Higher-Order Krylov State Complexity in Random Matrix Quenches

Abstract

In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the $\text{Krylov subspace}$. The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position $⟨n⟩$ defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments $⟨n^p⟩$ for $p>1$, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.

31 pages, 9 figures; v2: tables updated with 10 realizations, figure 2 improved, reference added, discussions added for the physical meaning of the generalized complexity on Krylov chain, $r_n$ in figure 3 , and $r$-parameter when $h$ approaches $0$ in appendix C, match published version in JHEP