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Journal of Modern Physics

Interaction of Dot Clasters

Alexander S. Chikhachev

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Models of the pointed interactions approximately describing real interactions of nuclear particles in quantum mechanics are considered. The concept of “a dot cluster”—a complex of charges which at the zero size create possibility of localization of a trial particle in the field of the final size is entered. States in one-dimensional systems, and also in three-dimensional systems with “a local isotropy” are studied. The conditions of dot systems characterized by the nonzero, including fractional, orbital moment were studied.

DOI
10.4236/jmp.2015.611165

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Models of the pointed interactions approximately describing real interactions of nuclear particles in quantum mechanics are considered. The concept of “a dot cluster”—a complex of charges which at the zero size create possibility of localization of a trial particle in the field of the final size is entered. States in one-dimensional systems, and also in three-dimensional systems with “a local isotropy” are studied. The conditions of dot systems characterized by the nonzero, including fractional, orbital moment were studied.
Models of point interactions that approximate the real interactions of atomic particles in quantum mechanics are considered. In a one-point one-dimensional system, states are studied taking into account the eigenfield and a non-stationary problem is solved. The concept of a "point cluster" is introduced - a complex of charges that, at zero size, create the possibility of localization of a test particle in a region of finite size. An integral model of point interaction is proposed. States in three-dimensional two-point systems with "local isotropy" are studied.
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We study how Sachdev-Ye-Kitaev (SYK) interactions can arise from localized single-particle states on a system that is effectively one dimensional. If a local interaction is projected onto coarse localized orbitals, the resulting couplings do not immediately follow the standard SYK distribution. Instead, they have a finite probability of being exactly zero, a broad non-Gaussian distribution for the nonzero values, and strong correlations coming from the geometry of the localized states. We then show that this changes when each localization volume is resolved into M>1M>1M>1 smaller microscopic pieces with random phases. As MMM increases, the distribution of the nonzero couplings moves toward the complex-Gaussian SYK form. At the same time, the large-MMM limit is a sparse but asymptotically canonical SYK network: the nonzero couplings create SYK clusters, while the pattern of missing or very weak couplings is still determined by the real-space overlap of the localized orbitals. Finally, we map the interaction tensor to a graph in pair space. This makes it possible to follow the formation, merger, and growth of SYK clusters, which we characterize using connected components and clique/simplex counts. The result is a minimal real-space phenomenological theory of SYK-cluster formation, providing clear experimental criteria.