Geometry and non-adiabatic response in quantum and classical systems (1602.01062v5)

Abstract: In these lecture notes, partly based on a course taught at the Karpacz Winter School in March 2014, we explore the close connections between non-adiabatic response of a system with respect to macroscopic parameters and the geometry of quantum and classical states. We center our discussion around adiabatic gauge potentials, which are the generators of unitary basis transformations in quantum systems and generators of special canonical transformations in classical systems. In quantum systems, eigenstate expectation values of these potentials are the Berry connections and the covariance matrix of these gauge potentials is the geometric tensor, whose antisymmetric part defines the Berry curvature and whose symmetric part is the Fubini-Study metric tensor. In classical systems one simply replaces the eigenstate expectation value by an average over the micro-canonical shell. For complicated interacting systems, we show that a variational principle may be used to derive approximate gauge potentials. We then express the non-adiabatic response of the physical observables of the system through these gauge potentials, specifically demonstrating the close connection of the geometric tensor to the notions of Lorentz force and renormalized mass. We highlight applications of this formalism to deriving counter-diabatic (dissipationless) driving protocols in various systems, as well as to finding equations of motion for slow macroscopic parameters coupled to fast microscopic degrees of freedom that go beyond macroscopic Hamiltonian dynamics. Finally, we illustrate these ideas with a number of simple examples and highlight a few more complicated ones drawn from recent literature.

• The paper establishes that adiabatic gauge potentials generate both unitary and canonical transformations, linking geometric tensors to non-adiabatic corrections.
• It utilizes a variational principle to derive approximate gauge potentials that facilitate dissipationless, counter-diabatic driving protocols.
• The study reveals that geometric structures, such as the Fubini–Study metric and Berry curvature, lead to observable phenomena like mass renormalization and Lorentz-like forces.

Overview of "Geometry and Non-Adiabatic Response in Quantum and Classical Systems"

In "Geometry and Non-Adiabatic Response in Quantum and Classical Systems," the authors explore the intricate connections between geometry and dynamics in quantum and classical systems, centering on the role of gauge potentials. The paper discusses adiabatic gauge potentials as pivotal tools in understanding the relationship between non-adiabatic responses to external parameters and the underlying geometric structures of system states. Through a detailed analysis, the authors explore how these gauge potentials manifest in both quantum and classical contexts, examining their implications for non-adiabatic corrections to physical observables.

Key Concepts and Analytical Framework

The paper introduces adiabatic gauge potentials as generators of unitary transformations in quantum systems and as generators of special canonical transformations in classical systems. In quantum systems, these transformed variables relate to the Berry connections and the geometric tensor, which has both symmetric and antisymmetric components corresponding to the Fubini-Study metric and the Berry curvature, respectively. In classical systems, similar transformations help derive canonical forms that elucidate the impact of non-adiabatic transitions.

To tackle complex interacting systems, the article presents a variational principle for determining approximate gauge potentials. This approach enables the authors to derive dissipationless (counter-diabatic) driving protocols, optimizing state preparation and providing insight into emergent dynamics beyond classical Hamiltonian formulations.

Results: Non-Adiabatic Response and Geometric Implications

The paper extensively examines the geometric tensor and its implications for non-adiabatic responses. The symmetric part of the geometric tensor, encapsulating the Fubini-Study metric, provides insights into the distance between quantum states, while the Berry curvature describes Lorentz-like forces in parameter space, manifesting in quantized responses analogous to the quantum Hall effect. These non-adiabatic corrections translate into observable phenomena, such as mass renormalization due to quantum fluctuations and the emergence of forces akin to the Coriolis effect.

By applying these concepts to various models, including quantum spin systems, the article demonstrates the practical utility of the theoretical framework in predicting and explaining observable phenomena in both quantum and classical realms.

Implications and Future Directions

The insights gained from this paper have significant implications for future research and applications. Understanding the geometric structure of states and the corresponding adiabatic and non-adiabatic dynamics can inform the design of advanced protocols for quantum computation and information processing. The methodology presented can potentially be adapted for use in high-dimensional systems where direct analytical solutions are challenging.

Moreover, this work sets the stage for exploring scenarios involving open systems and environmental interactions, as well as examining the interplay between quantum criticality and geometric topology. Such investigations could yield deeper understanding in quantum simulation and complex system dynamics, providing pathways to harnessing geometric and topological features for technological advancements in quantum engineering.

Conclusion

Through meticulous analysis and theoretical development, this paper offers a robust framework connecting geometric insights with dynamical behavior in quantum and classical systems. By leveraging the concept of gauge potentials, the authors unlock powerful tools to address non-adiabatic responses and emergent dynamics, paving the way for innovative research and technological developments in understanding complex quantum phenomena.