Entanglement Entropy and Twist Fields

TL;DR

This work analyzes entanglement entropy via the replica method in 1+1D CFTs, focusing on the role of branch-point twist fields and the validity of multi-interval formulae. It exposes inconsistencies in the conventional Calabrese–Cardy expressions for disjoint intervals by highlighting missing Schwarzian singularities in conformal mappings, and it introduces a practical numerical approach to compute tr ρ_A^n as a vacuum expectation value in an n-fold replicated system. The authors implement the method in 2D Potts models, confirming predicted twist-field correlator behavior for both unitary and non-unitary cases and demonstrating the approach yields direct access to the central charge through scaling. The results reinforce the reliability of single-interval predictions while outlining the limitations of multi-interval analytic formulas and suggesting broad applicability of the numerical technique to higher dimensions and gauge theories.

Abstract

The entanglement entropy of a subsystem of a quantum system is expressed, in the replica approach, through analytic continuation with respect to n of the trace of the n-th power of the reduced density matrix. This trace can be thought of as the vacuum expectation value of a suitable observable in a system made with n independent copies of the original system. We use this property to numerically evaluate it in some two-dimensional critical systems, where it can be compared with the results of Calabrese and Cardy, who wrote the same quantity in terms of correlation functions of twist fields of a conformal field theory. Although the two calculations match perfectly even in finite systems when the analyzed subsystem consists of a single interval, they disagree whenever the subsystem is composed of more than one connected part. The reasons of this disagreement are explained.

Figures (10)

• Figure 1: Three branch points generated by the fusion of two oriented intervals $(u_1,v_1)\,(u_2,v_2)$ in the limit $\vert u_1-u_2\vert\to 0$. A general symmetry of the replica approach described in Section \ref{['usy']} allows to deform the intervals as indicated on the right.
• Figure 2: The branch points in a square lattice are located on the sites of the dual lattice. The links intersecting the cut $\lambda$ connect two consecutive planar lattices labelled by $k$ and $k+1~ ({\rm mod}\, n)$.
• Figure 3: The $k$ labels of the nodes of the stack of $n$ cylinders after the transformation (\ref{['transfid']}).
• Figure 4: A graphical proof of the identity (\ref{['symuv']}).
• Figure 5: A graphical proof of the equality ${\rm tr}\,\rho_A^n={\rm tr}\,\rho_B^n$.