Scaling of entanglement in $2+1$-dimensional scale-invariant field theories

TL;DR

This paper investigates the universal entanglement entropy scaling in 2+1D critical fermionic theories, comparing numerical results with theoretical models including conformal field theory, Lifshitz, and holography, revealing high-precision agreement.

Contribution

It provides a detailed numerical study of entanglement entropy in fermionic models and tests various theoretical scaling functions, including holographic predictions, for the subleading term.

Findings

Entanglement entropy follows the area law with a scaling function of aspect ratios.

The subleading term matches Lifshitz and holographic models across a range of aspect ratios.

High-precision agreement between numerical data and theoretical scaling functions was achieved.

Abstract

We study the universal scaling behavior of the entanglement entropy of critical theories in $2+1$ dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision.