[**Majorana Fermions on a Lattice and a Matrix Problem**]{}

[**Yorick Hardy$^\ast$ and Willi-Hans Steeb$^\dag$**]{}\
$\ast$ Department of Mathematical Sciences,\
University of South Africa, Pretoria, South Africa,\
e-mail: [hardyy@unisa.ac.za]{}\
$\dag$ International School for Scientific Computing,\
University of Johannesburg, Auckland Park 2006, South Africa,\
e-mail: [steebwilli@gmail.com]{}\

[**Abstract**]{} We investigate matrix problems arising in the study of Majorana Fermions on a lattice. 

Majorana Fermions on a lattice have been studied by many authors [@1; @2; @3; @4; @5; @6; @7; @8; @9] in particular in connection with Fermionic quantum computing.  Here the following matrix problem arises: Find a pair of $n \times n$ hermitian matrices $A$ and $B$ such that $A^2=I_n$, $B^2=I_n$ and $[A,B]_+=0_n$ (i.e. the anticommutator vanishes).  Here $I_n$ is the $n \times n$ identity matrix and $0_n$ is the $n \times n$ zero matrix.  An example of such a pair of matrices is given by $$A = \sigma_1 = \pmatrix { 0 & 1 \cr 1 & 0 }, \qquad
B = -\sigma_2 = \pmatrix { 0 & i \cr -i & 0 }$$ where $\sigma_1$, $\sigma_2$, $\sigma_3$ are the Pauli spin matrices.  Note that the Pauli spin matrices satisfy $[\sigma_1,\sigma_2]_+=0_2$, $[\sigma_2,\sigma_3]_+=0_2$, $[\sigma_2,\sigma_1]_+=0_2$ and $\sigma_1^2=\sigma_2^2=\sigma_3^2=I_2$.

Here we derive properties of such pairs of $n \times n$ matrices $A$ and $B$ and give some applications. 

First we show that the conditions cannot be satisfied if $n$ is odd.  From $AB=-BA$ we obtain $\det(AB)=\det(-BA)$. Thus $\det(AB)=(-1)^n\det(AB)$. Since $\det(AB) \ne 0$ this implies that the condition $[A,B]_+=0_n$ cannot be satisfied if $n$ is odd. 

Thus in the following we assume that $n$ is even and $A$, $B$ satisfy the conditions $A^2=I_n$, $B^2=I_n$ and $AB+BA=0_n$. From the conditions $A^2=I_n$, $B^2=I_n$ it follows that $A=A^{-1}$, $B=B^{-1}$. From the condition $AB+BA=0_n$ we also find that $\mbox{tr}(AB)=0$ and therefore $\det(e^{AB})=1$. It also follows that $(AB)^2=(BA)^2=-I_n$.

Next we show that half of the eigenvalues of $A$ (and of $B$) must be $+1$ and the other half must be $-1$. Let $\mathbf{v}$ be an eigenvector of $A$ corresponding to the eigenvalue 1.  Then from $$(AB+BA)\mathbf{v}=AB\mathbf{v}+B\mathbf{v}=(A+I_n)B\mathbf{v}=
\mathbf{0}$$ and the fact that $\mathbf{v}\neq \mathbf{0}$, $B\mathbf{v}\neq \mathbf{0}$ it follows that $B\mathbf{v}$ is an eigenvector of $A$ corresponding to the eigenvalue $-1$. Similarly if $\mathbf v$ is an eigenvector of $A$ corresponding to the eigenvalue $-1$, then $B\mathbf{v}$ is an eigenvector of $A$ corresponding to the eigenvalue 1.  Since $B$ is invertible it follows that the eigenspaces of $A$ corresponding to the eigenvalues 1 and $-1$ have the same dimension. 

The above argument is also constructive, i.e. given $A$ and an orthonormal basis $$\{\,\mathbf{v}_{-1,1},\,\mathbf{v}_{-1,2},\,\ldots,\mathbf{v}_{-1,n/2},
\,\mathbf{v}_{1,1},\,\mathbf{v}_{1,2},\,\ldots,
\mathbf{v}_{1,n/2}\,\}$$ composed of eigenvectors $\mathbf{v}_{-1,j}$ corresponding to the eigenvalue $-1$ of $A$ and eigenvectors $\mathbf{v}_{1,j}$ corresponding to the eigenvalue 1 of $A$, we can construct a $B$ satisfying the properties above as $$B=\sum_{j=1}^{n/2}(\mathbf{v}_{1,j}
\mathbf{v}_{-1,j}^*+\mathbf{v}_{-1,j}\mathbf{v}_{1,j}^*).$$ All $B$’s can be constructed in this way by an appropriate choice of an orthonormal basis. 

For any $n \times n$ matrices $X$ and $Y$ over $\mathbb C$ we have the following expansion utilizing the anticommutator [@10; @11] $$e^X Y e^X = Y + [X,Y]_+ + \frac1{2!} [ X,[X,Y]_+]_+ +
\frac1{3!} [ X,[X,[X,Y]_+]_+]_+ + \cdots\,.$$ Consequently we find $$e^X Y e^{-X} = \left(Y + [X,Y]_+ + \frac1{2!} [ X,[X,Y]_+]_+ +
\frac1{3!} [ X,[X,[X,Y]_+]_+]_+ + \cdots\right)e^{-2X}$$ $$e^X Y e^{-X} = e^{2X}\left(Y - [X,Y]_+ + \frac1{2!} [ X,[X,Y]_+]_+ -
\frac1{3!} [ X,[X,[X,Y]_+]_+]_+ + \cdots\right)\,.$$ Thus since $[A,B]_+=0_n$ for the matrices $A$ and $B$ we have $$e^A B e^{-A} = Be^{-2A} \quad \mbox{and} \quad
e^A B e^{-A} = e^{2A} B\,.$$ Utilizing $B^2=I_n$ we have $e^{-A}=Be^{A}B$. Note that $(z \in {\mathbb C})$ $$e^{zA} = I_n \cosh(z) + A\sinh(z), \quad
e^{zB} = I_n \cosh(z) + B\sinh(z)$$ and $$e^{zAB} = I_n \cos(z) + AB\sin(z)\,.$$ Let $\otimes$ be the Kronecker product [@12; @13; @14; @15].  Then we have $$e^{z(A \otimes B)} = (I_n \otimes I_n)\cosh(z) + (A \otimes B)\sinh(z)$$ and for the anticommutators $$[A \otimes I_n,B \otimes I_n]_+ = 0_{n^2}, \qquad 
[I_n \otimes A,I_n \otimes B]_+ = 0_{n^2}$$ where $(A \otimes I_n)^2=I_n \otimes I_n$, $(B \otimes I_n)^2=I_n \otimes I_n$. Thus using the Kronecker product and the identity matrix $I_n$ we can construct matrices in higher dimensions which satisfy the conditions.  This can be extended.  Let $C$ be an $n \times n$ matrix with $C^2=I_n$. Then the pair $C \otimes A$ and $C \otimes B$ also satisfies the conditions $(C \otimes A)^2=I_n \otimes I_n$, $(C \otimes B)^2=I_n \otimes I_n$, $[C \otimes A,C \otimes B]_+=0_{n^2}$. An example for $n=2$ would be $A=\sigma_3 \otimes \sigma_1$, $B=-\sigma_3 \otimes \sigma_2$. However note that $$\begin{aligned}
[A \otimes A,B \otimes B]_+ &=& 2((AB) \otimes (AB)), \cr
[A \otimes B,A \otimes B]_+ &=& 2I_n \otimes I_n, \cr
[A \otimes B,B \otimes A]_+ &=& -2((AB) \otimes (AB))\,.\end{aligned}$$ Let $\oplus$ be the direct sum and $A$, $B$ be a pair of hermitian matrices satisfying $A^2=B^2=I_n$ and $[A,B]_+=0_n$. Then $A \oplus A$ and $B \oplus B$ is also such a pair (in $2n$ dimensions).  Let $A$, $B$ be such a pair with $n=2$. Then the $4 \times 4$ matrices $$A \star A := 
\pmatrix { a_{11} & 0 & 0 & a_{12} \cr 0 & a_{11} & a_{12} & 0 \cr
           0 & a_{21} & a_{22} & 0 \cr a_{21} & 0 & 0 & a_{22} }$$ and $B \star B$ are also such a pair. 

From the pair $A$, $B$ we can also form the $n \times n$ matrix $A+iB$ which is non-normal, i.e. $(A+iB)^*(A+iB) \ne (A+iB)(A+iB)^*$. Such matrices play a role for the study of non-hermitian Hamilton operators [@16].  Since $(A+iB)^2=0_n$ we obtain $$e^{z(A+iB)} = I_n + z(A+iB)\,.$$

[**Acknowledgment**]{}

The authors are supported by the National Research Foundation (NRF), South Africa.  This work is based upon research supported by the National Research Foundation.  Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and therefore the NRF do not accept any liability in regard thereto. 

[99]{}

A. Kitaev and C. Laumann, “Topological phases and quantum computation”, arXiv:0904.2771v1

S. B. Bravyi and A. Yu.  Kitaev, “Fermionic quantum computation”, Annals of Physics [**298**]{}, 210-226 (2002)

S. Esposito, “Transition from Majorana to Weyl fermions and anapole moments”, arXiv:1304.3669v1

L. H. Kaufman, “Knot Logic and Topological Quantum Computing with Majorana Fermions”, arXiv:1301.6214v1

A. Jaffe and F. L. Pedrocchi, “Reflection Positivity for Majorana”, arXiv:1305.1792v1

Ya-Jie Wu, Jing He and Su-Peng Kou, “Realizing Universal Majorana Fermionic Quantum Computation”, arXiv:1305.0114v1

E. T. Campbell, M. J. Hoban and J. Eisert, “Majorana fermions and non-locality”, arXiv:1305.1953v1

Shuo Mi, D. I. Pikulin, M. Wimmer and C.W. J. Beenakker, “Proposal for the detection and braiding of Majorana fermions in a quantum spin Hall insulator”, arXiv:1304.1685v1

W.-H. Steeb and Y. Hardy, “An Eigenvalue Problem for a Fermi System and Lie Algebras”, Quantum Inf.  Process DOI 10.1007/s11128-013-0650-5

I. Mendaš and Milutinović, “Anticommutator analogue of the Baker-Hausdorff lemma”, J. Phys.  A: Math.  Gen. [**22**]{}, L687-L689 (1989)

W.-H. Steeb, I. Tanski and Y. Hardy, [*Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications*]{}, World Scientific, Singapore (2012)

W.-H. Steeb, [*Problems and Solutions in Theoretical and Mathematical Physics*]{}, Third edition, Volume II: Advanced Level, World Scientific, Singapore (2009)

W.-H. Steeb and Y. Hardy, [*Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra*]{}, 2nd edition, World Scientific, Singapore (2011)

W.-H. Steeb and Y. Hardy, [*Problems and Solutions in Quantum Computing and Quantum Information*]{}, third edition, World Scientific, Singapore (2011)

W.-H. Steeb and Y. Hardy, [*Quantum Mechanics using Computer Algebra*]{}, 2nd edition, World Scientific, Singapore (2010)

W.-H. Steeb and Y. Hardy, “Exceptional Points, Nonnormal Matrices, Hierarchy of Spin Matrices and an Eigenvalue Problem”, arXiv:1301.2900v2
