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Physical review. B./Physical review. B

Nonintegrability in the PXP model: A graph-theoretical approach

HaRu K. Park, SungBin Lee

Published Feb 4, 2025• 3 citations
Publisher

Abstract

Establishing a rigorous proof of integrability or nonintegrability in quantum many-body systems remains one of the most challenging tasks, as it requires demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we introduce a graph-theoretical approach to address this issue. This framework not only provides a comprehensive method for analyzing the nonintegrability of quantum systems but also brings advantages for integrable systems by enabling the classification of local conserved quantities through a straightforward count of isolated loops in the associated graphs. As an illustrative example, we focus on the PXP model, a system widely believed to be nonintegrable, and rigorously confirm its nonintegrability by demonstrating the absence of local conserved quantities through graph-theoretical analysis. This proof for the PXP model conveys several key insights: It not only confirms the model's nonintegrability but also indicates that the quantum many-body scarring properties of the model is not associated with local conserved quantities. Our innovative approach is broadly applicable for proving (non-)integrability in other quantum many-body systems, significantly simplifying the process and offering numerous potential applications.

DOI
10.1103/physrevb.111.085104

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Establishing a rigorous proof of integrability or nonintegrability in quantum many-body systems remains one of the most challenging tasks, as it requires demonstrating the presence or absence of local conserved quantities and deciphering the complex dynamics of the system. In this paper, we introduce a graph-theoretical approach to address this issue. This framework not only provides a comprehensive method for analyzing the nonintegrability of quantum systems but also brings advantages for integrable systems by enabling the classification of local conserved quantities through a straightforward count of isolated loops in the associated graphs. As an illustrative example, we focus on the PXP model, a system widely believed to be nonintegrable, and rigorously confirm its nonintegrability by demonstrating the absence of local conserved quantities through graph-theoretical analysis. This proof for the PXP model conveys several key insights: It not only confirms the model's nonintegrability but also indicates that the quantum many-body scarring properties of the model is not associated with local conserved quantities. Our innovative approach is broadly applicable for proving (non-)integrability in other quantum many-body systems, significantly simplifying the process and offering numerous potential applications.
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.
The Yang-Baxter method is a powerful tool for studying integrable systems, but proving whether a system possesses local conserved quantities remains challenging. Shiraishi's proof of the XYZ model's nonintegrability with a magnetic field inspired the authors' demonstration of the PXP model's nonintegrability, previously only conjectured via level statistics. As the PXP model is a key example of quantum many-body scars exhibiting exotic long-time behavior, the present work highlights that such behavior is unrelated to local conserved quantities and introduces a graph-theoretical approach, suggesting universality in nonintegrability proofs.
The integrability of a quantum many-body system, which is characterized by the presence or absence of local conserved quantities, drastically impacts the dynamics of isolated systems, including thermalization. Nevertheless, a rigorous and comprehensive method for determining integrability or nonintegrability has remained elusive. In this paper, we address this challenge by introducing rigorously provable tests for integrability and nonintegrability of quantum spin systems with finite-range interactions. Our results significantly simplify existing proofs of nonintegrability, such as those for the S=1/2S=1/2S=1/2 Heisenberg chain with nearest-and next-nearest-neighbor interactions, the S=1S=1S=1 bilinear-biquadratic chain and the S=1/2S=1/2S=1/2 XYZ model in two or higher dimensions. Moreover, our results also yield the first proof of nonintegrability for models such as the S=1/2S=1/2S=1/2 Heisenberg chain with a non-uniform magnetic field, the S=1/2S=1/2S=1/2 XYZ model on the triangular lattice, and the general spin XYZ model. This work also offers a partial resolution to the long-standing conjecture that integrability is governed by the existence of local conserved quantities with small support. Our framework ensures that the nonintegrability of one-dimensional spin systems with translational symmetry can be verified algorithmically, independently of system size.
Spin-1 chain models have been extensively studied in condensed matter physics, significantly advancing our understanding of quantum magnetism and low-dimensional systems, which exhibit unique properties compared to their spin-1/21/21/2 counterparts. Despite substantial progress in this area, a rigorous proof of nonintegrability for the bilinear-biquadratic chain model has been lacking. While integrable solutions are known for specific parameter values, a comprehensive understanding of the model's general integrability has remained elusive. In this paper, we present a rigorous proof of nonintegrability for general spin-1 bilinear-biquadratic chain models, by extending the proofs of nonintegrability on spin-1/21/21/2 models. Our proof not only confirms the nonintegrability of widely studied models but also provides offer deeper insights into several areas. These include the identification of the absence of local conserved quantities in quantum many-body scar systems with perfect fidelity revivals, such as the AKLT model. This paper marks a significant step toward understanding the complex dynamics of spin-1 systems.
Absence of local conserved quantities, or nonintegrability, is often assumed when discussing various phenomena in quantum many-body systems, such as thermalization and transport. However, no concrete proof of this property is known in electron--phonon coupled systems, a typical setting for condensed matter physics. In this paper, we show that the one-dimensional Holstein model has no nontrivial local conserved quantities other than the Hamiltonian itself and the total fermion number operator. We further show that the absence of nontrivial local conserved quantities also holds for the more general Holstein--Hubbard model. We also establish the absence of nontrivial local conserved quantities in these models on higher-dimensional hypercubic lattices. Our result has accomplished an advance in nonintegrability proofs by expanding their scope to systems in which particles with different statistical properties are mixed.
Local conserved quantities are key to understanding macroscopic phenomena, and their absence indicates quantum nonintegrability. Here, the author rigorously proves that quantum Ising models on two- and higher-dimensional lattices possess no local conserved quantities. The proof holds for arbitrary parameter values, thereby theoretically confirming that there are no integrable systems other than the trivial case. Their result establishes this widely studied model as a definitive example of a nonintegrable quantum many-body system.