- The paper presents a comprehensive review using imaginary time path integrals to study quench dynamics in 1+1 dimensional CFTs.
- It shows that global quenches lead to linear growth and saturation of entanglement entropy while local quenches cause logarithmic increases.
- Comparisons with numerical simulations and experiments validate CFT predictions in capturing non-equilibrium behaviors in quantum many-body systems.
This paper provides a comprehensive review of the theoretical framework and methodologies used to understand quantum quenches in 1+1 dimensional Conformal Field Theories (CFTs). The authors, Pasquale Calabrese and John Cardy, focus on the imaginary time path integral approach to analyze quench dynamics, which has been instrumental in determining the time-evolution of correlation functions and entanglement entropy following both global and local quenches. The paper highlights the effectiveness of CFTs as a theoretical playground for studying non-equilibrium dynamics due to their analytical tractability and rich mathematical structure.
Key Concepts and Methodologies
- Imaginary Time Path Integral Approach: The authors utilize this approach to map complex time-dependent quantum problems onto more tractable Euclidean geometries, such as cylindrical or strip-shaped spaces. This method is particularly useful in studying the evolution of correlation functions and entanglement entropy post-quench.
- Global and Local Quenches:
- Global Quenches: These involve sudden changes in the Hamiltonian at a global scale, affecting the entire system uniformly. The analysis in CFTs reveals phenomena such as light-cone spreading of correlations and linear growth of entanglement entropy, which saturates at a value reflecting the thermalization process.
- Local Quenches: These involve perturbations at localized regions, such as the "cut and glue" protocol, where a system initially divided into disconnected parts is allowed to evolve as a single entity. This leads to logarithmic growth in entanglement entropy, showcasing the universality and independence from microscopic details that CFTs provide.
- Entanglement Dynamics: The review highlights how entanglement entropy behaves differently depending on the nature of the quench (global vs. local). In global quenches, the entropy initially grows linearly before saturating, while in local quenches, it exhibits a logarithmic increase.
- Comparison with Numerical and Experimental Results: The theoretical predictions made using CFTs are validated against numerical simulations and experiments in condensed matter systems. This comparison underscores the applicability and accuracy of CFT predictions in realistic scenarios.
- Limits of Applicability: The authors also discuss the limitations of CFT methods in describing quenches in non-critical systems and systems with non-linear dispersion relations, reflecting on scenarios where additional features such as quasi-particle dispersion and model-specific constraints play a significant role.
Implications and Future Directions
The insights gained from this paper have profound implications for understanding quantum many-body systems far from equilibrium. The methods and results are particularly relevant for ultra-cold atomic systems, which can closely simulate the conditions and assumptions inherent in CFTs.
Future research directions include extending these techniques to higher dimensions and non-integrable systems. Furthermore, there remains an open question regarding the extent to which thermalization and relaxation dynamics observed in CFTs generalize to more complex, interacting quantum systems.
In essence, this review serves as a key reference for researchers interested in the theoretical exploration of non-equilibrium dynamics, providing both a solid foundation in CFT approaches and a bridge to applications in experimental and numerical studies.