Complete Generalized Gibbs Ensemble in an interacting Theory (1507.02993v2)

Abstract: In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady state predictions. Here we solve this long-standing problem for the case of the spin-1/2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behaviour after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasi-local charges. As a test, we reproduce the exact post-quench steady state of the Neel quench problem obtained previously by means of the Quench Action method.

• The paper demonstrates that incorporating quasi-local charges completes the GGE framework, accurately predicting steady-state properties in interacting quantum systems.
• It applies the solution to the N{é}el-to-XXZ quench problem, with results validating the approach against the Quench Action method.
• The findings refine theoretical predictions of correlation functions and entanglement growth, deepening our understanding of quantum dynamics.

Complete Generalized Gibbs Ensemble in an Interacting Theory

The paper "Complete Generalized Gibbs Ensemble in an Interacting Theory" addresses a fundamental problem in understanding the non-equilibrium dynamics of quantum integrable systems. Specifically, it provides a comprehensive solution to the long-standing issue of identifying a complete set of conserved charges necessary for constructing a Generalized Gibbs Ensemble (GGE) that can accurately predict the steady-state properties of systems, particularly the spin-1/2 Heisenberg chain after a quantum quench.

Overview of the Research

The Generalized Gibbs Ensemble is a theoretical framework utilized for describing the stationary states of integrable systems post-quench. In integrable models, a complete thermalization, as encountered in generic systems, does not occur due to the existence of numerous conserved quantities. The GGE is introduced to account for these conserved quantities, extending the standard Gibbs ensemble approach by associating a 'chemical potential' to each conserved charge.

In previous studies, GGEs constructed with known local conserved charges have failed to encapsulate the correct equilibrium states for interacting models, like the Heisenberg XXZ chain. The discrepancy raises two potential hypotheses: either the GGE framework is fundamentally deficient for interacting theories, or there may be undiscovered conserved charges crucial for an accurate GGE representation.

Main Contributions

The paper makes a significant advancement by demonstrating that the latter hypothesis holds true. The authors constructed a complete GGE for the spin-1/2 Heisenberg chain by incorporating recently discovered quasi-local charges, which satisfy a form of locality weaker than strict ultra-local charges but are crucial for maintaining the integrity of the GGE in interacting systems.

1. Quasi-Local Charges: The authors emphasize the importance of quasi-local charges, which, unlike the extensively studied local charges, seem to fill the gap left by traditional methods in understanding steady-state properties. These charges are shown to be linearly independent from the ultra-local charges and are integral in accounting for all necessary constraints in the GGE.
2. Exact Quench Solutions: The explicit solution of the N{é}el-to-XXZ quench problem using their completed GGE aligns perfectly with predictions made by the Quench Action method, indicating the correctness of the proposed methodology.
3. Practical and Theoretical Implications: The successful incorporation of quasi-local charges within the GGE suggests a paradigm shift in understanding quantum quenches in integrable models. It refines the theoretical predictions related to correlation functions and entanglement growth in spin chains, paving the way for a deeper comprehension of nonequilibrium phenomena.

Implications on AI and Future Directions

The results of this paper hold implications beyond the specific model discussed, suggesting a general framework for dealing with integrable quantum systems rigorously. For artificial intelligence, particularly in designing quantum algorithms and understanding quantum dynamics, incorporating this holistic approach to conservation laws might lead to improved methodologies for simulating quantum systems efficiently.

Moreover, future work may focus on the extension of this framework to other integrable models, inclusion of the effects on finite systems, and exploration of truncated GGEs for practical applications where computational resources are constrained. The evolution towards integrating such quasi-local charges in practical computational models can aid in accurately predicting and utilizing quantum states in quantum computing and information processing tasks.

In conclusion, this paper presents a vital development in quantum statistical mechanics, confirming that quasi-local charges must be considered to fully understand and predict the post-quench dynamics of integrable systems such as the spin-1/2 Heisenberg model.