Log to log-log crossover of entanglement in $(1+1)-$ dimensional massive scalar field

TL;DR

This paper investigates how entanglement measures in a (1+1)-dimensional massive scalar field exhibit a crossover in their behavior near the zero-mode regime, influenced by boundary conditions and IR cutoff effects.

Contribution

It provides an analytical and numerical study of the entanglement spectrum, entropy, and negativity, revealing a boundary-condition-dependent crossover in their asymptotic behavior.

Findings

Crossover from log to log-log behavior in entanglement measures

Boundary conditions determine the nature of the crossover

Zero-mode effects dominate near the critical point

Abstract

We study three different measures of quantum correlations -- entanglement spectrum, entanglement entropy, and logarithmic negativity -- for (1+1)-dimensional massive scalar field in flat spacetime. The entanglement spectrum for the discretized scalar field in the ground state indicates a cross-over in the zero-mode regime, which is further substantiated by an analytical treatment of both entanglement entropy and logarithmic negativity. The exact nature of this cross-over depends on the boundary conditions used -- the leading order term switches from a $\log$ to $\log-\log$ behavior for the Periodic and Neumann boundary conditions. In contrast, for Dirichlet, it is the parameters within the leading $\log-\log$ term that are switched. We show that this cross-over manifests as a change in the behavior of the leading order divergent term for entanglement entropy and logarithmic negativity close to the zero-mode limit. We thus show that the two regimes have fundamentally different information content. Furthermore, an analysis of the ground state fidelity shows us that the region between critical point $\Lambda=0$ and the crossover point is dominated by zero-mode effects, featuring an explicit dependence on the IR cutoff of the system. For the reduced state of a single oscillator, we show that this cross-over occurs in the region $Nam_f\sim \mathscr{O}(1)$.