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  Approximation scheme in Kitaev's paper "Unpaired Fermions in 1D Wires" (2000)?

+ 3 like - 0 dislike

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.   

asked Aug 6, 2017 in Theoretical Physics by David B Roberts (135 points) [ revision history ]
edited Aug 6, 2017 by David B Roberts

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