Complexity growth in integrable and chaotic models

We use the SYK family of models with $N$ Majorana fermions to study complexity time evolution, formulated as shortest geodesic length on unitary group manifold between identity and evolution operator, in free, integrable, chaotic systems. Initially, follows trajectory, hence grows linearly time. how this linear growth is eventually truncated by appearance accumulation conjugate points, which signal presence shorter geodesics intersecting trajectory. By explicitly locating such "shortcuts" through analytical numerical methods, we demonstrate that: (a) free theory, encounters points at a polynomial time; consequently truncates $O(\sqrt{N})$, find an explicit operator "fast-forwards" $N$-fermion complexity, (b) class interacting integrable theories, upper bounded $O({\rm poly}(N))$, (c) argue that do not occur until exponential times $O(e^N)$, after it becomes possible infinitesimally nearby approximate operator. Finally, explore notion eigenstate models.