⚛️ high-energy theory

Pole skipping from universal hydrodynamics of (1+1)d QFTs

This paper demonstrates that the pole-skipping points of the stress tensor thermal two-point function in holographic (1+1)d QFTs align with predictions from a universal hydrodynamic theory, even beyond its standard regime of validity, thereby correctly reproducing the butterfly velocity.

Original authors: Richard A. Davison, Hanzhi Jiang

Published 2026-11-03

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Richard A. Davison, Hanzhi Jiang

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 vast landscape of theoretical physics, there is a persistent puzzle regarding how the chaotic, jittery motion of individual particles settles into the smooth, predictable flow of a fluid. When a system of particles is heated, they eventually reach a state of thermal equilibrium, where their collective behavior can be described by the laws of hydrodynamics, much like water flowing down a river. However, understanding exactly how this transition happens, especially in the simplest possible setting of one spatial dimension and one time dimension, has been difficult. In a perfectly symmetric world, this fluid-like behavior would never emerge; the particles would simply bounce around forever without losing energy. It is only when this perfect symmetry is slightly broken that the system begins to dissipate energy and behave like a fluid. The question remains: can we predict the exact properties of this emerging fluid, such as how fast disturbances travel through it, based solely on the microscopic rules of the particles?

A team of researchers has tackled this problem by studying a specific class of quantum theories that are mathematically linked to a theory of gravity in a higher dimension. They focused on how energy moves through these systems at high temperatures, looking for a specific mathematical signature known as "pole skipping." This signature appears in the way the system responds to disturbances, marking a point where the rules governing the flow of energy become ambiguous because two different mathematical behaviors—one representing a wave and the other a decay—intersect. By calculating exactly where these intersections occur, the researchers were able to determine a fundamental speed limit for how fast information spreads through the system, a quantity known as the butterfly velocity. Their work confirms that a proposed universal theory of fluid dynamics, which suggests that all such systems behave similarly at high temperatures, is correct. They found that the speed at which information scrambles in these complex quantum systems matches the predictions derived from a simplified, universal description of fluid flow, bridging the gap between the microscopic world of particles and the macroscopic world of fluids.

The researchers began by examining a theoretical framework where a two-dimensional quantum system is deformed by a small amount, breaking its perfect symmetry. In the absence of this deformation, the system would remain a conformal field theory, a state where energy travels at the speed of light without ever slowing down or spreading out. By introducing a small disturbance, the system begins to thermalize, meaning it eventually reaches a state where it can be described by fluid dynamics. The team wanted to see if the properties of this fluid, specifically the speed at which it carries information, could be predicted without solving the incredibly complex equations for every single particle. Instead, they turned to a method called the "holographic principle," which allows physicists to study these quantum systems by looking at a corresponding gravitational system in a higher dimension. In this gravitational picture, the quantum system is represented by a black hole, and the way energy moves in the quantum world is mirrored by the way waves move in the space around the black hole.

To find the answer, the researchers looked for the "pole skipping" points in the mathematical description of the system's energy flow. These points are special locations in the system's momentum and energy space where the usual rules for how waves propagate break down. At these specific points, a wave that would normally be absorbed by the system (a pole) and a wave that would normally be blocked (a zero) meet and cancel each other out, leaving the system's response undefined. The location of these points is not random; it is directly tied to the fundamental properties of the system, including the speed at which information spreads. The researchers calculated these locations for the gravitational black hole model and found that they matched the predictions of a universal theory of hydrodynamics proposed by other scientists. This universal theory suggests that at high temperatures, the details of the specific particles in the system do not matter; only the dimension of the operator that breaks the symmetry matters.

The study revealed that the speed at which information spreads, the butterfly velocity, is slightly slower than the speed of light but faster than the speed of sound in the fluid. This result is significant because it shows that even though the system is complex and quantum mechanical, its large-scale behavior is governed by simple, universal rules. The researchers found that for one specific type of pole skipping point, the location depends only on the dimension of the symmetry-breaking operator, confirming the universality of the hydrodynamic theory. However, they also discovered that other pole skipping points, which occur at different frequencies, depend on more detailed properties of the system, such as the specific interactions between the particles. This distinction is crucial: it shows that while the universal theory works perfectly for describing the most fundamental aspects of how the system thermalizes, there are still deeper layers of complexity that require a more detailed understanding of the specific particles involved.

The researchers achieved this by performing a careful mathematical resummation of the equations that describe the system. In standard physics, when a system is heated, the equations that describe its behavior often break down because the interactions between particles become too strong to handle with simple approximations. The team developed a way to reorganize these equations, effectively adding up an infinite number of small corrections to get a clear picture of what happens near the lightcone, the boundary where information travels at the maximum speed. By doing this, they were able to extend the validity of the universal hydrodynamic theory to earlier times and higher energies than previously thought possible. Their calculations showed that the universal theory correctly predicts the butterfly velocity and the location of the most important pole skipping point, providing strong evidence that this theory captures the essential physics of thermalization in these systems.

The findings have implications for our understanding of how chaos and order coexist in the universe. The butterfly velocity is a measure of how quickly a small change in a system, like a butterfly flapping its wings, can grow to affect the entire system. In the context of these quantum systems, this speed determines how fast information is scrambled and becomes inaccessible. The fact that this speed can be predicted by a universal theory suggests that there is a deep, underlying order to how complex systems evolve, even when they appear chaotic. The researchers' work demonstrates that by looking at the right mathematical signatures, we can uncover these universal laws without needing to know every detail of the microscopic world.

While the study confirms the validity of the universal hydrodynamic theory for a specific class of systems, it also highlights the limits of this universality. The researchers found that not all aspects of the system's behavior can be captured by the universal theory. Some of the pole skipping points, which correspond to more complex modes of energy flow, depend on the specific details of the particle interactions. This means that while the universal theory is powerful for understanding the broad strokes of thermalization, it cannot replace the need for detailed calculations when we want to understand the finer points of a specific system. The study serves as a reminder that in physics, there are often layers of description, from the universal to the specific, and that understanding how these layers connect is key to unlocking the secrets of the universe.

The researchers also noted that their work opens up new avenues for exploring the connection between quantum mechanics and gravity. By showing that the pole skipping points in the quantum system match the predictions of the gravitational model, they provide further evidence for the holographic principle, which suggests that the information contained in a volume of space can be represented by a theory on the boundary of that space. This connection is not just a mathematical curiosity; it offers a practical way to study complex quantum systems by using the simpler language of gravity. The ability to calculate the butterfly velocity and other transport coefficients using this method could lead to new insights into the behavior of materials and fluids in extreme conditions, such as those found in the early universe or in the cores of stars.

In conclusion, this paper provides a clear and concrete demonstration that the emergence of hydrodynamics in quantum systems is governed by universal laws that can be predicted from first principles. By identifying the locations of pole skipping points and calculating the butterfly velocity, the researchers have shown that the universal hydrodynamic theory is not just an approximation, but a fundamental description of how these systems behave at high temperatures. Their work bridges the gap between the microscopic world of quantum particles and the macroscopic world of fluids, offering a new perspective on how order emerges from chaos. The study stands as a testament to the power of theoretical physics to reveal the hidden structures that govern the natural world, even in the most complex and seemingly unpredictable systems.

Technical Summary: Pole Skipping from Universal Hydrodynamics of (1+1)d QFTs

Problem Statement
The emergence of dissipative hydrodynamics from the microscopic dynamics of generic interacting Quantum Field Theories (QFTs) remains a challenging problem, particularly in determining how macroscopic transport coefficients relate to underlying constituents. In (1+1) dimensions, conformal symmetry prevents the emergence of dissipative hydrodynamics; thus, its appearance is tied to the weak breaking of this symmetry via a relevant scalar operator deformation. While conformal perturbation theory works for small symmetry-breaking parameters ( $\bar{\lambda} \ll 1$ ) at high temperatures, it breaks down at late times ( $t \sim \bar{\lambda}^{-2}$ ) near the lightcone, necessitating a resummation to access the hydrodynamic regime. A recent proposal [4] suggests a "universal" hydrodynamic theory for the stress tensor thermal two-point function in (1+1)d QFTs with large central charge $c$ and small $\bar{\lambda}$ , where transport coefficients depend only on the operator dimension $\Delta$ .

This paper investigates the validity of this universal hydrodynamic proposal by examining "pole skipping" phenomena in holographic QFTs. Pole skipping occurs at specific points in complex momentum space $(\omega, k)$ where the retarded thermal two-point function $G(\omega, k)$ is undefined due to the intersection of a pole and a zero. These points encode fundamental thermalization properties, such as the butterfly velocity $v_B$ , and are computable in holographic theories via the near-horizon geometry. The central question is whether the locations of these pole skipping points in holographic theories match the predictions derived from the proposed universal hydrodynamics, specifically when the hydrodynamic derivative expansion is resummed near the lightcone.

Methodology
The authors employ a dual approach, combining field-theoretic resummation techniques with holographic calculations:

Resummed Universal Hydrodynamics (Field Theory Side):
- The authors review the universal hydrodynamic theory proposed in [4], which expresses the stress tensor two-point function in terms of transport coefficients determined to order $\bar{\lambda}^2$ by the dimension $\Delta$ of the breaking operator.
- Standard hydrodynamic dispersion relations are Taylor series in $k$ with a finite radius of convergence ( $k_{eq} = \Delta \pi T$ ). The pole skipping points of interest (specifically those related to $v_B$ ) lie outside this radius.
- To access these points, the authors take the high-temperature, near-lightcone limit ( $\omega \pm k \sim \bar{\lambda}^2$ ) of the hydrodynamic two-point function.
- They perform a resummation of the derivative expansion in $k$ by utilizing the explicit form of the CFT correlator for the operator $O$ . This yields a closed-form expression for the two-point function valid near the lightcone, extending the regime of validity beyond the standard hydrodynamic radius.
Holographic Calculation (Gravity Side):
- The authors consider (1+1)d QFTs dual to (2+1)-dimensional gravity theories with a scalar field $\phi$ and potential $V(\phi)$ , deforming an AdS $_3$ background.
- They construct the equilibrium black hole solution perturbatively in the high-temperature limit ( $\bar{\lambda} \ll 1$ ) up to quadratic order in the scalar field amplitude.
- Using this equilibrium geometry, they compute the butterfly velocity $v_B$ directly from the near-horizon metric.
- To locate pole skipping points, they analyze the linearized equations of motion for metric and scalar perturbations. They introduce a "master field" formalism ( $H$ ) to simplify the identification of extra ingoing solutions at the horizon, which correspond to pole skipping points.
- They solve the resulting recursion relations for the near-horizon expansion of the master field to determine the specific wavenumbers $k_n$ where pole skipping occurs at frequencies $\omega_n = -i 2\pi T n$ .

Key Contributions and Results

Butterfly Velocity Prediction:
The resummed universal hydrodynamics predicts a specific correction to the butterfly velocity $v_B$ at order $\bar{\lambda}^2$ . The holographic calculation confirms this prediction exactly. The result is universal, depending only on the operator dimension $\Delta$ :
$v_B = 1 - \alpha_\Delta \left( \frac{\pi \Delta (\Delta-2) \cot(\frac{\pi \Delta}{2})}{4(\Delta-1)} - 1 \right) \bar{\lambda}^2 + \dots$
This agreement holds for general $\Delta$ and matches previous results for specific cases ( $\Delta=3/2, 1, 2$ ).
Pole Skipping in the Upper Half-Plane:
The authors confirm the existence of a pole skipping point at $\omega_* = +i 2\pi T$ with wavenumber $k_*^2 = -(2\pi T / v_B)^2$ . The holographic derivation of $k_*$ matches the value predicted by the resummed hydrodynamic theory, validating the connection between the pole skipping location and the butterfly velocity in these (1+1)d holographic theories.
Pole Skipping in the Lower Half-Plane:
The study identifies pole skipping points at $\omega_n = -i 2\pi T n$ .
- Universal Point ( $n=1$ ): One of the two possible wavenumbers at $\omega = -i 2\pi T$ is found to be universal (dependent only on $\Delta$ ) and lies near the lightcone. Its location matches the prediction from the resummed hydrodynamics.
- Non-Universal Points: The other wavenumber at $\omega = -i 2\pi T$ , as well as all wavenumbers for $n \geq 2$ , are found to be non-universal. Their locations depend on the specific coefficients of the scalar potential $V(\phi)$ (e.g., $V_3, V_4$ ) and the Operator Product Expansion (OPE) coefficients of the operator $O$ . These points are far from the lightcone and cannot be accessed by the near-lightcone resummation of hydrodynamics.

Significance and Claims
The paper claims that the exact agreement between the resummed universal hydrodynamics and the holographic calculation for the butterfly velocity and the near-lightcone pole skipping points provides "very non-trivial evidence" that the proposed universal hydrodynamics correctly describes the late-time dynamics of these (1+1)d QFTs.

Crucially, the authors argue that the resummation of the hydrodynamic derivative expansion near the lightcone effectively extends the validity of the hydrodynamic description to early times. This allows the hydrodynamic theory to capture information about operator scrambling (encoded in $v_B$ ) and the specific structure of the two-point function near the lightcone, which would otherwise be inaccessible via standard derivative expansions.

The work highlights a distinction between universal and non-universal features of thermalization:

Universal: Features near the lightcone (like $v_B$ and specific pole skipping points) are governed by the dimension $\Delta$ alone and are captured by the resummed hydrodynamic theory.
Non-Universal: Features far from the lightcone (other pole skipping points) are sensitive to the detailed microscopic structure (OPE coefficients, potential details) and are not captured by the universal hydrodynamic limit.

The authors modestly note that while the resummed expression (5.2) captures the leading order behavior, determining higher-order corrections or extending these results to non-holographic QFTs remains an open challenge. They also acknowledge that the derivation of the pole skipping frequencies themselves (as opposed to just the wavenumbers) in the hydrodynamic framework currently relies on assumptions derived from holographic expectations rather than a first-principles derivation from the resummed correlator alone.

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