Remarks on the notion of quantum integrability (1012.3587v1)

Abstract: We discuss the notion of integrability in quantum mechanics. Starting from a review of some definitions commonly used in the literature, we propose a different set of criteria, leading to a classification of models in terms of different integrability classes. We end by highlighting some of the expected physical properties associated to models fulfilling the proposed criteria.

• The paper critiques traditional definitions of quantum integrability and highlights their limitations in addressing many-body quantum dynamics.
• The paper introduces a novel classification based on operator density, categorizing models into linear, quasi-polynomial, and other integrability classes.
• The framework offers new insights into ergodicity and non-equilibrium behavior by correlating the structure of conserved charges with physical phenomena.

Overview of the Notion of Quantum Integrability

In this paper, Jean-Sebastien Caux and Jorn Mossel address the concept of quantum integrability, a notion less well-defined and understood than its classical counterpart. The paper critiques existing definitions found in the literature and proposes a novel framework to categorize quantum models into different integrability classes, ultimately aiming to correlate these classes with distinct physical behaviors.

Background and Motivation

Classical integrability is well-established through the Liouville integrability criterion; however, quantum integrability lacks such clear delineation. The authors argue that this vagueness has led to confusion, particularly as integrability frequently appears in discussions on the dynamics and thermalization of many-body quantum systems. The absence of a precise quantum analogue to classical integrability means potentially important properties of quantum systems may remain unprobed.

Existing Definitions Analyzed

The paper provides a thorough examination of several definitions of quantum integrability:

1. QI:N (Naive Definition): A system is deemed integrable if it possesses a maximal set of independent commuting quantum operators. Although this mirrors the classical definition, it is too inclusive, as it applies trivially to all quantum systems with finite-dimensional Hilbert spaces.
2. QI:ES (Exactly Solvable): Defines integrability in terms of solvability, allowing explicit construction of eigenstates, akin to action-angle variables in classical mechanics. The authors find this unsatisfactory, as not all physically relevant features are accounted for.
3. QI:ND (Nondiffractive Scattering): Based on the factorization of scattering into two-body events, this is seen as a strong indicator of integrability but is considered by the authors more as a feature than a foundational definition.
4. QI:ELS and QI:LC (Energy Level Statistics and Level Crossings): These relate integrability to specific statistical properties of energy levels, notably the absence of level repulsion or Poisson statistics. While insightful, these properties can manifest in both integrable and non-integrable models, rendering them insufficient as overarching definitions.

Proposed Framework

The authors propose a new definition centered around the concept of operator density character within a specific basis, effectively examining the structure and independence of conserved charges. The classification into linear, polynomial, quasi-polynomial, or sub-exponential integrability classes allows for a more nuanced understanding of model behaviors:

• Linear Integrability: Models like free particles and fundamental lattice models display linear integrability, where conserved charges are typically associated with simple Fourier transforms or local operators in real space.
• Quasi-Polynomial Integrability: Models like the Haldane-Shastry chain fall into this category, characterized by more complex conserved charges that increase exponentially with system size.

Implications and Applications

The paper suggests that the proposed classification could better correlate with physical phenomena such as ergodicity and non-equilibrium dynamics. For instance, it builds connections with Mazur's inequality, offering insights into how the structure of conserved charges influences ergodic properties. The authors anticipate using this refined understanding to improve predictions about quantum system behaviors following non-equilibrium events like quantum quenches.

Conclusion and Future Directions

While addressing a complex and partially unresolved topic, the paper attempts to provide clarity and rigor to the notion of quantum integrability, suggesting a meaningful way to partition models. The proposed framework allows for practical assessments and could refine theoretical tools like generalized Gibbs ensembles, impacting ongoing research in quantum many-body dynamics. The authors suggest that further investigation into specific classes of models and observables is needed to substantiate and elaborate on the ramifications of this definition. Overall, this paper serves as a stepping stone toward a more comprehensive characterization of integrability within quantum mechanics.