Holographic Krylov Complexity for Charged, Composite and Extended Probes

This paper investigates holographic Krylov complexity for various probes in AdS$_5\times S^5$, demonstrating that while charged and composite pointlike probes retain universal leading growth with informative subleading corrections, genuinely extended operators exhibit qualitatively distinct intermediate behaviors that reveal a finer sensitivity to spatial structure.

The Gist

Imagine the universe is a giant, complex computer program. In this program, every particle, force, and interaction is a line of code. Complexity, in this context, isn't about how hard a problem is to solve; it's about how much the "code" of a specific object changes and spreads out as time goes on.

Think of a single drop of ink falling into a glass of water. At first, it's a tight, simple drop. As time passes, it swirls, stretches, and mixes with the water until it's everywhere. Krylov Complexity is a way to measure exactly how fast and how far that ink spreads through the "Hilbert space" (the mathematical room where all possible states of the universe exist).

This paper asks a fascinating question: Does the shape or "internal structure" of the ink drop change how it spreads?

To answer this, the authors use Holography. This is a magical principle in physics that says a 3D universe (like our bulk space) can be described entirely by a 2D surface (like a hologram). They use this to translate the messy math of "complexity" into the simpler language of falling objects.

Here is the breakdown of their experiments, using everyday analogies:

1. The Basic Idea: The Falling Rock

In previous studies, scientists imagined a simple, featureless rock falling down a deep, infinite well (representing the universe). They found that the speed at which the rock falls (its momentum) perfectly matches the speed at which the "complexity" of the system grows.

• The Rule: How fast the object falls = How fast the complexity spreads.

2. Experiment A: The Rock with a Secret (R-Charge)

First, the authors imagined a rock that isn't just a rock; it's a rock spinning rapidly while it falls. In physics terms, this is a particle carrying a "charge" (like an electric charge or a spin).

• The Analogy: Imagine a figure skater falling down a well while spinning. The spin adds a new dimension to their motion.
• The Result: The spinning changes the early stages of the fall. The complexity grows differently at the start because of the spin. However, once the skater has been falling for a long time, the spin becomes less important, and the complexity growth settles back into the standard pattern.
• Takeaway: If you have a "charged" object, the complexity of the universe remembers that charge at the beginning, but eventually, the universal laws take over.

3. Experiment B: The Composite Rock (Baryon Vertex)

Next, they looked at objects that look like single points from far away but are actually made of many smaller parts glued together. In the paper, these are "Baryon Vertices" (like protons made of quarks).

• The Analogy: Imagine a heavy backpack made of 100 tiny bricks tied together with strings, falling down the well. From a distance, it looks like one rock. But up close, it's a messy bundle.
• The Result: Even though this "backpack" is a complex bundle of strings and branes, as long as it falls as a single unit, the complexity grows exactly like the simple rock. The internal messiness doesn't change the main story of how fast complexity spreads.
• Takeaway: Being a "composite" object (made of parts) doesn't change the universal rule for how complexity grows, as long as the object stays together.

4. Experiment C: The Giant Balloon (Giant Gravitons)

They also studied "Giant Gravitons," which are huge, inflated objects that expand as they fall.

• The Analogy: Imagine a giant, expanding balloon falling down the well. It has its own internal pressure and shape.
• The Result: Similar to the backpack, even though this object is huge and has complex internal physics, its complexity growth follows the same "universal script" as the simple rock in the long run. The internal details only cause small, subtle ripples in the data.

5. Experiment D: The Stretching Rope (Extended Operators)

Finally, they did something different. Instead of a point-like object, they imagined a long string falling down the well, stretched out horizontally like a tightrope.

• The Analogy: Imagine a long rope being dropped into the well. Unlike the rock or the backpack, the rope has length. One end might be falling faster than the other, or it might be stretching and vibrating.
• The Result: This is where things get weird. The rope behaves differently. While the main speed of complexity growth is still similar to the rock, the details are completely different. The "subleading terms" (the fine print) show that the rope's complexity is sensitive to its shape and how it stretches.
• Takeaway: If an object is "extended" (spread out in space), it carries a unique signature. The complexity of a non-local object (like a long string) is fundamentally different from a local object (like a point particle).

The Big Picture

The authors conclude with a beautiful insight:

1. The Universal Law: For almost any object that looks like a "point" (even if it's spinning, charged, or made of many parts), the universe has a standard, universal way of growing complexity. It's like a heartbeat that everyone shares.
2. The Unique Signature: However, if you look closely at the details (the subleading terms), you can tell the difference between a spinning rock, a backpack, and a long rope.
3. The Extended Exception: If the object is truly "extended" (like a string), the rules change more significantly. The complexity of non-local things carries a finer, more detailed map of their structure.

In summary: The universe has a standard rhythm for how things get complicated, but the "internal structure" and "shape" of the object leave a unique fingerprint on that rhythm. By studying how different "probes" fall in a holographic universe, the authors are learning how to read those fingerprints to understand the true nature of quantum complexity.

Technical Summary

1. Problem Statement

The paper addresses the holographic realization of Krylov complexity (or spread complexity), a measure of how an operator spreads across the Hilbert space under time evolution. While previous work established a dictionary linking the rate of spread complexity ($\dot{C}$) to the proper momentum of an infalling point-like particle in the bulk (AdS), this correspondence was largely untested for operators with non-trivial internal structures or those that are genuinely non-local.

The authors investigate three specific classes of probes to determine the universality of this dictionary:

1. Charged Point Particles: Particles carrying conserved R-charges (moving in internal spaces).
2. Composite Point-like Probes: Objects that appear point-like to the boundary field theory but possess internal structure in the bulk (e.g., Baryon vertices and Giant Gravitons).
3. Extended Probes: Genuinely non-local operators modeled by fundamental strings stretched along spatial directions.

The core question is whether the "proper momentum = complexity growth" proposal holds for these complex systems, and if so, how internal structure and spatial extension modify the leading and subleading behaviors of complexity.

2. Methodology

The authors employ the holographic prescription proposed in [15], where the rate of change of Krylov complexity is identified with the negative of the generalized proper momentum ($P_y$) of an infalling probe:
$$\dot{C}(t) = -P_y$$
The analysis involves:

• Background: Primarily $AdS_5 \times S^5$ (dual to $\mathcal{N}=4$ SYM), with extensions to $AdS_3 \times S^3 \times CY_2$ and $AdS_7 \times S^4$ discussed in appendices.
• Dynamics: Solving the equations of motion for various probes using the Born-Infeld (BI) and Wess-Zumino (WZ) actions for D-branes, and the Nambu-Goto action for strings.
• Conserved Quantities: Utilizing conserved charges (energy $H$, angular momentum $J$, R-charge) to derive first-order equations of motion.
• Generalized Momentum: Defining a generalized proper coordinate $y$ such that $dy^2$ captures the metric contributions from both radial and internal/angular motions. The proper momentum is then calculated as $P_y = \sum P_i \frac{\partial \dot{x}^i}{\partial \dot{y}}$.
• Comparative Analysis: Comparing the time evolution of $\dot{C}(t)$ for structured/extended probes against the baseline "bare" massive particle ($\dot{C} \sim t$).

3. Key Contributions and Results

A. Charged Particles (Symmetry-Resolved Complexity)

• Setup: A massive particle falling radially in $AdS_5$ while rotating in the equator of $S^5$, carrying a conserved R-charge $J$.
• Result: The internal motion modifies the generalized proper momentum.
  • Late Time: The complexity growth remains linear in time ($\dot{C} \sim t$), matching the uncharged case.
  • Early Time: The growth is dominated by the conserved charge $J$. The rate $\dot{C}$ approaches a constant determined by $J$ rather than vanishing.
• Significance: This provides a holographic realization of symmetry-resolved Krylov complexity, showing that conserved quantum numbers dictate the early-time spread of the operator.

B. Composite Point-like Probes

The authors analyze two types of composite objects that are point-like from the boundary perspective but have bulk structure:

1. Baryon Vertices:

   • Setup: A D5-brane wrapping $S^5$ connected to $N$ fundamental strings (F1) stretching to the boundary.
   • Dynamics: The system is modeled as a composite particle with an effective potential arising from the string tension.
   • Result: The complexity growth is quadratic in time ($C \sim t^2$) at late times, identical to a bare massive particle. The internal structure (strings and WZ terms) only modifies the effective Hamiltonian (mass/energy), not the universal scaling law.

2. Giant Gravitons:

   • Setup: A D3-brane wrapping an $S^3$ inside $S^5$, rotating and falling in $AdS$.
   • Dynamics: The system involves coupled radial ($r$) and angular ($\theta, \phi$) motions with non-trivial Born-Infeld and Wess-Zumino couplings.
   • Result: Similar to the baryon vertex, the late-time complexity growth follows the standard $C \sim t^2$ law. However, the early-time behavior is non-zero ($\dot{C}(0) \neq 0$) due to the initial angular velocity, mirroring the behavior of the R-charged particle.

C. Extended Probes (Non-Local Operators)

• Setup: A fundamental string (F1) falling in $AdS$ while stretched along a spatial direction, modeling a non-local line operator.
• Dynamics: The string is treated as a rigid object with a specific embedding. The authors calculate the proper momentum of the string's center of mass.
• Results:
  • Leading Order: The rate of change $\dot{C}$ is still linear in time at both early and late limits, consistent with the point-particle universality.
  • Subleading Order: This is the critical finding. The subleading terms and intermediate regimes differ qualitatively from point-like probes.
  • Ratio Analysis: The ratio of the string's complexity growth rate to that of a particle ($\Pi = \dot{C}_{string}/\dot{C}_{particle}$) is a constant $>1$ at early times and approaches a different constant at late times.
  • Second Derivative: The second derivative of complexity ($\ddot{C}$) shows distinct behavior for the string compared to the particle, sensitive to the string's spatial extension.

4. Significance and Conclusions

• Universality of Leading Behavior: The paper establishes that for a broad class of probes (charged, composite, or extended), the leading large-time behavior of Krylov complexity is universal ($C \sim t^2$ or $\dot{C} \sim t$). This suggests that the overall "scrambling" speed is determined by the geometry of the bulk (AdS) rather than the specific nature of the operator.
• Sensitivity of Subleading Terms: While the leading order is universal, the subleading corrections and early-time dynamics encode specific information about the operator:
  • Charges control early-time growth (Symmetry-resolved complexity).
  • Compositeness (Baryons/Giants) modifies the effective mass but preserves the late-time scaling.
  • Spatial Extension (Strings) introduces distinct qualitative differences in the intermediate and subleading regimes, offering a "finer" notion of complexity sensitive to non-locality.
• Holographic Dictionary: The work validates and extends the dictionary relating bulk kinematics (proper momentum) to boundary complexity. It confirms that this dictionary is robust even for complex, structured, and extended objects, provided one uses a suitably generalized definition of momentum.
• Future Directions: The authors propose that these results pave the way for field-theoretic calculations of Krylov complexity in $\mathcal{N}=4$ SYM, specifically for symmetry-resolved sectors and operators with compositeness or non-locality. They also suggest relaxing the rigid approximation for strings to include fluctuations to further refine the understanding of non-local complexity.

In summary, the paper demonstrates that while the rate of complexity growth is largely universal for local and non-local operators in holographic CFTs, the structure of the growth (subleading terms and early-time behavior) serves as a precise diagnostic for the operator's internal charges, compositeness, and spatial extension.