Quadratic fermion Hamiltonian

A quadratic fermion Hamiltonian is bilinear in creation and annihilation operators. In a Majorana basis it is determined, up to a constant, by a real antisymmetric matrix.

For distinct fermionic orbitals obeying the full canonical anticommutation relations, fix the Fock state order $|11⟩=a^{†}{b}^{†}|0⟩$. With real $\epsilon ,\Delta >0$, the even-fermion parity matrix is $\left(\begin{array}{cc}0& -\Delta \\ -\Delta & 2\epsilon \end{array}\right)$, since $ab|11⟩=-|0⟩$. The two odd-fermion parity energy eigenvalues are both $\epsilon$. The even-fermion parity energy eigenvalues are $\epsilon \pm \sqrt{{\epsilon }^2+{\Delta }^2}$. This finite model isolates the relative sign and constant shift required in a fermionic Bogoliubov transformation without any momentum sum or particle-hole doubling factor.

A Bogoliubov--de Gennes Hamiltonian is the matrix representation of a quadratic fermion Hamiltonian in a Nambu spinor combining annihilation operators and creation operators. It treats ordinary one-particle motion and fermionic pairing in one eigenvalue problem.

For a column $c$ of $N$ fermionic annihilation operators, let $h=h^{†}$ and $\Delta =-{\Delta }^T$. With $\Psi =(c,c^{†}{}^T{)}^T$, defineThe canonical anticommutation relations give the identityThe factor $1/2$ compensates for the full particle-hole doubling; the trace constant follows when the hole block is normal-ordered.

A two-by-two opposite-spin pairing blockhas eigenvalues $E_{\pm }=\pm \sqrt{{\xi }^2+|\Delta {|}^2}$, as follows from its characteristic polynomial. Pairing opens a spectral gap at $\xi =0$. Negative and positive partners in a fully doubled representation are related by Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian; they are not independent species of physical quasiparticles. A reduced block and a full doubled basis therefore require consistent counting conventions.

Particle-hole symmetry relates $H_{\mathrm{BdG}}(k)$ to $-H_{\mathrm{BdG}}^{\ast }(-k)$. Consequently every energy $E$ at momentum $k$ is paired with energy $-E$ at momentum $-k$.

A Majorana fermion operator is a self-adjoint operator in a fermionic algebra satisfying $\{{\gamma }_i,{\gamma }_j\}=2{\delta }_{ij}I$. A single complex fermionic mode $c$ can be written using two such operators:The canonical anticommutation relations verify their Hermiticity and normalization, and the occupation is $c^{†}c=(I+i{\gamma }_1{\gamma }_2)/2$.

This algebraic rewriting is always possible for a complex mode. A spatially separated Majorana zero mode requires additional physical properties, including localization and near-zero commutator with the Hamiltonian. The existence of two Majorana operators in a formula alone therefore does not establish protected boundary states or a topological phase.

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 $2|J|$.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

Two counterpropagating Majorana fields coupled by a real mass $m(x)$ bind a Majorana zero mode when $m$ changes sign. Its envelope is proportional to $\exp [-{\int }^{x}m(s)ds/(\hslash v)]$ up to orientation conventions, so a constant asymptotic mass gives localization length $\hslash v/|m|$.

For the long-wavelength Bogoliubov--de Gennes Hamiltonian $H=-\mu (x){\tau }_z-i\hslash \Delta {\tau }_x{\partial }_x$, a sign change from $\mu >0$ in a one-dimensional p-wave superconductor to $\mu <0$ in the vacuum binds a zero mode of width $\xi =\hslash \Delta /|\mu |$. Neglecting the quadratic kinetic term is valid when $|\mu |\ll m{\Delta }^2$ up to an inessential numerical factor.

$2M$ Majorana zero modes define $M$ complex fermionic modes. Fixing total fermion parity leaves a $2^{M-1}$-dimensional ground space, encoding $M-1$ qubits.

When Majorana zero modes have disjoint exponentially localized supports, every local parity-even observable acts as a scalar on the fixed-parity ground space. Finite overlap changes this by terms exponentially small in separation divided by localization length.

The unitary $R_{ab}=\exp (\pi {\gamma }_a{\gamma }_b/4)$ exchanges two Majorana operators by conjugation, sending one to the other and the other to its negative, with signs fixed by braid orientation.

Pairwise fermion-parity measurements involving an ancillary Majorana mode can reproduce a Majorana braiding operator without physically moving the zero modes. Projecting the ancilla back to its initial parity sector turns the product of projectors into the braid unitary up to normalization and known outcome-dependent signs.

A Forced Majorana parity measurement alternates an undesired pairwise measurement with a reference parity measurement until the desired outcome occurs. Anticommuting parity observables give either result with probability one half, so the protocol terminates almost surely with a geometric waiting time.

Fermion parity is $(-1{)}^F$. It acts as plus one on bosonic states and minus one on fermionic states. Even operators commute with it and odd operators, including supersymmetry generators, anticommute with it. Local observables are parity even because observable algebras of spacelike separated regions must commute.

When a quadratic fermion spectrum was derived in a fixed fermion parity sector, filling all negative-energy modes is permitted only if the resulting particle number has that parity. If it does not, toggle the occupation of a mode with smallest absolute energy. Compare the minimum energies of the physical sectors. Zero-energy modes can create degeneracy without energy cost.

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 $(1\pm i{\gamma }_a{\gamma }_b)/2$.