Lanczos Coefficient Ratios Scale with Lattice Size and Hydrodynamic Tails

Understanding the behaviour of complex quantum systems presents a significant challenge in modern physics, and researchers continually seek improved methods for simulating these systems. Luca Capizzi, Leonardo Mazza, and Sara Murciano, all from Université Paris-Saclay, investigate the fundamental properties of a widely used computational technique called the Lanczos algorithm, specifically when applied to finite-sized quantum systems. Their work focuses on how certain measurable quantities, known as autocorrelation functions, evolve over time within these systems, and proposes a precise relationship between the algorithm’s internal workings and the underlying physics. This research establishes a crucial link between the computational method and the behaviour of quantum systems, potentially leading to more efficient and accurate simulations of complex materials and phenomena

Understanding many-body quantum dynamics presents a significant challenge; predicting whether and how a setup will eventually reach equilibrium is a problem of exceptional theoretical interest with ample practical applications in the context of synthetic quantum matter. Several theoretical frameworks attempt to describe this complex behaviour. This work investigates the behaviour of systems as their size increases, where the ratios between successive coefficients in a mathematical process called the Lanczos algorithm exhibit specific scaling patterns. These patterns depend on the ‘hydrodynamic tail’ of the autocorrelation function, a measure of how a system’s properties change over time, and the associated scaling with strong or approximate zero-modes is also discussed. The research supports these conjectures with numerical studies of different models, aiming to refine understanding of thermalization processes in quantum systems

Lanczos Algorithm Stability and Coefficient Behaviour

This supplemental material addresses concerns about the numerical stability of the Lanczos algorithm used in the main study. Specifically, it demonstrates that the standard version of the algorithm, without complex corrections, is sufficient to capture the universal features of the coefficients it generates. Even though the algorithm can accumulate small numerical errors, these errors do not significantly affect the key results. The material also shows that the observed behaviour of the coefficients is not an artifact of the algorithm’s instability and that finite system size is the primary reason for imperfect data alignment in some analyses.

The authors compared results obtained using the standard algorithm with those obtained using a more complex version that includes corrections to maintain accuracy. The results were remarkably similar, even when analysing the autocorrelation functions, demonstrating that the numerical limitations of the standard algorithm do not significantly impact the observed behaviour. The team also examined the scaling of cumulative products of the coefficients, finding that data alignment improved as the system size increased, consistent with theoretical predictions. They investigated the impact of finite system size on the data, finding that these effects were the primary reason for imperfect alignment in some analyses, with plateaus in the autocorrelation functions decaying faster than expected for larger systems.

Lanczos Scaling Reveals Universal Quantum Dynamics

Researchers have investigated the behaviour of the Lanczos algorithm when applied to complex quantum systems, focusing on how it reveals universal properties of these systems. By examining the algorithm’s coefficients, researchers aimed to uncover fundamental characteristics of quantum systems independent of specific details. The team discovered that the ratios between successive Lanczos coefficients exhibit predictable scaling behaviour as the system size increases, linked to the ‘hydrodynamic tail’ of the autocorrelation function. Specifically, the scaling depends on the presence of ‘zero-modes’, which represent special states within the system, and the strength of these modes significantly influences the observed behaviour.

Numerical studies across various models supported this proposed relationship between the scaling of the coefficients and the system’s underlying properties. In systems with an ‘approximate zero mode’, a state that closely resembles a true zero mode, the cumulative product of the Lanczos coefficients plateaus at a finite value, indicating a stable, unchanging state. Conversely, in systems without such a mode, the cumulative product continues to grow, suggesting ongoing dynamic activity. This difference provides a way to distinguish between systems with and without these special states, even when other characteristics are similar. Furthermore, the research demonstrates that the finite-size behaviour of the Lanczos coefficients contains universal information about the quantum system. The team observed that the data consistently aligned with theoretical predictions, even for systems of varying complexity, suggesting that the Lanczos algorithm can be a powerful tool for extracting fundamental insights into the behaviour of quantum many-body systems.

Lanczos Coefficients Reveal System Autocorrelation Decay

This research investigates the behaviour of the Lanczos algorithm when applied to complex quantum systems, focusing on how certain coefficients within the algorithm scale with system size. The study proposes a relationship between these coefficients and the decay of autocorrelation functions, which describe how a system evolves over time. The findings demonstrate that the Lanczos algorithm can reveal universal information about a system’s properties, even when dealing with finite-sized models. The authors acknowledge that the erratic behaviour of the Lanczos coefficients at higher iterations requires further investigation, and that the precise mechanisms governing the transition from decay to a stable state in the autocorrelation functions remain unclear. Future research directions include exploring the reasons behind observed bi-exponential behaviour and investigating how the algorithm’s coefficients relate to algebraic decay in the autocorrelation function.