⚛️ high-energy theory

Spread Complexity for Local Operator Quenches with Conserved Momentum

This paper investigates the spread complexity of locally excited states with net momentum in two-dimensional conformal field theories across various geometries and temperatures, utilizing both analytical and numerical methods to compute Lanczos coefficients and complexity evolution while observing a mismatch with the proposed holographic dual related to infalling particle momentum.

Original authors: Ali Fatemiabhari

Published 2026-09-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ali Fatemiabhari

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 modern physics, there is a growing fascination with a concept called complexity. This is not the everyday sense of something being difficult to understand, but a precise measure of how much a system changes as it evolves over time. Imagine a simple, quiet room. If you leave it alone, it stays the same. But if you introduce a single disturbance, like a pebble dropped into a still pond, ripples spread out, interacting with the walls and each other. In the quantum world, where particles and energy behave in strange ways, this spreading is not just a visual ripple; it is a fundamental transformation of the system's state. Physicists have long sought a way to quantify this transformation, asking how "spread out" a quantum state becomes as it moves through its possible configurations. This measure, known as spread complexity, acts like a ruler for the growth of quantum information. It helps scientists distinguish between systems that are chaotic and unpredictable and those that are orderly and predictable. Furthermore, a deep and mysterious connection exists between these quantum systems and the fabric of space-time itself, suggesting that the growth of complexity in a quantum world might be mirrored by the motion of objects falling through a gravitational universe.

A researcher has recently taken a significant step forward in understanding this phenomenon by studying a specific type of quantum disturbance. They focused on a scenario where a local operator, essentially a tool used to create an excitation or a "kick" in a quantum system, is applied in a way that gives the resulting state a net momentum. In simpler terms, they looked at what happens when a quantum system is disturbed not just randomly, but with a specific push in one direction. To do this, they examined these systems in three different settings: a flat, infinite plane; a cylinder, which represents a system wrapped around a loop; and a system held at a constant, warm temperature. In each case, they created a state where the disturbance carried a conserved momentum, meaning the system had a definite direction of movement that could not be lost.

The researcher began by calculating the earliest moments of this evolution. They determined the first few steps of a mathematical sequence that describes how the system spreads out. This sequence acts like a set of instructions for how a quantum particle hops from one state to the next. By analyzing these early steps, they found a clear pattern for how the complexity grows right after the disturbance is applied. They discovered that the rate at which the complexity increases depends on the specific parameters used to create the disturbance, particularly the difference between the "chiral" and "anti-chiral" regulators. These are technical terms for the specific ways the disturbance is smoothed out in different directions. When these regulators are equal, the system behaves in a well-known, symmetric way. However, when they are different, creating a net momentum, the behavior changes, and the complexity grows in a slightly different, more intricate pattern.

To see what happens over longer periods, the researcher turned to powerful computer simulations. Because the mathematical equations become incredibly difficult to solve by hand as time goes on, they used numerical methods to calculate the sequence of steps up to very high numbers. This allowed them to watch the spread complexity evolve for a much longer duration than was previously possible. In the flat, infinite setting and the warm temperature setting, they observed that the complexity continued to grow without bound, eventually rising exponentially. This rapid growth is a hallmark of systems with a continuous spectrum of energy. In contrast, for the system wrapped on a cylinder, the complexity did not grow forever. Instead, it rose and fell in a repeating cycle, returning to zero after a certain amount of time. This periodic behavior is a direct result of the finite size of the system; because the space is limited, the quantum information eventually loops back on itself, causing the system to return to its original state.

A major goal of this research was to test a bold idea from the field of holography, which suggests that the growth of quantum complexity in a system is directly linked to the physical momentum of a particle falling through a gravitational universe. Specifically, a proposal existed that the rate at which complexity grows should match the "proper momentum" of a particle falling into a black hole or a similar gravitational structure. The researcher applied this idea to their specific setup, where the falling particle carries a conserved momentum, just like their quantum states. They carefully calculated the momentum of the falling particle using advanced techniques that account for the particle's charge and motion. However, when they compared the rate of change of the quantum complexity with the momentum of the falling particle, the numbers did not match. The two quantities, which were expected to be two sides of the same coin, told different stories.

This mismatch is a significant finding. It suggests that the current proposal for the holographic dual of spread complexity is incomplete or needs adjustment when dealing with states that carry conserved momentum. The researcher did not find a simple error in their calculations; rather, the discrepancy appears to be a fundamental issue with the existing theoretical framework. The study confirms that while the connection between quantum complexity and gravity is a powerful tool, it requires refinement to handle more complex scenarios involving momentum. The work provides a detailed benchmark for future studies, offering precise data on how these quantum states behave and highlighting where our understanding of the bridge between quantum mechanics and gravity needs to be strengthened. By mapping out the behavior of these excited states with such precision, the researcher has provided a clearer picture of the limits of our current theories and pointed the way toward a more complete understanding of the quantum universe.

Technical Summary: Spread Complexity for Local Operator Quenches with Conserved Momentum

Problem Statement
This paper investigates the spread complexity (also known as Krylov complexity) of a specific family of locally excited states in two-dimensional Conformal Field Theories (2d CFTs). These states are constructed by applying a local primary operator with independent chiral ( $\epsilon_+$ ) and anti-chiral ( $\epsilon_-$ ) regulators. Unlike standard local quenches where regulators are equal, this asymmetry imparts a net conserved momentum to the excitation. The study aims to compute the Lanczos coefficients and the resulting spread complexity for these states across three distinct geometries: the plane, the cylinder, and the finite-temperature cylinder. Furthermore, the paper tests the validity of recent holographic proposals that identify the rate of change of spread complexity with the proper momentum of an infalling particle in the dual bulk geometry, specifically for states carrying conserved momentum.

Methodology
The author employs a combination of analytical and numerical techniques:

Analytical Calculation:
- Return Amplitude: The computation begins with the exact return amplitude (autocorrelation function) $S(t)$ , derived using the Operator Product Expansion (OPE) and conformal mapping techniques for the plane, cylinder, and thermal cylinder.
- Lanczos Coefficients: The first few Lanczos coefficients ( $a_n, b_n$ ) are derived analytically using the Toda-chain method (or Toda hierarchy), which relates the coefficients to the moments of the return amplitude.
- Early-Time Expansion: Using the first few coefficients, the early-time behavior of the spread complexity $C(t)$ is expanded explicitly.
Numerical Calculation:
- Spectral Method: To overcome the numerical instability inherent in the moment problem (where Hankel determinants become ill-conditioned), the author utilizes a spectral method. This involves discretizing the spectral measure of the return amplitude (via its Fourier transform) and applying the Gragg-Harrod (or RKPW) algorithm. This approach allows for the stable computation of Lanczos coefficients up to high Krylov indices ( $n \approx 3000-4000$ ).
- Complexity Evolution: The computed coefficients are used to construct the truncated Jacobi matrix, which is diagonalized to obtain the time-evolved amplitudes and the spread complexity over a long time range.
Holographic Comparison:
- The author reviews the "proper momentum" proposal, where the rate of change of complexity $\dot{C}(t)$ is dual to the proper radial momentum of a massive particle falling in the bulk.
- They apply the "Routhian method" to handle the conserved momentum of the probe particle. This involves performing a Legendre transform to fix the conserved charge (momentum) before calculating the proper momentum, ensuring the initial momentum vanishes to satisfy the universal early-time even-function behavior of complexity.

Key Contributions and Results

Analytic Coefficients and Early-Time Behavior: The paper provides explicit analytic expressions for the first few Lanczos coefficients ( $a_0, b_1, a_1, b_2$ ) and the early-time expansion of $C(t)$ for all three setups (plane, cylinder, finite temperature).
- The results show that when $\epsilon_+ \neq \epsilon_-$ , the coefficients deviate from the standard $sl(2, \mathbb{R})$ algebraic form (where $a_n \propto n$ and $b_n \propto \sqrt{n}$ ).
- The early-time complexity $C(t)$ exhibits a quadratic term $t^2$ followed by a $t^4$ correction that depends on the asymmetry $(\epsilon_+ - \epsilon_-)$ .
Numerical Asymptotics: Numerical results reveal that for large Krylov indices $n$ , the Lanczos coefficients $a_n$ and $b_n$ asymptote to the values determined by the smaller of the two regulators, $\epsilon_{\min} = \min(\epsilon_+, \epsilon_-)$ . This suggests that the long-time dynamics are dominated by the "sharpest" regulator.
Complexity Dynamics:
- Plane and Finite Temperature: The spread complexity grows unbounded. At finite temperature, it exhibits exponential growth, consistent with thermal systems.
- Cylinder: The complexity is periodic, reflecting the finite dimension of the accessible Hilbert space due to the compact spatial circle.
- Upper Bounds: In all cases, the spread complexity for asymmetric regulators is bounded from above by the complexity of the symmetric case with regulator $\epsilon_{\min}$ .
Holographic Mismatch: A central finding is the mismatch between the field theory results and the holographic proposals.
- When comparing the early-time expansion of the CFT spread complexity with the integrated proper momentum of the dual infalling particle (calculated via the Routhian method), the author finds that while the leading $t^2$ term can be matched by tuning a proportionality constant, the higher-order terms (specifically the $t^4$ coefficient) do not agree.
- This discrepancy persists even after applying the Routhian method to correctly account for conserved momentum, suggesting that the proposal identifying $\dot{C}(t)$ directly with the proper momentum of a probe particle may require amendment for states carrying conserved charges.

Significance and Claims
The paper claims to provide a rigorous benchmark for spread complexity in 2d CFTs with conserved momentum, utilizing high-precision numerical methods to access regimes beyond analytic reach. The primary significance lies in the detailed characterization of how independent chiral regulators affect the Krylov structure and complexity growth.

Crucially, the author modestly but firmly concludes that their results challenge the current holographic dictionary for spread complexity. The observed mismatch in the higher-order time evolution terms implies that the simple identification of complexity growth rate with the proper momentum of a falling particle is insufficient for states with non-zero conserved momentum. The paper suggests that a refined holographic dual is necessary to capture the full dynamics of these excited states. It does not propose a new holographic dual but rather highlights the need for one, serving as a testbed for future developments in the gauge/gravity correspondence regarding complexity.

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