Probing Krylov complexity in scalar field theory with general temperatures

Krylov complexity characterizes the operator growth in the quantum many-body systems or quantum field theories. The existing literatures have studied the Krylov complexity in the low temperature limit in the quantum field theories. In this paper, we extend and systematically study the Krylov complexity and Krylov entropy in a scalar field theory with general temperatures. To this end, we propose a new method to calculate the Wightman power spectrum which allows us to compute the Lanczos coefficients and subsequently to study the Krylov complexity (entropy) in general temperatures. We find that the Lanczos coefficients and Krylov complexity (entropy) in the high temperature limit will behave somewhat differently from those studies in the low temperature limit. We give an explanation of why the Krylov complexity does not oscillate in the high-temperature region. Moreover, we uncover the transition temperature that separates the oscillating and monotonic increasing behavior of Krylov complexity.