Symmetric Component of Krylov Complexity

• The symmetric component of Krylov complexity is defined as the smooth, averaged behavior of the Lanczos coefficients that drives the exponential growth of operator complexity.
• It dictates a universal exponential increase in operator spreading, following a rate of 2π/β, and separates model-independent trends from microscopic fluctuations.
• This concept links operator dynamics to quantum chaos diagnostics, emphasizing both the strength and limitations of using Krylov complexity to distinguish chaotic behavior.

The symmetric component of Krylov complexity refers to the contribution to operator growth and the resulting complexity trajectory that stems from the smooth, averaged behavior of the Lanczos coefficients, as opposed to model-specific fluctuations or finite-size staggering effects. Krylov complexity is a quantitative probe of operator growth under time evolution, especially relevant in the study of conformal field theory (CFT) and quantum chaos. Its structure, especially the dominance of the symmetric component, encodes essential features of how quantum systems explore their operator space, the relationship to chaos diagnostics, and the universality of late-time growth in a wide class of models.

1. Definition of Krylov Complexity and Operator Growth

Krylov complexity (often denoted $K(t)$ or $\mathcal{K}(t)$) measures the average position of a time-evolved operator in a Krylov basis. Given a simple operator $\mathcal{O}$, the Krylov basis $\{ \mathcal{O}_n \}$ is recursively generated as

$$\mathcal{O}_0 = \mathcal{O}, \qquad \mathcal{O}_{n+1} = [H, \mathcal{O}_n] - \text{orthogonalization},$$

where $H$ is the system Hamiltonian. This recursion produces an orthonormal basis, with orthogonality maintained by positive Lanczos coefficients $\{b_n\}$.

The time-evolved operator is expanded: $$e^{iHt} \mathcal{O} e^{-iHt} = \sum_n \phi_n(t)\, \mathcal{O}_n,$$ and the complexity is

$K(t) = 1 + \sum_n n |\phi_n(t)|^2.$

This construction realizes operator dynamics as a "hopping" process of a fictitious particle on a one-dimensional chain with hopping amplitudes $b_n$; the complexity $K(t)$ is the expected position of this particle.

2. Symmetric Component: Averaged Asymptotic Growth of Lanczos Coefficients

The "symmetric component" refers specifically to the large-$n$ asymptotic behavior of the Lanczos coefficients, which, in all models studied in (Dymarsky et al., 2021), exhibits a linear or smooth envelope independent of model-specific short-range structure or finite-$n$ staggering. Formally, for primary operators in a 2d CFT,

$$b_n^2 = (n+1)(n+2\Delta)\left(\frac{\pi}{\beta}\right)^2,$$

where $\Delta$ is the conformal dimension. At large $n$, this implies

$b_n \approx \left(\frac{\pi}{\beta}\right) n,$

manifesting a universal, model-independent, symmetric profile.

Even in systems where $b_n$ displays staggering (alternating or dimerized sequences for even/odd $n$), the combined (averaged) envelope is smooth and well approximated by the linear behavior. This symmetric component dictates the late-time exponential growth of Krylov complexity, regardless of integrability or microscopic details: $K(t) \sim \exp(\lambda_K t),$ with $\lambda_K$ determined by the slope of the smooth part of $b_n$, mostly $2\pi/\beta$ in CFTs.

3. Relationship to Quantum Chaos and the Chaos Bound

Krylov complexity has been proposed as a probe of quantum chaos, with conjectured connections to out-of-time-ordered correlators (OTOCs) and the Lyapunov exponent. A key result from (Dymarsky et al., 2021) is that in all considered CFT models—whether free, rational, or holographic—the rate of exponential growth for Krylov complexity, $\lambda_K$, saturates the universal Maldacena–Shenker–Stanford (MSS) chaos bound: $\lambda_K = \frac{2\pi}{\beta}.$ This result directly follows from the symmetric component of the Lanczos sequence; exponential complexity growth is dictated by the smooth linear increase of $b_n$ at large $n$. This universality, however, implies that Krylov complexity cannot uniquely identify chaos: exponential growth with this rate is found even in integrable or free models (e.g., rational CFTs), indicating that $\lambda_K$ is a sharp upper bound but not a differentiator of chaos.

4. Examples, Cases, and the Role of Staggering versus the Symmetric Envelope

Model Summary Table

Model/Class	Asymptotic $b_n$	$K(t)$ Growth	Staggering/Envelope
2D CFT (primary)	$b_n^2 = (n+1)(n+2\Delta)(\pi/\beta)^2$	$1 + 2\Delta \sinh^2(\pi t/\beta)$	Linear, no staggering
Free fields ($d=4$)	$b_n \sim (\pi/\beta) n$	Exponential	Linear, no significant staggering
Free fields ($d=5,6$)	$b_n$ exhibits staggering; average $\sim (\pi/\beta) n$	Exponential	Even/odd staggering, symmetric envelope
Holographic CFT	Numerically $b_n \sim (\pi/\beta) n$	Exponential	Linear at large $n$
Composite $\phi^2$	Similar asymptotics as above	Exponential	More complex at small $n$
Free fermions	Similar large $n$ behavior	Exponential	Asymptotic linear envelope

The exponential growth rate is always set by the symmetric, averaged asymptotic envelope of $b_n$, regardless of any initial staggering.

5. Mathematical Structure and the Emergence of Universality

The explicit forms encountered in 2d CFTs illustrate how the symmetric structure arises:

• The two-point thermal correlator dictates the orthogonalization and subsequentially $b_n$.
• The late-time behavior is controlled by the location of the pole at $\tau = \beta/2$; this fixes the slope of $b_n$'s envelope and reproduces the MSS bound.

The discrete Schrödinger equation in Krylov space, solved with these $b_n$, gives

$$K(t) = 1 + 2 \Delta \sinh^2\left( \frac{\pi t}{\beta} \right ),$$

with the exponent set solely by the asymptotically symmetric part of $b_n$.

6. Interpretations, Limitations, and Open Directions

The robustness of the symmetric component's control over late-time dynamics highlights both a key insight and main limitation of Krylov complexity as a chaos diagnostic:

• Exponential growth is universal: Any CFT with a well-defined thermal two-point function structure, free or interacting, exhibits exponential K-complexity growth at the rate $\lambda_K=2\pi/\beta$ set by its symmetric component.
• Not a chaos discriminator: This universality means K-complexity is a measure of the potential for exponential operator growth, not of its uniqueness to chaotic dynamics.

Potential future directions include:

• Studying deviations from universality in strongly non-CFT, massive, or interacting theories.
• Examining whether non-symmetric components (e.g., small-$n$ staggering or fluctuations) can encode finer information about integrability or chaos.
• Connecting these findings to holographic complexity measures and circuit complexity frameworks, particularly in the context of their late-time linear or exponential growth regimes.

7. Summary

In CFT and related models, the symmetric component of Krylov complexity—the smooth, asymptotic envelope of the Lanczos coefficients controlling operator growth—dictates the late-time exponential growth rate of operator complexity. This component is universal, independent of fine structure or integrability, and reproduces the upper bound on chaos (MSS bound) but cannot uniquely identify chaoticity. Thus, the symmetric component is both a source of analytic control and a limitation: it reveals the universal structure of operator spreading in thermal quantum field theory while cautioning against overinterpreting exponential growth as an exclusive signature of chaos.