Entanglement Asymmetry and Symmetry Restoration

Entanglement asymmetry measures how far a subsystem state is from commuting with its symmetry charge. Under a symmetry-preserving quench from a symmetry-breaking initial state, it can decay as the subsystem approaches a charge-block-diagonal state. This is subsystem restoration in a stated scaling limit, not proof that the full many-body state has become exactly symmetric.

Required background. Charge-resolved entanglement supplies charge projectors, charged moments, and Fourier resolution.

Helpful background. Information measures along RG flows supplies the distinction between a finite crossover and a theorem.

Dephasing relative to the symmetry

For a compact group, let $\mathcal G_A$ be the subsystem twirl. The von Neumann entanglement asymmetry is

$$\Delta S_A =S(\mathcal G_A\rho_A)-S(\rho_A) =D(\rho_A\Vert\mathcal G_A\rho_A) \ge0.$$

For $U(1)$, $\mathcal G_A$ deletes off-diagonal blocks in the $Q_A$ basis. Hence

$$\Delta S_A=0 \quad\Longleftrightarrow\quad [\rho_A,Q_A]=0$$

for the specified finite-dimensional or suitably regulated state. Rényi asymmetries use ratios of charged moments; their positivity and operational interpretation depend on the definition and should not be inferred from the von Neumann relative-entropy identity.

The dephasing definition, relative-entropy identity, and Rényi construction are Ares, Murciano, and Calabrese 2023, Eqs. (1)–(4).

Entanglement asymmetry compares a subsystem state with its twirled state. Restoration means loss of off-diagonal sector coherence for the declared subsystem and limit, not reconstruction of the full state. Schematic and not to scale.

Symmetric quench from a broken state

Let $[H,Q]=0$ but choose $\rho(0)$ with $[\rho(0),Q]\ne0$. Unitary evolution preserves the amount of global asymmetry under the same group action, because $e^{-itH}$ is covariant. Regional asymmetry can nevertheless move into nonlocal correlations and outside the subsystem. Thus $\Delta S_A(t)$ may decrease.

The limits matter. At fixed finite $A$, one can ask whether

$$\lim_{t\to\infty}\Delta S_A(t)=0.$$

In a ballistic scaling limit one instead holds $t/\ell$ fixed as subsystem size $\ell\to\infty$. The two limits can produce different functions. A finite system has recurrences, and an integrable model can retain symmetry-breaking information in conserved modes.

Restoration criteria

A convincing restoration claim checks more than a falling curve:

1. the Hamiltonian or channel is exactly symmetric in the regulator;
2. $\Delta S_A$ is computed from the same subsystem charge and dephasing map at all times;
3. the late-time value decreases under increasing system size and time window;
4. charge-sector truncation and Fourier errors are controlled; and
5. local symmetry-odd observables and the reduced-state commutator give consistent conclusions.

Decay to a small finite number is not exact restoration. Conversely, $\langle O_{\rm odd}\rangle=0$ for one operator does not imply $[\rho_A,Q_A]=0$.

Free-model quench

In a free fermion or spin-chain realization, start from a tilted product or BCS-like state that breaks $U(1)$ and evolve with a number-conserving quadratic Hamiltonian. The correlation matrix determines $\rho_A(t)$; charged moments of the dephased state determine Rényi asymmetries.

Quasiparticle propagation predicts a subsystem-size-dependent restoration timescale. One can compare several $\ell$ at fixed $t/\ell$, include finite-size recurrences deliberately, and test whether a “faster” restoration for a more strongly broken initial state survives matched energy density and charge distribution. Otherwise the comparison changes several physical controls at once.

Evidence boundary as of 10 August 2026

Entanglement asymmetry is an active diagnostic with analytic and numerical results for spin chains, free fields, conformal settings, and newer gauge or higher-form applications. Restoration rates and Mpemba-like order reversals are model- and protocol-dependent. There is no general theorem that every symmetric QFT dynamics drives every finite subsystem to zero asymmetry, and anomaly, conserved-charge, finite-density, and infrared effects can obstruct the simplest picture.

A 2026 gauge-theory example and its regulator/temperature dependence are given in Florio and Murciano 2026, Eqs. (1)–(8).

Validity map for symmetry restoration. The reduced algebra, twirling operation, regulator, and scaling limit must stay fixed. A finite-size decay or an unresolved charge distribution is evidence, not exact restoration. Schematic and not to scale.

Common pitfalls

Inferring global restoration from a subsystem. Global asymmetry can persist while local coherence spreads outside $A$.

Changing limits silently. Fixed-$A$ late time, ballistic $t/\ell$, and thermodynamic limits are different claims.

Using one odd observable as a complete test. Vanishing of one expectation value does not imply a block-diagonal reduced state.

References

• Ares, Filiberto, Sara Murciano, and Pasquale Calabrese. “Entanglement Asymmetry as a Probe of Symmetry Breaking.” Nature Communications 14 (2023): 2036. DOI.
• Florio, Adrien, and Sara Murciano. “Entanglement Asymmetry in Gauge Theories: Chiral Anomaly in the Finite Temperature Massless Schwinger Model.” Physical Review D 113 (2026): L091901. DOI.