Restoring permutational invariance in the Jordan–Wigner transformation

Abstract

The Jordan–Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labelling of the spins. One consequence of this dependence is that approximate solutions of a Jordan–Wigner-transformed Hamiltonian may depend on the (physically inconsequential) labelling of the spins. In this work, we turn to an extended Jordan–Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree–Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan–Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. We show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and $J_1-J_2$ Heisenberg Hamiltonians.

GRAPHICAL ABSTRACT

KEYWORDS:

• Jordan–Wigner transformation
• lattice models
• spin systems

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences. The Jordan–Wigner aspects were supported under Award DE-SC0019374, and the strong correlation aspects under Award DE-FG02-09ER16053. G.E.S. is a Welch Foundation Chair (C-0036).