Does causal dynamics imply local interactions?

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.

9 pages

References in corpus (5)

• Absence of Thermalization in Nonintegrable Systems
• Reversible quantum cellular automata
• On the structure of Clifford quantum cellular automata
• Approximate locality for quantum systems on graphs
• Discretizing quantum field theories for quantum simulation

Cited by in corpus (9)

• Crystalline Quantum Circuits
• A converse to Lieb-Robinson bounds in one dimension using index theory
• Exactly solvable many-body dynamics from space-time duality
• Mixing and localisation in random time-periodic quantum circuits of Clifford unitaries
• Absence of localization in two-dimensional Clifford circuits
• A massless interacting Fermionic Cellular Automaton exhibiting bound states
• Brick Wall Quantum Circuits with Global Fermionic Symmetry
• A perturbative approach to the solution of the Thirring quantum cellular automaton
• Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems