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Finite simplex lattice models are used in different branches of science,e.g., in condensed matter physics, when studying frustrated magnetic systemsand non-Hermitian localization phenomena; or in chemistry, when describingexperiments with mixtures. An $n$-simplex represents the simplest possiblepolytope in $n$ dimensions, e.g., a line segment, a triangle, and a tetrahedronin one, two, and three dimensions, respectively. In this work, we show thatvarious fully solvable, in general non-Hermitian, $n$-simplex lattice models{with open boundaries} can be constructed from the high-order field-momentsspace of quadratic bosonic systems. Namely, we demonstrate that such$n$-simplex lattices can be formed by a dimensional reduction ofhighly-degenerate iterated polytope chains in $(k>n)$-dimensions, whichnaturally emerge in the field-moments space. Our findings indicate that thefield-moments space of bosonic systems provides a versatile platform forsimulating real-space $n$-simplex lattices exhibiting non-Hermitian phenomena,and yield valuable insights into the structure of many-body systems exhibitingsimilar complexity. Amongst a variety of practical applications, these simplexstructures can offer a physical setting for implementing the discretefractional Fourier transform, an indispensable tool for both quantum andclassical signal processing.

Fully solvable finite simplex lattices with open boundaries in arbitrary dimensions