Charge-Resolved Entanglement and Charged Moments

Charge-resolved entanglement conditions a symmetric reduced state on a subsystem charge. Charged moments provide a generating function, Fourier inversion gives sector moments, and normalization by the sector probability gives Rényi entropies within each sector. The number fluctuation $H(p_q)$ and the within-sector entanglement are different terms.

Required background. Symmetry-constrained operations fixes the charge action and invariant algebra.

Helpful background. Rényi analytic continuation supplies the replica and continuation cautions.

Charged and projected moments

Assume $[\rho_A,Q_A]=0$ and integer $U(1)$ charges. Define

$$Z_n(\alpha) =\operatorname{Tr}\!\left(\rho_A^n e^{i\alpha Q_A}\right), \qquad \alpha\sim\alpha+2\pi.$$

Fourier inversion gives

$$Z_n(q) =\operatorname{Tr}(\Pi_q\rho_A^n) =\int_{-\pi}^{\pi}\frac{d\alpha}{2\pi}, e^{-i\alpha q}Z_n(\alpha).$$

At $n=1$, $p_q=Z_1(q)$. The normalized sector density matrix is $\rho_{A,q}=\Pi_q\rho_A\Pi_q/p_q$, and its Rényi entropy is

$$S_n(q) =\frac{1}{1-n} \log\frac{Z_n(q)}{p_q^n}.$$

Omitting the factor $p_q^n$ confuses an unnormalized sector moment with an entropy.

The charged-moment construction, flux insertion, and Fourier projection are Goldstein and Sela 2018, Eqs. (1)–(6).

Charged moments are the Fourier-resolved branch of the sector decomposition. They determine both sector probabilities and conditional Rényi entropies, which must be normalized separately. Schematic and not to scale.

Replica interpretation

In a path integral, $Z_n(\alpha)$ inserts a symmetry flux or defect through the replica geometry. The insertion is topological only under the corresponding symmetry and away from regulator-sensitive contacts. Its normalization depends on whether $Z_n$ is divided by $(\operatorname{Tr}\rho_A)^n$, and on the periodicity convention for $\alpha$.

The von Neumann sector entropy requires $n\to1$. Values at positive integers do not uniquely determine an analytic function without growth and regularity assumptions. Fourier inversion and continuation also need not commute in an uncontrolled saddle approximation. A safe calculation checks sector normalization at $n=1$ and reproduces the total entropy identity

$$S(\rho_A)=H(p)+\sum_qp_qS(q).$$

Equipartition and its limits

In many 1+1-dimensional critical systems, the leading large-interval term in $S_n(q)$ is approximately independent of $q$ over the central charge window—entanglement equipartition. Subleading logarithms, finite size, interactions, boundaries, and large-deviation sectors break equipartition. It is an asymptotic statement, not an exact symmetry theorem.

The sector window itself scales with the charge variance. A fixed-$q$ expansion near the distribution center does not control tails with $q-\bar q$ of order the interval size. Numerical Fourier transforms need enough $\alpha$ samples to resolve the desired tail without aliasing.

Free-fermion interval

For a number-conserving free fermion, the restricted correlation matrix $C_A$ gives

$$Z_n(\alpha) =\det\!\left[(1-C_A)^n+e^{i\alpha}C_A^n\right].$$

This determinant provides an exact finite-lattice benchmark. Discrete Fourier transform yields $Z_n(q)$, and the checks are periodicity in $\alpha$, positivity of $p_q$, $\sum_qp_q=1$, and reconstruction of $\operatorname{Tr}\rho_A^n=Z_n(0)$.

The continuum free-Dirac and complex-scalar charged moments, including their asymptotic sector entropies, are derived in Murciano, Di Giulio, and Calabrese 2020, §§ 3–5.

The continuum limit holds the interval length and filling fixed while the lattice spacing decreases. Charge fluctuations contain regulator- and boundary-sensitive terms, so an apparent equipartition plateau should be tested across interval sizes and Fourier grids.

Validity map

Validity map for charge resolution. The subsystem charge, regulator, periodicity, Fourier resolution, and operation class must all be fixed. Unresolved tails or uncontrolled replica continuation do not license exact sector entropies. Schematic and not to scale.

Common pitfalls

Using number variance as sector entanglement. Variance and $H(p_q)$ describe the sector distribution; $S(q)$ describes the conditional state.

Forgetting sector normalization. $Z_n(q)$ is unnormalized. Divide by $p_q^n$ before taking the Rényi logarithm.

Calling equipartition exact. Leading large-size equality can be broken by subleading terms and large-deviation sectors.

References

• Goldstein, Moshe, and Eran Sela. “Symmetry-Resolved Entanglement in Many-Body Systems.” Physical Review Letters 120 (2018): 200602. DOI.
• Murciano, Sara, Giuseppe Di Giulio, and Pasquale Calabrese. “Entanglement and Symmetry Resolution in Two Dimensional Free Quantum Field Theories.” Journal of High Energy Physics 2020, no. 8 (2020): 073. DOI.