Quantum Integrable Models: Hydrodynamic Approximation Via the Repulsive Lieb-Liniger Model Enables Large Scale Dynamics Study

The behaviour of complex quantum systems, particularly those confined to one dimension, presents a significant challenge to physicists, yet understanding their large-scale dynamics is crucial for advancements in areas like materials science and condensed matter physics. Friedrich Hübner from King’s College London, along with colleagues, now offers new insights into these systems through a refined approach to generalized hydrodynamics, a theoretical framework used to describe their evolution. This work introduces a novel perspective by employing what the researchers term ‘semi-classical Bethe models’ as an intermediary step between the microscopic quantum world and the macroscopic behaviour predicted by generalized hydrodynamics. The team demonstrates the existence and uniqueness of solutions describing the dynamics of the repulsive Lieb-Liniger model, a key system in this field, and importantly, reveals that the diffusive corrections to this behaviour do not follow previously assumed patterns, offering a fundamentally new understanding of how these complex quantum systems evolve over time.

Friedrich Hübner September 2025 Acknowledgements First and foremost, I would like to thank my supervisor, Benjamin, for four years full of interesting scientific discussions and the exciting new developments that started.

Convergence of Approximations and Scaling Analysis

This collection of appendices provides rigorous mathematical support for a theoretical framework describing systems with many interacting components. The work focuses on demonstrating the convergence of approximations and analyzing how errors scale with various parameters, establishing a solid foundation for the calculations presented in the main body of the research. Appendix D details the push-forward of measures, a concept from mathematical analysis used to formally define how probability distributions change under transformations, ensuring probabilities remain consistent during calculations. Appendix E focuses on the convexity of the Bethe phase, a specific state of matter often found in disordered systems, demonstrating that a key function possesses a unique minimum, justifying the use of a stationary phase approximation for simplifying complex integrals.

Appendix F provides a detailed error analysis, carefully tracking how errors scale with system size and temperature, validating the approximations used and demonstrating their reliability under specific conditions. The researchers calculate variances and meticulously analyze the contributions of different terms to the overall error, demonstrating a commitment to mathematical rigor and providing a clear understanding of the limitations of the approximations. Throughout these appendices, the researchers emphasize a rigorous mathematical foundation, careful scaling analysis, and thorough error analysis, providing a complete and reliable description of the system’s behavior, particularly in the limit of very large systems. The appendices collectively demonstrate the validity of the theoretical framework and offer insights into the scaling behavior of the system.

Bethe Models Bridge Microscopic and Macroscopic Dynamics

This research presents a significant advance in understanding the large-scale dynamics of integrable models, systems possessing an infinite number of conserved quantities. Researchers introduced semi-classical Bethe models, providing a crucial link between microscopic dynamics and macroscopic generalized hydrodynamics, and allowing the construction of systems with arbitrary scattering shifts, opening new avenues for investigation. The team demonstrated the existence and uniqueness of solutions to the Euler generalized hydrodynamics equation for the repulsive Lieb-Liniger model, a system relevant to experiments with cold bosonic atoms. Importantly, the research reveals that these solutions remain stable and do not develop gradient catastrophes, a finding crucial for the reliability of hydrodynamic descriptions, and is supported by experimental results from cold atom systems.

Further analysis led to the derivation of a “space-time quadrature”, a powerful tool for efficient numerical solutions, enabling detailed simulations of complex behaviors, such as shocks and turbulence, within integrable systems. The research also challenges previous assumptions regarding diffusive corrections, demonstrating that they are not accurately described by traditional equations, necessitating a re-evaluation of existing models and providing a foundation for developing more accurate descriptions. This work establishes a deeper understanding of how hydrodynamics emerges in integrable systems, paving the way for advancements in understanding more complex, non-integrable systems.

Lieb-Liniger Model, Unique Hydrodynamic Solutions Demonstrated

This research presents a novel approach to understanding the dynamics of integrable models, systems possessing an infinite number of conserved quantities. Researchers developed semi-classical Bethe models, providing a crucial link between microscopic origins and macroscopic generalized hydrodynamics, and allowing for a refined analysis of complex systems, demonstrating the existence and uniqueness of solutions to the generalized hydrodynamics equation for the repulsive Lieb-Liniger model. Importantly, the research proves the absence of gradient catastrophes, or shock formation, within this framework. The investigation challenges conventional understanding of diffusive corrections in generalized hydrodynamics, revealing that these corrections are not accurately described by traditional equations, and demonstrates that long-range correlations significantly influence diffusive behaviour, leading to a revised hydrodynamic description that accounts for these effects. While acknowledging limitations related to the approximations used, such as potential violations of unitarity and the validity of the stationary phase approximation, the researchers suggest avenues for future work, including a more rigorous derivation of their results and exploration of their applicability to a wider range of models. This research provides new mathematical tools and insights into the behaviour of complex systems, offering a more complete picture of their dynamic properties.