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

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\hyp 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.

💡 Research Summary

The paper addresses a central bottleneck in the numerical computation of low‑energy spectra for strongly correlated quantum many‑body systems: the limited frequency resolution that arises when Green’s functions are obtained by Fourier transforming a finite‑time real‑time evolution. According to the Nyquist‑Shannon theorem, a time window of length T yields a frequency spacing Δω = 2π/T. In tensor‑network (TN) approaches, extending T is exponentially costly because the entanglement entropy grows rapidly under real‑time evolution, especially in two or higher dimensions. Consequently, conventional TN methods are effectively restricted to one‑dimensional models with modest quasiparticle velocities.

The authors propose to overcome this limitation by augmenting the real‑time data with a set of complex‑time Krylov states. They define a complex time step τ = (1 − i tan α) δt, where the angle α∈