Unveiling nontrivial fusion rule of Majorana zero mode using a fermionic mode

Introduction. It has been a long-cherished dream of condensed matter physicists to find Majorana zero mode (MZM) for its potential application in fault-tolerant topological quantum computations kitaev ; nayak . Despite overwhelming evidences in favor of MZMs being reported over the last decade Fu ; sau ; fujimoto ; sato ; alicea2 ; lut ; 2DEG1 ; 2DEG2 ; kou ; deng ; das1 ; hao1 ; Marcus ; deng2 ; perge ; Yaz2 ; Jia ; Fes1 ; Fes2 ; PJJ1 ; PJJ2 , there is still room for other possible interpretations in regards to experimental results of various studies, such as the existence of Andreev bound states (ABS)Jie1 ; brouwer ; Aguado ; ChunXiao1 ; Moore ; Wimmer ; Aguado2 ; Pan ; ABSM . While it’s possible for ABS to demonstrate a great deal of similarities to MZMs in transport experiments, the two are fundamentally different particles in terms of the exchange statistics they obey, which could be exploited to discriminate them Ivanov ; alicea3 ; NQP ; Jienew .

In contrast to ABS which are ordinary fermions, MZMs follow non-Abelian statistics. This non-Abelian property reveals itself in two distinct but closely related aspects. First, in a system of MZMs, their quantum state may experience a nontrivial rotation when two of them are exchanged, unlike ABS or other Abelian anyons where the state only acquires a global phase. More importantly, two consecutive rotations in differing sequences may incur totally different final states, hence the name non-Abelian. Such rotations, or braiding operations, constitute the basic logic gates of topological quantum computations, with various theoretical studies being devoted to in the last decadesTQC2 ; TQC3 ; TQC4 ; MSQ ; net1 ; net2 ; Yu1 ; Sato ; Roy ; QD1 . Besides the braiding, fusion is another equally important non-abelian property of MZMsnayak . For MZMs, two of them can fuse into a particle of trivial type $I$ or a fermion $\Psi$, with the fusion rule formally expressed as $\gamma\times\gamma=I+\Psi$. Since the outcome is multiple, the final fusion results would rely on the fusion order of MZMs. In principle, since fusion is actually a process that couple two MZMs with finite energy, it should be easier to realize in experiment compared with braiding. Several pioneer work have suggested that fusion would induce nontrivial charge transport fusion1 ; fusion2 ; fusion3 . However, the protocols are still complicated for further investigation.

In this paper, we propose to unveil the nontrivial fusion rule of MZMs by simply attaching a fermionic mode (FM) to a single MZM. Such a setup could be possibly realized in multiple systems, for instance, in a nanowire based topological superconductor (TS) that is coupled to a quantum dot deng2 (see Fig. 1(a)), or in a topologically nontrivial vortex with a molecule attached to scanning probe microscopy acting as the FM Jievortex . The FM can usually be deemed as two pre-fused MZMs $\gamma_{A}$ and $\gamma_{B}$ with the hybridization energy $E_{d}$. As such, these two MZMs can be effectively split by tuning $E_{d}$ to zero. The fusion and splitting process between $\gamma_{B}$ and another MZM $\gamma_{1}$ in the TS can be controlled by tuning the coupling between the FM and the superconductor, as schematically depicted in Fig. 1(a). This is because MZM is its own anti-particle, and $\gamma_{1}$ only couples to half of the FM $\gamma_{B}$ Jienew . Based on these observations, different fusion and splitting processes can be realized through tuning the energy level of FM or adjusting the coupling between MZM and FM. Recent experiments have demonstrated that these parameters can be easily modulated using gate voltage Pkitaev ; Pgate1 ; Pgate2 , making our fusion protocol feasible for practical systems. Furthermore, these nontrivial fusion processes could induce significant charge transfer events, which can be precisely detected using charge sensing methods charge .

Model-independent Hamiltonian of the setup and the current formula. To simulate the fusion process of MZMs, we start from a model-independent Hamiltonian:

Nontrivial fusion loop and the related charge transfer. Due to the nontrivial fusion rule of MZMs, the fusion outcomes would rely on their fusion order. To distinguish between fusion and splitting processes, we denote the fusion of MZM $i$ and MZM $j$ as $F_{ij}$, and the splitting process as $F_{ij}^{-1}$. It is clear that when fusion and splitting of two MZMs occur consecutively, the end result is trivial, i.e., $F_{ij}^{-1}F_{ij}=I$. A minimal model illustrating the nontrivial fusion rule requires at least four MZMs. The left loop in Fig. 1(c) depicts a prototypical trivial loop for this minimal model since $F_{AB}^{-1}F_{B1}F_{B1}^{-1}F_{AB}=I$. A nontrivial fusion loop, however, requires two consecutive processes to be non-commute, such that the final state cannot return to the initial one after a complete loop. Based on this rule, many nontrivial loops can be designed by manipulating the order of fusion and splitting processes. The right loop in Fig. 1(c) illustrates a representative nontrivial loop, denoted by $F_{AB}^{-1}F_{B1}F_{AB}F_{B1}^{-1}$, which is similar to the one introduced in Ref. fusion1, . However, in this new proposal we need not tune the parameters in the TS system, which may avoid unwanted adverse effects due to the manipulation of TS wires.

To validate our proposal, we numerically calculate the charge transfer during both the trivial and the nontrivial loops. The splitting of the two MZMs that constitute the FM is controlled by $E_{d}$, which we assume to vary with time according to $E_{d}(t)=E_{0}[a+\cos(\omega t)]$ with $a=1$. This suggests that the FM would initiate with energy $2E_{0}$ at $\omega t=0$ and split at zero energy at $\omega t=\pi$. The fusion process is just the reverse of this process with $E_{d}(t)=E_{0}[1-\cos(\omega t)]$. Similarly, $t_{c}(t)=|E_{0}|[1\pm\cos(\omega t)]$ controls the splitting and fusion process of $\gamma_{B}$ and $\gamma_{1}$. At initial time we have $E_{d}=2E_{0}>0$ and thus the system is in $|0_{\text{AB}}0_{12}\rangle$ state. We may calculate the charge transfer for the nontrivial loop denoted by $F_{AB}^{-1}F_{B1}F_{AB}F_{B1}^{-1}$. According to the nontrivial fusion rule of MZMs, two MZMs would fuse either as a vacuum state $I$ or an unpaired fermionic state $\Psi$. The system’s state thus shifts to a superposition of $|0_{\text{AB}}0_{12}\rangle$ and $|1_{\text{AB}}1_{12}\rangle$, implying a half charge transferring from TS to FM. As revealed by the yellow dashed line in Fig. 2(a), a half charge indeed transfers from TS to FM after a complete loop. However, a FM state is usually not stable at the zero energy. If $E_{d}$ remains positive, the splitting process would not happen and the system simply returns to $|0_{\text{AB}}0_{12}\rangle$ state after one period. In this case, there’s no charge transfer as revealed by the red solid line shown in Fig. 2(a), even though the deviation is small (the corresponding $E_{d}(t)$ is shown by the red solid line in Fig. 2(c)). If instead the energy of the FM crosses zero, say for $a=0.9$, the FM state would switch to the occupied state $|1_{\text{AB}}\rangle$, and the whole system would choose $|1_{\text{AB}}1_{12}\rangle$ finally due to the conservation of fermion parity. This results in an integer charge transferring from FM to MZMs after one complete loop as red solid line in Fig. 2(a) shows. Such a nontrivial pumping cannot happen in the trivial loop denoted by $F_{AB}^{-1}F_{B1}F_{B1}^{-1}F_{AB}$, where the net charge transfer is always zero as indicated by Fig. 2(b), regardless whether $E_{d}$ crosses zero level during the whole process as can be seen in Fig. 2(d).

Since the splitting of MZMs $\gamma_{A}$ and $\gamma_{B}$ takes place at $E_{d}=0$, the system parameters at this moment play a pivotal role in determining the occurrence of nontrivial pumping. As indicated by path 1 in Fig. 3(c), we set $E_{d}(t)=+(-)E_{0}\cos(\omega t)$ for $t\in[0,T](t\in[2T,3T])$. In this situation, the final fusion energy of MZM transitions from $2E_{0}$ to $E_{0}$, while $E_{d}(t)$ and $t_{c}(t)$ both stay at zero energy at $t=T/2$ (or $\omega t=\pi/2)$ and the parameters cross one time as indicated by the circle. Consequently, an integer charge is transferred as revealed by the red solid line in Fig. 3(a), which is independent of the final fusion energy. However, if we fuse $\gamma_{B}$ and $\gamma_{1}$ before $E_{d}(t)$ reaches zero (path 2 in Fig. 3(c)), the net charge transfer would be zero after the loop is completed. This suggests that the nontrivial charge transfer directly relate to the crossing between $E_{d}$ and $t_{c}$ at zero energy in the parameter space. Intriguingly, path 3 mirrors path 1 but with an extra intersection of $E_{d}(t)$ and $t_{c}(t)$ at zero energy (shown by the red dashed line in Fig. 3(d)). This results in the charge transfer reverting to zero. Conversely, by varying $t_{c}(t)$ as depicted in path 4 of Fig. 3(d), where only a single zero-energy crossing occurs, the final charge transfer would remain to be one, as seen in Fig. 3(c). These results suggest a simple criterion for nontrivial charge pumping: if $E_{d}(t)$ and $t_{c}(t)$ intersect at zero energy an odd number of times in a closed loop, the charge pumping becomes nontrivial. Conversely, an even number of such intersections within the interval results in trivial charge pumping. This observation holds potential to guide experimental manipulations. Practically, tuning $t_{c}$ to zero may pose a primary challenge in our proposal. However, we expect the nontrivial pumping to be observed when the time period $T\ll 1/\min(t_{c})$, where $\min(t_{c})$ represents the minimum value of $t_{c}$ at the moment of intersection. Note that $t_{c}$ is determined by the tunneling barrier and usually decays exponentially with barrier size, which can be modulated by gate voltage. Adiabatic evolution further requires $T\gg 1/\Delta$. As a result, the time window for the nontrivial pumping is $1/\Delta\ll T\ll 1/\min(t_{c})$.

Numerical simulation in the nanowire superconducting nanowire. We have shown that the nontrivial charge pumping would be induced through controlling the order of both fusion and splitting processes among the MZMs. Naturally, one may wonder how ABS and MZM differ during this process. To investigate this, we further simulate the non-Abelian fusion processes in a realistic nanowire/superconductor (NS) system as depicted in Fig. 4(a). In general, the ABS state may arise due to inhomogeneous potential or disorder at the interface of the NS system S3 . Here, we introduce a typical hard wall confinement at the end of the NS to model the inhomogeneous potential. This results in a nearly zero-energy ABS state being trapped at the left end of NS before the system is driven into topologically nontrivial phases by increasing Zeeman energy $V_{x}$, as shown in Fig. 4(b). In practice, the chemical potential of the FM can be modulated through gate voltage as demonstrated in recent experiment, so is the coupling between the FM and the NS system Pgate1 ; Pgate2 . Thus our fusion protocol could be readily implemented in real systems.

In Fig. 4(c), we show the final charge transfer from FM to TS for the nontrivial loop depicted in Fig. 4(f) when the Zeeman energy $V_{x}$ varies, which suggests that ABS states can also introduce nontrivial charge pumping. Representative charge transfer processes are demonstrated in Fig. 4(d) for $V_{x}=0.25$ (red line), $0.35$ (blue line) and $0.7$ (cyan line). Clearly, an integer charge is transferred when $V_{x}=0.25$ and $V_{x}=0.7$. For $V_{x}=0.35$, however, only a fractional charge is transferred after a complete loop. We may also set $E_{0}=-0.05\text{ mev}$ at $t=0$ as shown in Fig. 4 (g), where the FM is initially occupied. The resulting charge transfer as shown in Fig. 4(e) is solely determined by the fusion channel of the FM. If the fermionic state is initially unoccupied (corresponding to fusion channel $I$), the charge should be transferred from MZM to FM. This transfer process will be reversed if the fermionic state is initially occupied (fusion channel $\Psi$). Moreover, integer pumping is only robust in the topological region, while the quantized charge pumping for $V_{x}=0.25$ is failed for the occupied case and is quantized for $V_{x}=0.35$. This means that the quantized charge pumping is robust for MZMs, regardless whether the initial FM state is occupied or unoccupied. As for ABS, it can be divided into two kinds of states. One is the electron-like ABS which can support quantized charge transfer if the initial FM state is occupied. Another one is the hole-like ABS which can support quantized charge transfer if the initial FM state is unoccupied.

The distinct behaviors of MZMs and ABS can be further understood from transport processes. During the fusion, the charge transfers through either TE or TH process, not both (see Supplementary materials for more information), determined by the total parity of the system. To understand this, we first note that the total parity is conserved with the evolution of $E_{d}$ and $t_{c}$. If the initial state is $|0_{\text{AB}}0_{12}\rangle$ which means both the FM and MZM are unoccupied, it would gradually switch to state $|1_{\text{AB}}1_{12}\rangle$ after half a period. The TE process, which only transfers electrons between FM and MZM, cannot make it. Hence, the pumping has to be accomplished through TH process, during which a hole in the FM is transformed to an electron in MZM. However, if $E_{0}<0$, the initial state would be $|1_{\text{AB}}0_{12}\rangle$ instead and only TE process occurs. Since MZM is its own anti-particle, there’s no difference between TE and TH process. While for ABS one of the two processes would be preferred as it is not hermitian.

Conclusion. With the assistance of a fermionic mode, we unveil the fusion rule of MZMs, which manifests itself in the nontrivial charge transfer process. The FM can be easily attached either to the end of 1D TS wire or a vortex in 2D TS system. Moreover, by controlling the order of fusion and splitting processes among MZMs, one may design trivial and nontrivial fusion loops tailored for the convenience of specific platforms since each process can be individually adjusted. Therefore, our proposal can be applied to various TS systems without additional constraints. In our proposal, one only needs to tune the energy level of FM as well as its coupling to the Majorana modes through gate voltages, leaving the parameters of topological superconductor intact. This may avoid potential complications arising from meddling with TS wires. Finally, we emphasize that the charge transfer are distinct for different situation, we can expect that other realistic factors, such as finite temperature or interaction, only play insignificant roles in our setup. Thus, our proposal provides a feasible route for showcasing the nontrivial fusion rules of MZMs.

Acknowledgement. This work is financially supported by National Natural Science Foundation of China (Grants No. 11974271, and No. 92265103), the National Basic Research Program of China (Grants No. 2015CB921102 and No. 2019YFA0308403), and the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302400).

Supplemental Material for “Unveiling non-trivial fusion rule of Majorana zero mode using a fermionic mode”