Evolutions of helical edge states in disordered HgTe/CdTe quantum wells

We study the evolutions of the nonmagnetic disorder-induced edge states with the disorder strength in the HgTe/CdTe quantum wells. From the supercell band structures and wave-functions, it is clearly shown that the conducting helical edge states, which are responsible for the reported quantized conductance plateau, appear above a critical disorder strength after a gap-closing phase transition. These edge states are then found to decline with the increase of disorder strength in a stepwise pattern due to the finite-width effect, where the opposite edges couple with each other through the localized states in the bulk. This is in sharp contrast with the localization of the edge states themselves if magnetic disorders are doped which breaks the time-reversal symmetry. The size-independent boundary of the topological phase is obtained by scaling analysis, and an Anderson transition to an Anderson insulator at even stronger disorder is identified, in-between of which, a metallic phase is found to separate the two topologically distinct phases.


INTRODUCTION
Disorder effect is one of the most important problems in the subject of topological insulators (TI) 1 . Recently a nonmagnetic disorder-induced topological insulator state, named as the topological Anderson insulator (TAI), is found by computer simulations in both two-2 and three-dimensional (3D) 3 systems, and is further confirmed through independent simulations by studying the local currents 4 . This state is numerically characterized by precisely quantized conductance e 2 /h 2-4 , and is theoretically understood as generated by the negative renormalized topological mass 3,5 within the self-consistent Born approximation (SCBA). Phase diagrams for these systems on the energy-disorder plane at fixed mass parameter are obtained 2,3,5 , where the TAI phase appears as an island between the so-called weak-and strongdisorder boundaries 5 . The locations of these boundaries depend generally on the size of the system under simulation. Topological invariants in disordered systems are also discussed 6,7 . In particular, it is argued that the 2D TAI phase is not a distinct one, but is instead part of the quantum spin Hall phase 6 with nontrivial spin-Chern number 8 in the presence of disorder if the phase diagram is extended to the mass axis.
Despite all these achievements on understanding the surprising disorder-induced topological phase, there still exist some questions. First, it has been experimentally proved that the quantized conductance, 2e 2 /h, in clean 2D HgTe/CdTe quantum wells (QWs) 9 measured in transport studies 10 is the direct consequence of gapless conducting edge states. However in the disorderinduced TI, the existence of such edge states in energy- * E-mail: liuqin@mail.sim.ac.cn momentum space, their evolution with disorder strength, as well as how the evolution picture corresponds to the wave-function behaviors in real space have not been well investigated yet. Second, when deviating the TAI region, how these edge states terminate with the increase of the disorder strength and what is their difference from that of the usual bulk states are also not resolved so far. Finally, as mentioned in Ref. 5 , the weak-disorder boundary is not an Anderson transition at all as the name TAI might suggest, in the sense that it is more of a band-effect rather than the result of a mobility edge. Therefore the Anderson transition boundary, which is expected to be the true size-independent phase boundary of TAI in strongdisorder region, is still absent for now.
Motivated by these issues, in this paper, we study the disorder-induced conducting edge states and the phase boundaries of the TAI in the effective model of disordered HgTe/CdTe QWs. First, the existence of gapless helical edge states is demonstrated in energy-momentum space through a gap-closing phase transition when above a critical disorder strength. This result appraises the topological aspect of the TAI in a straightforward way, and is the fundamental reason of the observed quan-tized anomalous conductance plateaus in previous transport studies [2][3][4][5] . Second, the evolution of such disorderinduced edge states with disorder strength is also studied, where how the edge states decay in strong-disorder region is focused on in particular. It is shown that for a finite-width system, unlike the usual exponential decay in quasi-1D disordered systems 11 , the nontrivial edge states decay in a stepwise pattern due to the competition between the sample width L y and the localization length ξ(W ). Finally, we locate the true Anderson transition boundaries by scaling analysis for an infinite system. These boundaries indicate that the system undergoes a multiple transitions, first from metal to topological insulator, then to metal again, and finally localizes as an Anderson insulator. It is interesting to compare with the work by Yamakage 12 et al that our result shows that a spin-s z nonconservation term is not necessarily needed to have a metallic region which separates two topologically distinct insulating phases.
The rest of this paper is organized as follows. In section II, we introduce our simulating model and methods. In section III, we present our numerical results and theoretical analyses. Finally, this work is concluded in section IV.

II. MODEL AND METHOD
Our starting point is the 2D effective Hamiltonian of HgTe/CdTe QWs 9 with on-site nonmagnetic disorders. In tight-binding representation on a square lattice, it gives 4 where ψ i = (ψ is↑ , ψ ip↑ , ψ is↓ , ψ ip↓ ) T is the field for the four orbital states, |s, ↑ , |p x + ip y , ↑ , |s, ↓ , |−p x + ip y , ↓ , on site i = (m, n), and ǫ i and t x(y) are respectively the onsite energy as well as the overlap integral matrices along x(y) direction, which are explicitly In the above, a is the lattice constant, A,B,C,D,M are material parameters depending on the QW width 9 , ǫ s(p) = C ± M − 4(D ± B)/a 2 , and ∆ i is the on-site disorder energy for nonmagnetic impurities, which is identical for the four orbitals and uniformly distributed in [−W/2, W/2] with disorder strength W . To compare with the previous works, we use the same values for the parameters as in Refs. 2,4 , i.e., A = 364.5 meV·nm, B = −686 meV·nm 2 , C = 0, D = −512 meV·nm 2 , M = 1 meV and a = 5 nm. The method of twisted boundary conditions 13,14 is used in this work to directly diagonalize the disordered system, and the transfer matrix method 11 is taken to obtain the wave-function distributions.
Without disorder, the mass parameter M characterizes the effect of band inversion by changing its sign from positive to negative, and results in a topological phase transition from a conventional insulating state (M > 0) to a quantum spin Hall phase (M < 0) with a single pair of helical edge states at each boundary. This pair of helical edge states is clearly seen in energy-momentum space by diagonalizing the clean Hamiltonian (setting ∆ i = 0 in Eq.(1)). However, for disordered systems, the translational symmetry is generally broken and the momentum is not a good quantum number anymore. But if we view the disordered system of size L x ×L y as a cell of an infinite one, and fold it into a cylinder with longitudinal axis in y-direction, then the translational symmetry is restored, where a particle gains an extra phase factor e iφx whenever it goes across the boundary in the x-direction. In such a case, −π ≤ φ x ≤ π, plays the role of the generalized momentum, and we could diagonalize the disordered system in E − φ x plane and study the edge properties as a function of W . It should be noted that the choice of L x should be larger than the average decay length of the wave-functions. It is found that L x = 15a is good enough for diagonalization, and we set L y = 100a all throughout the paper unless mentioned in particular.
Wave-functions study takes its special advantages to gain insights of how a conventional insulating state transits into a topological nontrivial phase and vice versa. To obtain the wave-functions in real space, we consider a two-terminal setup as seen in Fig.1, and adopt the standard transfer matrix method 11 with the iteration equation along x direction written as where Ψ(m) = (ψ m1 , ψ m2 , · · · , ψ mLy ) T is the wavefunction vector at slice x = m, and M (m) is the corresponding transfer matrix. In the region of disordered HgTe/CdTe QWs, the transfer matrix reads explicitly where E f is the Fermi energy of the clean system, I and O are respectively the 4L y × 4L y unit and null matrices, and is the block diagonal overlap matrix with the same di-

mension, while
is the block-tridiagonal slice Hamiltonian with open boundary conditions in y-direction. By applying a small external bias, we can calculate the longitudinal conductance using the Landauer-Büttiker formulae 11,15 as well as the wave-functions on each site. Fig.2(a) shows the TAI phase diagram reproduced by the transfer matrix method, which is in good consistent with the previous works 2, 5 .
In the following, we first explore the evolution of the energy spectrum in the generalized momentum space with disorder strength W , and then study the topological phase transition in the viewpoint of real space wavefunctions. The obtained results agree with each other well, which also validates our numerical methods.

A. Overview of the phase diagram
Let us first review the numerical and theoretical results of the TAI phase diagram 2,5 , which is reproduced in Fig.2(a) with the stripe length L x = 500a.
Numerically the TAI phase is identified by the suddenly appeared quantized conductance, 2e 2 /h, as the in-creasing of W at fixed E f . See the crescent-shaped region in Fig.2(a), where the statistical fluctuations are vanishing small. This region has a clear-cut weak-disorder boundary at which the TAI phase begins but a blurred strong-disorder boundary where the TAI phase terminates.
Theoretically the transitions into the TAI phase at the weak-disorder boundary can be understood within the the SCBA 3,5 . It is demonstrated that the random potential renormalizes both the Fermi energy and the mass parameter as a function of disorder strength,Ē f (W ) andM (W ), so that starting from a metallic phase, the mass changes sign when acrosses some critical disorder strength, and the TAI phase is defined by the combination of conditionsM < 0 as well as −|M | <Ē f < |M |. In Fig.2(a), the boundaryM (W ) = 0, which separates the positive and negative effective mass is plotted as blue dashed line, and the band edges,Ē f = ±M , obtained from the SCBA are shown in cyan. It is seen that the results predicted from the SCBA agree very well with the weak-disorder boundary of the TAI phase obtained from our numerical simulations. However the SCBA fails to give the correct strong-disorder boundaries, as seen by the dotted parts of the cyan lines.
Physically the strong-disorder boundary is blurred because in this region there are impurity states enter into the renormalized bulk gap, which embodies in the property that the self-energy has a nonvanishing imaginary part 3 . In Fig.2(a) regions by whether the band edges are effectively defined or not. If they are not, the mobility edges should play the role. Finally we note that the strong-disorder boundary is expected to be an Anderson transition boundary 3,5,7 , however a direct scaling analysis of Anderson transitions is still absent, which we will present below.

B. Evolutions of edge states with disorder strength
As mentioned above, the observed quantized conductance plateau in TAI phase is attributed to the presence of conducting edge states induced by disorder. However, the existence of such edge states has never been shown directly in the TAI's energy spectrum. In this section, we first diagonalize the HgTe/CdTe QWs for a specific disorder realization with increasing disorder strength W (meV). The twisted and open boundary conditions are used respectively in x and y directions. The obtained energy spectrum is in E −φ x plane, and five representatives with W = 0, 40, 80, 120 and 200 are given in Fig.2(b) 16 .
In the clean limit where W = 0, we see that the spectrum exhibits a topological-trivial feature with a full bulk gap E g ≃ 2 |M | (the derivations are due to the finite size of L y ), which inversely proves the validity of our method. With the increasing of the disorder strength, say to W = 40, it is seen that though the system still behaves like a normal insulator, there are trivial edge states (see the dashed purple lines) grown up from the lower band and the bulk gap becomes much narrower. In fact, we have observed the closing of band gap around W c = 50 (for our particular disorder realization) and its reopening immediately at a bit larger disorder strength, but with the presence of two pairs of gapless edge states. Since the phase transition necessarily accompanies with gap-closing 17 , we speculate that the system transits into the TAI phase for W > W c , and a typical spectrum after this transition is exemplified at W = 80. The observation of the gapless edge states with nonvanishing disorders in generalized momentum space provides the most direct evidence so far that it is the disorder-induced edge transport which leads to the quantized conductance in TAI phase. Moreover, these gapless edge states persist to even stronger disorder strength, nevertheless the low energy states begin to localize and smear the band edges, see the W = 120 figure for example, which is in good consistent with the phase diagram. Interesting thing happens in the strong-disorder region. With strong-disorder, naively we would expect that all states are localized and the system becomes an Anderson insulator. However, the energy spectrum at W = 200 shows that although the bulk gap is completely smeared by the impurity states and the band edges are ill-defined, the edge states are still robust and winding around the projected Brillouin zone.
To compare with the helical properties 18 of the edge states in clean HgTe/CdTe QWs, we have studied the edge states with disorder in detail by analyzing their eigen-functions. The results are shown in the sub-figure with W = 80. The edge states are labeled according to their density distributions ρ(y) = dxΨ † ( i)Ψ( i) as well as the spin polarization s(y) = dxΨ † ( i) σΨ( i), where the solid (empty) circles indicate the states exponentially localized near the y = 0 (L y ) boundary, and the color red (blue) represents the up (down) spin polarization. The helical character of these edge states for a given energy is clearly seen, where there is a single Kramer's pair at each edge with spin-momentum locking. The whole spectrum preserves the time-reversal symmetry by observing that E nα (φ x ) = E nᾱ (−φ x ), where n is the band index and α =↑, ↓, which is a result of nonmagnetic disorders. Whereas the spin degeneracy is lifted due to the breaking of inversion symmetry by disorder. At the time-reversal invariant points, φ x = 0, π, the edge states switch their Kramer's partners 19 , which leads to a nontrivial topology. This is also confirmed by calculating the Z 2 index ν 0 of this 2D system with the twisted boundary conditions on both directions following the method in Ref. 7 . All these results strongly support the statement that the transition from a normal insulator at W = 0 to TAI is a topological one, and the quantized conductance plateau originates from the disorder-induced conducting helical edge states which are intrinsic to TAI.

C. Edge states destruction at strong disorder
The robustness of the disorder-induced edge states shown in Fig.2(b) (the sub-figure with W = 200) naturally raises the questions that how will these edge states be destroyed in strong-disorder region, and what is their difference with the exponential localization of bulk states? Not like the case where the edge states themselves could be localized by magnetic perturbations, with nonmagnetic disorders, the edge states decay only through the coupling with each other. For a finite-width system, this generally depends on the competition between the localization length ξ(W ) of bulk states and the system width L y . Only if ξ(W ) ∼ L y , even localized bulk states can couple the edge states at opposite boundaries and destroy the edge states, thus we may expect something interesting to happen. While for an infinite system, only extended states can couple the edge states far away from each other, therefore a localization-delocalization Anderson transition is expected, which is the true strongdisorder boundary of TAI phase. In this section, we investigate the decay mechanism of the disorder-induced edge states for a finite-width system by studying its wavefunctions in real space. The scaling analysis of Anderson transitions for an infinite system is discussed in the next section. We first let L x = 10 3 a and fix E f = 10 meV. The logarithmic wave-function distributions before, on, and at the tail of conductance plateau are given respectively in Figs.3(a)-(b), (c)-(d), and (e). Before the TAI plateau, we see that the bulk states are weakened with the enhancement of disorder strength. When on the TAI plateau, the bulk states disappear completely and two propagating 1D edge states are seen clearly, which contribute a quantized conductance e 2 /h per edge. However, with the increase of disorder strength, the bulk states reappear and begin to localize but coexist with the conducting edge states, as seen in Fig.3(d). This is equivalent to the W = 120 case in Fig.2(b) in the generalized momentum space where the band edges are ill-defined but the mobility edge starts to count. We see that the wave-function behaviors in real space agree very well with the pictures presented above when studying the energy spectrum.
Interesting thing happens when we move to the tail of TAI plateau, as seen in Fig.3(e) where a longer length L x = 5 × 10 3 a is set. With periodic boundary conditions on both directions, only bulk states exist, which decay exponentially in the propagating direction 11 . Differently here, with open boundary conditions in one direction, we see that the edge states fade in three segments as indicated by the brown arrows, and decay much slower with localized bulk states in between. To check the interplay between these localized bulk states and the edge states, we choose two lines at n = 1 and 100 and plot the logarithmic intensities of their wave-functions in Fig.3(f). It is striking to find that the edge states decay in a stepwise pattern in contrast to the exponential way. This unusual behavior can be understood by the accidental coupling between the edge states at opposite boundaries assisted by the localized bulk states due to the fluctuations of the decay length L d . It is estimated that ξ ≃ L d ∼ 10a at this disorder strength, which is comparable to L y , and the intensities of edge states wave-functions drop off approximately at the interfaces of each segment, m = 1600 and 3800, where the local decay length, L d (m), is large. To check this, we enlarge the system width to L y = 300a,  and the corresponding edge states wave-function intensities at n = 1 and 300 are given in Fig.3(g). In this case, we have ξ ≃ L d ≪ L y , therefore a longer system length is needed for the accidental coupling between the edge states to happen. For the current simulation, it is seen that the edge states persist throughout the system, and no coupling occurs at the much larger scale of 5 × 10 3 a.

D. Scaling analysis of Anderson transitions
The above wave-function analysis shown in Figs.3(f)-(g) implies that the range of the TAI plateau is systemsize dependent. For a fixed length L x , the wider the width of the system is, the broader the conductance plateau will be. This is indeed true as seen in Fig.4 where the conductance plateaus for L y = 100a, 200a and 300a at L x = 500a are presented. What will happen as L y tends to infinite? For an infinite system, if the bulk states keep localized, where the localization length ξ has a finite value, then the edge transports, so that the conductance plateaus, will not be destroyed no matter how large the ξ is. Thus the TAI phase disappears in an infinite system only when the bulk states become delocalized and ξ is divergent. Therefore a localization-delocalization Anderson transition is expected, which kills the TAI phase in strong-disorder region.
To see the existence of such an Anderson transition, we carry out the scaling analysis of the bulk conductance using periodic boundary conditions 11 . The average logarithmic conductance, LnG , on E f − W plane is calculated for different system sizes L x = L y = L, and the average is performed under 2 × 10 3 disorder realizations.
For each E f , the Anderson transitions are recognized as the crossings of the LnG(W) lines with different L's, as seen in Fig.5. And in each region between two crossings, the system is judged to be in insulating (metallic) if LnG increases as the decrease (increase) of L at a fixed W . For example, there are three Anderson transitions at E f = 60 meV as indicated by the red lines in Fig.5, where the system first transits from metallic to insulating, which corresponds to the weak-disorder boundary of the TAI phase obtained by SCBA; then transits from this topological insulating phase to metallic again, which is the localization-delocalization strong-disorder boundary of the TAI phase we expect. After the third transition, the whole system is totally localized and becomes an Anderson insulator. The complete result is given in the inset of Fig.5, where the Anderson transitions are marked as white symbols on the phase diagram. The region enclosed by the white crosses is the TAI island, and we see that this region is much wider than the simulation result for a finite-width system. The conductance scaling at E f = 0 is also shown as blue lines in Fig.5. It is interesting to compare our results with the recent work of Ref. 12 , where only a special energy point E f = 0 is considered, and the kinetic term, Dk 2 , is omitted. It is concluded there that in the presence of an extra spin s z -conservation breaking term, the Rashba spin-orbit coupling, in the HgTe/CdTe QWs, a finite metallic region is found which partitions the two topological distinct insulating phases. However, our results demonstrate that the metallic region which separates the TAI and the Anderson insulator phases can exist by inclusion the quadratic kinetic term or tuning the E f to higher energies even with the spin s z -conservation, and the Rashba term is not necessary. This is the direct result of the localization-delocalization Anderson transition we predict. In Fig.5, the scaling behavior of conductance at E f = 0 by setting the parameter D = 0 is shown as green lines, where we see that the metallic region in-between the TAI and the Anderson insulator phases disappear indeed, which is the case discussed in Ref. 12 .

IV. CONCLUSIONS
In summary, the system of disordered HgTe/CdTe QWs is studied. The evolution of its energy spectrum in the generalized momentum space with the disorder strength is obtained, where the existence of topologically protected helical edge states in the TAI phase is demonstrated directly, which proves the conjecture that they are responsible for the observed quantized conductance plateau. With nonmagnetic perturbations which preserve the time-reversal symmetry of the system, the edge states can be destroyed only through the coupling with each other. For a finite-width system, it is shown that the edge states decay in a stepwise pattern assisted with the localized bulk states due to the fluctuations of the decay length, which is in extraordinarily contrast with the exponential decay of bulk states. While for an infinite system, the edge states can become decoherent only through the coupling with extended states, therefore a localization-delocalization Anderson transition is expected in strong-disorder region, which is confirmed by the scaling analysis. Moreover, multiple Anderson transitions are also obtained, where in particular, a metallic region which separates two topologically distinct insulating phases is found by inclusion of the quadratic kinetic term or turning E f to higher energies without an extra spin s z -nonconservation term. These results are also expected to be valid for 3D TAI 3 .