Hamiltonian operators and free fermionic lattice models
We will explore Hamiltonian operators from the “symmetric function” perspective favored by Lam (arXiv:0507341). In the same way that Schur polynomials can be built up from power sum symmetric functions, we can produce other symmetric functions by replacing the power sums with other functions. The symmetric functions we obtain, which include Macdonald and LLT polynomials, will have “Schur-like” identities such as Cauchy and Pieri rules.
A question we will ask is “When can a lattice model partition function also be obtained from a Hamiltonian operator?” In the six-vertex case, the condition turns out to be that the Boltzmann weights are free fermionic. If we consider lattice models with charge, we obtain a similar condition that suggests that the weights of Brubaker, Bump, Buciumas, and Gustafsson (arXiv:1806.07776) may be (essentially) the only set of weights that corresponds to a Hamiltonian operator.
Zoom link: https://stanford.zoom.us/j/91274693500?pwd=bmtHZnhTMG1HQ3pTOHYxUWJ2Z1ZjQT09
Zoom ID: 912 7469 3500
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Relevant papers: A combinatorial generalization of the Boson-Fermion correspondence