Symmetry-Resolved Entanglement Entropy

• Symmetry-resolved entanglement entropy is a refinement of bipartite entanglement that decomposes the reduced density matrix into distinct symmetry sectors based on conserved or non-abelian charges.
• It employs charged moments, Fourier transforms, and conformal mapping techniques to compute sector-specific Rényi and von Neumann entropies with controlled corrections.
• It has broad applications in quantum information, conformal field theory, and topological phases, revealing nuanced entanglement equipartition and finite-size or dynamical corrections.

Searching arXiv for recent and foundational papers on symmetry-resolved entanglement entropy. Using the available arXiv search capability to verify relevant literature and retrieve supporting papers. Symmetry-resolved entanglement entropy (SREE) is the refinement of bipartite entanglement by decomposing a reduced density matrix into sectors labeled by the symmetry carried by the subsystem. For a conserved additive charge, this amounts to resolving entanglement at fixed subsystem charge; for a non-abelian symmetry, it requires resolving into irreducible representations or isotypical components of an invariant observable algebra. The subject sits at the intersection of quantum information, conformal field theory, integrability, disorder physics, and representation theory, and it has developed into a framework for probing both the fine structure of entanglement spectra and the dynamical redistribution of entanglement among symmetry sectors (Fraenkel et al., 2019, Bianchi et al., 2024).

1. Basic definitions and entropy decomposition

For a bipartite pure state with reduced density matrix $\rho_A$ and a global $U(1)$ charge $Q=Q_A+Q_B$, symmetry resolution starts from the commutativity relation $[\rho_A,Q_A]=0$. The reduced density matrix is therefore block-diagonal,

$\rho_A=\bigoplus_q p(q)\,\rho_A(q),$

where $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$ is the probability of finding subsystem charge $q$, $\Pi_q$ is the projector onto the $Q_A=q$ sector, and $\rho_A(q)$ is the normalized block. The symmetry-resolved von Neumann entropy is

$U(1)$0

and the total entropy splits as

$U(1)$1

The last term is the Shannon entropy of the charge distribution and is commonly called the number entropy, while the first term is the configurational entropy (Bianchi et al., 2024, Chen et al., 2023).

The same structure extends to Rényi entropies. If

$U(1)$2

then the sector label $U(1)$3 resolves the usual Rényi entropy into fixed-charge contributions. This formulation is experimentally relevant because the probabilities $U(1)$4 are the full counting statistics of the subsystem charge, while the entropies $U(1)$5 quantify the internal entanglement remaining after conditioning on that charge sector (Arildsen et al., 7 Aug 2025, Yan et al., 2024).

In non-abelian settings, the analogous construction is subtler because the notion of subsystem must be compatible with the symmetry. For compact semisimple $U(1)$6, one defines subsystems operationally in terms of subalgebras of invariant observables, resolves the reduced state into isotypical sectors $U(1)$7, and defines

$U(1)$8

The corresponding $U(1)$9-local entropy again separates into an average of fixed-sector entropies plus a Shannon-type term for the sector weights (Bianchi et al., 2024).

2. Charged moments and computational formalisms

The central generating object of SREE is the charged moment

$Q=Q_A+Q_B$0

Its Fourier transform projects onto fixed-charge sectors,

$Q=Q_A+Q_B$1

and the symmetry-resolved Rényi entropy follows as

$Q=Q_A+Q_B$2

This charged-moment route underlies the replica construction in continuum QFT, free-fermion Toeplitz analyses, Gaussian correlation-matrix methods, and many numerical implementations (Fraenkel et al., 2019, Jones, 2022).

In $Q=Q_A+Q_B$3-dimensional CFT, one powerful reformulation is the boundary conformal field theory approach. For a single interval of length $Q=Q_A+Q_B$4, one excises disks of radius $Q=Q_A+Q_B$5 around the entangling points, imposes conformal boundary conditions, and conformally maps the punctured plane to an annulus of width

$Q=Q_A+Q_B$6

If the boundary conditions preserve the symmetry algebra to be resolved, the annulus partition function decomposes as

$Q=Q_A+Q_B$7

with $Q=Q_A+Q_B$8 the characters of irreducible representations of the preserved algebra. In this formulation, the character decomposition itself already resolves the entanglement spectrum, and symmetry-resolved Rényi entropies can be extracted without first computing charged moments (Giulio et al., 2022).

Other regimes admit distinct exact formalisms. In critical free-fermion chains, charged moments become Toeplitz determinants with Fisher-Hartwig singularities, permitting rigorous asymptotic expansions. In thermodynamic integrable models, the relevant objects are large-deviation functions derived from the thermodynamic Bethe ansatz. For spherical regions in higher-dimensional CFTs, the Casini-Huerta-Myers map turns the problem into a grand-canonical partition function on $Q=Q_A+Q_B$9 with imaginary chemical potential (Jones, 2022, Piroli et al., 2022, Huang et al., 12 Mar 2025).

3. Critical one-dimensional systems and the equipartition phenomenon

A recurring result in critical one-dimensional systems is entanglement equipartition: to leading order, the entropy in each symmetry sector coincides with the unresolved entropy. In CFT language, the charged moment is governed by a fluxed twist operator or its equivalent, and its $[\rho_A,Q_A]=0$0-dependence is Gaussian near $[\rho_A,Q_A]=0$1. In lattice free-fermion chains with $[\rho_A,Q_A]=0$2 massless Dirac fermions in the infrared, the rigorous large-$[\rho_A,Q_A]=0$3 asymptotics give

$[\rho_A,Q_A]=0$4

so that the leading term of $[\rho_A,Q_A]=0$5 is independent of the charge sector and the first sector dependence enters at order $[\rho_A,Q_A]=0$6 (Jones, 2022).

The scaling-limit statement is not, however, the whole story. Finite-size corrections depend strongly on the symmetry group. For discrete $[\rho_A,Q_A]=0$7 symmetries, deviations from equipartition decay algebraically with system size, with exponents fixed by operator scaling dimensions. For continuous $[\rho_A,Q_A]=0$8 symmetry, the corrections are only logarithmically small, of order $[\rho_A,Q_A]=0$9, and their coefficients are controlled by twisted overlaps or, equivalently, by the smooth $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$0-dependence of nonuniversal normalization factors (Estienne et al., 2020).

The BCFT approach sharpens this picture. If the factorization map associated with the entangling cut is compatible with the symmetry algebra, then the symmetry-resolved entanglement spectrum is obtained directly from the annulus character expansion. For the free massless boson with preserved $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$1, $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$2, or certain $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$3 structures, one finds sector-projected partition functions

$\rho_A=\bigoplus_q p(q)\,\rho_A(q),$4

and because $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$5, the conformal weight $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$6 cancels in

$\rho_A=\bigoplus_q p(q)\,\rho_A(q),$7

This yields exact equipartition at all orders in the UV cutoff expansion for the appropriate sectors, with

$\rho_A=\bigoplus_q p(q)\,\rho_A(q),$8

By contrast, for the $\rho_A=\bigoplus_q p(q)\,\rho_A(q),$9 resolution of the boson, equipartition holds only at leading order in $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$0 and is broken by $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$1 terms, producing a universal $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$2 fluctuation contribution (Giulio et al., 2022).

Long-range criticality modifies the breaking pattern further. In the dimerized long-range free-fermion chain, equipartition still holds at leading and double-logarithmic order, but the first charge dependence appears already at order $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$3, unlike the short-range critical case where the first nontrivial sector dependence arises only at order $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$4. The coefficient of this earlier breaking depends on the long-range hopping amplitudes (Ares et al., 2022).

4. Non-abelian symmetries, finite groups, and representation theory

For non-abelian symmetry groups, symmetry resolution cannot be reduced to simple additive charge conservation. The operational framework based on invariant observable algebras defines a $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$5-local subsystem $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$6 by a von Neumann subalgebra $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$7, where $p(q)=\mathrm{Tr}[\Pi_q\rho_A]$8 is the commutant of the symmetry generators. The Hilbert space then decomposes as

$p(q)=\mathrm{Tr}[\Pi_q\rho_A]$9

and sector projectors $q$0 isolate the isotypical component labeled by the subsystem irrep $q$1. In this setting, typical entropies of random pure states at fixed global irrep admit exact formulas for their averages and variances, and the resulting SU(2) Page curves are generically asymmetric under subsystem exchange $q$2, becoming symmetric only for global spin $q$3 (Bianchi et al., 2024).

Wess-Zumino-Witten models provide the controlled conformal realization of non-abelian SREE. For compact simple $q$4 at level $q$5, the leading term of the sector-resolved Rényi entropy is again the standard CFT logarithm,

$q$6

and the entanglement is equally distributed among irreducible sectors at this order. Equipartition is then broken by a representation-dependent constant $q$7, while a universal correction

$q$8

appears in every sector. For $q$9, this becomes the explicit term $\Pi_q$0 together with $\Pi_q$1 (Calabrese et al., 2021).

Boundary-CFT and orbifold methods extend the representation-theoretic viewpoint to arbitrary finite groups and compact Lie groups. For a finite group $\Pi_q$2, projection onto irrep $\Pi_q$3 gives

$\Pi_q$4

and in the ultraviolet limit the universal sector splitting is

$\Pi_q$5

The same framework resolves the entanglement spectrum: for finite groups,

$\Pi_q$6

while for continuous groups the spectrum acquires a Casimir-dependent damping. The orbifold formulation also makes explicit that anomalous symmetries obstruct symmetry resolution in the continuum limit, because one cannot construct a symmetric Cardy boundary state or a well-defined orbifold theory (Kusuki et al., 2023).

Recent numerical work on the bosonic Moore-Read state shows that approximate equipartition persists in a non-abelian fractional quantum Hall phase when one resolves both $\Pi_q$7 charge and fermion parity. In that setting, finite-size splittings are controlled by the distinct neutral and charged mode velocities, and the low-lying entanglement spectrum follows the Li-Haldane structure with sector-dependent corrections (Arildsen et al., 7 Aug 2025).

5. Dynamics, quenches, and thermodynamic macrostates

Symmetry resolution is not limited to ground states. In interacting integrable systems with a conserved $\Pi_q$8 charge, the thermodynamic Bethe ansatz combined with the Gärtner-Ellis theorem yields a large-deviation description of sector weights. If $\Pi_q$9 is the rate function for the charge density in the $Q_A=q$0-Rényi ensemble, then the extensive symmetry-resolved entropy density is

$Q_A=q$1

Its von Neumann limit simplifies dramatically:

$Q_A=q$2

namely, the sector-resolved entropy equals the Yang-Yang entropy density of an effective macrostate with shifted chemical potential chosen to enforce charge density $Q_A=q$3 (Piroli et al., 2022).

Zero-density excited states show a different universality. For a finite number of excitations above the vacuum, the ratio of charged moments to the ground-state charged moments depends only on the number, statistics, charges of the excitations, and the subsystem fraction $Q_A=q$4. For a single excitation of charge $Q_A=q$5,

$Q_A=q$6

and multiparticle states factorize or obey the multinomial generalization. This universality survives in interacting magnon states and extends, via branch-point twist-field arguments, to higher dimensions (2206.12223).

Real-time dynamics often preserves leading-order equipartition while generating informative subleading sector dependence. After an inhomogeneous quench in a critical free-fermion system, the total entropy grows as $Q_A=q$7 at long times, and the symmetry-resolved entropy obeys the same leading behavior; the first correction is a sector-dependent term decaying as $Q_A=q$8 (Chen et al., 20 Apr 2025). In bulk-driven and non-unitary compact-boson CFT, the resolved entropies are controlled by a width parameter $Q_A=q$9: when the driven scale grows without bound, equipartition is asymptotically restored, whereas finite $\rho_A(q)$0 produces visible $\rho_A(q)$1-dependence (Ares et al., 30 Mar 2026).

Disordered bosonic dynamics reveals phenomena not visible in the unresolved entropy. In the disordered Bose-Hubbard chain, weak-disorder low-energy quenches in the ETH regime show long-term entropy depletion in some channel-resolved number-entropy components even though the total entropy increases. At strong disorder, the total number entropy grows as $\rho_A(q)$2, but the channel analysis indicates that this ultraslow growth is dominated by reorganization within already occupied sectors rather than long-range transport. The same study identifies an “entanglement channel wave” and a high-energy cluster-MBL regime in which both total and sector-resolved entropies remain bounded at all times (Chen et al., 2023).

6. Higher dimensions, topological phases, and conceptual limits

For spherical regions in higher-dimensional CFTs with $\rho_A(q)$3 symmetry, the Casini-Huerta-Myers map converts the reduced density matrix into a thermal state on hyperbolic space. The charged moments become grand-canonical partition functions with imaginary chemical potential, and Laplace asymptotics give a universal large-volume expansion

$\rho_A(q)$4

with the same structure for the von Neumann entropy. All terms up to constant order are independent of $\rho_A(q)$5, so equipartition holds through $\rho_A(q)$6, and the first sector dependence is quadratic in $\rho_A(q)$7 at order $\rho_A(q)$8 when $\rho_A(q)$9 is fixed as $U(1)$00 (Huang et al., 12 Mar 2025).

Closely related behavior appears in de Sitter space. For the Bunch-Davies vacuum of free complex scalar and Dirac fields on hyperbolic de Sitter slices, the charged moments are Gaussian in the flux variable at large hyperbolic volume $U(1)$01, and the symmetry-resolved entropy is equipartitioned among charge sectors up to terms of order $U(1)$02, with the first explicit $U(1)$03-dependence suppressed by $U(1)$04 (Gaur et al., 2022). For gapless free Fermi gases in arbitrary dimension, the Widom formula implies that the leading flux dependence remains Gaussian with variance of order $U(1)$05, again producing leading equipartition after Fourier transform (Fraenkel et al., 2019).

Topological phases supply additional mechanisms. In the Kitaev chain, the topological phase exhibits exact parity-sector equipartition in the large-interval limit because virtual Majorana zero modes at the entanglement cut pair the even and odd entanglement spectra. In Gaussian symmetry-protected phases, majorization methods give rigorous lower bounds on total, number, and configurational entropies from protected mid-gap levels in the correlation spectrum (Fraenkel et al., 2019, Monkman et al., 2023).

Several common simplifications are therefore inaccurate. Equipartition is not a universal exact law; it may be exact to all orders in specially compatible BCFT constructions, only asymptotic in scaling limits, or broken by representation-dependent, finite-size, or dynamical corrections (Giulio et al., 2022, Estienne et al., 2020). Symmetry resolution is also not guaranteed for every formal symmetry: anomalous symmetries obstruct the continuum construction, and even non-anomalous symmetries require factorization maps or boundary conditions that preserve the algebra one wishes to resolve (Kusuki et al., 2023, Giulio et al., 2022). These constraints make SREE a probe not only of entanglement, but also of the compatibility between subsystem definitions, global symmetries, and the operator algebra of the theory.