Square-root growth of operator entanglement in an integrable brickwork circuit

Abstract

It is widely expected that the von Neumann entanglement of a local operator grows at most logarithmically in integrable many-body systems in infinite volume. We give a counterexample in a four-state brickwork circuit whose gate is a permutation matrix solving the constant Yang--Baxter equation. The von Neumann operator entropy grows as $(\log 2)\sqrt{t/\pi }+O(\log t)$, whereas fixed-index Rényi entropies grow linearly below index one and logarithmically above it. At fixed relative Hilbert--Schmidt error below one, matrix product operator simulations require a bond dimension growing at least as $\exp (c\sqrt{t})$ for some $c>0$.

4.5 pages and 17 pages of Supplemental Material, v2: references added