Lanczos-Pascal approach to correlation functions in chaotic quantum systems

Abstract

We suggest a method to compute approximations to temporal correlation functions of few-body observables in chaotic many-body systems in the thermodynamic limit based on the respective Lanczos coefficients. Given the knowledge of these Lanczos coefficients, the method is very cheap. Usually accuracy increases with more Lanczos coefficients taken into account, however, we numerically find and analytically argue that the convergence is rather quick, if the Lanczos coefficients exhibit a smoothly increasing structure. For pertinent examples we compare with data from dynamical typicality computations for large but finite systems and find good agreement if few Lanczos coefficients are taken into account. From the method it is evident that in these cases the correlation functions are well described by a low number of damped oscillations.

Figures (8)

• Figure 1: LP method for a toy model. Area estimator $\mathcal{A}_r$ and the autocorrelation function obtained by the LP method and benchmark (inset: log-plot) for $\alpha=0.1$ (left) and $\alpha=1$ (right).
• Figure 2: LP scheme for the autocorrelation function. For each system and each observable we depict the respective first $n_{\max}$ Lanczos coefficients $b_n$, the area estimate $\mathcal{A}_r$ along with the point of convergence $R$ and both the autocorrelation function obtained by the LP method and DQT. From left to right: Fast-mode observable in the chaotic Ising chain with $B_z=0.5$, $[(a),(b),(c)]$, slow-mode observable in the chaotic Ising chain with $B_z=0.5$, $[(d),(e),(f)]$, energy current in the two-dimensional Ising chain, $[(g),(h),(i)]$, local $x$ component in the spin-bath model, $[(j),(k),(l)]$, fast-mode observable in the tilted-field Ising model with $B_z=2.0$, $[(m),(n),(o)]$.
• Figure 3: LP scheme of higher correlators. Correlations functions between the first two Krylov states of the respective observable obtained by the LP method and DQT. From left to right: Fast mode and slow mode of the energy density-wave operator in the tilted-field Ising model with $B_z=0.5$, energy current in the two-dimensional Ising model, $x$-component of the local spin in the spin-bath model.
• Figure 4: Wave vector $\vec{\boldsymbol{\varphi}}$ related to the fast-mode of energy-density wave in the tilted-field Ising model with $B_z=0.5$ for various instances of time. Differently to the analysis in the main text, for simplicity we resort to solving the matrix equation (\ref{['eq_matrixODE']}). The curves are interpolated as the $\varphi_n(t)$ are only defined for $n\in\mathbb{N}$.
• Figure S1: DQT results for the autocorrelation function of the fast-mode observable $\mathcal{O}_{q=\pi}$ in the tilted field Ising model (TFIM) with $B_z=0.5$.Left: Results for different system sizes $L$ averaged over $N_p = 2^{28-L}$ different realizations of Haar-random states. Right: Results for fixed system size $L = 18$ and different $N_p$. The solid line indicates the exact diagonalization (ED) result as comparison.