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arXiv:2608.16674

Crossover from Fast Scrambling to Operator Confinement Tuned by an Auxiliary Qubit

Ryan Buechele, Joseph Szabo, Nandini Trivedi

quant-phcond-mat.otherSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

We demonstrate a static, disorder-free spin chain Hamiltonian which, by tuning the coupling to an auxiliary qubit, realizes a crossover between super-ballistic, ancilla-accelerated scrambling and sub-ballistic operator confinement. Our minimal model is the mixed-field Ising chain with a spin-1/2 ancilla coupled to its longitudinal magnetization. The ancilla mediates an effective all-to-all interaction which accelerates operator spreading and entanglement growth when weakly coupled, but rapidly saturates its entanglement and projects spin chain operators into effectively frozen subspaces when the ancilla coupling is strong. We locate this crossover independently through both a divergent peak in the mutual-information saturation time near $λ_c N/h \approx 8$ and an exponential suppression of the late-time OTOC growth rate, $\logα\propto 1/λ$. Through a Feshbach-Fano projection and Schrieffer-Wolff transformation, we reveal an effective hidden symmetry on the chain which confines operators on the chain for a time exponential in the coupling strength. This reconciles the fast, $\log(N)$ scrambling reported for random-unitary-circuit realizations of the star geometry with the confinement previously found in its time-independent Hamiltonian analog, showing both emerge from a single Hamiltonian family as a function of one dimensionless parameter.

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19 page, 9 figures

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We demonstrate a static, disorder-free spin chain Hamiltonian which, by tuning the coupling to an auxiliary qubit, realizes a crossover between super-ballistic, ancilla-accelerated scrambling and sub-ballistic operator confinement. Our minimal model is the mixed-field Ising chain with a spin-1/2 ancilla coupled to its longitudinal magnetization. The ancilla mediates an effective all-to-all interaction which accelerates operator spreading and entanglement growth when weakly coupled, but rapidly saturates its entanglement and projects spin chain operators into effectively frozen subspaces when the ancilla coupling is strong. We locate this crossover independently through both a divergent peak in the mutual-information saturation time near λcN/h≈8λ_c N/h \approx 8λc​N/h≈8 and an exponential suppression of the late-time OTOC growth rate, log⁡α∝1/λ\logα\propto 1/λlogα∝1/λ. Through a Feshbach-Fano projection and Schrieffer-Wolff transformation, we reveal an effective hidden symmetry on the chain which confines operators on the chain for a time exponential in the coupling strength. This reconciles the fast, log⁡(N)\log(N)log(N) scrambling reported for random-unitary-circuit realizations of the star geometry with the confinement previously found in its time-independent Hamiltonian analog, showing both emerge from a single Hamiltonian family as a function of one dimensionless parameter.
We introduce a minimal model for realizing a fast-to-slow scrambling transition mediated by an auxiliary central qubit (c-qubit). The c-qubit is coupled to a spin-1/21/21/2 Ising model with local Ising interactions and tunable c-qubit-spin coupling. Each spin becomes next-nearest neighbor to all others through the c-qubit, which mediates effective all-to-all interactions. As the interaction with the c-spin increases, we find a surprising transition from super-ballistic scrambling and information growth to continuously restricted sub-ballistic entanglement and operator growth. This slow growth occurs on intermediate timescales that extend exponentially with increasing coupling and system size, indicative of logarithmic entanglement growth. We find that in the slow-scrambling regime, the c-qubit Ising interaction allows commuting operators to grow support on all sites rapidly, while operators orthogonal to the interaction become echoed out. This projects local operators to lie in a restricted subspace and prevents extensive operator entanglement growth. We provide exact dynamics of small systems working with non-equilibrium, effective infinite temperature states, and additionally contribute analytic early-time expansions that support the observed rapid scrambling to quantum Zeno-like crossover. Tracing out the central qubit provides a unique translation from the full, closed unitary dynamics to a simple open system construction consisting of a typical spin-chain with hidden qubit degree of freedom.
Operator scrambling is commonly diagnosed by the growth of out-of-time-ordered correlators (OTOCs), yet a general symmetry principle underlying their effective dynamics has remained elusive. For Brownian or short-time-correlated large-NNN Majorana systems, we develop a symmetry-based effective field theory for operator scrambling, organized by a strong-to-weak U(1) symmetry breaking in operator space. The key observation is that, in the noninteracting fermion limit, the four-fold Keldysh contour representation of an OTOC admits an emergent strong U(1) symmetry in a doubled Hilbert-space description, even when the original system has no ordinary conserved quantity. The associated slow mode is the phase of the strong-charge creation operator, whose conjugate density is identified with the local operator size. Generic interactions explicitly break the strong symmetry and generate a mass term at lowest order for the would-be Goldstone mode, thereby converting diffusive operator spreading into chaotic growth. We further show that higher-order symmetry breaking terms are tightly constrained by an emergent duality that combines time reversal with contour permutation. This duality fixes the effective action up to quadratic order in the response field, relates the multiplicative noise strength directly to the Lyapunov exponent, and makes the positivity of the Lyapunov exponent a consequence of convergence of the real-time path integral. The resulting OTOC dynamics is governed by a noisy FKPP equation, which captures within a unified framework the early-time exponential growth, ballistic propagation, nonlinear saturation, and stochastic front broadening of operator scrambling. We verify this construction in a Brownian SYK chain, where a direct saddle-point expansion reproduces the symmetry-based effective action. Our results reveal a symmetry origin of operator-size hydrodynamics and scrambling.
Abstract We investigate many-body chaos and scrambling in the Hyperbolic Ising model, a mixed-field Ising model living in the background of AdS2AdS_{2}AdS2​ A d S 2 . The effect of the curvature is captured by site-dependent couplings obtained from the AdS2AdS_2AdS2​ A d S 2 metric applied to a flat nearest-neighbor spin chain. Using a combination of out-of-time-ordered correlators (OTOCs), Krylov complexity, and spectral statistics, we present consistent evidence that this model exhibits faster scrambling behavior relative to its flat counterpart. In particular, we observe signatures consistent with fast scrambling dynamics emerging from purely local interactions. At the system sizes accessible to tensor network simulations, the OTOCs display short-lived exponential growth regimes, from which we extract effective Lyapunov exponents. These effective finite-size exponents exhibit a temperature dependence broadly compatible with the Maldacena-Shenker-Stanford (MSS) bound within numerical uncertainty. Our results indicate that increasing spatial curvature can significantly decrease scrambling time in systems with only nearest-neighbor interactions, providing a minimal and computationally accessible platform for studying quantum chaos. This makes the model a promising test-bed for exploring scrambling and operator growth in near-term quantum simulation architectures.
We propose an experimentally realizable quantum spin model that exhibits fast scrambling, based on nonlocal interactions that couple sites whose separation is a power of 2. By controlling the relative strengths of deterministic, nonrandom couplings, we can continuously tune from the linear geometry of a nearest-neighbor spin chain to an ultrametric geometry in which the effective distance between spins is governed by their positions on a tree graph. The transition in geometry can be observed in quench dynamics, and is furthermore manifest in calculations of the entanglement entropy. Between the linear and treelike regimes, we find a peak in entanglement and exponentially fast spreading of quantum information across the system. Our proposed implementation, harnessing photon-mediated interactions among cold atoms in an optical cavity, offers a test case for experimentally observing the emergent geometry of a quantum many-body system.
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
For the disordered Ising chain with transverse and longitudinal fields, where couplings and fields are drawn from strictly positive distributions, Cao~\cite{Cao} has shown that the moments μ2k=∥[H,σ0z](k)∥22μ_{2k} = \|[H,σ^z_0]^{(k)}\|_2^2μ2k​=∥[H,σ0z​](k)∥22​ grow almost factorially, μ2k1/(2k)∼k/ln⁡kμ_{2k}^{1/(2k)}\sim k/\ln kμ2k1/(2k)​∼k/lnk, and thus asymptotically at the maximal allowed rate. We generalize this result by resolving the operator norm in support length and show that the weight at length ℓk∼k/ln⁡k\ell_k \sim k/\ln kℓk​∼k/lnk already exhibits almost factorial growth, ∥[H,σ0z]ℓk(k)∥2≳(k/ln⁡k)k\|[H,σ^z_0]^{(k)}_{\ell_k}\|_2 \gtrsim (k/\ln k)^k∥[H,σ0z​]ℓk​(k)​∥2​≳(k/lnk)k. This implies maximal spatial delocalization of local operators and, in particular, rules out dynamical locality---the strongest form of many-body localization---at any disorder strength. We further establish rigorously a finite-size crossover scale L∼(W/J)2L\sim (W/J)^2L∼(W/J)2, where WWW is the disorder and JJJ the coupling strength. For L≲(W/J)2L\lesssim (W/J)^2L≲(W/J)2 numerical studies only access a pre-asymptotic regime. Finally, we identify a structural path-entropy obstruction to perturbative LIOM constructions, based on the almost factorial branching of operator content and independent of resonance effects; the same mechanism strongly suggests ballistic real-time operator spreading, so sub-ballistic or localized dynamics would require a presently unidentified cancellation principle acting on almost factorially many disorder-dependent paths with random amplitudes.