The $S=\frac{1}{2}$ XY and XYZ models on the two or higher dimensional hypercubic lattice do not possess nontrivial local conserved quantities (2412.18504v4)

Abstract: We study the $S=\frac{1}{2}$ quantum spin system on the $d$-dimensional hypercubic lattice with $d\ge2$ with uniform nearest-neighbor interaction of the XY or XYZ type and arbitrary uniform magnetic field. By extending the method recently developed for quantum spin chains, we prove that the model possesses no local conserved quantities except for the trivial ones, such as the Hamiltonian. This result strongly suggests that the model is non-integrable. We note that our result applies to the XX model without a magnetic field, which is one of the easiest solvable models in one dimension.

• The paper rigorously shows that S=1/2 XY and XYZ hypercubic models lack any nontrivial local conserved quantities.
• It extends one-dimensional methodologies to higher dimensions through a linear equations approach to rule out local conservation laws.
• These findings imply non-integrability, offering insights into quantum chaos and thermalization in high-dimensional spin systems.

The $S=\frac{1}{2}$ XY and XYZ Models on Higher-Dimensional Lattices: A Study of Integrability

The discussion of integrability in quantum many-body systems often revolves around the presence or absence of local conserved quantities. In their work, Shiraishi and Tasaki investigate a class of quantum spin models on hypercubic lattices in dimensions two and higher, specifically focusing on $S=\frac{1}{2}$ spins with XY or XYZ interactions. Their primary result is the rigorous demonstration that these models lack nontrivial local conserved quantities, suggesting a non-integrable nature. This paper extends methodologies previously developed for one-dimensional systems to higher dimensions, yielding insights significant for both theoretical understanding and practical applications in quantum chaos and thermalization.

Key Contributions

1. Model Overview: The paper examines $S=\frac{1}{2}$ quantum spin systems on $d$-dimensional hypercubic lattices with $d \geq 2$. The interactions considered are of the XY or XYZ type with an arbitrary uniform magnetic field. The Hamiltonian includes terms for nearest-neighbor interactions, represented by the exchange constants $X$, $Y$, and $Z$.
2. Absence of Local Conserved Quantities: The central theorem of the paper establishes that for dimensions $d \geq 2$, the considered models do not possess local conserved quantities beyond trivial ones like the Hamiltonian itself. This extends earlier results from one-dimensional models to higher dimensions, reinforcing the models' non-integrable classification.
3. Integrability Assessment: By proving the absence of nontrivial local conserved quantities, the authors suggest that the discussed models are non-integrable, aligning with intuitions held by many researchers. They underscore that integrability is often linked to the existence of such conserved quantities.
4. Technical Approach: The authors expand and apply a method from prior work, which involves characterizing local conserved quantities through the structure of linear equations. They use this to show that possible candidates for conserved quantities must include terms of a width consistent with the dimension, leading to their elimination.
5. Implications for Non-Integrable Systems: The results highlight a key difference between integrable and non-integrable systems. Non-integrable models, as demonstrated here, do not support local symmetry operations that would preserve the state across dynamics.

Theoretical and Practical Implications

The findings have several implications for both the understanding of quantum many-body systems and potential future research directions:

• Non-Integrability Indicators: The lack of local conserved quantities is a strong indicator of non-integrability in these models. This absence also suggests the possible presence of quantum chaos, where energy levels exhibit non-trivial repulsion, and eigenstate thermalization, where individual states adopt thermal properties.
• Beyond Local Integrability: The paper opens avenues to explore quasi-local conservation laws and their role in high-dimensional quantum systems. While strictly local conserved quantities are absent, understanding the behavior of quasi-local ones could yield new insights into few-body quantum chaos.
• Technique Development: Extending the methods used to determine the non-existence of local conservations in higher dimensions marks a significant increase in the methodology's robustness. Future studies might focus on simplifying these approaches or extending them to other types of lattices and interactions.
• Quantum Thermalization: With the non-trivial local conserved quantities ruled out, these models may comply with the eigenstate thermalization hypothesis (ETH), a principle that asserts most quantum systems equilibrate to thermal states due to internal dynamics.

In conclusion, Shiraishi and Tasaki's work provides a crucial step toward understanding high-dimensional spin systems' integrability properties. Their approach not only solidifies the classification of these models as non-integrable but also proposes a general toolkit for investigating other models within and beyond two-dimensional frameworks. As quantum technologies advance, such theoretical insights become ever more relevant, bridging abstract theory with practical quantum computation and simulation.