Spin-1/2 operators exactly mapped to spinful canonical Fermi operators present in two bands

TL;DR

This work establishes an exact 1D mapping between $S=1/2$ spin operators and spinful canonical Fermi operators in two bands using an extended Jordan–Wigner framework. The construction relies on four on-site hybrid operators and a set of six free parameters to enforce $S=1/2$ while yielding two spinful fermion bands, enabling the translation of spin models into realistic two-band fermionic Hamiltonians. The Heisenberg model serves as a detailed test case, with the transformed Hamiltonian containing a rich array of terms and a density-dependent operator arising from the mapping; a fine-tuned instance demonstrates an SOC-enabled, correlated-hopping two-band system equivalent to the spin model. The results broaden the bridge between spin physics and itinerant multi-band fermions, with implications for SOC, correlated hopping, and higher-band generalizations, and provide a concrete roadmap for numerical implementations of the transformation.

Abstract

Recently it has been shown that the quantum spin-1/2 spin operators can be exactly transformed not only in spinless, but also in spinful canonical Fermi operators in 1D [\cite{JW1}], and 2D [\cite{JW2}] as well. In this paper, using the same technique based on an extended Jordan-Wigner transformation, we show in 1D, that the quantum spin-1/2 operators can be exactly transformed also in spinful canonical Fermi operators of spin-1/2 fermions that are belong to two bands.