Spectral decomposition and high-accuracy Greens functions: Overcoming the Nyquist-Shannon limit via complex-time Krylov expansion

Abstract

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green's functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution $Δω$ according to the numerically accessible time domain size $T$ via $Δω= 2π/T$. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasi-particle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. At the example of the critical $S-1/2$ Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.

Figures (2)

• Figure 1: fig:heisenberg:spec: Comparison between DSF of the $S-1/2$ Heisenberg model for different complex angles $\alpha$ (black) and a max. complex simulation time $\mathrm {Re}(T_\mathrm{cplx})=128$ with exact reference data (red). Real- time evolution was done until $T=8$ and $\alpha=0$ shows the uncorrected DSF. The absolute difference between the complex time and the exact reference data is indicated by the color- coded baselines. fig:heisenberg:convergence DSF for $\alpha=0.02\pi$ to illustrate the precision of the corrected Green's function. Left inset displays the bond dimensions of the complex- time evolution (green, blue), compared to the real case (purple). Right inset shows the error of the local time- dependent Green's function when propagating the time evolution beyond $T=8$ using the boost operator $\hat{S}(T,\omega)$ (solid line) and after incorporating the damping (dashed line) generated by the finite broadening $\eta=0.1$.
• Figure 2: fig:ssh:spec:alpha-small:s0: Comparison between uncorrected $S0$- bipolaron Green's function $\tilde{G}^{T=8}_{0,0}$ and complex- time Krylov space augmented $G^{\pi/50}_{0,0}$ with $\alpha=\pi/50$ on a $8\times 8$ cluster for $\lambda=0.025$, following a high- symmetry path through the Brillouin zone. Note the distinct peak structure indicating dressed two- electron states, which becomes visible only when taking into account $K^T_{AB}(\omega)$. Inset: The corrected $\vec{k}=\vec{0}$ Green's function exhibits a distinguished peak right at the ground- state energy $\omega=E^\mathrm{2e}_0$ computed via DMRG, which is completely washed out in $\tilde{G}^{T=8}_{0,0}$. fig:ssh:spec:alpha-large:bipolaron: Corrected Green's functions $G^{\pi/33}_{n,m}(\vec{0}, \omega)$ for different electron displacements $\vec{a}_{n,m}=n\vec{e}_x + m\vec{e}_y$ in the strong coupling regime $\lambda=0.2$ at small frequencies. Vertical blue curve indicates the two- electron ground- state energy $E^\mathrm{2e}_0$, which is well separated from the two- polaron continuum at frequencies $\omega>2E^\mathrm{1e}_0$. Above the noise threshold $R\approx 0.05$ (gray shaded area) no signal for a bipolaronic bound state can be found. Inset shows $G^{\pi/33}_{0,1}(\vec{0},\omega)$ along a high- symmetry path through the Brillouin zone.