Majorana Zero Modes

Majorana zero modes are the most experimentally pursued realization of non-abelian anyons. They are localized, zero-energy excitations of certain superconductors that behave as "half" of an ordinary fermion and carry the Ising anyon $\sigma$ . This lesson introduces them via Kitaev's exactly solvable chain.

A fermion split in two

An ordinary fermion mode has creation/annihilation operators $c^\dagger, c$ with $\{c, c^\dagger\}=1$ . Define two Hermitian Majorana operators

\gamma_1 = c + c^\dagger, \qquad \gamma_2 = -i\,(c - c^\dagger),

so that

\gamma_j^\dagger = \gamma_j, \qquad \gamma_j^2 = 1, \qquad \{\gamma_j, \gamma_k\} = 2\,\delta_{jk}.

Each $\gamma_j$ is its own antiparticle — a Majorana fermion. Ordinarily $\gamma_1$ and $\gamma_2$ sit at the same place and just repackage the fermion $c$ . The interesting physics is when they become spatially separated.

The Kitaev chain

Kitaev's model is a one-dimensional $p$ -wave superconducting wire of $N$ spinless fermionic sites. In the fine-tuned topological limit (chemical potential $\mu=0$ , hopping $=$ pairing $=t$ ), the Hamiltonian pairs Majoranas from neighboring sites, $\gamma_{2j}$ with $\gamma_{2j+1}$ , and leaves two Majoranas completely unpaired at the two ends of the wire:

\gamma_{1}\ (\text{left end}), \qquad \gamma_{2N}\ (\text{right end}).

These two end modes commute with the Hamiltonian, so they cost zero energy. Together they form one nonlocal fermionic mode

f = \tfrac12(\gamma_1 + i\,\gamma_{2N}), \qquad n_f = f^\dagger f \in \{0,1\},

whose occupation $n_f$ encodes a qubit. Because $f$ is built from operators at opposite ends of the wire, no local perturbation can read or flip it — the protection is geometric.

Even and odd fermion parity

The two states $n_f=0$ and $n_f=1$ differ by total fermion parity $P = (-1)^{n_f}$ , a quantity that local, parity-preserving noise cannot change. The ground-state degeneracy of a chain with $2m$ end modes is $2^{m-1}$ within a fixed total parity sector — exactly the $\sqrt2^{\,n}$ counting of $n$ Ising $\sigma$ anyons. Each Majorana end mode is a $\sigma$ .

Braiding Majoranas

In a network of wires (a "tri-junction" or T-junction geometry) Majorana end modes can be moved and exchanged. Exchanging two Majoranas $\gamma_a, \gamma_b$ implements the unitary

U_{ab} = \exp\!\left(\frac{\pi}{4}\,\gamma_a \gamma_b\right) = \frac{1}{\sqrt2}\left(1 + \gamma_a\gamma_b\right),

a noncommuting operation on the degenerate parity space — explicitly non-abelian. These braids generate the single- and two-qubit Clifford gates, matching the Ising anyon result: powerful and protected, but not universal by braiding alone.

What to take away

A Majorana zero mode is half of a fermion: a Hermitian self-conjugate operator that, when delocalized to the ends of a topological superconductor, stores a qubit in nonlocal fermion parity. Pairs of Majoranas realize Ising $\sigma$ anyons; braiding them gives protected Clifford gates. They are the leading experimental candidate for topological qubits, taken up again in the realizations lesson.