Krylov Complexity in Lifshitz-type Dirac Field Theories

TL;DR

This paper investigates how Krylov complexity behaves in Lifshitz-type Dirac theories, revealing the effects of UV cutoffs, lattice discretization, and the Lifshitz exponent on operator growth and information spreading.

Contribution

It provides the first detailed analysis of Krylov complexity in Lifshitz scaling theories, including continuum and lattice models, highlighting the influence of UV cutoffs and the Lifshitz exponent.

Findings

Krylov complexity shows initial exponential growth then linear saturation in continuum models.

Lattice models cause Krylov complexity to saturate due to finite basis size.

Increasing the Lifshitz exponent $z$ suppresses complexity and entropy in continuum models, but enhances them under UV cutoff.

Abstract

We study Krylov complexity in Lifshitz-type Dirac field theories with a generic dynamical critical exponent $z$. By computing the Lanczos coefficients for massless and massive cases, we analyze the growth and saturation behavior of Krylov complexity in different regimes. We incorporate a hard UV cutoff and investigate the effects of lattice discretization, revealing fundamental differences between continuum and lattice models. In the presence of a UV cutoff, Krylov complexity exhibits an initial exponential growth followed by a linear regime, with saturation values of the Lanczos coefficients dictated by the cutoff scale. For the lattice model, we find a fundamental departure from the continuum case: due to the finite Krylov basis, Krylov complexity saturates rather than growing indefinitely. Our findings suggest that Lifshitz scaling influences operator growth and information spreading in quantum systems. We further find that increasing the Lifshitz exponent $z$ suppresses Krylov complexity, entropy, and Lanczos growth in both massless and massive cases, while enhancing K-variance. This trend reverses under a hard UV cutoff, where complexity and entropy increase with $z$. In lattice models, early-time complexity and $b_n$ decay shift with $z$, echoing the continuum behavior of massive and massless regimes.