Prof. X. C. Xie and his collaborators publish a paper on Physical Review Letters reporting the non-Abelian braiding effect in spin superconductors

In an exciton (electron-hole pair) condensate with non-zero spin (usually induced by ferromagnetic order), the Coulomb attraction between the electron and hole plays the role as the superconducting pairing potential. In the meantime, such an exciton condensate couples with magnetic field gradient via its non-vanishing spin. In this way, such kind of exciton condensate can be viewed as a counterpart of the conventional charge superconductor. These phenomena discussed above was firstly raised by Prof. X. C. Xie and Prof. Qing-feng Sun et al. in 2011 (Physical Review B 84, 214501). Correspondingly, such a novel state was named by Prof. Xie and Prof. Sun as “spin superconductor”. After that, the concept of spin superconductor has been drawing attention, and some charming properties of spin superconductors including spin transport, electric Meissner effect, and Ginzburg-Landau-type theory has been put forward in the last decade.

In the last two decades, the researches on topological quantum computation mainly focused on the Majorana zero-mode (MZM) supporting non-Abelian braiding. Such non-Abelian braiding effect owes to the non-Abelian geometric phase of π as well as the degenerate ground state space spanned by the MZMs. The geometric phase of π here induced by the Aharonov-Bohm effect is further protected by the following two facts that (i) the vortex bound state satisfies the periodic boundary condition; and (ii) the effective charge carried by the MZM is exactly one-half of that carried by the Cooper pair in superconductor. Recently, Prof. X. C. Xie from Peking University and his collaborators investigated the possible effect of non-Abelian braiding in spin superconductor. They noticed that if a spin superconductor is topologically non-trivial, then the vortex bound state inside the spin superconductor will possess (i) degenerate ground states, (ii) “half-charge”, and (iii) periodic boundary condition at the same time. These three properties are exactly the same as those possessed by MZMs as we discussed above. It is worth noting that the “charge” here in spin superconductor is spin other than the electric charge, where the latter is the case for conventional topological superconductor and MZM. Correspondingly, the non-Abelian geometric phase of π here is induced by the Aharonov-Casher effect instead of the Aharonov-Bohm effect. Based on these ideas, Prof. X. C. Xie and his collaborators pointed out that the non-Abelian braiding operation can be conducted in spin superconductors utilizing the Aharonov-Casher effect. This work has been recently published online in Physical Review Letters (Physical Review Letters 128, 106804 ). Website link：https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.128.106804

Vortices of electric flux gradient can be formed in spin superconductors due to the electric Meissner effect. When spin superconductor interplays with non-trivial topology, such kind of vortex will bound a topological edge state which carries one-half effective spin compared with the spin of the exciton in the bulk state. Owing to the half-spin induced Aharonov-Casher effect, this topological edge state shares the similar non-Abelian braiding properties with the MZM in conventional topological superconductor. Furthermore, compared with its Majorana cousin, the topological edge state in spin superconductors can be experimentally more distinct for possessing non-vanishing electric charge. In the point of view of the electromagnetic multipole expansion, such a spin-superconductor-based braiding scheme could be the minimal scheme utilizing electromagnetic charge to realize non-Abelian braiding, in addition to the scheme of charge superconductor. Experimentally, the topologically non-trivial spin superconductor may be realized by ferromagnetic graphene with staggered potential. This theoretical proposal provides a new avenue investigating the non-Abelian braiding physics and the possible topological quantum computation without the assistance of MZM and charge superconductor.

Yijia Wu (Boya post-doctor in Prof. X. C. Xie’s group) is the first author, Jie Liu (Xi’an Jiaotong University) and Prof. X. C. Xie are the corresponding authors of this Letter. Other collaborators including Hua Jiang (Soochow University), Hua Chen (Zhejiang Normal University), and Haiwen Liu (Beijing Normal University). This work is financially supported by the National Basic Research Program of China, the National Natural Science Foundation of China, the Strategic Priority Research Program of Chinese Academy of Sciences, and China Postdoctoral Science Foundation.

A cartoon in the Facebook personal homepage of Prof. Xiao-Gang Wen from Massachusetts Institute of Technology.

Prof. Wen compared this cartoon with the fact that physicists separate an electron into two MZMs, where the latter support non-Abelian braiding. In the scheme raised by Prof. X. C. Xie and his collaborators, the non-Abelian braiding is implemented by separating a spin into two identical parts instead.