Tracing Through Scalar Entanglement

TL;DR

This work analyzes entanglement entropy for a gapped 1+1D massive scalar field at finite temperature by discretizing to a harmonic chain and studying the covariance-based matrix C^2. It develops parity-based decompositions into even and odd sectors to bound the two largest eigenvalues and relates these to the entanglement spectrum, providing analytic estimates for traces that control the largest eigenvalues. The key result is that in the low-temperature limit (T << m) the entanglement entropy changes as S(T) - S(0) ~ e^{-m/T} and the region-to-region entropy difference scales as S_A - S_B ~ e^{-m/T}, supported by numerical evidence and analytic bounds. This exponential sensitivity links to holographic ideas via the Ryu-Takayanagi proposal and suggests a generic feature of entanglement in gapped, finite-temperature systems with potential 1/N corrections in holographic contexts.

Abstract

As a toy model of a gapped system, we investigate the entanglement entropy of a massive scalar field in 1+1 dimensions at nonzero temperature. In a small mass m and temperature T limit, we put upper and lower bounds on the two largest eigenvalues of the covariance matrix used to compute the entanglement entropy. We argue that the entanglement entropy has exp(-m/T) scaling in the limit m >> T. We comment on the relation between our work and the Ryu-Takayanagi proposal for computing the entanglement entropy holographically.

Figures (6)

• Figure 1: The largest (a) and second largest (b) eigenvalue of $C^2$ plotted against the interval length for $mL=1/10$, $T=0$, and $N=1000$. The points are numerically computed. The curves above the points are the analytic upper bounds (\ref{['evenupper']}) and (\ref{['oddupper']}) computed from the traces. The solid curve below the points on the left is the lower bound (\ref{['lowerbound']}) computed from the variational principle.
• Figure 2: The largest eigenvalue of $C^2$ as a function of temperature for $mL = 1/50$: a) $\ell/L=1/5$; b) $\ell/L=4/5$; c) the difference between the two for a lattice with $N=200$. The points are numerical while the curve is the upper bound computed from $\operatorname{tr} C_e^2$.
• Figure 3: The contribution to $\delta S = S_A - S_B$ for the ten largest pairs of eigenvalues $\lambda_{j,A}$ and $\lambda_{j,B}$, arranged from largest to smallest. In this plot $mL = 0.02$, $m/T = 1$ and $\ell/L = 1/5$ for region $B$. Note that the odd parity eigenvalues do not contribute. ($N=200$ was used for this plot.)
• Figure 4: A log plot of the entanglement entropy $\delta S = S(T) - S(0)$ vs. $m/T$ with an interval size $\ell/L = 3/10$. The points are numerically computed, and the line $\log(\delta S) = -m/T$ is a guide to the eye: a) $m L = 5 \times 10^{-3}$; b) $m L = 5$. (For both plots, the points were computed with $N=50$, 100, 200, and 400. The data points for different values of $N$ all lie roughly on top of each other.)
• Figure 5: a) A log plot of the entanglement entropy difference $\delta S = S_A - S_B$ vs. $m/T$ for $m L = 5$ and $5 \times 10^{-3}$, and an interval B of size $\ell/L= 1/5$. At fixed $m/T$, the larger mass points lie below the smaller ones. The line $\log(\delta S / m L) = -m/T$ is a guide to the eye. (The lattice was taken to have size $N=200$, but there is no noticeable difference between this graph and a graph with $N=100$.) b) The entanglement entropy difference $\delta S$ vs. $\ell/L$ for (from bottom to top) $mL = 5 \times 10^{-3}$, $2$, and $5$. The mass to temperature ratio is $m/T = 10$. The line $e^{m/T} \delta S / m L = 3m / T - 3/2$ is a guide to the eye. (The lattice was taken to have $N=400$, but there is no difference between this graph and a graph with $N=200$.)