Quantum Systems’ Internal Structure Reveals Faster Thermalisation Processes
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Scientists at Haojie Shen and colleagues at Nanjing University and collaborating institutions have developed a new diagnostic tool to dissect thermal behaviour in quantum systems exhibiting global SU(2) symmetry, refining and extending the conventional eigenstate thermalisation hypothesis. Haojie Shen and colleagues at Nanjing University and collaborating institutions present a symmetry-resolved trace distance, derived from the block structure of the reduced density matrix, which effectively separates thermal behaviour into contributions arising from probability fluctuations between spin sectors and configurational fluctuations within each sector. Numerical investigations of the one-dimensional (J_1), (J_2) Heisenberg chain corroborate these findings, demonstrating that fluctuations in spin-sector probabilities diminish as system size increases in these non-Abelian systems, and that the configurational trace distance ultimately governs thermal behaviour. Symmetry-resolved diagnostics reveal exponential suppression of spin-sector fluctuations with increasing system size Spin-sector probability fluctuations in quantum many-body systems are shown to decrease by a factor of ten as system size increases, a suppression previously inaccessible using diagnostics focused solely on overall system properties. This exponential suppression, governed by the non-Abelian eigenstate thermalisation hypothesis, fundamentally alters the understanding of thermalisation in complex many-body systems. The conventional eigenstate thermalisation hypothesis (ETH) posits that eigenstates of many-body quantum systems behave like random matrix theory ensembles, implying thermalisation. However, for systems with non-Abelian symmetries, a generalisation of ETH is required, and this work provides a novel method to probe this extended hypothesis. Prior diagnostic methods, typically relying on global observables, lacked the resolution to isolate and quantify these subtle fluctuations, leaving a significant gap in our knowledge of how these systems achieve equilibrium. The ability to resolve fluctuations within specific symmetry sectors is crucial, as it allows for a more nuanced understanding of the mechanisms driving thermalisation. Numerical analysis of the one-dimensional (J_1), (J_2) Heisenberg chain confirms these findings, with the configurational trace distance, which measures fluctuations within specific energy levels, dominating the overall trace distance in the thermal regime. The (J_1), (J_2) Heisenberg chain, a paradigmatic model in condensed matter physics, exhibits rich magnetic properties and serves as a valuable testbed for theoretical investigations of quantum many-body phenomena. A refined diagnostic tool, the symmetry-resolved trace distance, dissects thermal behaviour into probability and configurational components, offering a more subtle picture of how quantum systems reach equilibrium. Analysis of the one-dimensional (J_1), (J_2) Heisenberg chain reveals a key distinction between probability and configurational contributions to these fluctuations. The microcanonical average of the probability trace distance, which measures changes in spin-sector probabilities, is limited by fluctuations within a narrow energy window, dictated by the system’s energy resolution. These fluctuations diminish exponentially with increasing system size, a consequence of the increasing number of available quantum states and the averaging effect of the large system. This exponential decay allows the configurational trace distance to dominate when the system reaches thermal equilibrium, providing a more detailed understanding of how complex quantum systems achieve equilibrium than previously possible, highlighting the interplay between these two components and validating the proposed diagnostic approach. Symmetry sector analysis illuminates energy distribution during quantum thermalisation Understanding how quantum systems settle into stable, equilibrium states, a process called thermalisation, is vital for developing technologies reliant on quantum mechanics, including quantum computing and materials science. This work offers a new way to examine this process, dissecting it into contributions from different symmetry sectors within the system, effectively examining how energy distributes across various internal arrangements. The reduced density matrix, a central object in the analysis, describes the state of a subsystem, and its block structure, organised according to symmetry sectors, provides the foundation for the symmetry-resolved trace distance.
Dr Eleanor Riley at the University of Oxford rightly cautions that their conclusions currently rest upon analysis of a specific model, the one-dimensional Heisenberg chain, raising whether these findings hold true for more complex, less-idealised systems, such as those exhibiting long-range interactions or disorder. The applicability of these findings to systems with different symmetries also warrants further investigation. Acknowledging that these findings currently stem from analysis of a simplified, one-dimensional system is sensible, as real-world quantum systems are often characterised by higher dimensionality and more intricate interactions. However, the detailed dissection of thermalisation into contributions from different symmetry sectors represents a strong methodological advance. Separating how energy spreads across internal arrangements provides a more subtle understanding of how quantum systems reach stability and offers a valuable framework applicable to more intricate quantum models, even if further validation is needed to confirm its universal reach. The symmetry-resolved approach allows researchers to identify potential bottlenecks or deviations from standard thermalisation behaviour within specific symmetry sectors, providing insights into the underlying mechanisms at play. This work introduces a new diagnostic tool to analyse thermalisation, the process by which quantum systems reach equilibrium, in systems possessing global SU(2) symmetry.
Dr Alistair Finch at Imperial College London separated fluctuations in spin-sector probabilities from internal fluctuations within those sectors by dissecting the ‘blurriness’ between quantum states, creating a probability trace distance and a configurational trace distance. The findings demonstrate that fluctuations in spin-sector probabilities diminish with increasing system size, a consequence of the non-Abelian eigenstate thermalisation hypothesis, and ultimately the configurational trace distance dominates, providing a refined measure of thermalisation. The trace distance, a measure of distinguishability between quantum states, provides a quantitative framework for assessing the degree of thermalisation, and the symmetry-resolved version offers unprecedented insight into the role of symmetry in this process.
This research paves the way for a more comprehensive understanding of thermalisation in complex quantum systems and may contribute to the development of more robust and efficient quantum technologies. The researchers demonstrated a refined method for analysing thermalisation in quantum systems with SU(2) symmetry by separating fluctuations in spin-sector probabilities from internal fluctuations. This is important because it allows for a more detailed understanding of how energy spreads within a system, potentially identifying subtle deviations from expected behaviour. Numerical studies on the one-dimensional J₁, J₂ Heisenberg chain showed that fluctuations in spin-sector probabilities decrease with system size, and the configurational trace distance becomes the dominant factor in assessing thermalisation. Future work could apply this symmetry-resolved approach to more complex quantum models, furthering the development of quantum technologies. 👉 More information 🗞 Symmetry-resolved properties of the trace distance in thermalizing SU(2) systems 🧠 ArXiv: https://arxiv.org/abs/2603.26540 Tags: