Fréchet Distributions Accurately Map Interactions Within Complex Quantum Systems

For operators constructed from products of at least four Majorana fermions, the statistics of off-diagonal matrix elements differ markedly from those observed in the Lieb-Liniger model. A generalised inverse Gaussian distribution now fits these statistics within the disorder-free Sachdev-Ye-Kitaev (SYK) model, advancing understanding of the Eigenstate Thermalization Hypothesis. Differences in how quantum systems distribute energy depend on the specific rules governing them.

Comparing the Sachdev-Ye-Kitaev (SYK) model with the Lieb-Liniger model, a change in the statistical behaviour of energy distribution is observed. The SYK model exhibits a generalised inverse Gaussian distribution, unlike the Fréchet distribution seen in the Lieb-Liniger model. Subtle differences in how quantum systems distribute energy reveal that the rules governing these systems profoundly impact their behaviour. Building on work with the Lieb-Liniger model, which demonstrated energy distribution statistics following a Fréchet distribution, Tingfei Li and Shuanghong Li at the Institute for Theoretical Physics, Chinese Academy of Sciences, investigated the disorder-free Sachdev-Ye-Kitaev (SYK) model, a theoretical set of tools for exploring complex quantum phenomena. The SYK model uses fixed interactions between its fundamental particles, known as Majorana fermions, simplifying calculations within the model. Specifically, the team found that the statistics of how different energy states interact, measured by off-diagonal matrix elements, align with a generalised inverse Gaussian distribution, a departure from the Fréchet distribution observed previously. This finding is significant because it demonstrates a clear distinction in energy distribution between the two models. These off-diagonal matrix elements represent the probability amplitude for a transition between two different energy eigenstates, and their statistical properties are crucial for understanding thermalisation. The SYK model, originally proposed as a holographic dual to certain gravitational systems, provides a simplified yet powerful framework for studying many-body quantum chaos and the emergence of spacetime.

Statistical transition reveals differing energy distributions between quantum models

A shift in the statistical distribution of off-diagonal matrix elements has occurred, moving from Fréchet distributions observed in the Lieb-Liniger model to a generalised inverse Gaussian distribution in the disorder-free Sachdev-Ye-Kitaev (SYK) model. This change in behaviour applies to operators with n ≥ 4 Majorana fermions. Previously, the complexity of many-body systems and the inability to accurately model their interactions limited the discernment of statistical properties of these matrix elements. The Lieb-Liniger model, a one-dimensional gas of bosons with delta-function interactions, serves as a benchmark for understanding integrable systems, where certain conserved quantities prevent thermalisation in the conventional sense. The SYK model, in contrast, is non-integrable and exhibits chaotic behaviour, making it a more realistic model for describing thermalising systems.

This transition signifies a fundamental difference in how quantum systems distribute energy, depending on the governing rules of the model, and opens avenues for further investigation into the Eigenstate Thermalization Hypothesis. Further calculations demonstrate that the shape of the distribution is largely independent of the specific arrangement of the Majorana fermions used to construct the operator, suggesting a universal characteristic of the SYK model. This divergence offers new insight into the Eigenstate Thermalization Hypothesis, confirming that even solvable systems exhibit distinct behaviours and allowing for variations between models as predicted. Specifically, analysis of operators involving four or more Majorana fermions, fundamental particles in the SYK model, revealed the generalised inverse Gaussian distribution for off-diagonal matrix elements, in contrast to the Fréchet distributions observed in the Lieb-Liniger model. Examining the statistical properties of these matrix elements within the same energy macrostate provides a clearer picture of energy distribution within these systems, and informs developments in areas like condensed matter physics. A macrostate, in this context, refers to a range of energy levels, allowing researchers to average over microscopic details and focus on the overall statistical behaviour.

Fréchet and inverse Gaussian distributions reveal differing thermalisation behaviours in solvable models

Increasing focus is being given to understanding how complex quantum systems distribute energy, a key step towards validating the Eigenstate Thermalization Hypothesis. Establishing universal rules governing this energy distribution proves challenging, yet a surprising divergence between two theoretical models, the Lieb-Liniger and Sachdev-Ye-Kitaev (SYK) models, has been revealed. The Lieb-Liniger model exhibits predictable Fréchet distributions for energy statistics, while the SYK model displays a generalised inverse Gaussian distribution instead. The Eigenstate Thermalization Hypothesis posits that, in a thermalising quantum system, the matrix elements of local operators are distributed in a specific way, ensuring that the system reaches thermal equilibrium. This hypothesis is central to our understanding of how quantum systems transition from isolated to interacting states.

Identifying these subtle differences is important for guiding future research into more complex physical scenarios. The statistical behaviour of quantum systems depends on their underlying rules, and this finding, applicable to operators constructed from products of at least four Majorana fermions, advances understanding of complex quantum phenomena. Recent studies explored the statistics of matrix elements of local operators in the Lieb-Liniger model, finding that the probability distribution function for off-diagonal matrix elements is well described by Fréchet distributions. By examining the disorder-free Sachdev-Ye-Kitaev (SYK) model, an investigation identified a generalised inverse Gaussian distribution governing the statistics of off-diagonal matrix elements, refining understanding of how energy thermalises in quantum systems. The generalised inverse Gaussian distribution is characterised by its heavy tails, indicating a higher probability of observing large fluctuations in the matrix elements. This suggests that energy transfer in the SYK model is more erratic and less predictable than in the Lieb-Liniger model. The implications of this research extend to the study of black holes, as the SYK model is believed to provide a simplified model for understanding the quantum properties of these enigmatic objects. Further research will focus on extending these findings to more complex models and exploring the connection between statistical distributions.

The research demonstrated that off-diagonal matrix elements in the disorder-free Sachdev-Ye-Kitaev model follow a generalised inverse Gaussian distribution, differing from the Fréchet distributions observed in the Lieb-Liniger model. This distinction in statistical behaviour is significant because it refines understanding of how energy thermalises within quantum systems. The findings suggest energy transfer in the SYK model exhibits greater fluctuations compared to the Lieb-Liniger model, and contribute to the Eigenstate Thermalization Hypothesis. Researchers intend to extend these observations to more complex models and further investigate related statistical distributions.