Krylov fractality and complexity in generic random matrix ensembles

Abstract

Krylov space methods provide an efficient framework for analyzing the dynamical aspects of quantum systems, with tridiagonal matrices playing a key role. Despite their importance, the behavior of such matrices from chaotic to integrable states, transitioning through an intermediate phase, remains unexplored. We aim to fill this gap by considering the properties of the tridiagonal matrix elements and the associated basis vectors for appropriate random matrix ensembles. We utilize the Rosenzweig-Porter model as our primary example, which hosts a fractal regime in addition to the ergodic and localized phases. We discuss the characteristics of the matrix elements and basis vectors across the three (ergodic, fractal, and localized) regimes and introduce tools to identify the transition points. The exact expressions of the Lanczos coefficients are provided in terms of $q$-logarithmic function across the full parameter regime. The numerical results are corroborated with analytical reasoning for certain features of the Krylov spectra. Additionally, we investigate the Krylov state complexity within these regimes, showcasing the efficacy of our methods in pinpointing these transitions.

Figures (9)

• Figure 1: (Left) The Lanczos coefficients for the RP model with $N = 4096$ (with $100$ ensemble averages) as a function of $x = n / N$, where $n$ is the index of the Lanczos coefficient. The black dashed lines are the fitted results given by \ref{['bngoe']} (top) and \ref{['bnpois']} (bottom) at the two extremes. (Right) The behavior of the fitting parameters $u$ and $v$ as functions of $\gamma$ and $N$.
• Figure 2: Left: Scaling of the $\mathrm{IPR}^{2}_{\text{K}}$ for $\varphi_{N-1}$ for different values of $N$ as a function of $\gamma$. The scaling with system size is nearly unaffected by $\gamma$. The numerical instability at low $\gamma$ arises due to the finite size of the system, which reduces with increasing system size. Right: Scaling exponent $\mathcal{D}_{2}$ of $\mathrm{IPR}^{2}_{\text{K}}$ for $\varphi_{N - 1}$ as a function of $\gamma$. In both cases, the exponent is nearly constant in the ergodic regime and decreases linearly in the fractal regime to settle at zero in the localized regime.
• Figure 3: Top: Spread complexity for the RP model \ref{['RPmodel']} in different regimes, initialized by a TFD state at infinite temperature. Bottom, left: The variation of the peak value of the spread complexity, and (bottom right) the peak time in different phases of the Hamiltonian \ref{['RPmodel']}, initialized by a TFD state at infinite temperature. The system size is $N = 500$, variance $\sigma^2 = 1/N$ with $800$ Hamiltonian realizations.
• Figure 4: (Main) The $\braket{r}$ value statistics with varying $\gamma$ for the RP model for different system sizes, $N = 100 \,(50000), N = 1000\, (5000)$, and $N = 5000 \,(500)$, where the respective number of averages are mentioned in the brackets. A clear transition is visible at $\gamma_c = 2$. (Inset) The same $\braket{r}$ value statistics against $(\gamma - \gamma_c) \log N$ to exhibit the data collapse.
• Figure 5: The variation of $p$ and $q$ for the Anstaz\ref{['anz2']}. Compare this to Fig. \ref{['fig:lanczosinitialfull']} (right) for the Anstaz\ref{['anz1']}.