Krylov Complexity in Lifshitz-type Scalar Field Theories

TL;DR

This work analyzes operator growth in Lifshitz-type free scalar field theories by computing Lanczos coefficients in Krylov space at finite temperature, across masses, UV cutoffs, and lattice discretizations with integer dynamical exponent $z$. Using the Wightman inner product and the moments of the power spectrum, the authors show that in the massless continuum, the Lanczos coefficients grow linearly as $b_n \simeq \alpha n + \gamma$ with $\alpha \simeq \pi/\beta$, while the intercept depends on $d$ and $z$. The Krylov complexity $K_{\mathcal{O}}(t)$ grows exponentially with rate $\lambda_K = 2\alpha = 2\pi/\beta$, and the K-entropy increases linearly with the same rate, with mass and UV scales modifying the early-time behavior and reducing late-time growth as $z$ increases. Introducing a hard UV cutoff or lattice discretization yields a linear growth phase in $b_n$ followed by saturation $b_s \approx (\Lambda^z \pm m^z)/2$ and a crossover time $n_s$, with corresponding phases in $K_{\mathcal{O}}(t)$ and $S_{\mathcal{O}}(t)$; overall, the results echo the relativistic case and support a universal operator-growth framework in Lifshitz theories.

Abstract

We investigate various aspects of the Lanczos coefficients in a family of free Lifshitz scalar theories, characterized by their integer dynamical exponent, at finite temperature. In this non-relativistic setup, we examine the effects of mass, finite ultraviolet cutoff, and finite lattice spacing on the behavior of the Lanczos coefficients. We also investigate the effect of the dynamical exponent on the asymptotic behavior of the Lanczos coefficients, which show a universal scaling behavior. We carefully examine how these results can affect different measures in Krylov space, including Krylov complexity and entropy. Remarkably, we find that our results are similar to those previously observed in the literature for relativistic theories.

Figures (12)

• Figure 1: Lanczos coefficients in the massless regime for different values of $z$ and $d$. As we see, although the slope is the same for all $z$, the $y$-intercept depends on $z$. In particular, as one increases $z$, the $y$-intercept decreases and the difference of $y$-intercepts for odd and even $n$ becomes less pronounced.
• Figure 2: Evolution of K-complexity in the massless regime for various values of the dynamical exponent. The complexity decreases as one increases $z$, though for the late time, the slope is the same for all cases, which is given by $\frac{2\pi}{\beta}$.
• Figure 3: K-variance (left) and K-entropy (right) as a function of time in the massless regime for various values of the dynamical exponent.
• Figure 4: Lanczos coefficients in the large mass regime. In this case the $y$-intercept is affected by non-zero mass.
• Figure 5: K-complexity in the large mass limit for different values of $m$ and $z$ with $d=5$. Here we set $\beta=1$. At early times the results are independent of the critical exponent, though at late times it has significant effects.