Nucleation of Sachdev-Ye-Kitaev Clusters in One Spatial Dimension

TL;DR

This paper explores how SYK interactions emerge from localized states in one dimension, showing how cluster formation depends on the microscopic structure and overlaps of localized orbitals.

Contribution

It introduces a real-space model for SYK cluster formation, revealing how local interactions evolve into SYK-like couplings as the system's microscopic resolution increases.

Findings

Couplings have a broad non-Gaussian distribution with finite zero probability.

Increasing microscopic resolution M makes couplings approach the complex-Gaussian SYK form.

SYK clusters form and grow, characterized by graph connectivity and clique counts.

Abstract

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 $M>1$ smaller microscopic pieces with random phases. As $M$ increases, the distribution of the nonzero couplings moves toward the complex-Gaussian SYK form. At the same time, the large-$M$ 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.