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arXiv:2603.19363 · Physical review. B./Physical review. B

Logarithmic growth of operator entanglement in a clean non-integrable circuit

Mao Tian Tan, Tomaž Prosen

cond-mat.stat-mechquant-phSubmitted Mar 19, 2026• Announced Mar 23, 2026• 0 citations
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Abstract

We study a so-called semi-ergodic brickwork dual-unitary circuits where, in the infinite volume limit, the two-point correlation functions of single-site operators exhibit ergodic behavior along one light ray and non-ergodic behavior along the other light ray. Here, however, we study intermediate and long-time dynamics of a system in a finite, large volume. Under such dynamics, the Heisenberg evolution of a single traceless single-site operator lies within a restricted subspace, and this time evolution can be mapped to a simpler problem of a single qutrit scattering with a bunch of qubits sequentially. Despite the model being non-integrable and free from any quenched disorder, the operator entanglement grows at most logarithmic in time, contrary to prior expectations. The auto-correlation function can be written in terms of a sum of products of $SO(3)$ matrices, allowing for a random matrix prediction for the auto-correlation function at late times. The operator size distribution also becomes bimodal at certain times, displaying intermediate behavior between chaotic and free systems.

Journal reference
Phys. Rev. B 113, 224316 (2026)
DOI
10.1103/sqyj-5spd

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We study a so-called semi-ergodic brickwork dual-unitary circuits where, in the infinite volume limit, the two-point correlation functions of single-site operators exhibit ergodic behavior along one light ray and non-ergodic behavior along the other light ray. Here, however, we study intermediate and long-time dynamics of a system in a finite, large volume. Under such dynamics, the Heisenberg evolution of a single traceless single-site operator lies within a restricted subspace, and this time evolution can be mapped to a simpler problem of a single qutrit scattering with a bunch of qubits sequentially. Despite the model being non-integrable and free from any quenched disorder, the operator entanglement grows at most logarithmic in time, contrary to prior expectations. The auto-correlation function can be written in terms of a sum of products of SO(3)SO(3)SO(3) matrices, allowing for a random matrix prediction for the auto-correlation function at late times. The operator size distribution also becomes bimodal at certain times, displaying intermediate behavior between chaotic and free systems.
We compute the dynamics of entanglement in the minimal setup producing ergodic and mixing quantum many-body dynamics, which we previously dubbed boundary chaos. This consists of a free, non-interacting brickwork quantum circuit, in which chaos and ergodicity is induced by an impurity interaction, i.e., an entangling two-qudit gate, placed at the system’s boundary. We compute both the conventional bipartite entanglement entropy with respect to a connected subsystem including the impurity interaction for initial product states as well as the so-called operator entanglement entropy of initial local operators. Thereby we provide exact results in a particular scaling limit of both time and system size going to infinity for either very small or very large subsystems. We show that different classes of impurity interactions lead to very distinct entanglement dynamics. For impurity gates preserving a local product state forming the bulk of the initial state, entanglement entropies of states show persistent spikes with period set by the system size and suppressed entanglement in between, contrary to the expected linear growth in ergodic systems. We observe similar dynamics of operator entanglement for generic impurities. In contrast, for T-dual impurities, which remain unitary under partial transposition, we find entanglement entropies of both states and operators to grow linearly in time with the maximum possible speed allowed by the geometry of the system. The intensive nature of interactions in all cases causes entanglement to grow on extensive time scales proportional to system size.
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)t/π+O(log⁡t)(\log2)\sqrt{t/π}+O(\log t)(log2)t/π​+O(logt), 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⁡(ct)\exp(c\sqrt t)exp(ct​) for some c>0c>0c>0.
The entanglement in operator space is a well established measure for the complexity of quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study the dynamics of local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of ``completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a ``phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits.
We study the operator entanglement of local operators in one-dimensional brickwork circuits whose two-site gate satisfies the braid relation; throughout this work, we call such a gate a Yang--Baxter gate. We establish upper bounds for several structured, overlapping classes of Yang--Baxter gates. We show that the operator Schmidt rank remains uniformly bounded in time for all qubit Yang--Baxter gates and, in arbitrary local dimension, for permutation gates obtained from non-degenerate Yang--Baxter maps. We also show that it grows at most polynomially for involutive dual-unitary Yang--Baxter gates and for arbitrary phase dressings of permutation gates obtained from non-degenerate Yang--Baxter maps. These results imply, respectively, constant and logarithmic upper bounds on the operator entanglement. Conversely, we construct a seven-state involutive Yang--Baxter gate without dual unitarity and a one-site operator whose exact operator Schmidt rank grows exponentially, although the corresponding operator entropies remain undetermined. Entanglement growth in the general Yang--Baxter case remains open. All proofs and selected examples were constructed by ChatGPT 5.6 Sol.
We explore the long-time behavior of Local Operator Entanglement entropy (LOE) in finite-size interacting integrable systems. For certain operators in the Rule 54 automaton, we prove that the LOE saturates to a value that is at most logarithmic in system size. This bound extends previous work [Phys. Rev. Lett. 122, 250603 (2019); Commun. Math. Phys. 371, 651 (2019)] showing LOE grows logarithmically in the early time regime, t\ll L t ≪ L , to the late time regime, $tL $. However, the late-time logarithmic bound relies on a feature of Rule 54 that does not generalize to other interacting integrable systems: namely, that there are only two types of quasiparticles, and therefore only two possible values of the phase shift between quasiparticles. We present a heuristic argument, supported by numerical evidence, that for generic interacting integrable systems (such as the Heisenberg spin chain) the saturated value of the LOE is volume-law in system size.
The entanglement in operator space is a well established measure for the complexity of quantum many-body dynamics. In particular, that of local operators has recently been proposed as dynamical chaos indicator, i.e. as a quantity able to discriminate between quantum systems with integrable and chaotic dynamics. For chaotic systems the local-operator entanglement is expected to grow linearly in time, while it is expected to grow at most logarithmically in the integrable case. Here we study the dynamics of local-operator entanglement in dual-unitary quantum circuits, a class of "statistically solvable" quantum circuits that we recently introduced. We identify a class of ``completely chaotic" dual-unitary circuits where the local-operator entanglement grows linearly and we provide a conjecture for its asymptotic behaviour which is in excellent agreement with the numerical results. Interestingly, our conjecture also predicts a ``phase transition" in the slope of the local-operator entanglement when varying the parameters of the circuits.