Complex Systems Display Hidden Fractal Patterns Despite Energy Loss

Shu Hamanaka and colleagues at Kyoto University present a thorough review of the multifractal characteristics of the non-Hermitian skin effect, a peculiar localisation phenomenon driven by dissipation and asymmetry. Multifractality emerges in the complex Hilbert space of many-body systems, a feature absent in simpler, single-particle scenarios. The many-body skin effect can occur alongside random-matrix spectral statistics, distinguishing it from many-body localisation. A solvable model based on a Cayley tree provides analytical insights into multifractal dimensions and offers a unifying framework for understanding these structures in open quantum systems and their connection to ergodicity. Quantifying state fragmentation using multifractal dimensions in many-body Hilbert space Multifractal analysis has proven central to quantifying the localization within complex quantum systems, extending beyond traditional methods of mapping particle distributions. The technique dissects how quantum states occupy the many-body Hilbert space, a complete inventory of every arrangement a group of interacting particles could be in, revealing patterns that repeat at different scales but with varying complexity, much like a coastline appears equally rough whether viewed from space or up close. This is particularly relevant in understanding systems where interactions between particles are strong, and traditional single-particle descriptions fail to capture the full behaviour. Calculating multifractal dimensions quantifies the fragmentation of these states, effectively measuring how densely, or sparsely they are distributed within this vast mathematical space. A higher multifractal dimension indicates a more uniform distribution, while a lower value suggests a highly fragmented, localized state. The concept builds upon the established field of fractal geometry, extending it to encompass the more nuanced behaviour of multifractals, where scaling properties vary across different regions of the system. A Cayley tree model effectively describes the Hilbert space and obtains analytical results, offering a pathway to understanding the underlying structure. This model, a hierarchical and infinitely branching tree-like structure, allows researchers to approximate the complex connectivity of the many-body Hilbert space, simplifying calculations while retaining essential features. Directly quantifying localization within the many-body Hilbert space builds upon this foundation, unlike previous work focusing solely on particle distributions. The analytical model developed allows for a deeper understanding of the fragmentation of quantum states, revealing repeating patterns at different scales. This approach moves beyond identifying localized states; it provides a quantitative measure of how localized they are, and how this localization manifests across the entire Hilbert space. The ability to analytically determine these dimensions is crucial, as it provides a benchmark for comparison with numerical simulations and, potentially, experimental observations. Multifractal dimensions reveal Hilbert space fragmentation and quantum chaos in nonreciprocal systems The average multifractal dimension, a key indicator of Hilbert space occupation, decreased from values close to unity under periodic boundary conditions to levels indicating progressively smaller occupation with increasing nonreciprocity. This reduction signifies a previously unattainable level of quantitative insight into how many-body skin effects fragment quantum states within the complex mathematical space defining all possible configurations. Nonreciprocity, the asymmetry in how a system responds to stimuli from different directions, is the driving force behind the non-Hermitian skin effect, and its impact on Hilbert space fragmentation is now quantifiable through these multifractal dimensions. Single-particle skin effects exhibit a vanishing multifractal dimension by comparison, highlighting the fundamentally different behaviour of many-body systems. This establishes that the many-body skin effect uniquely coexists with random-matrix spectral statistics, unlike many-body localisation which lacks ergodicity, demonstrating a novel dissipative quantum-chaotic regime. Ergodicity, in this context, refers to the system’s tendency to explore all accessible states over time; the absence of ergodicity in many-body localisation leads to a qualitatively different behaviour than the chaotic dynamics observed in the many-body skin effect. Detailed analysis of singular-value statistics revealed an averaged spacing ratio of 0.5299 under open boundary conditions, closely matching the random-matrix prediction of 0.5359. Similar agreement was found for complex level-spacing ratios with ⟨|z|⟩= 0.7365 and ⟨cos arg z⟩= −0.2355. These values provide strong evidence that the energy levels of the system exhibit the same statistical properties as those found in chaotic quantum systems described by random matrix theory. The close agreement between the calculated and predicted values reinforces the claim of a novel dissipative quantum-chaotic regime. Unlike many-body localized phases which typically show Poisson statistics, these spectral properties coexist with the multifractal structure, reinforcing the suggestion of a novel dissipative quantum-chaotic regime. Poisson statistics indicate a lack of correlations between energy levels, characteristic of localized systems, while the observed random-matrix statistics suggest strong correlations and chaotic behaviour. The solvable model on a Cayley tree, a hierarchical graph representing the many-body Hilbert space, enabled the analytical calculation of these dimensions, providing a crucial theoretical tool for understanding these complex phenomena. Experimental Validation of Multifractal Dimensions in Quantum Systems Remains an Open Challenge Vital for developing next-generation technologies, understanding how energy behaves in complex materials remains a key scientific goal. This is particularly important in areas such as energy harvesting, where maximising energy absorption and minimising dissipation are crucial, and in the development of novel electronic devices. This work offers a new way to map that energy distribution, focusing on the ‘many-body Hilbert space’, a complete accounting of all possible states within a quantum system. This approach allows researchers to move beyond simply observing macroscopic properties and delve into the underlying quantum mechanics governing energy flow. However, the authors acknowledge their work largely presents analytical modelling, leaving an important gap; experimental confirmation of these predicted multifractal dimensions remains elusive. The challenges in experimentally probing the many-body Hilbert space are significant, requiring precise control over quantum systems and sophisticated measurement techniques. Despite the current lack of experimental validation, this theoretical advance provides a powerful new analytical framework for mapping energy distribution within materials using the concept of the ‘many-body Hilbert space’. This allows for a better understanding of complex phenomena like the non-Hermitian skin effect, where energy is dissipated in unusual ways, and potentially design novel materials with tailored properties. The approach establishes a new method for characterising the many-body skin effect, a peculiar form of localization arising from asymmetry in quantum systems. Examining how quantum states occupy the many-body Hilbert space, the complete description of a system’s possible configurations, revealed a complex, fragmented structure termed multifractality. This multifractal signature distinguishes it from both single-particle scenarios and the related phenomenon of many-body localization, which lacks a similar pattern. The analytical model provides quantifiable insight into this fragmentation, offering a valuable tool for future investigations. Future research will likely focus on bridging the gap between theory and experiment, potentially through the use of engineered quantum systems or advanced spectroscopic techniques to directly measure the multifractal dimensions predicted by this model. The research demonstrated that the many-body skin effect, a form of energy localisation in quantum systems, exhibits a complex, fragmented structure called multifractality within the system’s complete state description, the many-body Hilbert space. This is significant because understanding how energy distributes in these systems could lead to the design of new materials with specific, tailored properties. By analytically modelling this effect on a Cayley tree, researchers were able to quantify the degree of fragmentation, distinguishing it from other similar phenomena. Future work aims to experimentally verify these predicted multifractal dimensions, potentially utilising engineered quantum systems for precise measurement. 👉 More information 🗞 Multifractal Analysis of the Non-Hermitian Skin Effect: From Many-Body to Tree Models 🧠 ArXiv: https://arxiv.org/abs/2603.26185 Tags: