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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.

Lattice models, also known as generalized Ising models or cluster expansions, are widely used in many areas of science and are routinely applied to alloy thermodynamics, solid-solid phase transitions, magnetic and thermal properties of solids, and fluid mechanics, among others. However, the problem of finding the true global ground state of a lattice model, which is essential for all of the aforementioned applications, has remained unresolved, with only a limited number of results for highly simplified systems known. In this article, we present the first general algorithm to find the exact ground states of complex lattice models and to prove their global optimality, resolving this fundamental problem in condensed matter and materials theory. We transform the infinite-discrete-optimization problem into a pair of combinatorial optimization (MAX-SAT) and non-smooth convex optimization (MAX-MIN) problems, which provide upper and lower bounds on the ground state energy respectively. By systematically converging these bounds to each other, we find and prove the exact ground state of realistic Hamiltonians whose solutions are completely intractable via traditional methods. Considering that currently such Hamiltonians are solved using simulated annealing and genetic algorithms that are often unable to find the true global energy minimum, and never able to prove the optimality of their result, our work opens the door to resolving long-standing uncertainties in lattice models of physical phenomena.