Role of helical edge modes in the chiral quantum anomalous Hall state

Although indications are that a single chiral quantum anomalous Hall(QAH) edge mode might have been experimentally detected. There have been very many recent experiments which conjecture that a chiral QAH edge mode always materializes along with a pair of quasi-helical quantum spin Hall (QSH) edge modes. In this work we deal with a substantial ‘What If?’ question- in case the QSH edge modes, from which these QAH edge modes evolve, are not topologically-protected then the QAH edge modes wont be topologically-protected too and thus unfit for use in any applications. Further, as a corollary one can also ask if the topological-protection of QSH edge modes does not carry over during the evolution process to QAH edge modes then again our ‘What if?’ scenario becomes apparent. The ‘how’ of the resolution of this ‘What if?’ conundrum is the main objective of our work. We show in similar set-ups affected by disorder and inelastic scattering, transport via trivial QAH edge mode leads to quantization of Hall resistance and not that via topological QAH edge modes. This perhaps begs a substantial reinterpretation of those experiments which purported to find signatures of chiral(topological) QAH edge modes albeit in conjunction with quasi helical QSH edge modes.

Although, the experiment depicted in ref. 1 is most probably a detection of a single topological chiral quantum anomalous Hall(QAH) edge mode. There have been some other quite recent experiments [2][3][4] where it has been reported that QAH edge modes occur in conjunction with quasi helical quantum spin Hall(QSH) edge modes 5 . The latter are prone to backscattering and are nothing but QSH edge modes which occur in a trivial insulator. These experiments which "see" QAH edge modes are in fact designed out of QSH edge mode setups in a topological insulator. By applying an extra Ferromagnetic layer or otherwise, an energy gap is sought to be created between the pair of helical edge modes in a QSH sample splitting these modes away from each other and suppressing one of these leads to a single chiral QAH edge mode in a sample. However, contrary to expectation it is not just a chiral QAH mode which was seen in those experiments [2][3][4] but it always comes with the additional baggage of quasi-helical QSH edge modes in the trivial phase 5 .
Helical QSH edge modes from which these chiral QAH edge modes evolve not only occur in topological insulators but they also do occur in a trivial insulator. Now applying a similar technique as before or attaching a ferromagnetic layer to a trivial insulator, we can again make the trivial helical edge modes evolve into chiral QAH edge modes. But in the latter case, the chiral QAH edge mode so produced wont have a topological character and therefore this chiral QAH edge mode won't be protected against backscattering. Now this begs the question how can one be sure of the topological character of QAH edge modes.
Another question which can crop up is, does the topological nature of the QAH edge modes which evolve from helical QSH edge modes in a topological insulator survive the evolution. This "evolution" from helical QSH to chiral QAH edge mode as has been described in refs [2][3][4] is via addition of magnetic impurities or a ferromagnetic layer. This may destroy their topological character since helical QSH edge modes are susceptible to spin flip scattering in presence of magnetic impurities. In this context our work becomes relevant, since in those QAH experiments what is quite evident is that the quantization of Hall resistance is attributed to chiral topological QAH edge modes which exist in combination with quasi helical QSH edge modes. What our work reveals is that a chiral trivial QAH edge mode which exists in combination with quasi helical QSH edge modes gives the quantization of Hall resistance and not the chiral topological QAH edge mode when combined with trivial QSH edge modes. Thus a shadow of doubt creeps up regarding the interpretation of those experiments 5 .

Two terminal and three terminal QAH samples
The Landauer-Buttiker formalism 10-12 relating currents with voltages in a multi terminal device has been extended to QSH edge modes in refs 13,14 as well as QAH samples in ref. 5 : In the above equation, V i is the voltage at i th probe/contact/terminal (we will be using the term probe or contact or terminal interchangeably for the same thing, i.e., a reservoir of electrons at some fixed potential) and I i is the current passing through the same terminal. T ij is the transmission probability from the j th to i th probe and G ij is the conductance. N denotes the number of terminals in the system.

Chiral(topological) QAH edge mode (2 terminal).
The case of a single chiral(topological) QAH edge mode is represented in Fig. 1(a). The relations between currents and voltages at the two terminals are derived from conductance matrix (2): Chiral QAH edge mode with trivial QSH edge modes (2 terminal). Here we have considered the general case, where the spin-flip scattering parameter f 0 between QAH chiral and trivial helical edge modes, will decide whether the QAH edge mode is topological (if f 0 = 0) or trivial (if f 0 ≠ 0). While f defines the spin-flip scattering between the trivial helical edge modes. This situation of topological chiral plus trivial helical QSH is shown in Fig. 1(b) while case of trivial chiral QAH plus trivial helical QSH is shown in Fig. 1(c). The current-voltage relations can be easily derived from conductance matrix below:  Fig. 1(d). The relations between currents and voltages at the various terminals are derived from the conductance matrix below: Since probe 2 is the voltage probe, the current through probe 2-I 2 is zero. We further choose reference potential  Fig. 1(e). The current-voltage relations can be derived from the conductance matrix below: . Thus from looking at just 2T and 3T samples it is quite evident that the trivial QAH case crosses over to the the single chiral QAH case and not the topological QAH.

Four Terminal quantum anomalous Hall bar
The four terminal sample is shown in Fig. 2. We calculate the Hall resistance R H = R 13,24 , the local (two probe) resistance R 2T = R 13,13 and the non-local resistance R NL = R 14,23 for various cases starting with just a single chiral(topological) QAH edge mode, then the chiral(topological) QAH edge mode with trivial quasi-helical QSH edge modes and finally the case of chiral(trivial) QAH edge mode with trivial quasi-helical QSH edge modes. Chiral topological QAH edge mode. Effect of disorder. Herein we consider two of the contacts (2, 4) to be disordered, see Fig. 2(a). Relations between the currents and voltages at the various terminals can be deduced from the conductance matrix, given below:   Disorder has no effect on the topological chiral QAH edge mode, the Hall resistance and local resistance remain the same as in the ideal (zero disorder) case. Finally, to calculate the non-local resistance R NL we consider 2, 3 as voltage probes and 1, 4 as current probes, we get V 2 = V 3 which gives R NL = 0. Thus disorder has no effect on a single chiral(topological) QAH edge mode.
Effect of disorder and inelastic scattering. Similar to before, we consider two of the contacts (2, 4) are disordered, see Fig. 2 . Similarly electrons coming from probe 3 are equilibrated with the electrons entering from probe 4 to a new energy as shown below- The currents and voltages at the contacts from 1 to 4 are related by the equations- Choosing reference potential V 3 = 0 and I 2 = I 4 = 0, since 2 and 4 are voltage probes, we thus derive V 2 = V 1 and . So inelastic scattering too, like disorder at voltage probe has no effect on the a single chiral(topological) QAH edge mode.

Chiral (topological) QAH edge mode with trivial QSH edge modes. Effect of disorder.
Herein, as before we consider two of the contacts 2 and 4 to be disordered, see Fig. 2(b). The relations between currents and voltages at the various terminals can be obtained from the conductance matrix below: T  T  T  T  T  T  T  T  T  T  T  T  T  T  T . By interchanging R 2 and R 4 in the above expressions for T 11 , T 12 , …, T 23 rest of the transmission probabilities T 31 to T 44 can be deduced. The transmission probabilities are calculated in this way-say T 23 , the transmission probability of electron from terminal 3 to 2 can be explained as the sum of paths available from 3 to 2 for one chiral topological edge mode and one pair of trivial helical edge modes. An electron in the topological edge mode coming out of probe 3 has probability zero to reach probe 2. But an electron in the trivial helical edge mode has finite probability to reach probe 2 from 3. An electron coming out of probe 3 can reach probe 2 with probability T 2 (1 − f), but that is just one path, it can also reach 2 with probability fR 2 T 2 (1 − f) following a second path due to spin flip scattering, similarly a third path is . Thus, we can form an infinite number of paths from probe 3 to 2, these can be summed to get the total transmission prob- . Similarly, the other transmission probabilities in Eq. 10 are obtained. Choosing reference potential V 3 = 0, and since 2 and 4 are voltage probes, we derive the local (two probe) resistance in absence of disorder ( . Similarly, as before non-local resistance is deduced as = For general case (i.e., with disorder) the expressions for R H , R 2T and R NL are too large to be reproduced here, so we will analyze them via plots, see Fig. 3

(a-d).
Effect of disorder and inelastic scattering. Herein we consider the effect of both disorder and inelastic scattering on topological QAH edge modes as shown in Fig. 4(a). Here the inelastic scattering is shown by starry blobs as in Fig. 4(a). As the QAH edge mode is topological, it will not equilibrate its energy with trivial helical edge modes. Thus, topological chiral edge modes equilibrate only between themselves, these equilibrate to energy ″ V i where i is the contact index from 1 to 4. The trivial helical edge modes equilibrate with other trivial helical edge modes and these equilibrate their energy to ′ V i . The contacts 2 and 4 are disordered as in the previous case. The currents and voltages at the contacts from 1 to 4 are related by the equations- where the potentials ′ V i and ″ V i are related to V i by-  Chiral (trivial) QAH edge mode with trivial QSH edge modes. Effect of disorder. Herein as before we consider two of the contacts 2 and 4 to be disordered, see Fig. 2(c). The current-voltage relations are derived from the conductance matrix below: T  T  T  T  T  T  T  T  T  T  T  T  T  T  T  = . T 1 0 0 . By interchanging R 2 and R 4 in the above equation (15) rest of the transmission probabilities T 31 to T 44 can be deduced. The transmission probabilities can be explained as before. T 23 , the transmission probability of electron from terminal 3 to 2 can be explained as the sum of probabilities from 3 to 2 for all the edge modes over all possible paths. An electron coming out of probe 3 at upper edge can reach probe 2 with probability T 2 (1 − f − f 0 ), but that is just one path, it can also reach 2 with probability (f + f 0 )R 2 T 2 (1 − f − f 0 ) following a second path due to spin flip scattering, similarly probability for a third path is

R NL
Triv (5 6( ) 2( ) ) (7 9( ) 3( ) ) 0 0 0 0 0 2 0 0 2 for zero disorder. For, general case the expressions for R H , R 2T and R NL are again too large to be reproduced here, so we will examine them via plots, see Fig. 3(a-d).  Effect of disorder and inelastic scattering. Herein, we consider the trivial QAH edge modes with both disorder and inelastic scattering as shown in Fig. 4(b). Here the QAH chiral as well as helical both edge modes are in the trivial phase, i.e., they are all prone to intra edge back scattering due to spin-flips. All the edge modes interact among themselves leading to their energies being equilibrated to the potential ′ V i ('I' is from 1 to 4). The contacts 2 and 4 are disordered as in the previous case. The currents and voltages at the contacts from 1 to 4 are related by the equations- 14,23 . The expressions for R H , R 2T and R NL are large, so again we will analyze them via plots as in Fig. 5(a-d).
In Table 1 we tabulate the results obtained so far. One important thing left out of our discussion so far has been the role of spin in QAH edge mode. A single chiral (topological) QAH should satisfy the following symmetry relations for R H (↑) = −R H (↓) and R NL = 0. We see that R NL (↑) ≠ R NL (↓) for topological QAH (with quasi helical QSH) while this isn't case for trivial QAH (with quasi helical QSH) edge modes, see Fig. 6 . In fact we see that R NL Triv for the case of disorder and inelastic scattering approaches zero similar to a single chiral QAH edge mode, while for the R NL Topo this doesn't, again leading to a contradiction with the way the experiments of ref. 2 have been interpreted as in ref. 5 .

Six terminal quantum anomalous Hall bar
In this section we analyze a six terminal QAH bar, we especially focus on the longitudinal resistance R L . For a single chiral(topological) QAH edge mode R L = 0, but the experiments 2-4 revealed a finite longitudinal resistance. This result prompted the interpretation of the experiments 2-4 as seeing not just a chiral(topological) QAH edge mode but in addition also a pair of quasi-helical QSH edge modes 5 . Since a non zero R L is the hallmark of helical QSH edge modes. Here we probe further by comparing as in sections before the three cases and try to find out if a topological QAH edge mode or a trivial QAH edge mode occurring with quasi-helical edge modes results in a non-zero R L .

Effect of disorder and inelastic scattering.
Herein we consider the effect of disorder and inelastic scattering on the various resistances for the sample as shown in Fig. 7(c). The contacts 1 and 4 are disordered as in the previous case. The currents and voltages at the contacts from 1 to 6 are related by the equations- , The expression for longitudinal resistance in the general case of arbitrary disorder are quite large so we examine it via plots as in Fig. 8(b). One thing is quite clear from Fig. 8(b), the case of trivial QAH edge mode with QSH quasi-helical goes over to single chiral(topological) QAH edge mode rather than the topological QAH edge mode with QSH quasi-helical edge modes. This behavior replicated in the four terminal case too calls for a reinterpretation of the experimental results 2,3,5 . In Table 2 we tabulate the results obtained for various cases for the longitudinal resistance. We also focus on the change due to change in magnetization from ↑ to ↓. There is a symmetry R L (↑) = R L (↓) for trivial QAH edge mode which does not hold for a topological QAH edge mode with quasi helical QSH edge modes. This response of the trivial QAH edge mode is again in line with what was experimentally seen.

Conclusion
We conclude by analyzing the Tables 1 and 2. We see that the trivial(chiral) QAH edge mode with trivial quasi-helical QSH edge modes is more closer to the experimental situation, as interpreted in ref. 5 than the topological(chiral) QAH edge mode with trivial quasi-helical QSH edge modes is. This implies a reevaluation of the consensus regarding those quantum anomalous Hall experiments [2][3][4] . Perhaps, something else is happening and maybe these are not true chiral(topological) quantum anomalous Hall edge modes which were seen, but rather what could be described as chiral(trivial) QAH edge modes.

Data availability statement.
As this is an analytical work, all the data needed to plot the Figs 3, 5, 6 and 8 are generated from the expressions shown in the manuscript.