Out-of-time-order correlator, many-body quantum chaos, light-like generators, and singular values

TL;DR

This paper investigates out-of-time-order correlators in quantum circuits using light-like generators, revealing universal decay behaviors and proposing conjectures on eigenvalue degeneracies, with analytical and numerical validation across various models.

Contribution

It introduces light-like generators as a tool to analyze OTOCs, derives universal decay forms, and proposes conjectures on eigenvalue degeneracies in many-body quantum chaos.

Findings

OTOCs can be approximated by the leading singular value of LLG.

Universal decay form of OTOC near the light cone is derived.

Eigenvalues of LLG of any size relate to the smallest size, enabling efficient analysis.

Abstract

We study out-of-time-order correlators (OTOCs) of local operators in spatial-temporal invariant or random quantum circuits using light-like generators (LLG) -- many-body operators that exist in and act along the light-like directions. We demonstrate that the OTOC can be approximated by the leading singular value of the LLG, which, for the case of generic many-body chaotic circuits, is increasingly accurate as the size of the LLG, $w$, increases. We analytically show that the OTOC has a decay with a universal form in the light-like direction near the causal light cone, as dictated by the sub-leading eigenvalues of LLG, $z_2$, and their degeneracies. Further, we analytically derive and numerically verify that the sub-leading eigenvalues of LLG of any size can be accessibly extracted from those of LLG of the smallest size, i.e., $z_2(w)= z_2(w=1)$. Using symmetries and recursive structures of LLG, we propose two conjectures on the universal aspects of generic many-body quantum chaotic circuits, one on the algebraic degeneracy of eigenvalues of LLG, and another on the geometric degeneracy of the sub-leading eigenvalues of LLG. As corollaries of the conjectures, we analytically derive the asymptotic form of the leading singular state, which in turn allows us to postulate and efficiently compute a product-state variational ansatz away from the asymptotic limit. We numerically test the claims with four generic circuit models of many-body quantum chaos, and contrast these statements against the cases of a dual unitary system and an integrable system.