Operator Local Quench Dynamics

• Operator local quench is a protocol where a local operator perturbs a quantum system’s ground state, inducing non-equilibrium evolution and spatial propagation of excitations.
• It employs analytical methods like CFT and integrability alongside numerical techniques such as DMRG and tensor networks to track changes in entanglement and operator growth.
• The investigation reveals signatures such as entanglement plateaus, operator scrambling, and spectral resonances that are crucial for interpreting experiments with cold atoms and superconducting qubits.

An operator local quench is a paradigm in quantum many-body and field theory dynamics wherein a well-defined local operator, often a primary field or its lattice implementation, acts instantaneously on the ground state or an equilibrium state at a precise point in space. This triggers a non-equilibrium evolution distinct from global quenches, leading to rich spatiotemporal dynamics of observables, entanglement, and operator growth. The concept, originally and most comprehensively developed in conformal field theory (CFT) and integrable models, has been generalized to a wide array of quantum systems including non-conformal, disordered, open, and topological systems. Research in this domain leverages analytical (e.g., CFT, integrability, holography), numerical (DMRG, tensor networks), and experimental (cold atoms, Rydberg arrays, superconducting qubits) approaches to probe the dynamics resulting from these localized perturbations.

1. Theoretical Framework and Definitions

The operator local quench protocol involves initializing the system in a ground state |0⟩ (or a finite energy density state) and acting with a local operator 𝒪̂(x₀) at time t = 0, producing a new initial state |ψ₀⟩ = 𝒪̂(x₀)|0⟩ (Zhang et al., 2019, Villa et al., 2020). This operator insertion may correspond to a physical process such as a local spin flip, particle addition/removal, or the injection of quasiparticles. In 1+1D CFTs, typically 𝒪̂ is a primary field of scaling dimension Δ𝒪, though more general insertions (e.g., sums of primaries or derivative operators) are also considered (Zhang et al., 2019, Ageev et al., 2022).

The time-evolved state after the quench is |ψ(t)⟩ = e^{-iHt}|ψ₀⟩. The subsequent measurement or calculation of observables—entanglement entropy, correlation functions, energy density, or operator entanglement—tracks the impact of the localized excitation as it propagates through the system.

Two main physical mechanisms are activated:

• Quasiparticle radiation: The operator excites left- and/or right-moving quasiparticles, whose trajectories determine the spatial and temporal spread of correlations and entanglement.
• Operator/spreading growth: The Heisenberg time evolution of a local operator (𝒪̂(t)) can exhibit growth in its operator ‘size’, especially in non-integrable or strongly interacting models (Carrega et al., 2020, Barch et al., 2023).

2. Entanglement and Subsystem Distances

The most prominent signatures of operator local quenches appear in the dynamical behavior of entanglement entropies and distance measures between reduced density matrices (RDMs) of subsystems (Zhang et al., 2019). For a subsystem A, the nth Rényi entropy after a quench by 𝒪̂ is given by

$$ΔS_A^{(n)}(t) = -\frac{1}{n-1} \log \left( \frac{\text{Tr}\,\rho_A(t)^n}{\text{Tr}\,\rho_{A,0}^n} \right)$$

where ρ_A(t) is the RDM of subsystem A at time t.

Principal results include:

• Entanglement plateaus: For nonchiral primaries, ΔS_A^{(n)}(t) exhibits a plateau of height log d_𝒪 (d_𝒪 quantum dimension) in the interval where the corresponding quasiparticle is within A, and is zero before and after (Zhang et al., 2019).
• Chiral selectivity: Purely holomorphic (right-moving) or anti-holomorphic (left-moving) operator insertions do not change the entropy, i.e., ΔS_A^{(n)}(t) = 0 at all times.
• Subsystem trace and Schatten distances: Distances D_A^{(n)}(t₁, t₂) between RDMs at different times are sensitive to the specific structure of the local excitation, and in the limit where an excited quasiparticle is (or is not) in A, these RDMs become orthogonal.

For mixed operator insertions 𝒪 = μ𝒫(w) + ν𝒬(𝑤̄), the ratio q = |ν|²/(|μ|² + |ν|²) parameterizes a family of entropies, with the plateau formula for the von Neumann entropy: $ΔS_A(t) = -q \log q - (1-q)\log(1-q)$ while the subsystem trace distance can distinguish between chiralities, capturing differences not present in entanglement alone (Zhang et al., 2019).

These predictions are quantitatively verified in numerical studies of the critical Ising and XX chains by computing time-dependent entanglement and subsystem distances using Gaussian correlation-matrix methods.

3. Operator Growth and Information Scrambling

Operator local quenches provide a setting to investigate operator scrambling, operator growth, and quantum chaos. In models where a local or finite-size operator is evolved in time (𝒪̂(t)), its ‘size’ in the Heisenberg picture increases under chaotic dynamics, reflecting the system's ability to spread local perturbations nonlocally (Carrega et al., 2020, Barch et al., 2023).

Key features:

• Operator growth (q > 2 SYK, nonintegrable systems): The commutator [H, 𝒪̂(t)] generates increasingly nonlocal operators, resulting in the rapid decay of local return amplitudes and the saturation of operator entanglement measures (Carrega et al., 2020, Barch et al., 2023).
• Operator hopping (q = 2 SYK, integrable systems): Operators retain their size and locality under time evolution, leading to slow or negligible growth in operator entanglement.

Operator entanglement—defined for the time evolution operator U(t) as

$$E_{\text{op}}(U(t)) = -\log\left( \frac{\text{Tr}[\mathbb{S}_{AA'} U^{\otimes 2}(t) \mathbb{S}_{AA'} (U^\dagger)^{\otimes 2}]}{\|U(t)\|_2^4} \right)$$

—quantifies this dynamical complexity and acts as a robust diagnostic of entanglement phase transitions, even in non-Hermitian systems where out-of-time-ordered correlators (OTOCs) can be unreliable (Barch et al., 2023).

4. Spectroscopic Protocols and Global Information Extraction

Operator local quenches serve as powerful probes for extracting global spectral properties—most notably excitation spectra—via "quench spectroscopy" protocols (Villa et al., 2020). Here, a local operator is applied at a specific site or region, and the time-resolved expectation value of an observable is monitored. The space-time Fourier transform,

$$G(k, \omega) = \int dx\, dt\, e^{-i(k x - \omega t)} G(x; t)$$

exhibits resonances at spectral energies and momenta of the Hamiltonian, revealing elementary excitation dispersions directly from local one-point measurements.

This methodology is applicable across integrable and nonintegrable models and has immediate experimental relevance for ultracold atoms, trapped ions, and superconducting qubit arrays, where local manipulations and measurements are feasible (Villa et al., 2020).

5. Holographic, Non-Conformal, and Generalized Settings

The operator local quench framework extends beyond 1+1D CFTs:

• Holographic CFTs and gravity duals: Quenches correspond to localized excitations (e.g., falling massive particles, folded yo-yo strings) in AdS gravity duals, deforming minimal surfaces and entanglement wedges (Astaneh et al., 2014, Kawamoto et al., 2022). The holographic entanglement entropy after a local operator quench matches large-c CFT predictions, and its propagation features light-cone structures and, in higher dimensions or with inhomogeneous Hamiltonians, can exhibit information retention beyond unitary quantum channels (Rangamani et al., 2015, Mao et al., 23 Mar 2024).
• Non-conformal, massive, and higher-dimensional theories: The dynamical response in massive scalar field theories, including energy and charge density profiles, shows regimes—single-peak, double-peak, plateau—that depend on both mass and details of regularization, diverging from strictly power-law CFT results (Ageev et al., 2022). In finite-volume (cylinder) geometries, complex, erratic evolutions with partial revivals and chaotic-like energy oscillations can emerge.
• Open, non-Hermitian, and monitored quantum systems: In these systems, operator entanglement, rather than OTOCs, robustly signals dynamical phases, separating area-law (integrable/purification) from volume-law (chaotic) behavior despite a breakdown of the Lieb-Robinson bound (Barch et al., 2023).

6. Computational Complexity and Simulation Approaches

Operator local quenches provide distinct computational advantages in simulating nonequilibrium dynamics. By focusing on the expectation value of local operators, one can use temporal matrix product states (tMPS) to represent time evolution with resources determined by the operator entanglement rather than by the rapidly growing state entanglement (Carignano et al., 2023). The key is that for many integrable systems, operator entanglement grows sublinearly (e.g., logarithmically) with time, making tMPS-based evaluation of local observables substantially more efficient than state-based approaches.

Precisely, for a local operator evolved in the Heisenberg picture, the simulation’s bond dimension scales as the effective rank of the reduced transition matrix, which is upper-bounded by the operator entanglement. If operator entanglement grows no faster than polynomially, the computational complexity remains tractable even at long times (Carignano et al., 2023).

7. Information Lattice and Local Information Flow

Recent developments use the "information lattice" framework to construct a space- and scale-resolved accounting of local quantum correlations produced by an operator local quench (Bauer et al., 1 May 2025). By decomposing the von Neumann information into local densities and currents,

$$i^\ell_n = I(\rho^\ell_n) - I(\rho^{\ell-1}_{n-1/2}) - I(\rho^{\ell-1}_{n+1/2}) + I(\rho^{\ell-2}_{n}),$$

one can track the hydrodynamic flow of quantum information across interfaces and through entanglement hierarchies, uncovering e.g., signatures of Majorana zero modes or topological edge states in quench-induced fractional entropy profiles. This approach generalizes to interacting, disordered, and open systems.

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Operator local quenches thus provide a conceptually sharp, technically tractable, and broadly applicable framework for probing out-of-equilibrium quantum dynamics. They unify insights into entanglement spreading, chaos, operator growth, spectral analysis, and information transport—spanning theoretical, computational, and experimental frontiers of many-body quantum physics.