We give general conditions for the existence of a Hamiltonian operator whose discrete time evolution matches the partition function of certain solvable lattice models. In particular, we examine two classes of lattice models: the classical six-vertex model and a generalized family of $(2n+4)$-vertex models for each positive integer $n$. These models depend on a statistic called charge, and are associated to the quantum group $U_q(\widehat{\mathfrak{gl}}(1|n))$. Our results show a close and unexpected connection between Hamiltonian operators and the Yang-Baxter equation. The six-vertex model can be associated with Hamiltonians from classical Fock space, and we show that such a correspondence exists precisely when the Boltzmann weights are free fermionic. This allows us to prove that the free fermionic partition function is always a (skew) supersymmetric Schur function and then use the Berele-Regev formula to correct a result from of Brubaker, Bump, and Friedberg on the free fermionic domain-wall partition function. In this context, the supersymmetric function involution takes us between two lattice models that generalize the vicious walker and osculating walker models. Then, we prove a sharp solvability criterion for the six-vertex model with charge that provides the proper analogue of the free fermion condition. Building on results by Brubaker, Buciumas, Bump, and Gustafsson, we show that this criterion exactly dictates when a charged model has a Hamiltonian operator acting on a Drinfeld twist of $q$-Fock space. The resulting partition function is then a (skew) supersymmetric LLT polynomial, and almost all supersymmetric LLT polynomials appear as partition functions of our lattice models. We also prove a Cauchy identity for skew supersymmetric LLT polynomials. (see 2024 author's note below)