Publication

Fully solvable finite simplex lattices with open boundaries in arbitrary dimensions

Fabrizio Minganti
2023
Journal paper
Abstract

Finite simplex lattice models are used in different branches of science, e.g., in condensed-matter physics, when studying frustrated magnetic systems and non-Hermitian localization phenomena; or in chemistry, when describing experiments with mixtures. An n-simplex represents the simplest possible polytope in n dimensions, e.g., a line segment, a triangle, anda tetrahedron in one, two, and three dimensions, respectively. In this work, we show that various fully solvable, in general non-Hermitian, n-simplex lattice models with open boundaries can be constructed from the high-order field-moments space of quadratic bosonic systems. Namely, we demonstrate that such n-simplex lattices can be formed by a dimensional reduction of highly degenerate iterated polytope chains in (k > n)-dimensions, which naturally emerge in the field-moments space. Our findings indicate that the field-moments space of bosonic systems provides a versatile platform for simulating real-space n-simplex lattices exhibiting non-Hermitian phenomena, and it yields valuable insights into the structure of many-body systems exhibiting similar complexity. Among a variety of practical applications, these simplex structures can offer a physical setting for implementing the discrete fractional Fourier transform, an indispensable tool for both quantum and classical signal processing.

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Ontological neighbourhood
Related concepts (32)
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment, a 2-dimensional simplex is a triangle, a 3-dimensional simplex is a tetrahedron, and a 4-dimensional simplex is a 5-cell. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices.
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by Ernst Witt in 1940. The Leech lattice Λ24 is the unique lattice in 24-dimensional Euclidean space, E24, with the following list of properties: It is unimodular; i.e., it can be generated by the columns of a certain 24×24 matrix with determinant 1. It is even; i.e.
Line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
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