Restoring permutational invariance in the Jordan-Wigner transformation

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 labeling of the spins. One consequence of this dependence is that approximate solutions of a Jordan-Wigner--transformed Hamiltonian may depend on the (physically inconsequential) labeling 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.