Effective Hamiltonian Method used in Kitaev Chain

Question

Kitaev's paper Unpaired Majorana Fermions in 1D Quantum Wires (https://arxiv.org/abs/cond-mat/0010440) is famous as a promising experimental proposal for realizing topologically-robust zero-energy fermionic quasiparticles in standard superconductor-semiconductor heterostructures. However, after reading the paper many times through, the reader gets the impression that Kitaev does not give a rigorous mathematical origin of the "effective low-energy Hamiltonian"

$$ H_{eff} = tb'b'' ~~~~~~~~~~~~~~~~~~~~~~~~~~~~(15)$$

which Kitaev writes down in equation (15). The assertion of this effective Hamiltonian is central to the arguments of the paper, and is ultimately used to characterize and show the existence of Majorana Zero Modes (MZM's) in the proposed system. This effective Hamiltonian is only approximate, because the exact solutions of the Bogoliubov-deGennes equations for (15) are not solutions of the Bogoliubov-deGennes equations for the full Hamiltonian.

This confuses me, so I wonder: is there a standard approximation scheme which is being invoked here? Perhaps a Schrieffer-Wolff transformation? My personal hunch: Kitaev's note regarding the physical intuition for $t$ as the "tunneling amplitude for a quasiparticle to tunnel across the chain" smells a lot like discrete WKB is being invoked here. Indeed, in the WKB method, the scaling of the energy gap between low-energy states (spatially separated by a large energy barrier) is proportional to the scaling of the overlap of their corresponding wavefunctions. However, I am not yet able to make this statement any more precise.

Answer 1

Yes, it's a Schrieffer-Wolff transformation. Recall that a Schrieffer-Wolff transformation of a Hamiltonian $H = H_0 + V$, such that the unperturbed Hamiltonian $H_0$ has low-energy subspace projected onto by the projector $P$, is a unitary transformation $W$ such that $[W H W^{\dagger},P] = 0$. The "effective low-energy Hamiltonian" is then defined as $H_{\mathrm{eff}} = P(WHW^{\dagger})P$. Its eigenvalues are a subset of the eigenvalues of $H$ and the low-energy eigenstates of $H$ are related to the eigenstates of $H_{\mathrm{eff}}$ by the unitary $W$. (In principle, this can all be done exactly -- so it's not really an "approximation scheme" as you suggest).

Here you can think of the $H_0$ as being the Hamiltonian where the edge Majoranas $b$ and $b'$ are uncoupled, which occurs when (in the notation of the paper) $|\Delta| = w > 0$, $\mu=0$. In that case there is an exact two-fold degeneracy, which is the subspace projected onto by $P$. Kitaev doesn't bother to actually do any Schrieffer-Wolff calculations because in this case locality uniquely determines the possible form of $H_{\mathrm{eff}}$, up to the value of the coefficient $t$. But in general there are standard ways to compute $W$ and $H_{\mathrm{eff}}$ order by order in perturbation theory.