Graph theoretical proof of nonintegrability in quantum many-body systems : Application to the PXP model

TL;DR

This paper introduces a graph-theoretical framework to rigorously prove non-integrability in quantum many-body systems, exemplified by the PXP model, and highlights its advantages over traditional methods.

Contribution

The work develops a novel graph-theoretical approach for proving non-integrability and classifying conserved quantities in quantum systems, simplifying the analysis process.

Findings

Proves the non-integrability of the PXP model by showing absence of local conserved quantities.

Highlights the advantage of graph counting for classifying conserved quantities even in integrable systems.

Demonstrates broad applicability of the method to other quantum many-body systems.

Abstract

A rigorous proof of integrability or non-integrability in quantum many-body systems is among the most challenging tasks, as it involves demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we establish a graph-theoretical analysis as a comprehensive framework for proving the non-integrability of quantum systems. Exemplifying the PXP model, which is widely believed to be non-integrable, this work rigorously proves the absence of local conserved quantities, thereby confirming its non-integrability. This proof for the PXP model gives several important messages not only that the system is non-integrable, but also the quantum many body scaring observed in the model is not associate with the existence of local conserved quantities. From a graph-theoretical perspective, we also highlight its advantage, even in integrable systems, as the classification of local conserved quantities can be achieved by simply counting the number of isolated loops in the graphs. Our new approach is broadly applicable for establishing proofs of (non-)integrability in other quantum many-body systems, significantly simplifying the process of proving nonintegrability and giving numerous potential applications.