⚛️ general relativity

The quadratic growth of Krylov spread complexity in the BTZ black hole

This paper establishes a dimension-independent boundary reconstruction of Krylov spread complexity for thermofield-double states, demonstrating that in the BTZ black hole dual, the complexity exhibits intermediate deviations from early-time quadratic growth before returning to asymptotic quadratic behavior, which is matched to a generalized bulk "complexity=anything" object constructed from an infinite series of extrinsic-curvature invariants.

Original authors: Aranya Bhattacharya, Mario Flory, Michal P. Heller, Emiliano Rizza, Tim Schuhmann

Published 2026-08-11

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CC BY 4.0

Original authors: Aranya Bhattacharya, Mario Flory, Michal P. Heller, Emiliano Rizza, Tim Schuhmann

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

Imagine the universe as a giant, cosmic library. Inside this library, every possible state of matter and energy is a unique book. For decades, physicists have been trying to figure out how to measure the "size" or "complexity" of these books. One popular idea, called the "Complexity = Volume" (CV) hypothesis, suggested that as a black hole ages, the complexity of its interior grows like a straight line on a graph—steady, predictable, and never-ending, much like a car cruising at a constant speed on a highway. This idea became a cornerstone of modern physics, linking the messy quantum world of particles to the smooth, curved geometry of space-time.

However, there's a new player in town: "Krylov spread complexity." Think of this not as a car on a highway, but as a drunkard's walk through a maze. Instead of moving in a straight line, the quantum state spreads out, exploring more and more paths. In the simplest models of gravity (specifically, 2D gravity), scientists found that this "drunkard" actually speeds up, moving in a curve that looks like a parabola (a quadratic growth). The big question that has puzzled physicists is: Does this speeding-up behavior happen in the real, 3D universe we live in, or was it just a quirk of the simple models? If it does happen, it would mean our old "straight-line" map of black holes is wrong, and we need a new way to understand how these cosmic monsters evolve.

This paper takes a bold step to answer that question by looking at the BTZ black hole, a 3D version of a black hole that lives in a universe with two spatial dimensions and one time dimension. The authors developed a clever new method called "Complexity from Partition Function" (CfZ). Imagine the partition function as a master recipe book for a black hole's heat and energy. The authors realized that if you take the derivatives (the mathematical equivalent of checking how the recipe changes with temperature) of this recipe, you can reconstruct the entire "drunkard's walk" of the quantum state without needing to simulate the whole universe. It's like being able to predict the entire path of a rolling ball just by knowing the shape of the hill it started on.

When they applied this method to the BTZ black hole, they found something surprising. The complexity didn't grow in a straight line like the old "Complexity = Volume" theory predicted. Instead, it started with a quadratic curve (speeding up), dipped slightly, and then seemed to turn back toward that quadratic shape. This suggests that for a long time, the black hole's interior is getting complex much faster than a simple linear growth would allow. The authors argue that the old "straight-line" models fail here because they assume the complexity measure stays finite and well-behaved at the very end of the black hole's evolution.

To fix this, the authors built a new "bulk" object—a geometric shape inside the black hole that matches the boundary's behavior. They constructed this shape by taking the standard volume and adding an infinite series of "extrinsic curvature" terms. Think of this as taking a simple balloon and adding an infinite number of tiny, weighted strings to it. If you cut the series short (use a finite number of strings), the balloon still grows in a straight line. But if you include the entire infinite series, the weight becomes so intense at the very end that the balloon's growth rate accelerates, matching the quadratic curve seen on the boundary.

The paper suggests that the "Complexity = Volume" idea, in its standard form, is likely incorrect for these types of black holes. Instead, the true geometric dual of Krylov spread complexity is a much more exotic object: a volume weighted by an infinite tower of geometric corrections that only reveals its true, accelerating nature when all the pieces are put together. While the authors can't prove the behavior continues forever (the math gets tricky at very late times), their simulations and analytical tools strongly suggest that the universe's complexity might be accelerating in a way we haven't fully appreciated before, rewriting the rules of how black holes grow.

Technical Summary: The Quadratic Growth of Krylov Spread Complexity in the BTZ Black Hole

Problem Statement
The microscopic definition of holographic complexity remains unsettled. While geometric candidates like "Complexity = Volume" (CV) and the broader "Complexity = Anything" (CAny) class predict a universal late-time linear growth regime in eternal black holes, the specific boundary quantity computing these observables is unknown beyond two-dimensional dilaton gravity. Recent work in the double-scaled Sachdev–Ye–Kitaev (DSSYK) model suggested a link between bulk geometry and Krylov spread complexity, but it is unclear if this relation persists in higher-dimensional holography or if the standard linear growth behavior is a universal feature of all holographic complexity measures. This paper investigates whether Krylov spread complexity, defined via the Krylov basis of a thermofield-double (TFD) state, can be reconstructed from semiclassical holographic partition functions in dimensions higher than two, and what its late-time behavior implies for bulk duals.

Methodology
The authors employ a "Moment Method" (MM) combined with a partition-function reconstruction (CfZ) to bridge black hole thermodynamics and Krylov dynamics. The core procedure involves:

Partition Function to Moments: For a TFD reference state $|TFD(\beta)\rangle$ , the moments of the Hamiltonian $H$ are derived as derivatives of the thermal partition function $Z(\beta)$ : $m_k = (-\partial_\beta)^k Z(\beta) / Z(\beta)$ .
Moments to Lanczos Coefficients: These moments determine the Lanczos coefficients ( $a_n, b_n$ ) via the Lanczos recursion, which define the Krylov basis.
Lanczos to Complexity: The Krylov spread complexity $C_K(t)$ is calculated as the mean position of the time-evolved state in this basis.
Semiclassical Limit Analysis: The authors apply this to the DSSYK model and the BTZ black hole (dual to a 2D CFT). A critical technical insight is the "order-of-limits" rule: the classical limit (e.g., large central charge $c$ or $q \to 1$ ) must be taken after assembling the full complexity series, not at the level of individual Lanczos coefficients. Truncating coefficients before assembly discards essential subleading terms that cancel to produce the correct leading-order classical behavior.
Analytic Continuation: Since the early-time expansion of $C_K(t)$ has a finite radius of convergence, the authors use diagonal Padé approximants to analytically continue the series beyond the thermal crossover regime ( $t/\beta \sim O(1)$ ) to probe late-time behavior.
Bulk Construction: To find a bulk dual, the authors construct a generalized CAny functional. They relax the standard CAny assumption that the evaluated functional must be finite on the accumulation surface (the final slice inside the horizon). Instead, they propose an infinite series of extrinsic-curvature invariants weighted by coefficients determined by matching the boundary early-time series.

Key Contributions and Results

Validation in DSSYK: The authors demonstrate that the CfZ prescription applied to the semiclassical saddle of DSSYK correctly reproduces the known Krylov complexity results, provided the classical limit is taken only after the complexity is assembled. They show that individual Lanczos coefficients possess expansions where subleading terms are crucial; truncating them leads to incorrect complexity predictions (e.g., failing to reproduce the wormhole length at finite temperature).
BTZ Black Hole Prediction: Applying CfZ to the BTZ saddle (Cardy scaling of a 2D CFT above the Hawking-Page temperature), the authors derive an explicit early-time series for Krylov spread complexity. The series exhibits the universal quadratic growth $C_K(t) \sim t^2$ at early times.
Intermediate and Late-Time Behavior: Unlike standard CV or CAny proposals which predict linear growth ( $C \sim t$ ) at late times, the Padé continuation of the BTZ Krylov series shows an intermediate departure from quadratic growth followed by a return toward quadratic asymptotics ( $C_K(t) \sim t^2$ ) in the semiclassical pre-saturation window ( $1 \ll t/\beta \ll c$ ). This behavior is incompatible with the linear growth of finite-functional CAny observables.
Bulk Dual Construction: The authors construct a candidate bulk observable that matches the boundary Krylov complexity. This observable is defined on maximal-volume slices but weighted by an infinite series of extrinsic-curvature invariants ( $K_{ab}K^{ab}$ $K_{ab} K^{ab}$ ).
- The coefficients of this series are fixed order-by-order by matching the boundary early-time expansion.
- Crucially, the infinite resummation of these coefficients leads to a logarithmic divergence on the final accumulation slice ( $r = r_f$ ).
- This divergence is the mechanism that allows the bulk observable to support quadratic growth. Any finite truncation of the series would revert to the standard linear growth of CAny, but the full infinite series avoids this, sustaining the quadratic behavior. The authors emphasize this is an existence statement and a candidate dual, not a unique holographic dictionary.

Significance and Claims
The paper claims that Krylov spread complexity provides a dimension-independent boundary reconstruction of black hole interior growth that differs fundamentally from standard geometric proposals like CV.

Beyond Linear Growth: The results suggest that the linear late-time growth characteristic of CV and CAny is not a universal feature of all holographic complexity measures. Instead, Krylov spread complexity in the semiclassical limit exhibits quadratic growth, implying a different mechanism of state-space exploration (accelerating mean position along the Krylov chain).
Generalized Bulk Functionals: The work demonstrates that a bulk dual for Krylov complexity can exist but requires relaxing the finiteness condition of the CAny class. The dual is a "generalized complexity = anything" object involving an infinite tower of geometric invariants that diverge on the final slice. The authors view this as an existence proof that semiclassical Krylov complexity admits a natural bulk realization, rather than a unique construction.
Methodological Robustness: The "order-of-limits" rule established in DSSYK is presented as a general principle for semiclassical holography: one cannot simply take the classical limit of the building blocks (Lanczos coefficients) without losing the delicate cancellations required for the correct physical observable.
Scope: The authors position this as an existence proof that semiclassical Krylov complexity admits a natural bulk realization. They do not claim this is the unique holographic dictionary but argue it is a consistent candidate that naturally extends to higher dimensions, where the construction would involve higher-dimensional curvature invariants.

The paper concludes that the discrepancy between Krylov complexity and standard geometric proposals highlights a distinction between finite local geometric densities (linear growth) and resummed observables probing different aspects of state-space (quadratic growth), with the transition to saturation occurring only at finite central charge.

Technical Summary: The Quadratic Growth of Krylov Spread Complexity in the BTZ Black Hole

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