Krylov Complexity in Lifshitz-type Dirac Field Theories

TL;DR

This work analyzes Krylov complexity for Lifshitz-type Dirac field theories with dynamical exponent $z$, comparing continuum and lattice formulations and assessing the impact of a hard UV cutoff. By computing Lanczos coefficients and Krylov diagnostics (K-entropy, K-variance) for massless and massive fermions, it shows that Lifshitz anisotropy suppresses early-time operator growth, while thermal effects and mass gaps refine the growth pattern; a UV cutoff or lattice discretization induces saturation of complexity, contrasting with unbounded linear growth in the continuum. The results reveal how $z$, temperature, and discretization jointly govern information spreading, with fermionic statistics introducing nonzero diagonal Lanczos coefficients and alternating growth patterns absent in bosonic cases. Overall, Lifshitz scaling modulates operator growth in a way that depends crucially on regularization and temperature, offering insights into scrambling and complexity in non-relativistic QFTs.

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.

Figures (20)

• Figure 1: Lanczos coefficients in the massless regime for various dynamical exponents $z$ and spacetime dimensions $d$.
• Figure 2: Slope of the Lanczos coefficients in $d=90$ for fermionic and bosonic fields, illustrating dependence on $z$.
• Figure 3: Left: Intercept $\gamma$ of $b_n$ vs. $z$ in massless Dirac field. Right: same for massive Dirac field. $\gamma$ decreases rapidly with $z$, suppressing early-time growth.
• Figure 4: Evolution of K-complexity in massless bosonic and Dirac fields for various $z$ values in $d=21$.
• Figure 5: Slope of $\log K(t)$ versus dynamical exponent $z$ in the massless Dirac field at $m = 0$ and $d = 21$, extracted over the time interval $t \in [1.5, 2]$. The blue dashed line indicates the asymptotic limit $\lambda_K = 2\pi$.