Factorial Fock free fermions

We use a double shifted power analog of free fermion fields to introduce current operators, Hamiltonians, and vertex operators which are deformed by two families of parameters and satisfy analogous formulas to the classical case. We show that the deformed half vertex operators correspond to the row transfer matrices of a solvable six vertex model recently given by Naprienko [arXiv:2301.12110], which under a specialization yields the factorial Schur functions (up to a reindexing of parameters). As a consequence, we show that under the boson-fermion correspondence using our deformed half vertex operators, the natural basis (under this specialization) maps to the double factorial Schur functions. Furthermore, the image of the natural basis vectors are tau function solutions to the 2D Toda lattice.