Abstract
We study the properties of the set of marginal distributions of infinite translation-invariant systems in the two-dimensional square lattice. In cases where the local variables can only take a small number d of possible values, we completely solve the marginal or membership problem for nearest-neighbours distributions (d = 2, 3) and nearest and next-to-nearest neighbours distributions (d = 2). Remarkably, all these sets form convex polytopes in probability space. This allows us to devise an algorithm to compute the minimum energy per site of any TI Hamiltonian in these scenarios exactly. We also devise a simple algorithm to approximate the minimum energy per site up to arbitrary accuracy for the cases not covered above. For variables of a higher (but finite) dimensionality, we prove two no-go results. To begin, the exact computation of the energy per site of arbitrary TI Hamiltonians with only nearest-neighbour interactions is an undecidable problem. In addition, in scenarios with d≥2947, the boundary of the set of nearest-neighbour marginal distributions contains both flat and smoothly curved surfaces and the set itself is not semi-algebraic. This implies, in particular, that it cannot be characterized via semidefinite programming, even if we allow the input of the programme to include polynomials of nearest-neighbour probabilities.
Highlights
The distribution of stars at large scales, the vacuum state of a quantum field theory, the thermal state of any solvable spin model: these are examples of systems with infinitely many sites where the description of a bounded environment does not depend on its location within the lattice
Despite a long history of research on TI systems, driven by the needs of statistical physics (e.g. [1]), it is far from clear what those constraints exactly are. This conundrum is at the essence of the TI MARGINAL problem, where a number of probability distributions of finitely-many variables are provided and the task is to certify if they correspond to the marginals of a TI system
We have studied the problem of deciding whether a number of distributions correspond to the marginals of a 2D TI system, what we called the MARGINAL problem
Summary
The distribution of stars at large scales, the vacuum state of a quantum field theory, the thermal state of any solvable spin model: these are examples of systems with infinitely many sites where the description of a bounded environment does not depend on its location within the lattice. Contrary to the solvable cases, for local random variables of high cardinality, the set of nearest-neighbour distributions admitting a TI extension is no longer a convex polytope in the space of probabilities or even a semi-algebraic set.. Contrary to the solvable cases, for local random variables of high cardinality, the set of nearest-neighbour distributions admitting a TI extension is no longer a convex polytope in the space of probabilities or even a semi-algebraic set.3 This implies, in particular, that standard tools from convex optimization, such as linear programming [13] or semidefinite programming [14] cannot be used to fully characterize Bell non-locality in large 2D condensed matter systems. (b) for random variables with support d greater than or equal to 2947, the set of nearest-neighbour distributions is not a semi-algebraic set
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