Dynamical response and time correlation functions in random quantum systems

Time-dependent response and correlation functions are studied in random quantum systems composed of infinitely many parts without mutual interaction and defined with statistically independent random matrices. The latter are taken within the three Wigner-Dyson universality classes. In these systems, the response functions are shown to be exactly given by statistical averages over the random-matrix ensemble. Analytical results are obtained for the time dependence of the mean response and correlation functions at zero and positive temperatures. At long times, the mean correlation functions are shown to have a power-law decay for GOE at positive temperatures, but for GUE and GSE at zero temperature. Otherwise, the decay is much faster in time. In relation to these power-law decays, the associated spectral densities have a dip around zero frequency. The diagrammatic method is developed to obtain higher-order response functions and the third-order response function is explicitly calculated. The response to impulsive perturbations is also considered. In addition, the quantum fluctuations of the correlation function in individual members of the ensemble are characterised in terms of their probability distribution, which is shown to change with the temperature.

💡 Research Summary

The paper investigates the dynamical response and time‑correlation functions of a class of random quantum systems composed of an infinite number of non‑interacting subsystems. Each subsystem is described by a random Hermitian Hamiltonian (\hat H_{0p}) and a random perturbation operator (\hat V_{p}). These matrices are drawn independently from one of the three Wigner‑Dyson universality classes: the Gaussian Orthogonal Ensemble (GOE, (\nu=1)), the Gaussian Unitary Ensemble (GUE, (\nu=2)), or the Gaussian Symplectic Ensemble (GSE, (\nu=4)). The average density of states follows the Wigner semicircle law, while on small energy scales the level‑spacing distribution and the two‑point spectral correlation function exhibit the universal power‑law behavior characteristic of the chosen class.

The authors develop a time‑dependent perturbation theory for an external field (\lambda(t)) acting on the system initially prepared in a canonical thermal state at inverse temperature (\beta). The expectation value of an observable (\hat A) is expanded in powers of (\lambda(t)), leading to a hierarchy of response functions (R_{AV\cdots V}(t,t_{1},\dots)) expressed as nested commutators under the unperturbed dynamics. Because the subsystems do not interact, the total response factorises into a sum over subsystem contributions. In the limit of an infinite number of subsystems, the sum becomes an ensemble average over the random‑matrix distribution, allowing the mean response functions to be written solely in terms of matrix averages (\langle\cdot\rangle_{\text{ME}}).

The paper focuses first on the linear response ((n=1)) with (\hat A=\hat V). The mean first‑order response function (R_{VV}(t)) is shown to be exactly the thermal two‑time correlation function of the perturbation operator. Analytic expressions are derived for both the high‑temperature limit ((\beta=0)) and the low‑temperature limit ((\beta\to\infty)). At high temperature the correlation decays exponentially (or Gaussian) with a characteristic time set by the bandwidth, and the associated spectral density (\tilde C(\omega)) is flat near (\omega=0). At low temperature, however, the decay becomes algebraic. For GOE the mean correlation exhibits a (t^{-3/2}) power law at positive temperatures, while for GUE and GSE a (t^{-2}) decay appears only at zero temperature. These long‑time tails are directly linked to a dip (a suppression) in the low‑frequency part of the spectral density, reflecting the level‑repulsion physics of the underlying random‑matrix ensemble. Outside the asymptotic regimes the decay is much faster, confirming that the power‑law behavior is a genuine universal feature of the Wigner‑Dyson statistics.

To address higher‑order effects, the authors introduce a diagrammatic method. Each diagram represents a specific contraction pattern of matrix elements in the ensemble average and is associated with a polynomial that resembles, but is distinct from, Bell polynomials. Using this formalism they compute the third‑order response function (R_{VVV}(t_{1},t_{2},t_{3})) explicitly. The third‑order term is the leading non‑vanishing nonlinear contribution for Gaussian ensembles because all even‑order moments of the perturbation matrix vanish. The calculated third‑order response displays a rich dependence on the time ordering of the three arguments and on the temperature, offering predictions for nonlinear optical experiments and multi‑pulse spectroscopy.

The response to an impulsive perturbation (\lambda(t)=\lambda_{0}\delta(t)) is also examined. The authors show how the impulse generates a sudden change in the observable that subsequently evolves according to the previously derived correlation functions, providing a clear link between the theoretical formalism and experimentally accessible pump‑probe measurements.

Finally, the paper studies sample‑to‑sample fluctuations of the correlation function. By evaluating the full probability distribution (P(C)) of the correlation for a single realization of the random matrices, they find that at high temperature the distribution is essentially Gaussian, as expected from the central limit theorem. As the temperature is lowered, the distribution becomes increasingly skewed and develops heavy tails, reflecting the enhanced role of rare level spacings dictated by the Wigner‑Dyson statistics. This temperature‑dependent fluctuation analysis offers a quantitative tool for interpreting experimental data where disorder or randomness plays a crucial role.

In summary, the work provides a comprehensive analytical framework that combines random‑matrix theory with time‑dependent perturbation theory to obtain exact expressions for linear and nonlinear response functions in an infinite‑component random quantum system. The results illuminate how universal spectral statistics control long‑time dynamical behavior, how non‑Gaussian spectral features manifest as low‑frequency dips, and how higher‑order responses can be systematically constructed via diagrammatic techniques. The paper opens avenues for applying these methods to realistic disordered materials, many‑body localization problems, and chaotic quantum systems where random‑matrix universality is expected to dominate the long‑time dynamics.