Quantum Systems Maintain Order Even As They Appear to Randomise, Physicists Confirm

Scientists are increasingly focused on understanding how isolated quantum systems evolve to appear thermal, mirroring the behaviour predicted by statistical mechanics. Yuke Zhang and Pengfei Zhang, both from the State Key Laboratory of Surface Physics & Department of Physics at Fudan University (with Pengfei Zhang also affiliated with Hefei National Laboratory), and et al. present a detailed investigation into the finite-size scaling of the full eigenstate thermalization hypothesis (ETH). This research is significant because it distinguishes between different sources of corrections to the full ETH, energy fluctuations and fluctuations within energy windows, and establishes a systematic methodology for validating this hypothesis in complex many-body systems, resolving previously observed anomalous growth in finite-size corrections.

Finite-size corrections to the full eigenstate thermalization hypothesis in quantum spin chains remain a challenging theoretical problem

Researchers have uncovered subtle corrections to a fundamental hypothesis governing how quantum systems reach thermal equilibrium. This work details a systematic analysis of finite-size effects within the full eigenstate thermalization hypothesis (full ETH), a cornerstone of quantum statistical mechanics.
Despite the seemingly paradoxical behaviour of isolated quantum systems, which evolve unitarily and therefore preserve information, they often exhibit thermal characteristics remarkably similar to classical systems. The full ETH extends the original eigenstate thermalization hypothesis by accounting for complex correlations between energy eigenstates, offering a more complete description of this thermalisation process.

This study presents a detailed exact-diagonalization analysis of these corrections, performed on quantum spin chains within the canonical ensemble. Crucially, the research distinguishes between two distinct sources of finite-size corrections: those stemming from energy fluctuations, which diminish polynomially with increasing system size, and those arising from fluctuations within specific energy windows, which decay exponentially.

This decomposition provides a refined understanding of how system size impacts the validity of the full ETH. The analysis resolves a long-standing puzzle concerning anomalous growth of finite-size corrections observed in certain observables, even within chaotic systems. Specifically, the team focused on multitime correlation functions, examining how operators evolve over time and interact with each other.

By separating the summation over energy eigenstates, they were able to identify and quantify the contributions from different energy scales. The researchers demonstrate that the observed corrections can be accurately predicted using thermal density matrices, validating the underlying principles of the full ETH.

This refined methodology offers a practical approach for verifying the full ETH in a wide range of many-body quantum systems, paving the way for more accurate simulations and a deeper understanding of quantum thermalisation. The work introduces free cumulants as fundamental objects governing higher-order correlation functions, building upon previous developments connecting the full ETH to free probability theory.

By carefully analysing the behaviour of these cumulants, the researchers were able to disentangle the various contributions to the finite-size corrections and establish a clear connection between the theoretical predictions and the numerical results. This decomposition clarifies the validity of the full ETH in chaotic spin chains, even when corrections to certain correlation functions exhibit unexpected behaviour at moderate system sizes. The findings provide a systematic and practical methodology for validating the full ETH in complex quantum systems.

Finite-size scaling of multitime correlation functions in quantum spin chains reveals universal critical behavior

Exact diagonalization served as the primary technique for a detailed study of finite-size corrections to the full eigenstate thermalization hypothesis (ETH) in the canonical ensemble. The research focused on quantum spin chains, employing a systematic finite-size scaling analysis of correlation functions rather than direct examination of matrix element relations.

Multitime correlation functions, defined as C(q)(t1, t2, · · · , tq−1) ≡⟨O(t1) · · · O(tq−1)O(0)⟩, were calculated to probe the thermalization behaviour of local operators. These calculations were performed using the infinite-temperature ensemble, represented by a density matrix ρ = 1/D, where D denotes the dimension of the Hilbert space.

The study decomposed finite-size corrections into two distinct sources to understand their scaling behaviour with system size. Polynomial decay with system size characterised corrections arising from energy fluctuations, while exponential decay described those originating from fluctuations within each energy window.

This decomposition allowed for a precise determination of how these corrections influence the validity of the full ETH. Specifically, the contribution from fluctuations within each energy window was predicted using thermal density matrices, providing a theoretical benchmark for comparison with numerical results.

Researchers distinguished between energy fluctuations and fluctuations within each energy window to refine the understanding of finite-size effects. The analysis revealed that certain correlation functions, particularly those involving multiple summations with differing scalings, can exhibit anomalous growth with increasing system size at moderate system sizes.

Despite this, the work demonstrates the underlying validity of the full ETH in chaotic spin chains, offering a systematic and practical methodology for validating it through exact diagonalization techniques. This approach provides a refined understanding of the full ETH and its limitations in finite-size systems.

Finite size scaling of many-body eigenstates reveals polynomial and exponential corrections to thermalisation dynamics

Researchers detail a systematic methodology for validating the full eigenstate thermalization hypothesis (full ETH) in quantum many-body systems through an exact-diagonalization study of finite-size corrections. Analysis distinguishes between corrections stemming from energy fluctuations, which diminish polynomially with system size, and those arising from fluctuations within specific energy windows, exhibiting exponential decay with system size.

This work resolves a previously observed anomaly where finite-size corrections to certain observables displayed unexpected growth alongside increasing system size, even within chaotic systems. The study focuses on multitime correlation functions of operators at infinite temperature, defined as C(q)(t1, t2, · · · , tq−1), and utilizes the full ETH ansatz to decompose these functions.

Specifically, for a two-point function C(2)(t), the research demonstrates that it can be expressed as the sum of free cumulants k2(t) and a term proportional to the diagonal elements of the operator, represented as D−1 Σi O2ii. The free cumulants, kq(t), are defined as averages over individual energy levels within a narrow energy window, effectively capturing the statistical behavior predicted by the full ETH.

Decomposition of finite-size corrections reveals that contributions from energy fluctuations scale polynomially with system size, while fluctuations within each energy window decay exponentially. This distinction is crucial for understanding the validity of the full ETH, as the exponential decay confirms the expected behavior in chaotic spin chains.

Furthermore, the research clarifies that anomalous growth observed in certain correlation functions at moderate system sizes does not invalidate the full ETH, but rather arises from specific summation characteristics. The analysis provides a refined understanding of the full ETH and a practical approach for its validation using exact diagonalization techniques.

Finite size scaling of error terms validates eigenstate thermalisation in many-body quantum systems

Researchers have established a systematic methodology for validating the full eigenstate thermalization hypothesis in many-body systems. This work presents a detailed exact-diagonalization study of finite-size corrections to relations underpinning the full ETH, focusing on how these corrections manifest in the canonical ensemble.

The analysis distinguishes between corrections arising from energy fluctuations, which diminish polynomially with system size, and those stemming from fluctuations within specific energy windows, which decay exponentially. Specifically, the investigation resolves instances of anomalous growth in finite-size corrections for certain observables, even within chaotic systems.

Results demonstrate that the longitudinal and transverse components of specific error terms decay exponentially and polynomially with system size, respectively, consistent with expectations from the ETH. Analysis of the random-field XXZ model, using both single-site and two-site operators, confirms these decay behaviours and validates the full ETH for the chosen observables.

The authors acknowledge that their study is limited to finite-size systems and specific observables. Furthermore, the decomposition method employed relies on fitting parameters to the last four data points, which may introduce some degree of uncertainty. Future research could extend these findings to larger system sizes and a wider range of observables to further strengthen the validation of the full ETH and explore its limitations in more complex many-body systems.