Symmetry Choices Determine Thermal Behaviour in Complex Systems

A thorough investigation into the mathematical basis of eigenstate thermalization reveals its dependence on the chosen basis. Lennart Dabelow and colleagues at Queen Mary University of London, collaborating with researchers from Bielefeld University, found that the proportion of states exhibiting thermalization can fundamentally depend on the basis used. The study shows a scenario where eigenstate thermalization vanishes in one basis, but remains in another, challenging the assumption of basis independence. It establishes a connection between system symmetries, specifically spatial translation and reflection, and the prevalence of degeneracies, alongside quantifiable limits on basis dependence and its implications for system relaxation over time.

Demonstrated basis dependence reveals symmetry’s role in eigenstate thermalization

The fraction of basis states exhibiting eigenstate thermalization plummeted to zero in one chosen basis, a stark contrast to the persistent, non-zero value observed in another. It represents the first instance of a system where the weak eigenstate thermalization hypothesis, a cornerstone of statistical mechanics, is demonstrably basis-dependent; previously, it was believed to be an intrinsic property of the system itself. Eigenstate thermalization (ETH) describes the property whereby energy eigenstates of a many-body quantum system, when subjected to observation of local observables, statistically resemble a thermal equilibrium ensemble at the same energy. This allows for the application of statistical mechanics to isolated quantum systems, bypassing the need for a true external heat bath. The observed basis dependence fundamentally questions this established understanding. The researchers employed a one-dimensional spin-1/2 chain, a model frequently used in condensed matter physics to study interacting quantum systems, as their test case. This system, while relatively simple, exhibits sufficient complexity to demonstrate the phenomenon and allows for detailed analytical and numerical investigation.

Such degeneracies are intrinsically linked to systems exhibiting both spatial translation and reflection symmetries, providing a quantifiable connection between symmetry and basis dependence. In a translationally invariant basis, the ETH quantifier approached zero as system size increased, with linear fits yielding uncertainties of no more than 0.002 for all inverse temperature values tested. This means that as the system became larger, the probability of finding a thermalized eigenstate in this basis diminished rapidly, effectively vanishing in the thermodynamic limit, the limit as the system size approaches infinity. A second, maximally ETH-violating basis, however, exhibited a persistent, non-zero ETH quantifier even with larger systems; extrapolations to infinite system size revealed β-dependent values ranging from 0.012 to 0.229. The ETH quantifier, in this context, is a measure of how closely an eigenstate resembles a thermal distribution. Values closer to one indicate strong thermalization, while values closer to zero indicate a deviation from thermal behaviour. The β parameter is related to the inverse temperature, allowing the researchers to examine the basis dependence at different energy scales. Dynamical typicality methods, which assess whether a single initial state evolves to resemble a thermal ensemble over time, confirmed this behaviour, with error bars generally too small to discern on the presented scales. Analytical work further revealed that a finite number of energies and a bounded observable norm were sufficient conditions for this basis-dependent ETH violation, suggesting the effect is durable as the total ETH quantifier remained above zero for all system sizes. This robustness implies that the violation of ETH is not merely a finite-size effect but a genuine property of the system in certain bases.

Mathematical choices fundamentally alter predictions of quantum system behaviour

Understanding how complex systems settle into equilibrium is vital for modelling everything from materials science to cosmology; the concept of eigenstate thermalization offers a pathway to predict this behaviour. However, this work reveals a surprising fragility within that framework, demonstrating that the very definition of a ‘thermalized’ state can shift depending on the mathematical tools employed. This isn’t merely an academic point, as it challenges the reliability of simulations routinely used to explore quantum systems, particularly those with inherent symmetries that promote multiple, equally valid, descriptions of energy levels. The implications extend to various fields, including the study of many-body localized phases, where ETH is expected to break down, and the development of quantum technologies, where precise control over system dynamics is crucial.

This discovery doesn’t invalidate years of simulations entirely, but rather refines our understanding of them. Accurate results require selecting the correct mathematical ‘basis’, essentially a way of describing energy levels. The choice of basis dictates how the quantum system is represented mathematically, and different bases can lead to drastically different interpretations of the same physical system. Seemingly stable quantum systems can exhibit surprising behaviour depending on the chosen analytical method, meaning simulations used to model materials and even cosmology may yield differing results with alternative mathematical descriptions. For instance, in materials science, accurately predicting the thermal conductivity of a material relies on understanding how energy is transported through the system, which is directly linked to the ETH properties of its constituent energy eigenstates. A mischosen basis could lead to significant errors in these predictions.

The principle explaining how systems reach equilibrium depends sharply on the chosen mathematical basis, and this sensitivity is the core of the findings. The research reveals a fundamental link between a system’s symmetries, such as translational and reflection symmetry, and the presence of energy level degeneracies, where multiple states share the same energy. These symmetries constrain the possible energy levels of the system, leading to the formation of degenerate states. This basis dependence challenges established methods for evaluating thermalization, particularly in systems exhibiting these degeneracies, and necessitates careful consideration of the chosen basis when interpreting simulation results. The researchers demonstrated that the vanishing of the ETH quantifier in the translationally invariant basis was directly linked to the presence of these degeneracies and the specific way the system was represented mathematically. Further research will likely focus on identifying criteria for selecting appropriate bases in different physical systems and developing methods to account for basis dependence in simulations, ensuring more accurate and reliable predictions of quantum system behaviour.

The research demonstrated that eigenstate thermalization, how a quantum system reaches equilibrium, can depend heavily on the mathematical basis used to describe it. This matters because inaccurate basis choices in simulations of materials could lead to errors in predicting properties like thermal conductivity, potentially impacting materials science and other fields. The team found that for systems with both translational and reflection symmetries, energy level degeneracies arise, and in one particular basis, the fraction of states exhibiting thermalization vanished entirely. Future work will concentrate on establishing guidelines for selecting appropriate bases and refining simulation methods to account for this dependence, improving the reliability of quantum system predictions.