Krylov complexity of deformed conformal field theories

TL;DR

This paper investigates how integrable deformations like $T{ar{T}}$, $J{ar{T}}$, and $J{ar{J}}$ affect Krylov complexity in 2D conformal field theories, revealing unexpected growth behavior and potential bounds violations under $T{ar{T}}$ deformation.

Contribution

It provides a first-order perturbative analysis of Krylov complexity under these deformations, highlighting deviations from expected growth patterns especially in $T{ar{T}}$ deformed theories.

Findings

$T{ar{T}}$ deformation causes non-linear growth in Lanczos coefficients.

Krylov exponent exceeds that of the undeformed theory for positive deformation parameter.

$J{ar{J}}$ and $J{ar{T}}$ deformations show no first-order correction to complexity.

Abstract

We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of $T{\bar{T}}$, $J{\bar{T}}$, and $J{\bar{J}}$ deformations, focusing on first-order corrections in the deformation parameter. Under $T\bar{T}$ deformation, we demonstrate that the Lanczos coefficients $b_n$ exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both $J{\bar{J}}$ and $J{\bar{T}}$ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large-$n$ or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.