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Approximation theory for Green's functions via the Lanczos algorithm

Gabriele Pinna1, Oliver Lunt1,2, and Curt von Keyserlingk1

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Phys. Rev. B 112, 054435 – Published 13 August, 2025

DOI: https://doi.org/10.1103/lsl4-4lb4

Abstract

It is known that Green's functions can be expressed as continued fractions, the content at the nth level of the fraction is encoded in a coefficient bn, which can be recursively obtained using the Lanczos algorithm. We present a theory concerning errors in approximating Green's functions using continued fractions when only the first N coefficients are known exactly. Our focus lies on the stitching approximation (also known as the recursion method), wherein truncated continued fractions are completed with a sequence of coefficients for which exact solutions are available. We assume a standard conjecture about the growth of the Lanczos coefficients in chaotic many-body systems, and that the stitching approximation converges to the correct answer. Given these assumptions, we show that the rate of convergence of the stitching approximation to a Green's function depends strongly on the decay of staggered subleading terms in the Lanczos coefficients. Typically, the decay of the error term ranges from 1/poly(N) in the best case to 1/poly(logN) in the worst case, depending on the differentiability of the spectral function at the origin. We present different variants of this error estimate for different asymptotic behaviors of the bn, and we also conjecture a relationship between the asymptotic behavior of the bn's and the smoothness of the Green's function. Lastly, with the above assumptions, we prove a formula linking the spectral function's value at the origin to a product of continued fraction coefficients, which we then apply to estimate the diffusion constant in the mixed field Ising model.

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Phys. Rev. B 112, 054435– Published 13 August, 2025
  • Received 31 May 2025
  • Revised 1 August 2025
  • Accepted 5 August 2025
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