Kitaev-chain Hamiltonian

Kitaev states the model as Eq. (4) of Unpaired Majorana fermions in quantum wires:

$H_1 = \sum_j \left( -w(a^\dagger_j a_{j+1} + a^\dagger_{j+1} a_j) - \mu\left(a^\dagger_j a_j - \tfrac{1}{2}\right) + \Delta a_j a_{j+1} + \Delta^* a^\dagger_{j+1} a^\dagger_j \right)$

and glosses it in the same sentence: $w$ is a hopping amplitude, $\mu$ a chemical potential, and $\Delta = |\Delta|e^{i\theta}$ the induced superconducting gap.

The form this catalog quoted is the community's, not the paper's. Two differences, both small and both worth stating rather than smoothing over: the paper writes the hopping amplitude as $w$ where the conventional restatement uses $t$, and it carries the $-\tfrac{1}{2}$ offset on the number term, which the conventional form drops. Neither changes the physics — the offset is a constant shift — but a record that quotes a paper should quote it.

Two limits, and the whole point sits between them. Kitaev works the model at two special parameter choices before the general case. At $|\Delta| = w = 0$, $\mu < 0$, the two Majorana operators of a site pair with each other and the chain is trivial. At $|\Delta| = w > 0$, $\mu = 0$, the Hamiltonian collapses to Eq. (7), $H_1 = iw\sum_j c_{2j}c_{2j+1}$ — Majorana operators now pair across sites, which leaves $b' = c_1$ and $b'' = c_{2L}$ appearing in no term of the Hamiltonian at all. Those are the unpaired Majorana fermions the title is about, and they are a consequence of the pairing pattern rather than an added ingredient.

Where the phases are. The bulk spectrum is Eq. (13), $\epsilon(q) = \pm\sqrt{(2w\cos q + \mu)^2 + 4|\Delta|^2\sin^2 q}$, and the paper places the trivial phase at $2|w| < |\mu|$ and the topological one at $2|w| > |\mu|$ with $\Delta \neq 0$. At finite length the two boundary modes interact through Eq. (15), $H_{\mathrm{eff}} = \tfrac{i}{2}t\,b'b''$ with $t \propto e^{-L/l_0}$ — so the two ground states differ in energy by an amount exponentially small in the chain length, and in fermionic parity, which is the abstract's own claim.

Why the catalog holds it. This is a quadratic fermionic Hamiltonian with a closed-form spectrum and an exactly-known phase boundary, which makes it a benchmark whose right answer is known at every size — the same property that makes the transverse-field Ising model one. It is joined to Hamiltonian you can query because it is a Pauli sum after a Jordan–Wigner mapping, not because any route in this map is about topological order.