Dynamical response and time correlation functions in random quantum systems

TL;DR

The paper analyzes the dynamical response and time-correlation functions of random quantum systems built from infinitely many noninteracting parts, each described by random Hermitian matrices from the GOE, GUE, or GSE. By proving that mean response and correlation functions are exact ensemble averages and introducing a diagrammatic approach, it computes first- and higher-order responses, including the third-order term, and examines responses to impulsive perturbations. At long times, the results reveal universal algebraic decay for GOE at finite temperature and faster decays for GUE/GSE, accompanied by a dip in the spectral density near zero frequency; zero-temperature behavior shows different decay patterns across classes. The work also characterizes quantum fluctuations of correlation functions within individual ensemble members, showing temperature-dependent probability distributions and scaling with system size, thereby connecting universal spectral statistics to measurable dynamical properties.

Abstract

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.

Figures (14)

• Figure 1: Mean correlation function (\ref{['Mcorrel-Z']}) versus time for (a) GOE, (b) GUE, and (c) GSE $100\times 100$ matrices with $a_{H_0}=a_V=1$ at inverse temperature $\beta=1$. The real part is depicted by squares and the imaginary part by circles. The statistics is carried out over $10^3$ realisations. The time steps are $\Delta t = 0.01$. For comparison, the solid lines show the theoretical predictions by Eq. (\ref{['Mcorrel-Z-intermediate']}) in terms of Bessel functions.
• Figure 2: Mean correlation function (\ref{['Mcorrel-Z']}) versus time for GOE $100\times 100$ matrices with $a_{H_0}=a_V=1$ at inverse temperatures (a) $\beta=5$, (b) $\beta=0.5$, (c) $\beta=0.05$, and (d) $\beta=0$. The real part is depicted by squares and the imaginary part by circles. The statistics is carried out over $10^3$ realisations. The time steps are $\Delta t = 0.01$. For comparison, the solid lines show the theoretical predictions by Eq. (\ref{['Mcorrel-Z-intermediate']}) in terms of Bessel functions.
• Figure 3: Mean correlation function (\ref{['Mcorrel-Z']}) versus time for GOE $2\times 2$ matrices with $a_{H_0}=a_V=1$ at inverse temperature $\beta=0.2$. The real part is depicted by squares and the imaginary part by circles. The statistics is carried out over $10^8$ realisations. For comparison, the solid lines show the power-law decays predicted by Eq. (\ref{['Re-Im-Mcorrel-GOE-N=2']}).
• Figure 4: Mean correlation function (\ref{['Mcorrel-Z']}) versus time for GOE $3\times 3$ matrices with $a_{H_0}=a_V=1$ at inverse temperature $\beta=0.2$. The real part is depicted by squares and the imaginary part by circles. The statistics is carried out over $10^8$ realisations. For comparison, the solid lines show the power-law decays predicted by Eq. (\ref{['Mcorrel-long-time-GOE']}) with fitted prefactors.
• Figure 5: Mean correlation function (\ref{['Mcorrel-Z']}) versus time for GUE $2\times 2$ matrices with $a_{H_0}=a_V=1$ at inverse temperature $\beta=0.2$. The real part is depicted by squares and the imaginary part by circles. The statistics is carried out over $10^8$ realisations. For comparison, the solid lines show the predictions of Eq. (\ref{['Re-Im-Mcorrel-GUE-N=2']}).