Optimal Frobenius light cone in spin chains with power-law interactions

TL;DR

This paper establishes the optimal Frobenius light cone in one-dimensional quantum systems with power-law interactions, revealing how information propagates and saturates bounds in such long-range interacting systems.

Contribution

It derives the tight Frobenius light cone bound for power-law interactions and constructs a protocol that saturates this bound, advancing understanding of information dynamics in long-range quantum systems.

Findings

Optimal Frobenius light cone scales as t ~ r^{min(α-1,1)} for α > 1

Constructed a random Hamiltonian protocol saturating the bound

Bound on operator growth and butterfly velocity in long-range systems

Abstract

In many-body quantum systems with spatially local interactions, quantum information propagates with a finite velocity, reminiscent of the ``light cone" of relativity. In systems with long-range interactions which decay with distance $r$ as $1/r^\alpha$, however, there are multiple light cones which control different information theoretic tasks. We show an optimal (up to logarithms) ``Frobenius light cone" obeying $t\sim r^{\min(\alpha-1,1)}$ for $\alpha>1$ in one-dimensional power-law interacting systems with finite local dimension: this controls, among other physical properties, the butterfly velocity characterizing many-body chaos and operator growth. We construct an explicit random Hamiltonian protocol that saturates the bound and settles the optimal Frobenius light cone in one dimension. We partially extend our constraints on the Frobenius light cone to a several operator $p$-norms, and show that Lieb-Robinson bounds can be saturated in at most an exponentially small $e^{-\Omega(r)}$ fraction of the many-body Hilbert space.