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A fermion-parity measurement projects onto an eigenspace of the parity of a chosen collection of fermionic modes. For two Majorana fermion operators ${\gamma }_a$ and ${\gamma }_b$, the projectors are $(1\pm i{\gamma }_a{\gamma }_b)/2$.

Two distinct Majorana fermion operators with $\{{\gamma }_i,{\gamma }_j\}=2{\delta }_{ij}$ define the displayed fermionic annihilation operator. Direct multiplication gives $f^2=0$ and $\{f,f^{†}\}=1$. Its number operator is a projection operator with eigenvalues zero and one. An individual Hermitian Majorana operator has ${\gamma }_i^{†}{\gamma }_i={\gamma }_i^2=1$, so that expression is not a nontrivial occupation observable. The occupation belongs to a chosen pair. A quadratic coupling $iJ{\gamma }_a{\gamma }_b=J(2f^{†}f-1)$ splits the pair's two occupations by $2|J|$.

The continuum Hamiltonian is that of a spinless one-dimensional p-wave topological superconductor. Its topological phase has $\mu >0$, whereas the empty vacuum may be modeled as the trivial phase with $\mu <0$. With superconductor at $x<0$ and vacuum at $x>0$, this is

Near the transition $\mu =0$, momenta are small and the quadratic term may be neglected. In Pauli-matrix notation, the long-wavelength Bogoliubov--de Gennes Hamiltonian isThe zero-energy equation becomesChoose a constant spinor ${\chi }_{+}$ with ${\tau }_y{\chi }_{+}={\chi }_{+}$. Thendecays on both sides because $\mu$ changes from positive to negative. For asymptotically constant $|\mu |$, the Continuum p-wave Majorana interface mode has widthIts characteristic momentum is $|p|\sim |\mu |/\Delta$. The neglected kinetic energy is small compared with $|\mu |$ when $|\mu |\ll m{\Delta }^2$ up to a factor of two, equivalently when $\xi \gg \hslash /(m\Delta )$.

In the Nambu basis, Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian lets a zero-energy eigenvector be chosen self-conjugate, $v(x)=u(x{)}^{\ast }$. The corresponding quasiparticle operator isTaking the adjoint and interchanging the two terms gives ${\gamma }^{†}=\gamma$. With the usual normalization it obeys $\{\gamma ,\gamma \}=2$, so it is a Majorana fermion operator localized at the interface.

For each complex fermion define two Majorana fermion operatorsThey obey $c_r^{†}=c_r$, $\{c_r,c_s\}=2{\delta }_{rs}$, and $a_j=(c_{2j-1}+i{c}_{2j})/2$. Substituting these inverse relations expands every hopping and pairing monomial as a bilinear in the $c_r$. Diagonal terms $c_r^2=1$ contribute only a constant.

For $r\ne s$, $c_r{c}_s$ is anti-Hermitian. Hermiticity of the original quadratic fermion Hamiltonian therefore makes its coefficient purely imaginary, so it can be written $i{A}_{rs}$ with $A_{rs}$ real. Sincethe symmetric part again changes only the constant. Absorbing conventional factors into $A$ gives

Let ${{\rm Y}}_{123}=i{\gamma }_1{\gamma }_2{\gamma }_3$. Reversing three mutually anticommuting Majoranas changes sign, soMoving ${\gamma }_4$ through the three factors givesThus ${{\rm Y}}_{123}$ and ${\gamma }_4$ obey exactly the algebra of two Majorana fermion operators. Their exchange is implemented bywhich conjugates one into the other, up to the orientation sign, just like a Majorana braiding operator.

Ordinary braids permute the elementary ${\gamma }_j$ with signs. Part (d) implements $e^{i\pi {\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4/4}$ by braids and parity measurements, and conjugation by this unitary maps an elementary Majorana to a Hermitian cubic monomial of the other three. Repeating this operation grows or shrinks an odd monomial by two factors, while ordinary braids place the desired indices in the active positions. By induction, every Hermitian odd monomial in ${\gamma }_1,\dots ,{\gamma }_{2M-2}$ can be reached from ${\gamma }_1$, with factors of $i$ inserted according to its degree to make it Hermitian. Hence braids plus suitable fermion-parity measurements map, in their action on $|\psi ⟩$,

The inverse definitions areSubstitution into the hopping and pairing terms expresses the Hamiltonian as a bilinear in the Majorana fermion operators. Hermiticity makes its off-diagonal coefficients purely imaginary in the Majorana bilinear, whileremoves the symmetric part; diagonal terms are constants because $c_j^2=1$. Therefore, up to that additive constant,whereConversely every real antisymmetric $A$ makes this expression Hermitian, so this is the general Majorana form of a quadratic fermion Hamiltonian.

Expand the anticommutators using the relations in part (a). The terms with two creation or two annihilation operators vanish, leavingEach Majorana fermion operator is Hermitian and squares to the identity. In particular, $({\gamma }_i^A{)}^{†}{\gamma }_i^A=1$, so this expression cannot distinguish an empty mode from an occupied one. A single Majorana operator is its own adjoint and does not supply distinct creation and annihilation operators for such an occupation.

An occupation can instead be assigned after pairing Majorana operators into a fermionic mode. For two distinct Majorana operators ${\gamma }_a,{\gamma }_b$,This is a nontrivial projection operator with eigenvalues zero and one. Thus occupation belongs to a chosen pair of Majorana operators, not to an individual Majorana operator.

For any two distinct normalized Majorana fermion operators $\eta ,\chi$, let $f=(\eta +i\chi )/2$. ThenandThe same applies to $f^{†}$. Both specified pairs have this form. Moreover, the Majorana operators entering $f_1$ are all distinct from those entering $f_2$, so every cross anticommutator is zero. ThereforeThus $f_1,f_2$ are ordinary fermionic annihilation operators, constructed by a different pairing of the four Majorana operators.