Real-space decomposition of $p$-wave Kitaev chain

TL;DR

This paper introduces a real-space Bogoliubov transformation for spinless fermions, enabling the decomposition of complex Kitaev chains into simpler sub-chains, and explores its implications for topological phases and dynamics.

Contribution

It presents a novel real-space decomposition method for Kitaev chains, linking complex systems to simpler models and analyzing their topological and dynamical properties.

Findings

Decomposition maps a $2N$ chain to two $N$ chains at zero chemical potential.

The method reveals groundstate topology and dynamics through sub-chain analysis.

Approximate validity persists with finite chemical potential in gapped regions.

Abstract

We propose an extended Bogoliubov transformation in real space for spinless fermions, based on which a class of Kitaev chains of length $2N$ with zero chemical potential can be mapped to two independent Kitaev chains of length $N$. It provides an alternative way to investigate a complicated system from the result of relatively simple systems. We demonstrate the implications of this decomposition by a Su-Schrieffer-Heeger (SSH) Kitaev model, which supports rich quantum phases. The features of the system, including the groundstate topology and nonequilibrium dynamics, can be revealed directly from that of sub-Kitaev chains. Based on this connection, two types of Bardeen-Cooper-Schrieffer (BCS)-pair order parameters are introduced to characterize the phase diagram, showing the ingredient of two different BCS pairing modes. Analytical analysis and numerical simulations show that the real-space decomposition for the ground state still holds true approximately in presence of finite chemical potential in the gapful regions.