Spectral Gaps and Condition Numbers Determine Steady States in Spin Systems

Quantum systems constantly interact with their environment, leading to a loss of quantum information and a process called dissipation, which is often modelled using Lindblad dynamics. Kenji Shimomura from Kyoto University, alongside Nagisa Hara from the University of Ottawa and Seiichiro Kusuoka from Kyoto University, investigate how these dynamics establish a stable, non-equilibrium state in infinite quantum spin systems. This research addresses a fundamental challenge in understanding how the behaviour of a quantum system extends from finite, manageable sizes to infinitely large systems, demonstrating that the steady state of an infinite system reliably reflects the behaviour of its finite counterparts under specific, well-defined conditions. The team’s work establishes crucial links between the system’s inherent properties, such as the ‘normality’ of its dynamics and the energy gaps within its components, and the consistency between finite and infinite system behaviour, offering valuable insight for the development of robust quantum technologies and a deeper understanding of complex quantum phenomena.

Researchers have established a rigorous understanding of how quantum systems settle into stable states over time, even when those systems are infinite in extent. They investigated what happens as a system approaches equilibrium, defining both a steady state and a time-averaged steady state, and demonstrated that these are not always the same, particularly for infinite systems. The team identified conditions under which the long-term behavior of a finite system can accurately predict the behavior of an infinite system, relating this to a measure of the system’s “normality” and the presence of energy gaps within the system.

Non-Hermitian Topology and Open Quantum Systems

This body of work explores the fascinating world of open quantum systems, non-Hermitian physics, and topological phases of matter. Researchers are investigating how quantum systems interact with their environment, leading to phenomena like dissipation and decoherence, and how these interactions affect the system’s fundamental properties. A central theme is the study of non-Hermitian systems, where the usual rules of quantum mechanics are modified, leading to unusual behaviors like non-Hermitian skin effects and exceptional points. The research also delves into topological phases of matter, focusing on how quantum systems can exhibit robust, protected states that are insensitive to local disturbances.

This includes investigations of topological insulators, superconductors, and related materials. This work is underpinned by a strong mathematical foundation, utilizing operator theory, functional analysis, and topology. Researchers are exploring how these concepts apply to a variety of systems, including those with strong interactions and complex geometries. They are also developing and applying numerical methods to simulate the behavior of these systems and analyze their properties. Specific research directions include investigating how dissipation affects topological invariants, exploring the interplay between non-Hermitian physics and many-body localization, and developing master equations to describe the dynamics of open quantum systems. The work also investigates the limits on the speed of information propagation in quantum systems, the entanglement properties of quantum systems, and the classification of gapped phases in dissipative systems.

Infinite Systems, Normality, and Stable States

Researchers have established that the long-term behavior of a quantum system depends critically on how well-behaved the system is, mathematically described by a “condition number”. A system with a poorly-behaved condition number will not converge to a single, predictable equilibrium state, even if its individual components appear stable. The team showed that even if the system’s components have stable energy gaps, the overall system can still exhibit unpredictable long-term behavior, highlighting the importance of considering the system as a whole. The study further establishes that the time-averaged behavior of the system converges to a unique, stable state, regardless of the initial conditions, provided certain criteria are met.

This convergence is demonstrated through a detailed analysis of how the system evolves over time, showing that the long-term behavior is independent of the specific starting point. The researchers proved that this unique stable state is consistent across different methods of calculation, confirming its robustness and reliability. Finally, the team investigated a specific model to illustrate the importance of the “condition number”. They found that in certain scenarios, the predicted equilibrium state does not match the actual long-term behavior of the system, due to an unbounded condition number. This demonstrates that simply understanding the individual components of a system is not enough; the overall “health” of the system, as measured by the condition number, is critical for predicting its long-term stability.

Predictable Behaviour in Infinite Quantum Systems

Researchers are investigating the behavior of complex quantum systems as they approach a steady state, focusing on infinite lattices of interacting spins. They define a unique, long-term average state, termed the time-averaged nonequilibrium steady state (TANESS), and demonstrate conditions under which this state aligns with the system’s overall nonequilibrium steady state (NESS). This alignment is not guaranteed and depends on properties of the system’s dynamics, specifically a measure of how “normal” the system’s evolution is and the presence of energy gaps within the system. The study establishes that under certain conditions, the long-term behavior of the system is predictable and independent of the initial size of the considered region.

This is crucial because calculations on infinite systems are impossible; instead, scientists typically model finite systems and extrapolate. The findings suggest that, given specific characteristics of the system’s dynamics, this extrapolation is valid, and the behavior of a finite system accurately reflects the infinite system’s long-term average. Future work could explore how these conditions apply to more complex systems and investigate the impact of different types of interactions between the spins. The research provides a foundation for understanding the long-term behavior of quantum systems and offers insights into the validity of using finite-size models to approximate infinite systems, which is essential for practical calculations and simulations in quantum physics.