Entanglement Evolution in Lifshitz-type Scalar Theories

TL;DR

This work analyzes entanglement propagation after a global mass quench in free scalar Lifshitz theories with dynamical exponent $z$. By combining a Lifshitz lattice model with an extended Alba–Calabrese quasi-particle framework, it derives the $z$-dependent group velocities, mode occupations, and entropy densities, and validates them against lattice numerics, including a zero-mode analysis. The study reveals three growth regimes for entanglement entropy—initial rapid, main linear, and tortoise saturation—and shows that while a widened light-cone governs early-time propagation, slow tortoise modes prevent exact saturation for $z>1$, though an effective saturation time remains definable. Larger $z$ enhances spatial correlations and accelerates early growth but strengthens slow-mode dominance, challenging straightforward thermalization to a Gibbs ensemble and pointing to an effective causal structure akin to a Lieb-Robinson bound in these non-relativistic QFTs. The results offer benchmarks for non-equilibrium dynamics in Lifshitz systems and implications for holographic comparisons.

Abstract

We study propagation of entanglement after a mass quench in free scalar Lifshitz theories. We show that entanglement entropy goes across three distinct growth regimes before relaxing to a generalized Gibbs ensemble, namely 'initial rapid growth', 'main linear growth' and 'tortoise saturation'. We show that although a wide spectrum of quasi-particles are responsible for entanglement propagation, as long as the occupation number of the zero mode is not divergent, the linear main growth regime is dominated by the fastest quasi-particle propagating on the edges of a widen light-cone. We present strong evidences in support of effective causality and therefore define an effective notion of saturation time in these theories. The larger the dynamical exponent is, the shorter the linear main growth regime becomes. Due to a pile of tortoise modes which become dominant after saturation of fast modes, exact saturation time is postponed to infinity.

Figures (9)

• Figure 1: Numerical data showing entanglement entropy as a function of time in a theory with relativistic scaling. Left: CFT prediction vs. harmonic lattice simulation with periodic BC. The numerical results correspond to $\ell=100$ and $m_0=1,\;m=10^{-6}$. Right: Time derivative of EE during the thermalization process.
• Figure 2: Alba-Calabrese quasi-particle picture for a relativistic free boson and three distinct modes together with the zero mode. The saturation time for each mode is shown with $\{t^*_1,t^*_2,t^*_3\}$.
• Figure 3: Analytic versus numerical results for the evolution of EE in harmonic lattice. In the left panel we have set $m_0=1$ and $m=2$ and in the right panel we have set $m_0=1$ and $m=0$. In the right panel the effect of the zero mode which is not captured by the analytical picture causes small deviation for $t>\ell/2$.
• Figure 4: Mode occupation number and entropy density as a function of $k$ for different values of the dynamical exponent. Here we set $m_0=1$. We have shown two values of post-quench mass to see how fast the $n(k)$ and $s(k)$ diverge for small values of mass. Comparing these two values of $m$ gives a sense of how fast the occupation number (and entropy density) diverge in the massless limit. The same behaviour is correct for fixed post-quench mass and small pre-quench mass.
• Figure 5: Left: $v_g^{\rm max}$ as a function of $z$ for different mass parameters. Right: $v_g^{\rm max}$ as a function of $m$ for different values of the dynamical exponent.