Correspondence between Entanglement Growth and Probability Distribution of Quasi-Particles

TL;DR

The paper establishes a concrete link between the time evolution of Rényi entanglement entropy for locally excited states and the probability distribution of emergent quasi-particles in free field theories with d+1 ≥ 4. By combining the replica trick with a quasi-particle propagator picture, it shows that the entire ΔS_A^{(n)} dynamics, including late-time saturation, can be captured by a reduced density matrix that encodes particle-location probabilities. A simple toy model of freely propagating quasi-particles reproduces the replica-calculated excess entropies for several geometries, and a detailed analysis of finite/infinite subsystems and mutual information demonstrates how entanglement and correlations arise from particles’ spatial distributions. The results illuminate the physical meaning of entanglement growth in non-equilibrium settings and offer a tractable framework for predicting entanglement dynamics in free theories and beyond, with clear avenues for generalization and comparison to other quench protocols.

Abstract

We study the excess of (Renyi) entanglement entropy in various free field theories for the locally excited states defined by acting with local operators on the ground state. It is defined by subtracting the entropy for the ground state from the one for the excited state. Here the spacetime dimension is greater than or equal to 4. We find a correspondence between entanglement and a probability. The probability with which a quasi-particle exists in a subregion gives the excess of the entropy. We also propose a toy model which reproduces the excess in the replica method. In this model, a quasi-particle created by a local operator propagates freely and its probability distribution gives the excess.

Figures (13)

• Figure 1: The total Space divided into two subspaces $A$ and $B$ at $t=0$.
• Figure 2: Operator insertion points before taking the analytic continuation in a: $(\tau, x^1)$ and b:$(r, \theta)$.
• Figure 3: Sketch of a: $n$-sheeted Riemann surface, b: $n$-sheeted Riemann surface with operator insertions.
• Figure 4: A quasi-particle picture. At $t=-t$, quasi-particles appear at the point where $\phi$ is located. (a) shows that all of them are included in $B$ in $l \ge t>0$. Therefore entanglement between them can not contribute to $\Delta S^{(n)}_A$. (b) shows that their entanglement can contribute to $\Delta S^{(n)}_A$ in $t>l>0$ because some of them are included in $A$. In the late time, it can be interpreted in terms of entanglement between two quasi-particles.
• Figure 5: The ratios of diagrams. The diagram constructed of $G^{(n)}(\theta_1-\theta_2)$ is called $D^{(n)}_1$. The diagram constructed of $G^{(n)}(\theta_1-\theta_2+2\pi)$ and $G^{(n)}(\theta_1-\theta_2-2(n-1)\pi)$ is $D^{(n)}_2$. The one constructed of $G^{(1)}(\theta_1-\theta_2)$ is $D^{(1)}_1$. (a) is the ratio of $D^{(n)}_1$ to $(D^{(1)}_1)^n$. (b) is the ratio of $D^{(n)}_2$ to $(D~{(1)}_1)^n$.