⚛️ high-energy theory

Holographic Local Operator Quenches with Conserved Momentum and Spin

This paper establishes a precise holographic dictionary between local operator quenches in 2D CFTs and point particles carrying conserved momentum or spin in asymptotically AdS $_3$ spacetimes, demonstrating exact agreement between boundary and bulk calculations of energy density and entanglement entropy while introducing new quantum quench protocols.

Original authors: Pawel Caputa, Pedro Castellini Grand, Justin R. David, Rahul Metya

Published 2026-08-21

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CC BY 4.0

Original authors: Pawel Caputa, Pedro Castellini Grand, Justin R. David, Rahul Metya

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the deepest realms of theoretical physics, scientists are trying to understand how the universe behaves when it is pushed out of balance. Imagine a calm pond; if you drop a single stone into it, ripples spread outward, carrying energy and information across the surface. In the quantum world, which governs the behavior of atoms and light, similar disturbances happen constantly. When a tiny piece of energy is added to a system, it does not just sit there; it moves, interacts, and eventually spreads out, often leading the system to a state of thermal equilibrium, much like a hot cup of coffee cooling down to room temperature. However, the universe is not always so simple. Sometimes, systems possess hidden rules, known as conservation laws, that prevent this smooth spreading. Just as a spinning top resists falling over due to its angular momentum, quantum systems can carry specific quantities like momentum or spin that constrain how they evolve. Understanding how these conserved quantities guide the flow of energy and information is a central challenge, as it reveals the fundamental mechanics of how the universe processes change.

A powerful tool for exploring these mysteries is a theoretical framework called the holographic principle. This idea suggests that a complex, three-dimensional universe with gravity can be mathematically described by a simpler, two-dimensional surface without gravity, much like a hologram where a flat image encodes a three-dimensional shape. In this specific view, the behavior of particles moving through a curved space is mirrored by the behavior of operators, or mathematical instructions, acting on a flat surface. For years, researchers have used this mirror to study how energy moves, but they have mostly focused on particles that carry only energy. They have rarely looked at what happens when those particles also carry momentum or spin, even though these are common features in the real world.

In this new work, a team of physicists has built a detailed dictionary to translate between these two worlds for particles that carry momentum and spin. They started by imagining a massive particle moving through a specific type of curved space known as Anti-de Sitter space. This space acts like a gravitational trap where particles fall inward. The researchers calculated exactly how such a particle, moving with a specific speed or spinning as it falls, would warp the space around it. They found that the particle's motion creates a distinct pattern of energy and momentum that ripples through the geometry. Crucially, they discovered that the speed and direction of the particle are encoded in how the space is distorted, creating a precise map between the particle's physical properties and the shape of the universe it inhabits.

The team then turned to the other side of the mirror: the two-dimensional quantum world. They proposed that the complex state created by a falling particle corresponds to a very specific way of disturbing a quantum system. Instead of simply adding energy at a single point, they found that one must "smear" or spread the disturbance differently in two opposing directions, much like stretching a piece of dough unevenly. By adjusting this stretching, they could create a state that carries a specific amount of momentum. When they calculated the energy and momentum of this quantum state, it matched the gravitational calculations perfectly. This confirmed that the "stretching" of the disturbance in the quantum world is the direct equivalent of the particle's motion in the gravitational world.

The researchers extended this discovery to include particles with intrinsic spin, a property that makes them rotate like tiny tops. In the gravitational picture, a spinning particle creates a different kind of twist in the fabric of space. On the quantum side, this corresponds to a disturbance where the left-moving and right-moving parts of the energy wave are not identical. The team showed that by making these two parts unequal, they could generate a quantum state that carries spin. They calculated the entanglement entropy, a measure of how much information is shared between different parts of the system, for both the gravitational and quantum descriptions. The results were identical, proving that their new dictionary works not just for simple movement, but for the complex internal rotation of particles as well.

Beyond the immediate match between gravity and quantum mechanics, the work opens a new path for studying quantum systems that do not have a gravitational twin. The specific way of creating these states—by applying different regulators to the left and right sides of a quantum operator—defines a new class of experiments. These protocols allow scientists to study how energy and information move in systems that conserve momentum or spin, a scenario that was difficult to analyze before. The researchers demonstrated that these methods work even in simple, solvable models of quantum matter, suggesting that the insights gained from the complex gravitational calculations can be applied to real-world quantum materials. By establishing this clear link between moving particles and specific quantum disturbances, the study provides a robust foundation for understanding how conserved charges shape the evolution of the universe, from the smallest quantum scales to the largest cosmic structures.

Technical Summary: Holographic Local Operator Quenches with Conserved Momentum and Spin

Problem Statement
Understanding the non-equilibrium dynamics of interacting quantum many-body systems, particularly how local excitations propagate and how conserved quantities influence thermalization and information transport, remains a central challenge in theoretical physics. While the AdS/CFT correspondence has provided a powerful geometric framework for studying local operator quenches—typically modeled as massive particles falling into Anti-de Sitter (AdS) space—most existing studies focus on excitations carrying energy but no additional conserved charges. The precise dual description for configurations where the excitation carries conserved longitudinal momentum, angular momentum, or intrinsic spin has not been systematically detailed. This paper addresses the gap in the holographic dictionary relating point particles with these conserved charges in asymptotically AdS $_3$ spacetimes to specific states in the dual two-dimensional conformal field theory (2D CFT).

Methodology
The authors employ a dual approach, constructing exact solutions in the bulk gravity theory and corresponding states in the boundary CFT, then demonstrating their equivalence.

Gravity Side: The authors utilize embedding-space techniques and SO(2,2) isometries to construct fully back-reacted geometries for massive point particles in three distinct asymptotically AdS $_3$ backgrounds: Poincaré AdS (vacuum), Global AdS (finite size), and the BTZ black brane (finite temperature).
- They introduce two independent boost parameters, $\eta_1$ and $\eta_2$ , to map a particle with non-vanishing longitudinal velocity or angular momentum in the original coordinates to a static particle at the origin of an auxiliary global AdS frame.
- Using these maps, they derive the back-reacted metrics analytically. From these metrics, they extract the holographic stress tensors via the Fefferman-Graham (FG) expansion and compute entanglement entropies using the Hubeny-Rangamani-Takayanagi (HRT) prescription.
- The analysis is extended to particles with intrinsic spin by incorporating spinning conical defects, where the particle's worldline action is supplemented by a spin term.
CFT Side: The authors propose a universal family of locally excited states generated by local primary operators with independent chiral ( $\epsilon_+$ ) and anti-chiral ( $\epsilon_-$ ) Euclidean regulators.
- These regulators correspond to asymmetric smearing in Euclidean time and space, effectively generating states with finite longitudinal momentum or angular momentum.
- They compute the expectation values of the stress tensor using the Operator Product Expansion (OPE) and calculate entanglement entropies in the large- $c$ limit using the Heavy-Heavy-Light-Light (HHLL) approximation for four-point functions.
- For spinning excitations, they consider primary operators with unequal holomorphic and anti-holomorphic scaling dimensions ( $h \neq \bar{h}$ ).

Key Contributions and Results

Establishment of the Dictionary: The paper establishes a precise mapping between the boost parameters in the bulk ( $\eta_1, \eta_2$ ) and the regulator parameters in the CFT ( $\epsilon_\pm$ ).
- In Poincaré AdS, asymmetric smearing ( $\epsilon_+ \neq \epsilon_-$ ) produces a state with conserved momentum $P$ . The holographic stress tensor components $T_{\pm\pm}$ match exactly with CFT calculations derived from the OPE.
- In Global AdS, the setup describes particles with angular momentum $J$ . The authors derive the exact holographic stress tensor and match it to the CFT result on a cylinder, fixing the dictionary relations between the boost parameters and the CFT regulators.
- In the BTZ background, the correspondence is extended to finite temperature, reproducing known results for longitudinal momentum while generalizing the dictionary.
Entanglement Entropy Matching: The authors compute the time-dependent entanglement entropy for a single interval in all three backgrounds using the HRT prescription.
- They find exact agreement with large- $c$ CFT calculations.
- A key result is the identification of the asymmetry in the entanglement growth: the "height" and width of the left- and right-moving pulses in the entanglement entropy are determined by the distinct regulators $\epsilon_+$ and $\epsilon_-$ . This asymmetry directly reflects the conserved momentum or spin of the excitation.
- For spinning particles, the entanglement entropy depends on the combinations $\Delta \pm s$ (where $\Delta$ is the conformal dimension and $s$ is the spin), leading to distinct coefficients for the left- and right-moving contributions.
Spinning Excitations: The work generalizes the local quench framework to include intrinsic spin.
- In the bulk, this corresponds to infalling spinning conical defects.
- In the CFT, this corresponds to primary operators with $h \neq \bar{h}$ .
- The authors show that in the extremal limit ( $\Delta = |s|$ ), the excitation becomes purely chiral (either purely left- or right-moving), resulting in a vanishing contribution to the entanglement entropy from one side of the interval.
Applications Beyond Holography: The authors demonstrate that the identified CFT states are well-defined in arbitrary 2D CFTs, regardless of the existence of a classical gravity dual.
- They apply the formalism to Rational CFTs (RCFTs) to derive the evolution of Rényi entropies, showing that the approach to the plateau value is controlled by the specific regulator ( $\epsilon_-$ or $\epsilon_+$ ) depending on the interval's position relative to the excitation.
- They analyze the Free Fermion CFT, confirming that the finite-width corrections to the Rényi entropy factorize into a universal time-dependent profile and an operator-dependent prefactor.
- They propose a new class of local quantum quench protocols in generic 2D CFTs where the initial excitation carries conserved momentum, defined by conformal maps with asymmetric regulators.

Significance and Claims
The paper claims to provide the first detailed and general dual description of local quenches carrying conserved momentum and spin in both the bulk and boundary descriptions. The primary significance lies in the "exact agreement" found between the gravitational calculations (stress tensors and entanglement entropies) and the CFT computations across vacuum, finite-size, and finite-temperature settings.

The authors emphasize that their work extends the standard framework of local quenches (which typically assume symmetric smearing and zero momentum) to a broader class of non-equilibrium states. By identifying the CFT states dual to particles with conserved charges, they provide a new theoretical laboratory for studying the interplay between energy transport, conserved charges, and quantum entanglement. The work suggests that these setups offer simple yet non-trivial models for exploring information propagation and thermalization in systems where symmetries and conservation laws play a dominant role, potentially preventing standard thermalization and leading to generalized equilibrium states. The paper concludes by noting that these results serve as a foundation for future investigations into higher-dimensional holographic quenches, OTOC correlators in multi-sided black holes, and the role of conserved charges in the emergence of semiclassical spacetime.

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