Complexity growth in integrable and chaotic models

Vijay Balasubramanian,Yue Li,Matthew Decross,Onkar Parrikar,Arjun Kar

doi:10.1007/jhep07(2021)011

Abstract

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(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), 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.

Highlights

Gravity theory, or, equivalently, in its holographic field theory dual, if the latter exists [8, 9]
The shortest geodesic follows the time evolution trajectory, and 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; complexity growth truncates at O( 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(poly(N )), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN ), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator
We found all conjugate points along the linear geodesic in the complexity metric, and we determined the associated geodesic loops

Summary

Introduction

Gravity theory, or, equivalently, in its holographic field theory dual, if the latter exists [8, 9]. As physics generally happens in the continuum, it is advantageous to work with a naturally continuous notion of complexity for operators in physical quantum systems Such a notion was formulated in terms of minimal geodesic lengths on highdimensional manifolds of operators [20,21,22], and many recent results on complexity make use of this formalism [11,12,13,14, 17, 23,24,25,26,27,28,29,30,31,32,33,34].1. We will see that free and integrable models reproduce this expectation, with the first conjugate point signaling the end of complexity growth entirely and a transition to a plateau regime in the distance function within an O(1) time afterward

Objectives

Discussion

Conclusion