the temperature). (54) The dependence of on is shown in Figs. 3(a) and (b). In these figures the upper curves correspond to the logarithmically divergent with . In contrast, for finite (rest of the curves) the conductivity correction shows only weak dependence on the dephasing rate.

As to the 3D TIs, we focus on compounds BiSe and BiTe where the surface states can be described by the effective two-band Dirac Hamiltonian, accounting for the hexagonal warping of the bands [see, e.g., Ref. (55); (56)]. The warping term is cubic in momentum and enters formally as the Dirac mass term. However, since it preserves the time-reversal symmetry, we find that for weak warping the quantum-interference conductivity correction has the same form as for the conventional 2DES with spin-orbit interaction: (31); (32); (34)

 δσxx=−αe2πhlnτφτ,α=−12. (4)

The specific of the surface state shows up mainly in the dependence of the conductivity on magnetic field applied parallel to the TI surface:

 Δσxx(B)=−e22πhln(1+B2B2∥),B∥=ℏeλ√ℓtrℓφ. (5)

Such dependence reflects the finite penetration length, , of the surface state into the bulk, i.e. the magnetic flux through the effective width of the surface state [see Eq. (5) for , where and are the transport mean free path and dephasing length, respectively]. Quantum transport in the in-plane magnetic fields has been studied theoretically for disordered metallic films (57) and electron quantum wells. (58) The present case differs from the previous studies in that the topological surface states have a different microscopic profile of the transverse wave functions. We discuss the dependence of on the magnetic field orientation in Sec. IV in connection with recent experiments on BiTe (Ref. (52)).

## Iii HgTe quantum wells

### iii.1 Effective Hamiltonian

We use the effective 4-band Hamiltonian of HgTe wells (4) which can be written in the following form:

 HHgTe=(H+Hi00~H+Hi), (6)
 H=\boldmathσ(A^k+M^kz)+(D^k2−EF)σ0, (7) ~H=−\boldmathσ(A^k+M^kz)+(D^k2−EF)σ0, (8) Hi=V(r)σ0. (9)

The two diagonal blocks in describe pairs of states related to each other by time reversal symmetry (Kramers partners). In the upper block the Hamiltonian has a matrix structure with Pauli matrices and unit matrix acting in space of two lowest-energy subbands of the HgTe quantum well: (4) an s-like electron one and a p-like heavy hole one . For the lower block the basis states have the opposite spin projections: and . The linear term in (proportional to constant and momentum operator ) originates from the hybridization of the s- and p-like subbands. is the effective Dirac mass:

 M^k=M+B^k2, (10)

where constant determines the band gap at the gamma () point of the Brillouin zone (see Fig. 4). The positive quadratic terms and take into account the details of the band curvature in HgTe quantum wells. (4); (5) Finally, in Eq. (6) accounts for interaction with randomly distributed short-range impurities. The impuritity potential is characterized by the correlation function,

 ζ(r−r′)=⟨⟨V(r)V(r′)⟩⟩=ℏπNτ0δ(r−r′), (11) ζk=∫ζ(r)e−ikrdr=ℏπNτ0, (12)

parametrized in terms of the characteristic scattering time and the density of states (DOS) at the Fermi level, , for one Dirac cone [brackets denote averaging over impurity positions and is the Fourier transform of the disorder correlation function].

We emphasize that the mass term (10) is symmetric with respect to momentum inversion . Hence, Hamiltonian does not possess the symmetry under transformation

 ^k,\boldmathσ→−^k,−\boldmathσ. (13)

Within a given block of Eq. (6), i.e. in subband pseudospin space, such a transformation plays the role of ”time reversal”. At the same time, the real time-reversal symmetry is ensured by the matrix form of (6). Physically, this means that the Kramers partners reside on two Dirac cones superimposed at point. (16) Note that the zero off-diagonal elements in (6) imply conservation of the spin projections of and subbands, which is a good approximation for symmetric HgTe wells. (5); (16); (59); (60) In this case, each of the Dirac cones contributes independently to transport processes, which is accounted for by the factor of 2 in the expressions for the conductivity corrections discussed in subsection III.4.

### iii.2 Disorder-averaged single-particle Green’s functions and elastic life-time

We begin by calculating the disorder-averaged retarded/advanced Green’s functions for an n-type HgTe well under weak-scattering condition

 kFvFτ≫1, (14)

where is the elastic scattering time and and are the Fermi velocity and wave-vector, respectively. In the standard self-consistent Born approximation (see, e.g. Refs. (61); (62)) the equation for is shown diagrammatically in Fig. 5(a). In representation it reads

 GR/Ak=GR/A0k+GR/A0k∫dk′(2π)2ζk−k′GR/Ak′GR/Ak, (15)
 GR/A0k=12σ0+% \boldmathσekϵ−ξk,ek=sAk+Mkz√A2k2+M2k,|ek|2=1, (16)

Here the Green’s function describes a conduction-band carrier with dispersion in the absence of disorder [index labels the Kramers partners residing on the different Dirac cones]. The valence band contribution is neglected under assumption that the energy separation between the valence and conduction bands is much bigger than the disorder-induced band smearing,

 2√A2k2F+M2kF⋅n=2(EF−Dk2F)≫ℏ/τ. (17)

Because of the large band-structure constant meVnm [see, e.g. Ref. (16)] the requirement (17) can be satisfied simultaneously with the weak-scattering condition (14). We also note that the matrix structure of (16) accounts for the carrier chirality and is of primary importance throughout the paper.

The solution to Eq. (15) can be sought in the form , where and satisfies the equation

 ΣR/Ak=∫ζk−k′ϵ−ξk′−ΣR/Ak′1+ekek′2dk′(2π)2. (18)

Approximate solution (61); (62) of the latter equation near the Fermi surface, , yields the disorder-averaged Green’s function as

 GR/Ak=12σ0+\boldmath% σekF⋅nϵR/A−ξk,ϵR/A=ϵ±iℏ2τ, (19)

where is the elastic life-time given by

 1τ=∓2ℏImΣR/AkF⋅n=Nℏ× (20) ×∫dϕn′ζkF⋅|n−n′|1+e⊥n⋅e⊥n′+e∥n⋅e∥n′2,
 e∥n=sn√1−e2⊥n,e⊥n=sMkF⋅nz√A2k2F+M2kF⋅n. (21)

In Eq. (20) the unit vectors and specify the directions of the incident and scattered momentum states, respectively, and and are the in- and out-of-plane components of the unit vector . For isotropic (12) and (10) one finds the elastic time

 τ=τ01+e2⊥, (22)

which is shorter than the disorder-related time scale [see, Eq. (11)]. This is due to the allowed backscattering into an opposite momentum state () which is absent in the gapless case. (63) The backscattering is the consequence of the symmetry breaking due to the mass term. The strength of the symmetry breaking is controlled by the out-of-plane component of the unit vector [see, Eq. (21)].

### iii.3 Cooperon

The quantum-interference corrections to the Drude conductivity can be expressed diagrammatically by the Hikami boxes shown in Fig 5b. Apart from the single-particle Green’s functions (19) they involve the disorder-averaged two-particle Green’s function, , known as the Cooperon. In this subsection we will set up and solve the equation for .

For the potential disorder defined by Eq. (11) the diagram for the Cooperon equation (see, Fig 5c) is read off as follows

 Cαβα′β′(q)=τ2τ0δαα′δββ′+ℏπNτ0∫dk(2π)2 ×∑γ′δ′GRαγ′(k,ϵ+ℏω)GAβδ′(q−k,ϵ)Cγ′δ′α′β′(q), (23)

where the Greek indices label the states in pseudospin () space. The prefactor in the first term is due to the chosen normalization of . To solve Eq. (23) it is convenient to first expand the Cooperon in the orthonormal eigenfunctions of the pseudospin of a two-electron system:

 Cαβα′β′=∑ijCijΨiαβΨj∗α′β′,∑αβΨjαβΨi∗αβ=δji. (24)

The indices label the pseudospin-singlet () and pseudospin-triplet () states. The conductivity corrections will be eventually expressed in terms of the coefficients for which we derive the following algebraic equations (see, also, Appendix A):

 Cij(q)=τ2τ0δij+ (25) τ4τ0∑sTr⟨(σ0−% \boldmathσe−kF⋅n)σi(σ0+% \boldmathσekF⋅n)σs⟩Csj(q),

where the square brackets stand for integral over the directions of the momentum on the Fermi surface:

 ⟨...⟩=∫2π0dϕn2π...1−iτω+iτvFn⋅q. (26)

Evaluating the traces of the products of the Pauli matrices in Eq. (25) we find

 C0j=τ2τ0δ0j+τ2τ0⟨1−e+⋅e−⟩C0j +τ2τ0∑b=x,y,z⟨(e+−e−+ie+×e−)⋅b⟩Cbj, (27) Caj=τ2τ0δaj+τ2τ0⟨(e+−e−−ie+×e−)⋅a⟩C0j +τ2τ0∑b=x,y,z⟨(1+e+⋅e−)(a⋅b)−i(e++e−)⋅a×b −(e+⋅a)(e−⋅b)−(e−⋅a)(e+⋅b)⟩Cbj. (28)

We separated the singlet and triplet Cooperons with respect to the first index so that run over only. Respectively, vectors and run over the unit vector basis of the Cartesian system. We also introduced a convenient shorthand notation

 e±=e±kF⋅n. (29)

As discussed in subsection III.1, the specifics of the effective Hamiltonian for HgTe quantum wells consists in the symmetry of the mass term (10). Being an even function of , it breaks the symmetry of Hamiltonian in Eq. (7) under transformation . The symmetry breaking is encoded in the unit vectors (29) which have antiparallel in-plane and parallel out-of-plane components,

 e±=±e∥+e⊥, (30)

where and are defined by Eq. (21). In view of the identities

 e++e−=2e⊥,e+−e−=2e∥, e+⋅e−=1−2e2∥,e+×e−=2e∥×e⊥, (31)

Eqs. (27) and (28) reduce to

 C0j=τ2τ0δ0j+ττ0⟨e2∥⟩C0j (32) +ττ0∑b=x,y,z⟨(e∥+ie∥×e⊥)⋅b⟩Cbj, Caj=τ2τ0δaj+ττ0⟨(e∥−ie∥×e⊥)⋅a⟩C0j (33) +ττ0∑b=x,y,z⟨e2⊥(a⋅b)−ie⊥⋅(a×b)+(e∥⋅a)(e∥⋅b) −(e⊥⋅a)(e⊥⋅b)⟩Cbj,

or, explicitly,

 [τ0τ−⟨e2∥⟩]C0j − ⟨ex+ieye⊥⟩Cxj (34) − ⟨ey−iexe⊥⟩Cyj=τδ0j,
 −⟨ex−ieye⊥⟩C0j + [τ0τ−⟨e2⊥+e2x⟩]Cxj (35) − ⟨exey−ie⊥⟩Cyj=τδxj,
 −⟨ey+iexe⊥⟩C0j − ⟨eyex+ie⊥⟩Cxj (36) + [τ0τ−⟨e2⊥+e2y⟩]Cyj=τδyj,
 Czj=τ2τ0δzj. (37)

For the quantum-interference conductivity corrections we will only need the - and -dependent diagonal Cooperons , and . Each of them is obtained from coupled Eqs. (34) – (36) where index should be set to or , respectively. The coefficients in these equations are evaluated by expanding (65) the denominator in Eq. (26) in the small Cooperon momentum and frequency ,

 τvFn⋅q≪1,τω≪1. (38)

In doing so, we keep the lowest order terms that yield the nonzero angle average and compete with the small symmetry-breaking parameter [see, Eq. (21)]. Under these approximations we obtain the following expressions for the diagonal Cooperons:

 C00(q,ω)=1Dq2+τ−1M−iω,τ−1M=2e2⊥τ, (39)
 D=Dτ(2+5e2⊥+e4⊥)/e2∥,Dτ=v2Fτ/2, (40)
 Cxx(q,ω)=2τe2∥2−e4∥cos2ϕq2+5e2⊥+e4⊥ (41) +2τ(τ−1M−iω)e4∥(1+3e2⊥)2−e6∥sin2ϕq2+5e2⊥+e4⊥C00(q,ω),
 Cyy(q,ω)=2τe2∥2−e4∥sin2ϕq2+5e2⊥+e4⊥ (42) +2τ(τ−1M−iω)e4∥(1+3e2⊥)2−e6∥cos2ϕq2+5e2⊥+e4⊥C00(q,ω),

where angle in Eqs. (41) and (42) indicates the Cooperon momentum direction: .

Let us analyze Eqs. (39) – (42). The symmetry breaking has a three-fold effect on the Cooperons. First, it results in a relaxation gap in the singlet Cooperon (39), which implies suppression of the quantum interference for times larger than (even in the absence of the phase breaking, i.e. for ). Second, the symmetry breaking affects the diffusion constant in Eq. (40). The diffusion constant renormalization comes from the off-diagonal Cooperons. (41) In the absence of the symmetry breaking (i.e. for ) one finds (41); (63) with . Finally, the expressions for the triplet Cooperons (41) and (42) contain additional terms , remaining finite in the limit . Despite the smallness of the parameter , these terms give a noticeable contribution to the net conductivity correction. We will return to this point when discussing Eq. (53) in the next subsection.

### iii.4 Hikami boxes and net conductivity correction

We now turn to the evaluation of the Hikami boxes for the conductivity corrections, shown in Fig. 5b. With the Cooperon defined by Eq. (23) the first and second diagrams in Fig. 5b correspond to the following analytical expressions:

 δσ(1)xx=e2ℏπNτ2∫dϵ2πω[f(ϵ)−f(ϵ+ℏω)] ∫dq(2π)2Cββ′γγ′(q,ω) (43) ×∫dk(2π)2(GAk,ϵVxkGRk,ϵ+ℏω)γ′β(GRq−k,ϵ+ℏωVxq−kGAq−k,ϵ)γβ′,
 δσ(2)xx=e2ℏ2π2N2τ0τ2 ∫dϵ2πω[f(ϵ)−f(ϵ+ℏω)]∫dq(2π)2Cββ′γγ′(q,ω) (44) ×∫dk(2π)2∫dk′(2π)2(GAk,ϵVxkGRk,ϵ+ℏωGRq−k′,ϵ+ℏω)γ′β(GRq−k,ϵ+ℏωGRk′,ϵ+ℏωVxk′GAk′,ϵ)γβ′,

where is the matrix current vertex renormalized by disorder (11) in the usual ladder approximation (see, e.g. Ref. (62) and diagram in Fig. 5d) and is the Fermi distribution function. We will skip the details regarding the third diagram in Fig. 5a since it finally gives the same result as Eq. (44).

Evaluation of the integrals in Eqs. (43) and (44) yields (see, also, Appendix B):

 δσ(1)xx=e2Dτ2hv2F¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯[(σ0+\boldmathσekF⋅n)VxkF⋅n(σ0+\boldmathσekF⋅n)]γ′β[(σ0+\boldmathσe−kF⋅n)Vx−kF⋅n(σ0+\boldmathσe−kF⋅n)]γβ′∫dq(2π)2Cββ′γγ′(q), (45)
 δσ(2)xx=e2Dττ8hv2Fτ0 × ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯[(σ0+\boldmathσekF⋅n)VxkF⋅n(σ0+\boldmathσekF⋅n)]γ′β1(σ0+\boldmathσe−kF⋅n)γγ1 (46) × ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(σ0+\boldmathσe−kF⋅n′)β1β[(σ0+\boldmathσekF⋅n′)VxkF⋅n′(σ0+\boldmathσekF⋅n′)]γ1β′∫dq(2π)2Cββ′γγ′(q),

where the bar denotes averaging over the momentum directions on the Fermi surface: . We note that the correction is entirely due to the carrier chirality. If we omit the chirality matrix in Eq. (46), the independent averaging of the current vertices gives .

The renormalized vertex acquires the standard prefactor , where is the transport scattering time:

 1τtr = Nℏ∫dϕn′(1−n⋅n′)ζkF|n−n′| (47) × 1+e⊥n⋅e⊥n′+e∥n⋅e∥n′2 = 1+3e2⊥2τ0. (48)

and satisfies the identity

 (σ0+\boldmathσekF⋅n)VxkF⋅n(σ0+\boldmathσekF⋅n) = 2τtrτvFnx (49) × (σ0+\boldmathσekF⋅n),

which helps to perform the averaging in Eqs. (45) and (46). The conductivity corrections and can then be expressed in terms of the singlet and triplet Cooperons as follows

 δσ(1)xx = 2e2Dτπℏ(τtrτ)2∫dq(2π)2[2¯¯¯¯¯¯n2x(1−e2⊥)C00(q) (50) − 2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯n2x(1−e2y)Cxx(q)−2¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯n2x(1−e2x)Cyy(q)],
 δσ(2)xx = −2e2Dτπℏτ2trττ0e2∥(¯¯¯¯¯¯n2x)2∫dq(2π)2 (51) × [(1+e2⊥)C00(q)−Cxx(q)−e2⊥Cyy(q)],

where . We have also summed up the contributions of both Dirac cones of the HgTe QW spectrum, which yields the factor of in front of the integrals. Noticing further that on average over the directions of the triplet Cooperons (41) and (42) coincide, we express the net conductivity correction in the form:

 δσxx=δσ(1)xx+2δσ(2)