  Generalized Ising models, also known as cluster expansions, are an important
tool in many areas of condensed-matter physics and materials science, as they
are often used in the study of lattice thermodynamics, solid-solid phase
transitions, magnetic and thermal properties of solids, and fluid mechanics.
However, the problem of finding the global ground state of generalized Ising
model has remained unresolved, with only a limited number of results for simple
systems known. We propose a method to efficiently find the periodic ground
state of a generalized Ising model of arbitrary complexity by a new algorithm
which we term cluster tree optimization. Importantly, we are able to show that
even in the case of an aperiodic ground state, our algorithm produces a
sequence of states with energy converging to the true ground state energy, with
a provable bound on error. Compared to the current state-of-the-art polytope
method, this algorithm eliminates the necessity of introducing an exponential
number of variables to counter frustration, and thus significantly improves
tractability. We believe that the cluster tree algorithm offers an intuitive
and efficient approach to finding and proving ground states of generalized
Ising Hamiltonians of arbitrary complexity, which will help validate
assumptions regarding local vs. global optimality in lattice models, as well as
offer insights into the low-energy behavior of highly frustrated systems.
