Anomalous quantization trajectory and parity anomaly in Co cluster decorated BiSbTeSe2 nanodevices

Dirac Fermions with different helicities exist on the top and bottom surfaces of topological insulators, offering a rare opportunity to break the degeneracy protected by the no-go theorem. Through the application of Co clusters, quantum Hall plateaus were modulated for the topological insulator BiSbTeSe2, allowing an optimized surface transport. Here, using renormalization group flow diagrams, we show the extraction of two sets of converging points in the conductivity tensor space, revealing that the top surface exhibits an anomalous quantization trajectory, while the bottom surface retains the 1/2 quantization. Co clusters are believed to induce a sizeable Zeeman gap ( > 4.8 meV) through antiferromagnetic exchange coupling, which delays the Landau level hybridization on the top surface for a moderate magnetic field. A quasi-half-integer plateau also appears at −7.2 Tesla. This allows us to study the interesting physics of parity anomaly, and paves the way for further studies simulating exotic particles in condensed matter physics.

, we repeated the results which we discussed carefully for the anomalous quantization trajectory. Here, the anomalous RGFD trajectory is repeatable. These data are measured at T = 1.8 K.
Supplementary Figure 4. The RG flow derived from the topological non-linear Sigma model. Different from the Co decorated samples ( Fig. 2d and Supplementary Fig. 3), here the RGFD of Cu clusters decorated sample (sample D) shows the normal trajectory, which is similar to the undecorated sample ( Fig. 2a, In Supplementary Fig. 3 Another problem is that at what extent the disorder will destroy the dissipationless transport. As we all known, the QH state has robustness against disorder, but too much disorder will destroy QH state. Here, we cannot find out the amount of disorder brought by the Co clusters. But considering that we have repeated and obtained the similar anomalous RGFD behavior in three independent samples, it is believed that the phenomenon observed here is not a coincidence but is indeed within an experimentally realistic parameter region. The main purpose of this work is focused on the separation of the quantization behavior of the top and bottom surface states. In order to do so, we have to introduce certain amount of disorder that can indeed separate the two surface states, and on the other hand does not destroy the dissipationless transport. With regard to the question, i.e., under what parameter conditions will the QH behavior be preserved, is beyond the scope of the current work.

Supplementary
This is an interesting topic that we wish to give quantitative answer in our further research.

Supplementary Note 3. Topological θ-term in the effective non-linear sigma model.
The robustness of quantized Hall conductance against disorder in the two dimension electron gas (2DEG) has been well studied by previous literature. The robustness and the renormalization group flow can be understood through an effective topological field theory around the mean-field saddle point, which is a non-linear sigma model with a topological θ-term. However, the effective field theory to explain the QH states in the TI devices is still in absence. To make our work as complete as possible, we now give a derivation of the topological field theory of the TI surface with disorder and under magnetic field, the result of which can explain the RGFD satisfactorily in our experiment.
In our experiment, we first investigated the undecorated sample under magnetic field.
In this case, the Dirac fermion with a U(1) gauge field in the Landau gauge can be written as The corresponding Dirac equation can be solved which leads to the Landau levels Hence, we have arrived at the conclusion that both the n = 0 LL and the QAH state of the TI surface are associated with the bulk topology characterized by C = 1/2. Now we are willing to consider the effect of disorder on the TI surface with magnetic field. It is known from the QH effect in the 2DEG that the fluctuation around the mean-field saddle point is usually described by a nonlinear sigma model in terms of the sigma field Q. Furthermore, the nontrivial bulk topology discussed above will give rise to a topological θ-term. For the 2DEG, the θ-term is found to be 2, 3 σ xy r x , y . (S.4) From this topological θ-term, one can arrive at the RGFD in the low energy window.
We now consider the corresponding topological θ-term correction due to the zeroth LL in our undecorated sample. Since the zeroth LL shares the same bulk topology with the QAH state, we start with the QAH state in the TI surface. In our explanation of the anomalous RGFD, the Zeeman effect in the absence of Co clusters is ignored. This is based on our experimental observation. In our experiment, we have measured and obtained the renormalization group flow diagram (RGFD) for both the decorated and undecorated sample, where the anomalous quantization trace is found to exist only in the decorated sample. According to our analysis, a significant Zeeman gap will lead to the suppression of hybridization of neighboring LLs, and will result in the anomalous quantization trace. However, no anomalous quantization trace occurs in the undecorated sample, which suggests that the Zeeman effect in the undecorated sample should be insignificant.
Although the Zeeman gap in the Bi 2 Se 3 and Sb 2 Te 2 Se has been measured 6 to be around several meV. We still expect that the Zeeman gap in BSTS could be much smaller. This is due to the following reasons.
First, the Zeeman gap under magnetic field for the undecorated sample reads as where g s is the Lande g-factor. The g-factor of topological surface state is still a highly debated question. It is reported 6 that the estimated g-factor in Bi 2 Se 3 and Sb 2 Te 2 Se is +18 and -6 respectively. However, it is worthwhile to note that the g-factor of the topological surface state is completely different from that in the bulk and is highly material dependent. The different elements will induce different orbital characters of the wave function of the surface state, leading to completely different g-factors. Hence, even though Bi 2 Se 3 and Sb 2 Te 2 Se have been investigated 6 , the g-factor of the material BSTS studied in our work is still unknown. To be more comprehensive, we can resort to some relevant works. In the study of the shift of zeroth LL 7 in Sb 2 Te 3 , through the experimental scanning tunneling spectrum, it is found that, for the undecorated material, the zero-mode deviation at 7 Tesla is at most around 1 meV, which is relatively small and neglected compared to the exchange term due to magnetic decoration. On the other hand, while studying the quantum oscillation in topological surface state 8 of Be 2 Te 2 Se, it is found that the measured n-1/B curve does not show any significant deviation from a straight line, justifying that the Lande g-factor should be no larger than 2. The small g-factor leads to about 1meV shift of the zeroth LL due to magnetic field. More importantly, in BiSbTeSe 2 , it has shown that there is also no obvious shift of the zeroth LL 9 . Therefore, this is a strong suggestion that the g-factor in BSTS may be much smaller than that in Bi 2 Se 3 and Sb 2 Te 2 Se.
Second, from the experimental data of the RGFD, we can estimate the magnitude of the Zeeman gap due to Co decoration. The estimated magnitude has a safe lower bound which is 4.8meV. The actual Zeeman gap due to decoration could be much larger than the lower bound. Therefore, the Zeeman effect after decoration should be much more significant than that in the bottom surface (without decoration), which is around 1meV (or 3meV at most).
Due to the above reasons, we do not take the Zeeman gap in the bottom surface into account, which is believed to be not important to account for our experiment.