Quasilocal charges in integrable lattice systems

Abstract: We review recent progress in understanding the notion of locality in integrable quantum lattice systems. The central concept are the so-called quasilocal conserved quantities, which go beyond the standard perception of locality. Two systematic procedures to rigorously construct families of quasilocal conserved operators based on quantum transfer matrices are outlined, specializing on anisotropic Heisenberg XXZ spin-1/2 chain. Quasilocal conserved operators stem from two distinct classes of representations of the auxiliary space algebra, comprised of unitary (compact) representations, which can be naturally linked to the fusion algebra and quasiparticle content of the model, and non-unitary (non-compact) representations giving rise to charges, manifestly orthogonal to the unitary ones. Various condensed matter applications in which quasilocal conservation laws play an essential role are presented, with special emphasis on their implications for anomalous transport properties (finite Drude weight) and relaxation to non-thermal steady states in the quantum quench scenario.

• The paper introduces a formal framework for quasilocal conserved charges, extending traditional locality to include rapidly decaying correlations beyond adjacent sites.
• It employs quantum transfer matrices and Yang-Baxter relations to construct extensive families of quasilocal operators in the XXZ spin-1/2 model.
• The study reveals that incorporating quasilocal charges is essential for accurately describing non-equilibrium dynamics and transport phenomena in integrable lattice systems.

Quasilocal Charges in Integrable Lattice Systems: A Comprehensive Overview

The paper "Quasilocal charges in integrable lattice systems" provides an in-depth analysis of the conceptual and practical implications of quasilocal conserved quantities in integrable quantum lattice systems. Integrable systems, renowned for their exact solvability, possess a plethora of conserved charges arising from underlying symmetries often governed by quantum groups or Yang-Baxter structures. This study focuses on advancing the traditional understanding of locality to include quasilocal conserved operators, particularly within the framework of the anisotropic Heisenberg XXZ spin-1/2 chain.

Key Concepts and Methodologies

Fundamental to the paper's discussion is the introduction of quasilocality, a notion extending beyond strict locality to encompass conserved charges supported on configurations broader than finitely adjacent sites but exhibiting rapid decay in correlations. The paper outlines two primary methodologies for constructing quasilocal operators, leveraging quantum transfer matrices and exploring their applicability across different representational landscapes, including unitary and non-unitary classes.

1. Quasilocality and Pseudolocality: The paper establishes a rigorous framework for understanding quasilocal and pseudolocal operators, particularly how they manifest in conserved quantities that dictate non-equilibrium dynamics. Quasilocal operators are demonstrated to be pivotal in explaining the non-ergodic and ballistic transport phenomena observed in these systems.
2. Yang-Baxter Integrability: The paper explores the algebraic structures arising from the Yang-Baxter equation, exploring their role in generating a continuum of conserved quantities via quantum transfer matrices. Special attention is given to the foundational RLL and RTT relations, crucial for developing the hierarchy of conserved operators in the XXZ model.
3. Construction and Algebraic Properties: Utilizing higher-spin auxiliary spaces and non-standard representations, the paper constructs extensive families of quasilocal charges. The main algebraic toolset comprises Lax operators and the transfer matrix, with modifications allowing for non-unitary spins leading to unique conserved operators.
4. Applications and Implications: The constructed quasilocal charges find application in several frontier topics in condensed matter physics. They provide critical insights into the long-standing debate about transport properties – particularly the non-zero spin Drude weight at finite temperatures in one-dimensional spin chains. Notably, the paper rigorously critiques the adequacy of the Generalized Gibbs Ensemble (GGE) in completely describing equilibrium states after a quantum quench, stressing the importance of incorporating quasilocal charges for an exhaustive account.

Results and Theoretical Insights

The findings of the paper present a clear numerical and theoretical exposition of quasilocal charges. The developed formalism facilitates deep understanding of generalized hydrodynamics in integrable models and their relaxation properties. Additionally, the study underscores the necessity of including non-unitary representations to fully grasp the interplay of different symmetries in governing transport properties, shedding new light on phenomena such as ballistic transport and the anomalous scaling of response functions.

Future Directions

The pursuit of comprehensively generalizing these techniques to a wider class of integrable models remains an open and compelling research avenue. Extending the results to models like the Hubbard chain, higher-rank algebras, and superintegrable systems offers potential pathways to uncovering deeper insights. Furthermore, exploring the connections between quasilocality and classical integrability via the framework of the algebraic Bethe ansatz can yield promising results in both theoretical and applied physics.

In summary, the paper "Quasilocal charges in integrable lattice systems" contributes significantly to our understanding of integrable models in quantum mechanics, both by expanding the concept of locality to quasilocality and by offering insights into their implications for non-equilibrium dynamics and transport phenomena.