More about the doubling degeneracy operators associated with Majorana fermions and Yang-Baxter equation

A new realization of doubling degeneracy based on emergent Majorana operator Γ presented by Lee-Wilczek has been made. The Hamiltonian can be obtained through the new type of solution of Yang-Baxter equation, i.e. -matrix. For 2-body interaction, gives the “superconducting” chain that is the same as 1D Kitaev chain model. The 3-body Hamiltonian commuting with Γ is derived by 3-body -matrix, we thus show that the essence of the doubling degeneracy is due to . We also show that the extended Γ′-operator is an invariant of braid group BN for odd N. Moreover, with the extended Γ′-operator, we construct the high dimensional matrix representation of solution to Yang-Baxter equation and find its application in constructing 2N-qubit Greenberger-Horne-Zeilinger state for odd N.

T he Majorana mode [1][2][3][4] has attracted increasing attention in physics due to its potential applications in topological quantum information processing [5][6][7] . Specifically, the degenerate ground state in Majorana mode serves as topologically protected states which can be used for topological quantum memory. In the Ref. 8 and the Hamiltonian takes the form The algebra in equation (1) is conceptually incomplete. Besides the parity, the nonlinear operator C is introduced 8 to form the set where P implements the electron number parity, and P 2 5 1. The emergent Majorana operator C and parity operator P lead to the doubling degeneracy at any energy level, not only for the ground state.
On the other hand, based on the obtained new type of solution R i h ð Þ of Yang-Baxter equation (YBE), which is related to Majorana operators, the corresponding Hamiltonian can be found by following the standard way 9 , i.e.
the Hamiltonian H* LR i h ð Þ Lh h~0 j . We find that the Hamiltonian derived from R i h ð Þ is 1D Kitaev model 1 .
Moreover, because 1 1 1D 3-body S-matrix can be decomposed into three 2-body S-matrices based on YBE, we construct the 3-body Hamiltonian from 3-body S-matrix and find its doubling degeneracy. Hence, the advantage of parametrizing the braiding operator B i to R i h ð Þ is that the desired Hamiltonian associated with Majorana operators can be derived from R i h ð Þ. Now let us first give a brief introduction to the Majorana representation of braiding operator as well as the solution of Yang-Baxter equation.
The non-Abelian statistics 10 of Majorana fermion (MF) has been proposed in both 1D quantum wires network 7 The Yang-Baxter equation (YBE) 9,11,12 is a natural generalization of braiding relation with the parametrized form: where x, y stand for spectral parameters, The solutions of equation (7) was intensively studied by Yang, Baxter, Faddeev and other authors [11][12][13][14][15][16][17][18][19][20] in dealing with many body problems, statistical models, low-dimensional quantum field theory, spin chain models and so on. We call this type of solutions type-I.
then the YBE reads 26 : with the constraint for three parameters h 1 , h 2 and h 3 : i.e. the Lorentzian additivity by h~1 c u. It is well known that the physical meaning of h is to describe entangling degree, which is jsin 2hj for 2-qubit 23 . The type-II solution of YBE R i h ð Þ means the operation between two Majorana fermions, c i and c I 1 1 . Because c i 's satisfy Clifford algebraic relations: Then the solution R i h ð Þ~e hc i c iz1 transforms the Majorana fermions c i and c i 1 1 in the following way: Since the solution of Yang-Baxter equation can be expressed in Majorana form, the following problems arise: (i) How to understand the C-operator intuitively on the basis of the concrete MF model generated by YBE; (ii) How to obtain the 3-body Hamiltonian, which possesses the doubling degeneracy, from YBE; (iii) What is the relationship between C-operator (as well as extended C9) and the solution R i h ð Þ of YBE. In this paper, we show that the emergent Majorana operator C is a new symmetry of R h ð Þ as well as Yang-Baxter equation. Due to the symmetry, the 3-body Hamiltonian derived from YBE holds Majorana doubling. We also present a new realization of doubling degeneracy for Majorana mode. Moreover, we discuss the topological phase in the ''superconducting'' chain. The generation of Greenberger-Horne-Zeilinger (GHZ) state via the approach of YBE is also discussed.

Results
Topological phase in the derived ''superconducting'' chain. The topological phase transition in the derived ''superconducting'' chain based on YBE is discussed. We find that our chain model is exactly the same as 1D Kitaev model. Let us first give a brief introduction to 1D Kitaev model. 1D Kitaev's toy model is one of the simplest but the most representative model for Majorana mode 1,4 . The model is a quantum wire with N sites lying on the surface of three dimensional p-wave superconductor, and each site is either empty or occupied by an electron with a fixed spin direction. Then the Hamiltonian is expressed as the following form: Here a { j , a j represent spinless ordinary fermion, v is hopping amplitude, m is chemical potential, and D 5 jDje 2iQ is induced superconducting gap. Define Majorana fermion operators: which satisfy the relations: Then the Hamiltonian is transformed into the Majorana form: An interesting case is m 5 0, v 5 jDj. In this case, the Hamiltonian turns into Majorana mode corresponding to topological phase: The above Hamiltonian has two degenerate ground states, j0ae and j1ae 5 d { j0ae. Here d { 5 e 2iQ/2 (c 1 2 ic 2N )/2 is a non-local ordinary fermion. The degenerate states can be used for topological quantum memory qubits that are immune to local errors. Now let us construct the ''superconducting'' chain based on the we can express a state jy(t)ae as y t ð Þ j i~R i y 0 ð Þ j i. Taking the Then the HamiltonianĤ i t ð Þ related to the unitary operator (22), we have: This Hamiltonian describes the interaction between i-th and (i 1 1)th sites with the parameter _ h. Indeed, when h~n| p 4 , the unitary evolution corresponds to the braiding progress of two nearest Majorana fermion sites in the system, here n is an integer and signifies the times of braiding operation. If we only consider the nearest-neighbour interactions between MFs and extend equation (23) to an inhomogeneous chain with 2N sites, the derived ''superconducting'' chain model is expressed as: with _ h 1 and _ h 2 describing odd-even and even-odd pairs, respectively. Now we give a brief discussion about the above chain model in two cases (see Fig. 1): In this case, the Hamiltonian is: As defined in equation (16) and (17), the Majorana operators c 2k 2 1 and c 2k come from the same ordinary fermion site k, ic 2k{1 c 2k~2 a { k a k {1 (a { k and a k are spinless ordinary fermion operators).Ĥ 1 simply means the total occupancy of ordinary fermions in the chain and has U(1) symmetry, a j R e iw a j .
This Hamiltonian corresponds to the topological phase of 1D Kitaev model and has Z 2 symmetry, a j R 2a j . Here the operators c 1 and c 2N are absent inĤ 2 , which is illustrated by the crossing under the dashed line in Fig. 1 In Ref. 8, the concreted realization of the operators was presented in terms of Pauli matrices. On the other hand, as pointed out in Ref. 27, there is the transformation between the natural basis and Bell basis for W 0 j i~;; j i, :; j i, ;: j i, :: where Y z j i~1ffiffi ffi 2 p :: through the matrix B II : where B II~1 ffiffi ffi 2 p and which forms ''extra special 2-group''. Obviously, M is extension of i for i 2 5 21.
An interesting observation is 28 : M~{iĈ ð36Þ whereĈ is the charge conjugate operator in Majorana spinor. The eigenstates ofĈ take the formŝ where j + j i~1ffiffi ffi 2 p :: Here we would like to give an intuitive interpretation of the operator C in Ref. 8  We follow the concrete realization for c j given in Ref. 8, (in this paper I is 2 3 2 identity matrix) C~{ic 1 c 2 c 3~{ I6s 1 : In our notation, c 3~{Ĉ , i.e. (38) and (39) are eigenstates of c 3 . It is easy to find P j + j i~+ g + , P g + ~+ g + ; ð46Þ In the derivation of (43)-(47), the relations s 1 5 (S 1 1 S 2 ) and Þhave been used where S 6 5 S 1 6 iS 2 . To show the importance of C-operator we define new Clifford algebra {D i , D j } 5 2d ij , where D 1 5 c 2 , D 2 5 Cc 1 , D 3 5 c 3 . It is interesting to find that Namely, by acting D j on jjae or jgae, the representation is exactly Pauli matrices, i.e. belonging to SU(2) algebra. It can be checked that where S i form the reducible representation of SU (2): The introduced interacting Hamiltonian H B 5 2i(aD 1 D 2 1 bD 2 D 3 1 kD 3 D 1 ) can be recast to where a 1 5 2k, a 2 5 a, a 3 5 b. Noting that D 1 D 2 D 3 5 2iI fl I, i.e. trivial. The direct check gives: Then the H B can be written in the form: M 2~c os f s 3 6I{I6s 3 2 Now the meaning of H B is manifest: 4-dimension is quite different from 2-dimension. The ''edge block'' leads to M 1 with superconducting type of Hamiltonian whereas ''interior block'' M 2 is connected with the usual spin chain. It is easy to find the eigenstates of M 1 and M 2 : Acting C on (63) it yields So C transforms between jy 1 ae and jy 2 ae that holds for the same energy. It never occurs in 2 dimensions. Meanwhile, equation (53) shows that C commutes with the Hamiltonian H B , which means that C-transformation does not change the property of Hamiltonian H B . This example shows that operator C is crucial in leading to Majorana doubling in dimensions $4. With the new defination of D 2 , we should define a new parity operator: Direct check gives the complete set of algebra It is noteworthy that the introduced P B in equation (68) commutes with D j instead of the anti commuting relation between P and c j . And P B still anticommutes with C. Acting P B on the eigenstates jy 1 ae and jy 2 ae, it follows In such a concrete realization C plays the essential role. The Hamiltonian (54) formed by (52) looks a typical nuclear resonant model in 4 dimensions. Only the higher dimensions allow the operator C leading to the doubling degeneracy.

Majorana doubling in 3-body Hamiltonian based on YBE. Now we discuss the interaction of 3 Majorana fermions based on YBE.
It is well known that R i h ð Þ describes the 2-body interaction. And the physical meaning of Yang-Baxter equation is that the interaction of the three bodies can be decomposed into three 2-body interactions: Because of the constraint in equation (11), R 123 depends only on two free parameters and has the following form 29 : where cos g~cos h 2 cos h 1 zh 3 ð Þ, Here the parameters h 1 and h 3 are replaced by g and b. R 123 g,b ð Þ is also a unitary operator and describes the interaction of three Majorana operators.
We suppose that the parameter g is time-dependent and b is timeindependent in R 123 g,b ð Þ, then the desired 3-body Hamiltonian can be obtained from equation (22): The constructed Hamiltonian, which has been mentioned in Ref. 7, 8, describes the 2-body interactions among the three Majorana operators. It describes the effective interaction in a T-junction formed by three quantum wires. In Ref. 8, it has been shown that the above Hamiltonian, which commutes with emergent Majorana operator C 5 2ic 1 c 2 c 3 , holds Majorana doubling. From the viewpoint of YBE, the intrinsic commutation relation is between C and the solution of It is shown that: Indeed, the above commutation relation indicates that emergent Majorana operator C is a new symmetry of the solution R i h ð Þ of YBE. It is due to the decomposition of 3-body interaction into three 2-body interactions via the approach of YBE that the derived Hamiltonian holds Majorana doubling.
The extended emergent Majorana mode C9 supporting odd number N of Majorana operators 8 is, It is easy to check that: where B i~e Generation of 2n-qubit GHZ state via YBE. Quantum entanglement plays an important role in quantum information theory and has been discussed in both theoretical 30 and experimental [31][32][33] aspects for a long time. There are various ways in describing different types of entanglement. It is also well known that the relationship between Yang-Baxter equation and 2-qubit entangled state as well as 3-qubit entanglement has been discussed in Ref. 22, 23, 29, 34. Here we construct high dimensional matrix representation of solution to Yang-Baxter equation and discuss how it generates 2N-qubit GHZ state for odd N. In previous section, we present Clifford algebric relation for different Majorana operators, It can be used for constructing solution to YBE: The representation of c i in the Majorana form is given by: Then by constructing Yang-Baxter chain, we find its similarity to 1D Kitaev model. Indeed, the 4D-matrix representation is equivalent to the Majorana fermion representation under Jordan-Wigner transformation. In other words, we can express c i by matrix directly. For three operators. c 1 , c 2 and c 3 satisfying Clifford algebra, its 4D matrix representation has been presented in Ref. 8: here s i are Pauli matrices.
What we are interested in is constructing higher dimensional matrix representation of c i . Taking 8D representation as an example, c i is: Then the matrix form of emergent Majorana mode C 8 is, The Hamiltonian supporting three Majorana operators has been defined in equation (2): Obviously, C commutes with the Hamiltonian H int .
Let us extend C to N sites C i , which should also satisfy Clifford algebra {C i , C j } 5 2d ij . The C i has the following form: Then we have: It is easy to check that e hCiCiz1 is the 4 3  i h ð Þ on 6-qubit natural basis, such as j""""""ae, we have: This state represents a type of 6-qubit entangled states. In the case of h~p 4 , the generated state is 6-qubit GHZ state, and where the constraint of Clifford algebra C n i ,C n j n o~2 d ij leads to the odd number n. C n i can be expressed as: Then we have The 4 n 2D (n odd) matrix representation of solution to YBE is: Ĥ 123 . The commutation relation can also be generalized to the extended C9-operator(87) for odd N sites, [C9, B i ] 5 0 (i 5 1, 2, …N 2 1), hence C9 is an invariant of the braid group B N . We present a new realization of Majorana doubling based on emergent Majorana mode and show the role of C in leading to the doubling degeneracy of H B intuitively. We also make use of the extended C9-operator to construct high dimensional matrix representation of solution to YBE. By acting the high dimensional matrix representation of solution of YBE on natural basis, we generate the GHZ-type entangled state. Thus we conclude that the braiding process of the extended C9-operators corresponds to the generation of GHZ entangled state. These results may guide us to find much closer relationship between Yang-Baxter equation and quantum information as well as condensed matter physics.