Krylov complexity of fermion chain in double-scaled SYK and power spectrum perspective

TL;DR

This paper studies the early growth of Krylov complexity in the double-scaled SYK model, revealing exponential growth at low temperatures and analyzing the spectral properties influencing this behavior.

Contribution

It provides a systematic analysis of Krylov complexity and Lanczos coefficients in the DSSYK model, especially focusing on initial exponential growth and power spectrum effects.

Findings

Krylov complexity exhibits exponential growth at low temperatures.

Initial growth of Krylov complexity is independent of system size.

Sech-like power spectrum shape leads to linear growth of Lanczos coefficients.

Abstract

We investigate Krylov complexity of the fermion chain operator which consists of multiple Majorana fermions in the double-scaled SYK (DSSYK) model with finite temperature. Using the fact that Krylov complexity is computable from two-point functions, the analysis is performed in the limit where the two-point function becomes simple and we compare the results with those of other previous studies. We confirm the exponential growth of Krylov complexity in the very low temperature regime. In general, Krylov complexity grows at most linearly at very late times in any system with a bounded energy spectrum. Therefore, we have to focus on the initial growth to see differences in the behaviors of systems or operators. Since the DSSYK model is such a bounded system, its chaotic nature can be expected to appear as the initial exponential growth of the Krylov complexity. In particular, the time at which the initial exponential growth of Krylov complexity terminates is independent of the number of degrees of freedom. Based on the above, we systematically and specifically study the Lanczos coefficients and Krylov complexity using a toy power spectrum and deepen our understanding of those initial behaviors. In particular, we confirm that the overall sech-like behavior of the power spectrum shows the initial linear growth of the Lanczos coefficient, even when the energy spectrum is bounded.