Lowering Temperature Can Restore Order in Complex Quantum Systems

A surprising link between temperature and the fundamental property of integrability in complex systems, both classical and quantum, has been revealed. Hyeongjin Kim and colleagues at Boston University show that lowering the temperature of a system, calculated using established thermodynamic principles derived from statistical mechanics, can drive it towards a state of perfect order, even when strong interactions would normally cause chaotic behaviour. The counterintuitive finding is supported by observations of slowed relaxation dynamics and scaling relations reminiscent of continuous phase transitions, suggesting temperature can be used as a control parameter for chaos. The research further highlights differences in the relaxation behaviour between quantum and classical models depending on dimensionality, providing a new understanding of how order emerges from disorder, with potential implications for understanding the behaviour of complex materials and quantum simulations

Temperature reduction induces order from chaos via integrability change

A drop in average fidelity susceptibility of stationary states, from values indicative of chaotic behaviour to those mirroring continuous phase transitions, occurred as temperature decreased. This shift was previously unattainable without directly manipulating integrability-breaking interactions, typically achieved through careful tuning of Hamiltonian parameters. Reducing temperature steers systems towards an integrable point, even with strong disruptive forces present, revealing a new correspondence between temperature and integrability-breaking in both classical and quantum many-body systems. Integrability, in this context, refers to the existence of a sufficient number of conserved quantities that allow for exact solutions and predictable behaviour, contrasting with chaotic systems where even slight changes in initial conditions lead to drastically different outcomes. The ability to induce integrability through temperature control offers a novel approach to taming chaotic dynamics.

Auto-correlation functions exhibited slow relaxation dynamics, confirming a violation of ergodicity as the system approached this ordered state, irrespective of whether the system was classical or quantum. Ergodicity, a fundamental concept in statistical mechanics, assumes that a system will eventually explore all accessible states given sufficient time; the observed violation suggests that the system becomes trapped in a limited region of phase space as temperature decreases. Dynamical exponents, quantifying the rate of relaxation towards order, differed between classical and quantum models depending on the system’s dimensionality, highlighting a subtle interaction between these two areas. The analysis of the XXZ spin chain, a standard model in quantum physics, revealed that both nearest-neighbour interaction strength, denoted as ∆, and disorder, represented by W, acted as integrability-breaking perturbations, disrupting the system’s inherent order. The model became integrable when ∆′ equalled zero, or with infinitely strong disorder, localising the system’s energy states; this behaviour was confirmed through examination of the z-magnetization, Sz tot, which remained conserved throughout the process. The conservation of Sz tot indicates the presence of a hidden symmetry within the system, a hallmark of integrability. While these findings clarify a clear link between temperature and chaotic behaviour, optimising these parameters for specific, complex systems remains an open question, as do the vital engineering challenges in maintaining such low temperatures in practical applications, particularly in the context of quantum computing and materials science. Further research is needed to explore the limits of this temperature-induced integrability and its applicability to a wider range of physical systems.

Mapping Chaos to Order via Adiabatic Parameter Modulation and Fidelity Susceptibility

Employing adiabatic transformations, a technique borrowed from quantum mechanics, allowed researchers to carefully map the evolution of systems from chaotic to ordered states. Slowly altering the system’s parameters traced a path between different configurations without disrupting its fundamental properties, akin to gently reshaping a sculpture rather than shattering and rebuilding it. This method, rooted in the adiabatic theorem, ensures that the system remains in its ground state throughout the transformation, provided the changes are sufficiently slow. This established a connection between a system’s ‘integrability-breaking interactions’, the forces that cause disorder, and its temperature, revealing that decreasing temperature could counteract strong chaotic influences. Tracking changes in the system’s ‘fidelity susceptibility’, a measure of its sensitivity to external changes, pinpointed the transition towards a more predictable, integrable state and quantified the degree of order achieved. Fidelity susceptibility, calculated as the second derivative of the overlap between initial and time-evolved states, provides a sensitive probe of the system’s stability and its susceptibility to perturbations. Temperature, derived from the rate of entropy change with energy, a cornerstone of thermodynamics, proved key as a control parameter alongside integrability-breaking interactions, though no specific qubit counts or sample sizes were detailed in this analysis. The lack of detail regarding system size and computational resources limits the scope for independent verification and scaling analysis.

Temperature control of chaos via geometrical ordering and unquantified regularisation

Establishing temperature as a readily adjustable control for chaotic systems offers a compelling alternative to manipulating the intricate details of integrability-breaking interactions, the forces that introduce randomness. The calculations, however, rely heavily on a “small frequency cutoff μ”, a parameter lacking a numerical value, hindering direct comparison with other theoretical models and limiting the immediate potential for experimental verification of these findings. This parameter, often introduced to regularise divergent integrals in theoretical calculations, requires careful consideration and justification. Despite this limitation, the work demonstrates a geometrical link between temperature and a system’s move towards order, offering a potentially simpler route to study complex many-body systems. The geometrical interpretation suggests that the system’s trajectory in parameter space can be visualised as a path towards a region of higher symmetry and lower complexity.

Temperature serves as a control parameter for chaotic systems, equivalent to directly altering the forces that cause disorder. Slowly evolving a system’s parameters demonstrated that lowering temperature steers systems towards a predictable, integrable state, even with strong disruptive forces present. The observed slow relaxation dynamics and scaling relations suggest a connection to continuous phase transitions, indicating a fundamental shift in system behaviour as order emerges. This connection to phase transitions implies that the system exhibits critical behaviour near the integrable point, with long-range correlations and universal scaling laws. The ability to control chaos through temperature manipulation opens up new avenues for exploring the fundamental principles governing complex systems and potentially designing novel materials with tailored properties.

The research revealed a connection between temperature and the emergence of order in chaotic systems. Lowering the temperature was shown to guide systems towards an integrable state, despite the presence of disruptive interactions. This demonstrates that temperature can function as a control parameter for chaos, similar to directly altering the system’s disruptive forces. Researchers observed slow relaxation dynamics and scaling relations analogous to continuous phase transitions as the system approached this ordered state, suggesting a fundamental change in behaviour.