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We give general conditions for the existence of a Hamiltonian operator whosediscrete time evolution matches the partition function of certain solvablelattice models. In particular, we examine two classes of lattice models: theclassical six-vertex model and a generalized family of $(2n+4)$-vertex modelsfor each positive integer $n$. These models depend on a statistic calledcharge, and are associated to the quantum group$U_q(\widehat{\mathfrak{gl}}(1|n))$. Our results show a close and unexpectedconnection between Hamiltonian operators and solvability.  The six-vertex model can be associated with Hamiltonians from classical Fockspace, and we show that such a correspondence exists precisely when theBoltzmann weights are free fermionic. This allows us to prove that the freefermionic partition function is always a (skew) supersymmetric Schur functionand then use the Berele-Regev formula to correct a result of Brubaker, Bump,and Friedberg. In this context, the supersymmetric function involution takes usbetween two lattice models that generalize the vicious walker and osculatingwalker models. We also observe that the supersymmetric Jacobi-Trudi formula,through Wick's Theorem, can be seen as a six-vertex analogue of theLindstr\"{o}m-Gessel-Viennot Lemma.  Then, we prove a sharp solvability criterion for the six-vertex model withcharge that provides the proper analogue of the free fermion condition.Building on results by Brubaker, Bump, Buciumas, and Gustafsson, we show thatthis criterion exactly dictates when a charged model has a Hamiltonian operatoracting on a Drinfeld twist of q-Fock space. The resulting partition function isthen always a (skew) supersymmetric LLT polynomial.

Lattice Models, Hamiltonian Operators, and Symmetric Functions