  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)
