Spectral functions and time evolution from the Chebyshev recursion

TL;DR

This paper enhances Chebyshev-based spectral function calculations by linking them to time evolution, introducing a recursive algorithm, and improving resolution for many-body systems, with implications for DMRG and DMFT methods.

Contribution

It develops a modified Chebyshev expansion for higher precision, establishes a connection to time evolution, and proposes a new recursive algorithm for spectral computations.

Findings

Chebyshev recursion can be made equivalent to time evolution in a certain limit.

The modified Chebyshev series reduces expansion order by a factor of about 6.

Chebyshev recursion extracts less spectral information than time evolution algorithms for a fixed entanglement level.

Abstract

We link linear prediction of Chebyshev and Fourier expansions to analytic continuation. We push the resolution in the Chebyshev-based computation of $T=0$ many-body spectral functions to a much higher precision by deriving a modified Chebyshev series expansion that allows to reduce the expansion order by a factor $\sim\frac{1}{6}$. We show that in a certain limit the Chebyshev technique becomes equivalent to computing spectral functions via time evolution and subsequent Fourier transform. This introduces a novel recursive time evolution algorithm that instead of the group operator $e^{-iHt}$ only involves the action of the generator $H$. For quantum impurity problems, we introduce an adapted discretization scheme for the bath spectral function. We discuss the relevance of these results for matrix product state (MPS) based DMRG-type algorithms, and their use within dynamical mean-field theory (DMFT). We present strong evidence that the Chebyshev recursion extracts less spectral information from $H$ than time evolution algorithms when fixing a given amount of created entanglement.