Strong-Weak Duality via Jordan-Wigner Transformation: Using Fermionic Methods for Strongly Correlated $su(2)$ Spin Systems

TL;DR

This paper demonstrates that the Jordan-Wigner transformation can weaken strong correlations in $su(2)$ spin systems by mapping them to fermionic representations, simplifying certain computational approaches.

Contribution

It provides qualitative and numerical evidence that Jordan-Wigner transformation reduces correlation strength and complexity in spin systems, offering improved methods for solving challenging spin problems.

Findings

Fermionic representation weakens strong correlations in spin systems.

Jordan-Wigner transformation reduces Hamiltonian complexity in key phase diagram regions.

Low-cost techniques can mitigate string operator challenges in classical computations.

Abstract

The Jordan-Wigner transformation establishes a duality between $su(2)$ and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree-Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J$_1$-J$_2$ Heisenberg (in 1D and 2D) indicate that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams, and provides a better starting point for addressing challenging spin problems.