A widely used approach to modeling complex quantum systems relies on a potentially unphysical simplification: assuming all interactions between particles are equally likely. Random-matrix ensembles, a long-standing theoretical tool, typically posit nonzero matrix elements across the Hamiltonian, a departure from conventional models where many interactions are absent. This framework also struggles to accurately represent systems where particle location matters, due to its inherent flexibility under basis rotations. Researchers are now engineering models that combine the benefits of random-matrix theory with spatial structure and local interactions to better represent chaotic many-body quantum systems with time-independent Hamiltonians.

Random-Matrix Ensembles Model Generic Quantum Systems

Spectral correlations within suitably chosen energy windows consistently align with calculations derived from random-matrix theory, revealing a surprising universality across diverse physical systems. This agreement extends to systems where the statistical properties of the Hamiltonian’s spectrum remain consistent despite variations in underlying physical details, a phenomenon previously observed and now further substantiated by detailed modeling. This approach retains the core principle of averaging physical properties over an ensemble of systems, a technique central to understanding complex quantum behavior.

A key diagnostic for characterizing chaotic dynamics is the spectral form factor (SFF), the ensemble-averaged Fourier transform of the energy-level density, and its behavior provides key insights into the underlying quantum chaos. Standard random-matrix theory predicts specific features in the SFF as a function of a transform variable, often referred to as time, with enhancements observed when the variable is large.

Similar behavior has been observed in unmodified Floquet quantum circuits, where all sites are coupled to neighbors via quantum gates, and a crossover to the characteristic ramp predicted by random-matrix theory occurs at a generalized Thouless time. Paired Feynman paths in Fock space provide an exact treatment of the model at both large and small values, offering a powerful analytical tool for understanding the system’s dynamics.

“In addition to a characterization of energy dynamics, the other main objective of our calculations is the spectral form factor,” the paper states, highlighting the focus on understanding the statistical properties of energy levels as a window into the system’s chaotic nature.

Spatially Extended Systems & Limitations of Existing Models

Recent investigations into spatially extended chaotic systems use coupled SYK models and random quantum circuits, yet each approach introduces unique characteristics diverging from truly generic many-body systems. Specifically, the exotic low-temperature behavior inherent in single-site levels of coupled SYK models, and the elimination of energy conservation due to time-dependent Hamiltonians in random quantum circuits, create distinctions from systems exhibiting fully developed spatial extension and chaos.

These differences highlight the challenges in establishing a universal framework for understanding spectral correlations across diverse quantum systems. Calculations presented offer a microscopic comparison point for effective field theories examining spectral correlations in chaotic quantum many-body systems, revealing several shared features with existing models. Prior work has explored similar systems, those employing random matrices to represent local interactions within spatially extended configurations, focusing on the density of states and the validity of the eigenstate thermalization hypothesis.

However, Floquet circuits lack conserved densities, though variations incorporating U(1) symmetry and associated conservation laws have been designed, demonstrating diffusive timescales in the spectral form factor. This work aims to pinpoint similarities and differences between spectral correlations in time-independent Hamiltonian models and those found in Floquet systems, offering a refined understanding of energy dynamics.

The current analysis, exact within the large- and small-coupling limits, is a natural approximation for chaotic many-body lattice models defined through an ensemble of Hamiltonians. A notable parallel exists between the results and expressions for the spectral form factor derived for low-dimensional systems in the semiclassical limit, as well as single-particle models of diffusive conductors, specifically regarding the appearance of the return probability.

Simulations with moderately weak intersite coupling, as detailed in a recent publication, corroborate these effects for small coupling strengths. “For Floquet models, an equivalent dependence on boundary conditions and on system size to the one we discuss here is displayed,” the authors note, emphasizing the consistency of observed patterns across different system types.

Diffusive Energy Dynamics via Classical Master Equation

Calculations reveal energy dynamics within the modeled quantum system adhere to a diffusive pattern, describable by a classical master equation for small and large coupling. This analytical description, achieved for both weak and strong coupling limits, simplifies simulations significantly compared to directly modeling the underlying quantum system; numerical solutions to the master equation prove far less computationally intensive. The resulting equation governs the flow of energy density throughout the chain, with the symmetry of the equation indicating that nonlinear terms are irrelevant in the scaling sense.

Simulations corroborate the diffusive nature of energy transfer, demonstrating qualitative agreement with analytical derivations. The research details a method for calculating an approximate energy-diffusion constant, offering a quantifiable measure of how quickly energy spreads through the system. The master equation’s properties were examined to establish general characteristics.

A central result connects the spectral form factor, a measure of energy level distribution, to the classical master equation governing energy dynamics. Early-time analysis reveals the spectral form factor is dominated by individual sites, before energy exchange between sites becomes significant. This initial state is characterized by a lack of energy redistribution.

Spectral Correlations and the Diagonal Approximation

The current work extends treatment of energy transport and spectral correlations beyond earlier studies by focusing on weak intersite coupling, a condition enabling exact computation of both dynamics and correlations through paired Feynman paths. This approach uses a many-body adaptation of the diagonal approximation, a technique familiar from semiclassical systems and disordered conductor models, to achieve analytical tractability across a broad range of system sizes. This model allows for a more subtle exploration of energy flow within complex quantum systems.

Specifically, the master equation provides a pathway to understand how energy dynamics shape the overall spectral properties of the system. “At times later than the onset of the ramp in the SFF for a single site,” the paper details. This analytical description simplifies simulations and offers a quantifiable measure of energy diffusion, moving beyond mere observation of diffusive behavior to characterization of its rate.

Feynman Paths Enable Exact Analysis of Energy & Spectra

This analytical approach circumvents limitations inherent in earlier studies of energy transport and spectral correlations, which often relied on approximations valid only in specific regimes or for systems with large local Hilbert space dimensions. The current model achieves tractability by restricting analysis to conditions of weak intersite coupling, allowing for precise calculations of energy dynamics and spectral correlations. This relationship isn’t simply an observed numerical trend; it stems directly from the model’s capacity to decompose complex interactions into paired Feynman paths.

The model’s accuracy extends to defining timescales, with calculations performed assuming a specific energy scale and a dimensionless parameter governing the strength of intersite coupling. Researchers evaluated the spectral form factor, a key metric for characterizing quantum chaos, alongside the two-point correlation function of energy density to validate the approach.

The spectral form factor experiences a substantial enhancement when the coupling strength is significant, mirroring behavior seen in unmodified Floquet quantum circuits. The authors write, highlighting the model’s ability to maintain precision across a broad range of system parameters. the same calculations can be justified when disorder is weak, using the inverse of dimensionless conductance as an expansion parameter, suggesting a robustness beyond the initial assumptions. The team proposes that combining multiple sites with small local Hilbert space dimensions into effective sites with larger dimensions could extend the applicability of their methods, opening avenues for future research.

Spectral Form Factor as a Chaos Diagnostic

Researchers found this evolution by expanding the time-evolution operator and averaging over two-site couplings, then resumming contributions for specific conditions. This analytical approach allows for tracking how energy exchange between sites leads to equilibration over time, offering a detailed view of energy dynamics within complex quantum systems. Calculations demonstrate that, in the late-time regime, the spectral form factor exhibits a ramp, consistent with the diffusive nature of energy transport.

The team extended their studies to include nonzero inverse temperatures, showing the model’s adaptability to a broader range of system parameters. This flexibility allows for analysis at both large and small scales, providing a comprehensive understanding of energy transport. Comparison of numerical and analytical results for the spectral form factor and two-point correlation functions of energy density confirms the model’s accuracy across varying intersite coupling strengths.

Specifically, results were generated for coupling strengths of and, with additional analysis performed using reduced disorder by drawing from a narrower energy interval. An enhancement of the spectral form factor was observed even with reduced disorder, consistent with a larger energy-diffusion constant, indicating a strong correlation between disorder levels and spectral characteristics. “Ratio between the spectral form factor for open and periodic boundary conditions,” the authors note, was examined across 5000 realizations, further validating the model’s robustness.

The team’s method involved integrating over energy transfers and using factors linear in combinations of variables, ultimately yielding a contribution to the spectral form factor of a specific form. This process, detailed in their calculations, provides a quantifiable measure of chaotic behavior and offers insights into the underlying mechanisms governing energy exchange in these systems.

The resulting spectral form factor is real and non-negative, with a time-independent normalization ensuring consistency and reliability of the analysis. These findings contribute to a deeper understanding of how chaos manifests in quantum systems and provide a powerful diagnostic tool for characterizing their behavior.

Scaling Sense & Diffusive Dynamics Confirmed Numerically

While direct measurement of the return probability, the likelihood of the system returning to its initial energy state, proves challenging in larger systems due to exponential decay, the simulations consistently demonstrate diffusive scaling, validating the model’s core assumptions. This confirmation is particularly significant given the computational simplification inherent in using the master equation to approximate the underlying, more complex quantum behavior. The simulations reveal a ramp in the spectral form factor, exhibiting a specific slope consistent with the theoretical prediction, further solidifying the diffusive nature of energy transport.

Analysis of the two-point correlator of energy density, performed across varying system sizes and coupling strengths, provides additional evidence supporting this behavior; results for and demonstrate a perfect agreement with diffusive scaling. These findings demonstrate how the master equation accurately captures the essential physics of energy distribution within the complex system.

The approach employed offers a conserving description of the many-body system in Fock space, meaning the probability density of the wave function remains constant throughout the simulation. This conservation property, while fundamental, imposes a strong constraint on any approximation scheme used, and the researchers believe their method provides a robust approximation applicable beyond the limits of large or small systems.

“While this is an elementary requirement, it is also a strong constraint on approximation schemes,” the authors state, highlighting the method’s potential for broader applicability and suggesting challenges for alternative modeling techniques. The ability to model energy dynamics with a conserving approach offers a pathway to understanding chaotic quantum systems without sacrificing accuracy or physical realism.

👉 More information
🗞 Chaotic Many-Body Quantum Dynamics, Spectral Correlations, and Energy Diffusion
✍️ J. T. Chalker and Dominik Hahn
🧠 DOI: http://link.aps.org/doi/10.1103/bjj5-49pg

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