Local Joining Quench in Quantum Systems

• Local Joining Quench is a protocol where two decoupled quantum systems are suddenly joined, initiating ballistic entanglement and correlation front propagation.
• Analytical methods using CFT, orbifold constructions, and boundary operators predict logarithmic entanglement entropy growth and dynamical observable behavior.
• The approach extends to diverse systems like spin chains, colloidal gels, and holographic models, deepening insights into non-equilibrium dynamics and quantum information measures.

A local joining quench is a non-equilibrium protocol in which two initially decoupled quantum systems are suddenly connected at a point, triggering dynamical entanglement generation and transport phenomena. This process serves as a minimal model for non-equilibrium dynamics in a variety of quantum systems, with diverse theoretical frameworks capturing its universal and model-specific features.

1. Definition and Universal Features

A local joining quench involves preparing a composite system as a tensor product of two ground states, $|\psi(0)\rangle = |A\rangle \otimes |B\rangle$, and then coupling $A$ and $B$ at a point ($t=0$), such that the evolution is under a fully homogeneous Hamiltonian on the combined domain. This protocol initiates a local non-equilibrium event: the removal of a defect or barrier locally "glues" the two halves, producing ballistic fronts of correlations and entanglement that propagate outward from the junction.

Typical observables of interest are:

• Entanglement entropy $S_{\text{ent}}(t)$ of a spatial subregion,
• Local observables such as current and magnetization profiles,
• Higher-order quantities such as entanglement Hamiltonians, full counting statistics, and more recently, Krylov entropies and subsystem complexity.

A foundational result is that in conformal field theories (CFTs), the entanglement entropy following a local joining quench exhibits universal logarithmic growth,

$$S_{\text{ent}}(t) = \frac{c}{3}\log(t/\epsilon) + \text{const},$$

where $c$ is the central charge and $\epsilon$ is a UV regulator/interface smoothing parameter (0908.2622, Wen et al., 2015, Caputa et al., 26 Feb 2025).

2. Theoretical Methods: CFT, Orbifold Construction, and Boundary Operators

In (1+1)D systems, the field-theoretic approach leverages conformal mappings and replica techniques. The joining quench is mapped to a "pants" geometry (or its generalizations), which is then transformed to the upper half-plane or another canonical domain. In the presence of additional symmetries (e.g., U(1) in fractional quantum Hall edges), the theory is extended to include orbifold sectors and boundary condition changing operators (BCCOs). These operators, with scaling dimension $h=1/16$ for the Ising twist, mediate the transition from reflecting (disconnected) to transmitting (joined) boundary conditions.

For free fermions, re-fermionization techniques allow explicit calculations using Majorana modes; for Laughlin states, the appropriate orbifold CFT and twist fields must be used, capturing both charge and neutral excitations (0908.2622). In general, the entanglement entropy grows logarithmically, but structural details such as the full counting statistics (FCS) for transferred charge or energy also encode the underlying symmetries and the fusion rules of the CFT.

3. Dynamical Observables: Entanglement, Noise, and Higher-Order Diagnostics

A central finding of local joining quenches is the distinction between various dynamical observables:

• Entanglement Entropy: Exhibits universal logarithmic growth for both free and interacting systems at or near criticality. The scaling is set by the underlying central charge and the geometry of the quench (Wen et al., 2015, Caputa et al., 26 Feb 2025).
• Current/Charge Noise and FCS: In systems where U(1) charge is conserved (e.g., quantum Hall edges), the second cumulant of the transmitted charge, $S_{\text{noise}} \sim (\nu/\pi^2) \log(t/\delta)$, also grows logarithmically but with a prefactor sensitive to the filling factor $\nu$ and fusion data of the CFT. Notably, noise and entanglement entropy are not generically proportional—contrary to the noninteracting case—and must be interpreted as measures of dynamical, rather than static, entanglement (0908.2622).
• Entanglement Negativity: For mixed states and subsystems, the evolution of logarithmic negativity reveals more detailed aspects of correlation spreading, including light-cone effects and plateaus not visible in entanglement entropy. These features are robust to model details and verified numerically in noninteracting chains (Wen et al., 2015, Chen et al., 2023).
• Entanglement Hamiltonian: After a local joining quench, the entanglement Hamiltonian takes a local form, where the right- and left-moving energy densities are weighted by functions tied to the propagating fronts. This construction gives direct insight into the microscopic structure of emergent entangled subsystems and their modular flows (Bonsignori et al., 26 Aug 2025).
• Krylov Complexities and Entropies: The time-evolved wavefunction’s expansion in the Krylov (Lanczos) basis provides measures—spread complexity and K-entropy—with universal growth rates proportional to the central charge. These diagnostics are operator-independent and encode non-bipartite aspects of non-equilibrium dynamics (Caputa et al., 26 Feb 2025).

4. Holographic Dualities and Geometric Interpretation

Holographic duals of local joining quenches in AdS/CFT have been constructed, notably via the AdS/BCFT correspondence. Here, the joining of two boundary CFTs maps to the joining of tensionless boundary surfaces, which, after detachment, form a folded (yo-yo) string in the bulk. The tip of this string—moving along a null trajectory—generates a light front that deforms Ryu–Takayanagi geodesics, encoding the causal spreading of entanglement. The time-dependent entanglement entropy thus computed matches field-theoretic results (Astaneh et al., 2014, Rangamani et al., 2015, Shimaji et al., 2018).

In higher-dimensional or out-of-equilibrium settings (e.g., scalar injection in AdS4), additional effects such as competition between condensate formation (entanglement suppression) and propagating quasiparticles (entanglement growth) are revealed. Such configurations can violate the First Law of Entanglement Entropy, reflecting underlying symmetry breaking or condensate dynamics (Zenoni et al., 2021).

Holographically, the rate of complexity growth after the join is mapped to the proper radial momentum of the tip of an end-of-the-world brane (Caputa et al., 26 Feb 2025).

5. Model Extensions and Broader Contexts

• Spin Chains: Local joining quenches have been studied in Ising and Heisenberg quantum spin chains, where the evolution of observables after the join (e.g., magnetization, correlation functions) matches CFT criteria at criticality and is captured semiclassically in ordered or disordered phases via kink propagation or domain-wall scattering (Divakaran et al., 2011).
• Colloidal Gels: In classical soft matter, a local joining quench can refer to the controlled assembly of colloidal particles by gradually ramping up attractive interactions. Slow “joining” drives the system toward local crystallinity and increased gel stability compared to an abrupt quench, highlighting the generality of joining protocols across physical disciplines (Royall et al., 2012).
• Non-Equilibrium Quantum Spectroscopy: In many-body systems, local joining quenches serve as probes for the global excitation spectrum. Monitoring local observables after such a quench (quench spectroscopy) allows extraction of excitation dispersions and transition energies, even in interacting or complex systems (Villa et al., 2020).
• Quantum Information Measures: Hydrodynamic decompositions (e.g., information lattices) reveal scale- and site-resolved local information flows, enabling the distinction between topological and non-topological entanglement propagation in joining protocols (e.g., Majorana junctions) (Bauer et al., 1 May 2025). Keldysh techniques provide real-time methods valid beyond analytic continuation and accommodate arbitrary initial conditions, exposing the interplay between initial state, interaction region, and post-quench energy flow (Radovskaya et al., 26 Mar 2024).
• Lieb–Robinson Bounds: Local joining quenches are subject to velocity bounds on entanglement spreading; finite Lieb–Robinson velocities establish “light cones” bounding the non-equilibrium growth of entanglement and information, independent of subsystems’ volume (Drumond et al., 2017).

6. Controlling and Generalizing the Protocol

Variations of the joining quench—e.g., the Möbius quench, where homogeneous time evolution with an inhomogeneous Hamiltonian leads to equivalent dynamical behaviors—are related via conformal mappings and Weyl transformations. This equivalence extends to holographic duals, where the induced profile of an end-of-the-world brane mirrors the entanglement evolution and periodicity in the system (Kudler-Flam et al., 2023).

Slow or controlled joining (in classical and quantum contexts) allows tuning between highly non-equilibrium and quasi-adiabatic dynamics, and thus enables experimental control over local order formation or relaxation rates.

7. Limitations, Distinctions, and Outlook

While much of the universal phenomenology—ballistic entanglement growth, front propagation, and scaling of complexities—holds in both free and interacting systems (including holographic regimes at large $c$), specific observables (e.g., noise vs. entanglement entropy) can differ in the details of their dependence on symmetries and microscopic data, as seen in the contrast between Laughlin states and quantum Ising chains (0908.2622).

The non-universality of entropy growth in the absence of a U(1) conserved current, the existence of logarithmic or exponential regimes depending on interactions and dimensionality, and the emergent intricacies from charge-resolved or pseudo entropic measures (Shinmyo et al., 2023, Chen et al., 2023) point to a rich territory for exploration.

Advances in numerical and analytical tools (Krylov, Keldysh, hydrodynamic information decompositions) as well as experimental probes (ultracold atoms, quantum dots, engineered quantum materials) continue to expand the applicability and relevance of local joining quenches in uncovering the dynamical structures of quantum and classical systems.