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

This paper investigates whether the relationship between Krylov spread complexity and black hole geometry, previously established in two-dimensional dilaton gravity, persists in higher-dimensional holographic theories. Specifically, the authors seek a boundary quantity that captures the growth of black hole interiors for the Bañados-Teitelboim-Zanelli (BTZ) black hole, which is dual to a two-dimensional conformal field theory (CFT).

The authors employ a partition-function construction (CfZ) to derive Krylov spread complexity. By using the thermofield-double (TFD) state as a reference, they relate Hamiltonian moments to derivatives of the thermal partition function. A critical methodological contribution is the identification of an 'order-of-limits' rule: in semiclassical holography, the classical limit must be taken only after the complexity has been fully reconstructed from the Lanczos coefficients. Applying this to the BTZ black hole, they compute the early-time series of the spread complexity and use Padé approximants to analyze its behavior beyond the initial expansion.

The analysis reveals that the Krylov spread complexity for the BTZ black hole does not exhibit the late-time linear growth predicted by standard complexity-equals-anything (CAny) bulk observables like the maximal volume (CV). Instead, the growth departs from the universal early-time quadratic behavior and then trends back toward quadratic growth at later times. To account for this, the authors propose a generalized bulk observable constructed from an infinite series of extrinsic-curvature invariants. While any finite truncation of this series reverts to linear growth, the fully resummed infinite series becomes singular on the final slice, allowing it to support the observed quadratic asymptotics.

This work provides a systematic, dimension-independent route from black hole thermodynamics to Krylov dynamics. By demonstrating that Krylov spread complexity can be reconstructed from semiclassical partition functions, the authors offer a robust framework for testing holographic complexity in higher-dimensional black holes, challenging the universality of late-time linear growth in existing holographic complexity proposals.