How far from thermal? New measure maps quantum state ‘athermality

Researchers are quantifying how far quantum systems deviate from thermal equilibrium using a new measure called “athermality,” defined as the relative entropy between a quantum state and its closest thermal state. The work focuses on integrable quantum systems, like the free XY spin chain and the XXZ spin chain, which settle into “generalized Gibbs ensembles” rather than conventional thermal states after a disturbance. The team found that athermality can become anomalously small when the post-quench Hamiltonian is critical, even at finite energy density, and systematically develops a singularity at criticality inherited from the GGE’s entropy.

Athermality as a Quantum Resource for Measuring State Difference

This measure, rooted in quantum resource theory, identifies how resourceful a non-thermal state is, specifically assessing the difficulty of preparing a thermal state with the same average energy. The research demonstrates that athermality functions as a quantitative distinction between the local stationary states of integrable systems following disturbances and the thermal states they would reach without integrability. The analysis defines athermality mathematically as the relative entropy between a given state and the closest thermal state, possessing the same average energy and utilizes von Neumann entanglement entropy in its calculation.

Generalized Gibbs Ensembles and Thermalization Failure

Investigations into non-equilibrium quantum dynamics reveal that the accuracy of the generalized Gibbs ensemble (GGE), a tool for describing systems far from thermal equilibrium, is not guaranteed, even when the system appears to settle into a stationary state. This approach is particularly relevant for integrable quantum systems, where the GGE is expected to provide a complete description, yet discrepancies can still emerge. Accurately quantifying athermality, therefore, offers a powerful tool for validating and refining the GGE, ultimately leading to a more complete understanding of non-equilibrium quantum dynamics and the limits of thermalization in complex systems.

Defining Athermality: Relative Entropy to Closest Thermal State

This approach moves beyond assessing whether a system reaches a stationary state, instead focusing on the degree of difference between its actual state and a standard thermal distribution. The researchers define this “closeness” by minimizing the relative entropy, identifying the generalized Gibbs ensemble (GGE) with the same average energy as the state under examination, simplifying the calculation of athermality to the difference in their von Neumann entropies. Within quantum resource theory, this framework provides a way to characterize arbitrary states, not just those following quenches.

Athermality Calculations in the XY and XXZ Spin Chains

Calculations involving the XXZ spin chain reveal a connection between athermality and the entropy of the generalized Gibbs ensemble (GGE). This suggests a deeper relationship between these two quantities than previously understood, offering a more precise way to characterize deviations from thermal equilibrium. The researchers found that athermality often becomes anomalously small when the post-quench Hamiltonian is critical in its ground state, despite probing physics at a finite energy density.

Lieb-Liniger Gas and Harmonic Chain Athermality Analysis

This approach allows for new insights into how these systems settle into states that are not conventional thermal states, but rather generalized Gibbs ensembles. This is counterintuitive because generalized Gibbs ensembles describe highly excited states far from the ground state where quantum phase transitions occur, yet criticality appears to reduce the deviation from thermal equilibrium.

Analysis of the XY spin chain in a transverse field revealed consistent conclusions even when interactions were introduced in the XXZ chain, suggesting a robustness to this phenomenon. Further investigation extended to the Lieb-Liniger gas, where the athermality was analyzed for both repulsive and attractive interactions, and the generalized Gibbs ensemble entropy was calculated for both coupling types.

The absence of a quantum phase transition in the repulsive gas was reflected in the lack of striking features in the athermality, providing a clear link between macroscopic phase behavior and microscopic quantum properties. The study also considered the harmonic chain, examining how the bosonic statistics of the elementary modes influence athermality. “The most surprising outcome of our work concerns the role of quantum criticality,” the researchers state, highlighting the unexpected connection between criticality and reduced athermality.

Athermality Exhibits Anomalously Small Values at Critical Points

This reduction in athermality occurs even when examining systems with finite energy density, a result that challenges conventional expectations about the behavior of highly excited quantum states. This suggests a fundamental connection between the system’s approach to criticality and the structure of its post-quench state, where conserved quantities dictate the form of the GGE.

Extensive Nature and Local Minima of System Athermality

Integrable quantum Hamiltonians possess an extensive number of algebraically independent charges, fundamentally altering how they settle after a disturbance. These conserved quantities, expressed as sums or integrals of local densities, prevent the scrambling of quantum information and allow for a form of local memory retention even over extended periods. This contrasts sharply with typical thermalization, where disturbances lead to a loss of initial state information.

The resulting state after a disturbance isn’t a standard Gibbs ensemble, but a Generalized Gibbs Ensemble (GGE) that accounts for these conserved quantities. A newly defined measure, athermality, quantifies the deviation of this GGE from a traditional thermal state. Analysis reveals that the behavior of athermality around critical points can be unexpectedly complex, sometimes exhibiting local minimums or maximums.

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