Average Equilibration Time for Gaussian Unitary Ensemble Hamiltonians

Abstract

Understanding equilibration times in closed quantum systems is essential for characterising their approach to equilibrium. Chaotic many-body systems are paradigmatic in this context: they are expected to thermalise according to the eigenstate thermalisation hypothesis and exhibit spectral properties well described by random matrix theory (RMT). While RMT successfully captures spectral correlations, its ability to provide quantitative predictions for equilibration timescales has remained largely unexplored. Here, we study equilibration within RMT using the framework of equilibration as dephasing, focusing on closed systems whose Hamiltonians are drawn from the Gaussian unitary ensemble (GUE). We derive an analytical expression that approximates the average equilibration time of the GUE and show that it is independent of both the initial state and the choice of observable, a consequence of the rotational invariance of the GUE. Numerical simulations confirm our analytical expression and demonstrate that our approximation is in close agreement with the true average equilibration time of the GUE. We find that the equilibration time decreases with system size and vanishes in the thermodynamic limit. This unphysical result indicates that the true equilibration timescale of realistic chaotic many-body systems must be dominated by physical features not captured by random matrix ensembles -- the GUE in particular.

Figures (3)

• Figure 1: Plot of the average proportionality constant $c_\mathrm{Num}(L)$ as a function of the number of qubits $L$ (where $N=2^L$) for randomly sampled GUE Hamiltonian. Blue squares: $c_\mathrm{Num}(L)$ computed using \ref{['eq:unimodal_c_fit']}, where $t_\mathrm{fc}$ is obtained by averaging $n=1000$ randomly sampled GUE matrices for each value of $L$. Orange dashed line: fit of the raw data with respect to shifted exponential function $A e^{-B*L} + c_\mathrm{Num}$. Red dotted line: apparent asymptotic value $c_\mathrm{Num}$ for large $L$, obtained from the fit. The variance is set to $\sigma = 1/\sqrt{N}$. Further details regarding this choice and the fit can be found in App. \ref{['app:unimod-constant']}.
• Figure 2: Plot of average spacing distribution $P(s)$ as a function of the gap $s=\vert E_i - E_j\vert$ for different values of the system size $N$. The standard deviation of the sampled GUE Hamiltonians is set to $\sigma = 1/\sqrt{N}$. For large $N$ this distribution is unimodal, thus providing further justification for the equilibration time \ref{['eq:EquilibrationTimeHeuristic']}.
• Figure 3: The blue line corresponds to the fraction of crossings of the bound in \ref{['eq:boundavgtimsig']} for individual realization of $H_{\text{GUE}}$ as a function of $L$, where $N=2^L$; the red dashed line corresponds to the inverse of the average effective dimension $\tfrac{1}{d_\mathrm{eff}}$ as a function of $L$. For large $L$, both, the crossings and the effective dimension vanish, indicating that the system equilibrates and stays close to equilibrium, not only on average, but even for individual realisations.