Quantum Systems Reach Balance Even When Completely Isolated from Outside Forces

A thorough understanding of how quantum systems evolve and reach equilibrium is emerging. Rohit Patil and Marcos Rigol at The Pennsylvania State University introduce a pedagogical explanation of eigenstate thermalization, a key phenomenon for explaining thermalization in isolated quantum systems. Their work uses random matrix theory and examines entanglement entropy to illuminate behaviours observed in quantum many-body systems. The research offers valuable insight into the fundamental principles governing the dynamics of closed quantum systems and advances understanding of how complex systems approach thermal equilibrium

Establishing thermalisation baselines via random matrix comparisons

Random matrix theory served as a key framework for this investigation, providing a statistical perspective on the complex behaviours of quantum systems. This theory, originally developed in nuclear physics to describe energy levels of heavy nuclei, allows modelling of systems where detailed knowledge of individual components is unavailable, similar to predicting election outcomes from limited polling data. The core principle is that the statistical properties of the Hamiltonian, the operator describing the total energy of the system, can be captured by ensembles of random matrices, irrespective of the microscopic details. Applying this theory established a baseline expectation for energy level statistics and related properties in a thermalizing system, independent of specific quantum model details. This is crucial because many-body quantum systems are notoriously difficult to solve exactly, and random matrix theory provides a powerful tool for making predictions about their behaviour. The level spacing distribution, for example, is a key statistical measure that distinguishes between integrable and chaotic systems, with random matrix theory predicting a Wigner-Dyson distribution for chaotic systems.

The spin-1 XXZ model was investigated using eigenstate thermalization, examining systems with up to L sites and three states per site. Numerical results were obtained from 100 mid-spectrum Hamiltonian eigenstates, focusing on magnetization sectors M = 0 and M = L/2. These magnetization sectors represent different total spin orientations of the system. Analyses were performed at both quantum chaotic (λ = 0) and integrable (λ = 1) points to compare entanglement entropy, a parameter that dictates the system’s behaviour. Entanglement entropy quantifies the amount of quantum correlation between different parts of the system, and its volume-law scaling, meaning it grows proportionally to the system size, is a hallmark of thermalizing systems. The density of states, measuring the number of energy levels at a given energy, closely follows a Gaussian distribution regardless of whether the spin-1 XXZ model is integrable or nonintegrable. These results were obtained for chains ranging in size from L = 10 to L = 14. The model conserves total magnetization, denoted as M, and is subject to various discrete symmetries including lattice translation and space reflection. These symmetries constrain the possible eigenstates and influence the energy level structure.

Integrable to non-integrable transitions revealed through energy level scaling in the spin-1 XXZ

The standard deviation of energy levels, σEm/L, scales with L−γ, shifting from γ ≈0.526 at nonintegrable points (λ = 0) to γ ≈0.517 at integrable points (λ = 1). This scaling exponent, γ, provides a sensitive measure of the transition between integrable and non-integrable behaviour. Integrable systems possess an infinite number of conserved quantities, which constrain their dynamics and prevent thermalization, while non-integrable systems lack such constraints and exhibit chaotic behaviour. Quantifying this transition previously required extrapolating from increasingly large systems, a process hampered by computational limitations and the difficulty of isolating true thermal behaviour from finite-size effects. Finite-size effects arise because real simulations are always performed on systems of finite size, and these effects can mask the underlying physics. Error thresholds now stand at 1.2%, demonstrating that the spin-1 XXZ model exhibits predictable scaling behaviour even with relatively small system sizes. This finding offers a valuable insight into the limitations of finite-size scaling, as the subtle change in γ was previously undetectable, and highlights the power of the established methodology for accurately characterising thermalization transitions. The ability to observe this scaling with smaller systems provides a significant advantage for future research, reducing computational cost and enabling the study of larger and more complex systems. Furthermore, the consistent behaviour observed reinforces the reliability of the theoretical framework used in this study, bolstering confidence in the predictions made by random matrix theory and eigenstate thermalization theory.

Defining Genericity and Convergence in Eigenstate Thermalisation

Isolated quantum systems are increasingly predictable in how they settle into stable states, representing an important step towards unifying quantum and classical physics. The emergence of statistical mechanics from quantum mechanics has long been a central goal in physics, and eigenstate thermalization provides a mechanism for achieving this. This work introduces a set of tools for eigenstate thermalization, the process by which these systems reach equilibrium without external influence. Eigenstate thermalization posits that individual energy eigenstates of a quantum system already contain the information necessary to describe thermal behaviour, meaning that the system does not need to interact with an external reservoir to reach equilibrium. However, the research concentrates on ‘generic’ systems, and precisely defining the boundaries of this generality remains a challenge. Determining which systems are truly ‘generic’ and exhibit eigenstate thermalization is an active area of research, as certain types of systems, such as those with long-range interactions or disorder, may exhibit different behaviours.

Recent work explores the connection between thermalization and ‘Haar-random states’, a different mathematical approach, but it remains unclear how these perspectives fully converge. Haar-random states represent a completely random quantum state, and their properties can be used to predict the behaviour of thermalizing systems. This study provides a clear introduction to eigenstate thermalization, explaining how isolated quantum systems reach stability despite lacking external influence. By utilising random matrix theory, a statistical method for analysing complex systems, the work establishes a connection between theoretical predictions and observed behaviours in quantum many-body systems, bridging quantum mechanics and statistical mechanics. Demonstrating consistent scaling behaviours even in relatively small systems offers a valuable pedagogical tool for understanding the process by which a system settles into a stable state, and enables further investigation into the nuances of thermalisation. Understanding these nuances is crucial for applying these principles to real-world systems, such as condensed matter materials and quantum devices.

The research demonstrated that isolated quantum systems can reach equilibrium without external influence through a process called eigenstate thermalization. This means the information needed for stability is already present within the system’s energy states, rather than requiring interaction with a reservoir. Researchers used random matrix theory and analysis of Haar-random states to connect theoretical predictions with behaviours observed in quantum many-body systems. The study provides a valuable introduction to this phenomenon and establishes a foundation for further investigation into the conditions under which it occurs.