J. Stefan Institute Maps Information Compression at Criticality

Researchers at the J. Stefan Institute, Wrocław University of Science and Technology, Columbia University, and ICTP have demonstrated a principle governing quantum dynamics at criticality: efficient description of complex systems with limited information. The work, led by Simon Jiricek and Lev Vidmar, reveals that a small fraction of Hamiltonian eigenlevels, not a majority, is sufficient to accurately reproduce the power-law decay of survival probability in these quantum systems. This simplification is not random; the resulting truncated spectrum exhibits a power-law level-spacing distribution, indicating order at the edge of chaos. The spectral form factor displays the same asymptotic power-law decay as the survival probability, suggesting a connection between energy level distribution and the persistence of quantum states.

Information Compression at the Ergodicity Boundary

A small fraction of a quantum system’s energy levels, as the researchers demonstrate, is sufficient to accurately model its complex behavior at the boundary between order and chaos. This finding challenges conventional understanding regarding the need for complete descriptions of quantum dynamics and opens new avenues for simplifying calculations in complex systems. The work, involving Simon Jiricek at the J. Stefan Institute in Ljubljana, Slovenia, along with collaborators from Wrocław University of Science and Technology, Columbia University, and ICTP, focuses on understanding how information is compressed within the energy spectrum of systems at the edge of ergodicity. This means the arrangement of energy levels displays self-similar patterns across different scales, suggesting an underlying order even as the system approaches chaos.

Researchers achieved this simplification by systematically truncating the Hamiltonian spectrum, retaining only the most significant energy levels based on their contribution to the system’s wavefunction. “We identify and sample the subset of Hamiltonian eigenlevels that controls the dynamics, leading to an emergent compression of the energy spectrum,” the authors write, outlining their core methodology. The resulting truncated spectrum exhibits a fractal structure characterized by a level-spacing distribution with a power-law tail, and its spectral form factor displays the same asymptotic power-law decay as the survival probability. The implications of this work extend beyond theoretical physics, potentially informing the development of more efficient algorithms for simulating complex quantum systems and understanding the behavior of matter at its most fundamental level.

Hamiltonian Spectrum Truncation for Quantum Dynamics

The pursuit of simplifying complex quantum systems has long driven theoretical physicists, with techniques like tensor-network representations offering efficient descriptions of weakly entangled states. Now, researchers from the J. Stefan Institute, Wrocław University of Science and Technology, Columbia University, and ICTP are turning their attention to highly excited quantum states at the boundary of ergodicity, where conventional wavefunction compression methods falter. These researchers introduce a novel approach focused on compressing the energy spectrum itself, rather than the wavefunction, revealing efficiencies in describing quantum dynamics. The core of this work lies in systematically truncating the Hamiltonian spectrum, the set of all possible energy levels, while retaining the essential features of the system’s evolution. This suggests a high level of inherent compression within critical quantum systems. The truncation procedure involves selecting eigenlevels based on their overlap with the initial state, prioritizing those that most strongly influence the system’s evolution.

As detailed in the paper, the approximate survival probability derived from this truncated spectrum closely mirrors the exact survival probability calculated using the full Hamiltonian. This method, studied on both interacting and non-interacting models, consistently demonstrates that the critical dynamics are governed by a small fraction of the total energy levels.

Power-Law Decay of Survival Probability at Criticality

Researchers at the J. Stefan Institute, Wrocław University of Science and Technology, Columbia University, and ICTP detail a method for systematically simplifying the description of quantum systems undergoing transitions between order and chaos, focusing on the decay of survival probability, how long a quantum state persists, and its unexpected connection to the arrangement of energy levels within the system. The team’s core finding centers on a small fraction of Hamiltonian eigenlevels. They demonstrate that a small fraction of Hamiltonian eigenlevels suffices to reproduce the power-law decay of the survival probability, meaning that only a small fraction of all possible energy states governs the system’s evolution. This fractal arrangement is not random, but rather displays self-similar patterns at different scales, suggesting a fundamental organization underlying the dynamics. The researchers achieved this compression by identifying and sampling the specific Hamiltonian eigenlevels that most strongly influence the system’s dynamics.

This approach allows for a simplified representation of the system while preserving its essential features, and crucially, reveals a deep link between the distribution of energy levels and the persistence of quantum states. This equivalence is not coincidental; the study suggests that the way energy levels are distributed directly impacts how long quantum states remain coherent. The fractal dimension of the truncated spectrum coincides with the fractal dimension of the system’s eigenstates, reinforcing the idea that this compression is not an arbitrary simplification, but rather a reflection of the system’s inherent structure.

The ability to efficiently model complex quantum systems promises advancements across diverse fields, from materials science to fundamental physics. Recent work from researchers at the J. Stefan Institute, Wrocław University of Science and Technology, Columbia University, and ICTP reveals a principle governing how drastically these systems can be simplified without losing crucial dynamic information. This is not merely a mathematical trick; the simplification reveals a deeper order within the apparent chaos of critical quantum states. The core of this discovery lies in the fractal structure of the Hamiltonian spectrum itself. The team found that the arrangement of energy levels is not random, but instead exhibits a level-spacing distribution with a power-law tail. This means that patterns repeat across different energy scales, a surprising level of order emerging at the boundary between predictable and chaotic behavior. They explain, highlighting their approach. The truncation procedure results in large gaps in the spectrum, giving rise to the observed power-law distribution of energy level spacings.

Conventional understanding suggests that accurately modeling complex quantum dynamics requires accounting for a vast number of interacting energy levels. However, recent work challenges this assumption, revealing that a surprisingly small subset of these levels suffices to capture the essential behavior of the system.

Their work, submitted July 20, 2026, focuses on identifying and sampling only those energy levels that truly govern the system’s evolution, effectively compressing the Hamiltonian spectrum.