Topological superconductors and Majorana models

Take a spinless nearest-neighbour chain with hopping t, chemical potential mu, and real nonzero p-wave pairing Delta. In the Nambu basis $(c_k,c^\dagger_{-k})^T$, its Bogoliubov–de Gennes Hamiltonian has a diagonal normal-state term and an odd-in-momentum pairing term. The spectrum is symmetric because the Nambu representation redundantly includes particles and holes. That redundancy is not the same as an extra independently occupiable physical band or an ordinary charge-conjugation symmetry of a number-conserving solid.

For nonzero pairing, gap closings occur when the pairing vanishes at zero or pi momentum and the normal coefficient also vanishes. The familiar topological interval is chemical-potential magnitude below twice the hopping magnitude. The interval is separated from the trivial regimes by a bulk gap closing. Pairing cannot simply be set to zero while retaining the same gapped statement: in that limit the interior interval generally contains a gapless normal metal.

Resolve each physical fermion into two Hermitian Majorana operators with anticommutator two. At zero chemical potential and pairing equal to positive hopping, the open-chain Hamiltonian pairs one Majorana on a site with one on the next site. The first A operator and final B operator are absent. They commute with this ideal Hamiltonian and form one ordinary zero-energy fermion when combined. Their Hermitian character does not make each one an independently occupiable particle state.

Away from the exactly solvable point but within the gapped phase, end wavefunctions remain localized for a sufficiently long clean chain. Overlap across a finite sample generally gives an exponentially small splitting rather than an exact zero. Local perturbations can also couple to additional nearby degrees of freedom. The protecting class-D distinction survives without the extra real-parameter symmetry of the displayed special model, provided the appropriate gap and fermion-parity assumptions remain valid.

Two unpaired Majoranas define one fermionic mode with empty and occupied states of opposite fermion parity. If the total parity of an isolated system is fixed and no other zero modes are available, only one of those two occupations lies in the selected sector. Four separated Majoranas can encode two fermionic occupations; fixing their combined parity leaves a two-dimensional space. This is the minimal counting reason that a single end pair is not automatically one freely controllable fixed-parity qubit.

A grounded superconductor, a floating island with charging energy, and a system exchanging quasiparticles with a reservoir impose different parity and charge constraints. Poisoning events can change the sector and undermine an encoding. Adiabatic exchanges in suitable networks require additional controls and do not follow from merely plotting a two-level Bloch sphere. The ideal chain identifies a mechanism and a phase criterion; demonstrating protected operations requires an apparatus-specific model and evidence beyond a zero-bias spectral feature.