Entanglement and symmetry resolution in two dimensional free quantum field theories

TL;DR

This work analyzes symmetry-resolved entanglement in two-dimensional free quantum field theories with a U(1) symmetry by computing charged moments for Dirac and complex scalar fields using both modified twist fields and Green's-function methods. It derives exact and asymptotic expressions, including Painlevé V equations, for the charged moments and their Fourier transforms to symmetry-resolved entropies, confirming entanglement equipartition at leading order and identifying symmetry-breaking subleading corrections. The results are validated against lattice simulations and extended to massive regimes and arbitrary dimensions for hyperplane entangling surfaces. The study provides a robust framework for understanding how internal symmetries shape entanglement structure in free QFTs and outlines paths toward interacting theories and higher-dimensional generalizations.

Abstract

We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux $α$, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.

Figures (11)

• Figure 1: The universal constant $c_n(\alpha)$ extracted from the numerical solution of the Painlevé equation (\ref{['eq:PainleveFermion']}) for different values of $\alpha$ and $n$ as a function of $t=m\ell$ (full lines). The numerical data are obtained varying $\ell$ between $200$ and $400$ lattice points and properly choosing $m$ in such a way $t=m\ell \in (0,1)$. For larger $\alpha$ and $n$, we need larger subsystem size to have a good match between field theory and lattice calculation because lattice corrections become stronger.
• Figure 2: Leading scaling behaviour of the charged Rényi entropies with the insertion of a flux $\alpha$. The numerical results (symbols) for two different values of $\alpha$ and masses $m$ are reported as functions of $t=m\ell$ when $n=1$. The data match well the prediction in Eq. (\ref{['eq:totalsc']}) (solid lines) which includes lattice corrections as explained in the text.
• Figure 3: Subtracted universal charged entropy $\delta Z(\alpha,t)$ in Eq. \ref{['zeta']}. Left (right) panel is for $n=1$ ($n=2$). The dashed lines are the small $t$ expansion in Eq. (\ref{['eq:totalsc']}) for $n=1$ while the solid lines are the Painlevé exact solution. The tiny discrepancies observed in some cases are finite $\ell$ corrections.
• Figure 4: The probability $\mathcal{Z}_1(q)$. Top: As a function of $t=m\ell$ at fixed $q=0$ for mass $m=0.0005$ (left) and $m=0.001$ (right). The dashed green line is $\mathcal{Z}_1(q)$ obtained by the saddle point approximation, i.e. Eq. (\ref{['eq:SP-FTrans-step3']}). The solid green line is the exact Fourier transform without taking the quadratic approximation. For large $\ell$ (and $t$ as a consequence) the saddle-point approximation converges to the exact value, as expected. Bottom: The same at fixed $t$ as function of $q$.
• Figure 5: Symmetry resolved entanglement entropies for a few different values of $q$ and $n$ as functions of $\ell$. The field theory prediction is tested against exact lattice computations. The agreement with Eq. (\ref{['eq:SP-SRRE-v2Order']}), that includes lattice effects, is remarkable. For large $|q|$, the approximation at the order $q^2$ is no longer sufficient and neglected corrections to the scaling become important, as well known for the massless case riccarda.