The area and volume laws for entanglement of scalar fields in flat and cosmological spacetimes

TL;DR

This work investigates area and volume scaling in entanglement for free scalar fields in both flat (Minkowski) and curved (FLRW) spacetimes, introducing the modular capacity $C$ as a measure of entropy fluctuations alongside the von Neumann entropy $S$. It uses lattice discretization and covariance methods to compute $S$ and $C$ for spherical and strip entangling surfaces, analyzing constant-mass states and abrupt quenches, and then extends to de Sitter and radiation-dominated cosmologies with a focus on non-equilibrium dynamics. Key findings include that both $S$ and $C$ obey area laws in Minkowski for spherical and strip geometries, with a universal-like ratio $C/S$ that depends on mass but not on geometry; quasiparticle models capture early-time dynamics in some cases, while in curved spacetimes a transition to a volume term occurs after quenches or during RD-era evolution. The results illuminate universal features of entanglement fluctuations in quantum field theory, offer guidance for interpreting non-equilibrium processes in cosmology, and bridge lattice methods with cosmological backgrounds and holographic-inspired intuition. $S$ and $C$ thus provide complementary probes of entanglement structure with potential implications for thermalization and holographic perspectives in higher-dimensional QFTs.$

Abstract

We study the area and volume laws for entanglement of free quantum scalar fields. In addition to the entropy, we use the notion of the capacity of entanglement, which measures entropy fluctuations. We consider flat spacetimes as well as the curved ones relevant for cosmology. Moreover, we put special emphasis on quench phenomena and different geometries of the entangling surfaces. First, we show that, in the Minkowski spacetime, the capacity of entanglement, like entropy, exhibits the area law for two kinds of geometries of the entangling surfaces: the sphere and strip. Moreover, we show that the ratio of both quantities takes the same values for both surfaces. Next, we turn our attention to quenches. Namely, we analyse the dynamics of capacity; in particular, contribution of the volume and surface terms. Moreover, we compare these results with theoretical predictions resulting from the quasiparticles model. In the second part, we consider the above issues for the FLRW spaces; especially, for de Sitter space as well as a metric modeling the transition to radiation-dominated era. Finally, we analyse the abrupt quenches in de Sitter space.

Figures (16)

• Figure 1: The $(1+3)$-dimensional Minkowski spacetime - the spherical geometry. The left panel: $m=0$, entropy (blue data points) and capacity (orange data points) with respect to $R^2$. The right panel: the ratio $C/S=a_2/a_1$ with respect to $m$, for $m=0$ it is $5.2$.
• Figure 2: The abrupt quench in the $(1+3)$-dimensional Minkowski space - spherical geometry; $m_i=10$, $m_f=0$, $N=60$. The left panel: entropy - blue $n=15$, green $n=25$, black $n=30$. The right panel: capacity - yellow $n=15$, orange $n=25$, red $n=30$.
• Figure 3: The ratio $b_i/a_i$ for the abrupt quench in the $(1+3)$-dimensional Minkowski space - spherical geometry; $N=60$, $m_f=0$. Blue curve - $b_1/a_1$ for entropy, red curve - $b_2/a_2$ for capacity. The left panel: $m_i=10$. The right panel: $m_i=0.5$.
• Figure 4: The abrupt quench in the $(1+3)$-dimensional Minkowski space - spherical geometry; $N=60$, $m_f=0$. The left panel: $m_i=10$, $t=2$ slice; entropy (blue points) and capacity (red points) with respect to the area, $R^2$. The right panel: $m_i=10$, $t=30$ slice, the capacity with respect to the volume, $R^3$.
• Figure 5: The $(1+2)$-dimensional Minkowski space - the spherical geometry. The left panel: the ratio $C/S=a_2/a_1$ with respect to $m$; for $m=0$ it equals $2.92$. The right panel: the dynamics of the capacity for the abrupt quench, $m_i=10$, $m_f=0$, $N=60$; blue $n=10$, red $n=20$, green $n=30$ (vertical lines correspond to periods: $t=n$). The black vertical line corresponds to $t=N=60$ - a partial revival.