Symmetry-resolved Entanglement Entropy, Spectra & Boundary Conformal Field Theory

TL;DR

This work develops a comprehensive BCFT framework to study symmetry-resolved entanglement entropy (SREE) in 1+1D CFTs with finite or compact Lie group symmetries. It presents two complementary computational routes: (i) charged moments via defect lines and boundary-state actions, and (ii) an orbifold CFT approach that bypasses charged moments and connects SREE to orbifold partition functions, with explicit treatment of anomalies. The authors prove universal leading SR EE behavior, derive the first subleading corrections that depend on irrep data (e.g., for finite groups, a universal $\log(d_r^2/|G|)$ shift; for compact Lie groups, a $2\log d_r$ term and Casimir-related suppressions), and provide symmetry-resolved entanglement spectra with rigorous Tauberian bounds. They illustrate the theory through concrete examples (Ising $\mathbb{Z}_2$, $\mathbb{Z}_k$ in $u(1)_k$, and $\mathbb{Z}_2$ in $su(2)_{2k}$), and perform numerical checks in the abelian case, establishing consistency between BCFT predictions and lattice results. The work also clarifies how anomalies obstruct symmetry resolution and extends the analysis to continuous groups, enhancing understanding of entanglement structure in symmetry-laden QFTs and offering tools for higher-dimensional and non-invertible-symmetry contexts.

Abstract

We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a $1+1$D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, $G$. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for $U(1)$ resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.

Figures (3)

• Figure 1: Density of the (non-normalized) entanglement spectrum $n(\lambda, Q)$ for a subsystem of size $\ell=1000$ and $M=24$. The polygonal lines are the numerical points while the solid continue line is eq. \ref{['eq:nlambda']} in a given charge sector $Q$, and the total one is $n(\lambda)=I_0(x)$, where $x=2\sqrt{b\log(\lambda_{0}/\lambda)}$. This plot shows the matching between our result and the one in Ref. Goldstein2018 (where the authors used $\ell=10000$). Their prediction corresponds to the dashed lines, which completely overlaps with eq. \ref{['eq:nlambda']} as $x$ increases.
• Figure 2: Numerical check of eq. \ref{['EEspecu1']} for a free fermionic system with conserved particle number. The geometry is an interval of size $\ell$ on the infinite line. The symbols are the numerical points obtained using the procedure described in Section \ref{['app:spectrum']} to evaluate the (normalized) entanglement spectrum in each charge sector $Q$. The solid red lines correspond to $(2\delta \pm 1) \pi ^2 e^{x-\frac{\pi ^2Q^2}{6 x}} \sqrt{\frac{1}{108 x^4}}$ with $\delta=0.75$ and $x=2\sqrt{b\log(\lambda_{0,Q}/\lambda)}$, while the blue ones correspond to eq. \ref{['eq:f1pm']} with $c=1$.
• Figure 3: Same numerical check as in Fig. \ref{['fig:spectrum']} but now the entanglement spectrum in each charge sector is not normalized (i.e. $\sum_i\lambda_{i,Q}<1$) and the scaling variable is $x=2\sqrt{b\log(\lambda_0/\lambda)}$, as done in Ref. Goldstein2018.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Remark 5.1
• Theorem 5.2: Tauberian theorem for EE spectra:finite group
• Theorem 5.3: Tauberian theorem for EE spectra: $U(1)$
• Remark 5.4
• Remark 5.5
• Theorem 5.6: Tauberian Theorem in BCFT: finite group
• Remark 5.7
• Theorem 5.8: Tauberian Theorem in BCFT: $U(1)$