We use the SYK family of models with $N$ Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such "shortcuts" through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at $O(\sqrt{N})$, and we find an explicit operator which "fast-forwards" the free $N$-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by $O({\rm poly}(N))$, and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times $O(e^N)$, after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.
翻译:我们用SYK模型的SYK模型组来研究时间演化的复杂性,这种模型是作为身份和时间演化操作者之间在自由、不可磨灭和混乱系统中的单一组群中最短的大地测量长度而形成的,最初,最短的大地测量学在时间演化轨迹之后,因此复杂性会随着时间演化而逐渐增加。我们研究这种线性增长如何最终被共解点的外观和累积所截断,它表明存在较短的大地测量学在时间演化轨迹中相互交错。我们通过分析和数字方法明确定位这种“近距离”的单一组群,我们证明:(a) 在自由理论中,时间演化时会遇到交错点;因此,复杂性增长在美元(sqrrt{N})值上,我们发现一个明确的操作者,“快速前进”自由的美元-温度时间变异,(b)在一个可互动的理论类中,复杂度在美元(rrr)之后, 概率(n_rbilate) 之后,在时间点上(我们争论中)会争论(nroalalal) 和(n_rum_) 之后, 时间变的复杂度(n) 开始。