Does causal dynamics imply local interactions?

TL;DR

This paper investigates whether quantum cellular automata's causal dynamics imply local interactions, revealing examples of non-local generators and conditions under which local or quasi-local Hamiltonians exist.

Contribution

It provides the first examples of QCA with non-local generating Hamiltonians and characterizes conditions for local or quasi-local Hamiltonians in fermionic QCAs.

Findings

Some QCAs have fully non-local Hamiltonian generators.

One-dimensional fermionic QCAs have quasi-local Hamiltonians with exponential or algebraic decay.

Certain integrable systems lack local or quasi-local constants of motion.

Abstract

We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian. On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.