Entanglement Asymmetry in Non-Abelian Systems

• Entanglement asymmetry is a measure that quantifies how non-Abelian SU(N) symmetry is broken in a subsystem by comparing the reduced density matrix with its symmetrized counterpart.
• The WZW model framework uses Rényi entanglement asymmetry and four-point function evaluations to probe real-time symmetry restoration and to uncover phenomena like the quantum Mpemba effect.
• Analytic techniques based on KZ equations and conformal blocks yield detailed insights into initial asymmetry plateaus and power-law decay, linking dynamics to model parameters such as N and k.

Entanglement asymmetry quantifies the extent to which a global symmetry—here, non-Abelian SU$(N)$—is broken within a subsystem of an extended quantum system, beyond what can be captured by local order parameters. In the context of the $\widehat{su}(N)_k$ Wess–Zumino–Witten (WZW) conformal field theory, the real-time dynamics of symmetry restoration after explicit symmetry breaking are probed using the time evolution of the Rényi entanglement asymmetry. This framework provides a natural generalization of symmetry-resolved entanglement to non-Abelian groups, and directly exposes non-classical features like the quantum Mpemba effect, where states with greater initial symmetry breaking can exhibit faster subsequent symmetry restoration (Fujimura et al., 6 Sep 2025).

1. Definition of Non-Abelian Rényi Entanglement Asymmetry

For a subsystem $A$ with reduced density matrix $\rho_A$ in a system governed by a symmetry group $G$ (here $G = \mathrm{SU}(N)$), the $n$-th Rényi entanglement asymmetry is defined as

$$\Delta S_A^{(n)} = \frac{1}{1-n} \left( \log\operatorname{Tr}[\rho_{A,G}^{n}] - \log\operatorname{Tr}[\rho_A^{n}] \right)$$

where $\rho_{A,G}$ is the symmetrized reduced density matrix,

$$\rho_{A,G} = \int_G dg\, U_A(g)\, \rho_A\, U_A(g)^\dagger$$

for compact $G$, or a sum for finite $G$. For $\mathrm{SU}(N)$, $U_A(g)$ implements the group action within $A$ (using the appropriate representation). $\Delta S_A^{(n)}$ measures the extra entropy generated by projecting $\rho_A$ into the $G$-invariant sector, thus quantifying the degree to which $\rho_A$ fails to respect the symmetry. For a fully $G$-invariant state, $\Delta S_A^{(n)}=0$.

In practice, particularly for conformal field theories with non-Abelian symmetry, $\operatorname{Tr}[\rho_{A,G}^{n}]$ can be represented as a multivalued partition function or, specifically in the WZW context, as a four-point function involving symmetry-breaking insertions.

2. WZW Model Setup and Symmetry-Breaking Initial States

The $\widehat{su}(N)_k$ WZW model is a $(1+1)$-dimensional rational conformal field theory with an exact global $\mathrm{SU}(N)$ symmetry at level $k$. The spontaneous breaking of continuous symmetries is forbidden by the Coleman–Mermin–Wagner theorem; thus, to observe symmetry restoration dynamics, explicit symmetry-breaking must be engineered via the initial state.

Two classes of initial states are considered:

• Fundamental Primary Operator Insertion: The system is prepared by an insertion of a primary field in the fundamental representation of $\mathrm{SU}(N)$ at time $-i\tau_0$, generating an excited state that explicitly breaks the symmetry. The real-time evolution of entanglement asymmetry after such a quench is directly tractable via conformal blocks.
• Adjoint Representation (Current Insertion): The initial state is prepared using insertions of the conserved $\mathrm{SU}(N)$ currents, which transform in the adjoint representation. This provides a comparison channel to test universality and representation dependence in symmetry restoration.

Both cases permit analytic access to four-point functions entering the calculation of $\operatorname{Tr}[\rho_{A,G}^{n}]$ via the Knizhnik–Zamolodchikov (KZ) equations, with explicit dependence on system parameters $N$ and $k$.

3. Analytic Structure: Four-Point Function Evaluation

For the fundamental primary case, the explicit formula for the second Rényi entanglement asymmetry is

$$\Delta S_A^{(2)}(t) = -\log \left[ \frac{ 2G_{1,1}(t) + (N+1)(G_{1,2}(t) + G_{2,1}(t)) + N(N+1)G_{2,2}(t) }{ N(N+1)(G_{1,1}(t) + G_{1,2}(t) + G_{2,1}(t) + G_{2,2}(t)) } \right]$$

where $G_{A,B}(t)$ are blocks composed of conformal blocks $F_{A}^{(\pm)}$ and their antiholomorphic counterparts, satisfying the WZW KZ equations. The key time-dependent cross-ratios controlling the evolution are set by the physical geometry (e.g., $z = f(t, \ell, \tau_0)$).

In the large-interval ($\tau_0\to 0$) regime, the asymmetry exhibits a step-function profile with plateaus at

$$\log \left[ \frac{N(N+1)}{2} \right] \quad \text{or} \quad \log N$$

depending on the insertion point, before decaying at long times.

For the adjoint (current) insertion, a parallel form involving the corresponding current-current four-point functions is constructed.

4. Dynamical Restoration and Quantum Mpemba Effect

A central result is that for explicit symmetry breaking induced by fundamental primaries, the restoration of the $\mathrm{SU}(N)$ symmetry—i.e., the decay of $\Delta S_A^{(2)}(t) \to 0$ at long times—shows a marked quantum Mpemba effect:

• Mpemba Effect for Non-Abelian Symmetry: States with larger initial symmetry breaking (i.e., larger initial $\Delta S_A^{(2)}$) relax more rapidly to the symmetric steady state. For fixed $k$, increasing $N$ (which enhances the initial plateau $\propto \log N$) both increases the initial asymmetry and sharpens the subsequent decay.
• Dependence on WZW Level and Representation: By contrast, at fixed $N$, increasing $k$ decreases the initial breaking and makes restoration slower; higher $k$ leads to more gradual decay, as seen in the explicit time-evolution curves.
• Non-universality in the Adjoint Case: This pronounced Mpemba effect is absent when symmetry breaking arises from adjoint operators; varying $N$ or $k$ changes the initial and decay profiles, but the crossing or overtaking necessary for the effect is not observed.

The occurrence (or absence) of this effect is linked to the detailed representation structure of the symmetry-breaking operators and the fusion properties of WZW primaries.

5. Asymptotic and Scaling Behavior

Both initial plateau values and late-time decay rates can be extracted from the four-point functions:

• In the regime $\tau_0\to 0$, the initial asymmetry saturates analytical values, providing effective "symmetry-broken degrees of freedom" per fundamental representation.
• At long times ($t\to\infty$ or as the relevant cross-ratio approaches a boundary), the asymmetry decays as a power law with a rate governed by small parameters $\epsilon$ whose exponents depend parametrically on $N$ and $k$.
• These results hold for Renyi index $n=2$ but generalize (via replica continuation) to von Neumann asymmetry.

6. Significance and Implications

Entanglement asymmetry in the $\widehat{su}(N)_k$ WZW model serves as a diagnostic for real-time symmetry restoration processes in systems where spontaneous symmetry breaking is forbidden. This measure excels when non-local or entanglement-based probes are necessary, as in $1+1$-dimensional conformal dynamics, and its ability to quantify "how much" and "how quickly" a symmetry is restored directly reflects group-theoretic and dynamical information.

The demonstration of a non-universal quantum Mpemba effect—sensitive to representation and model parameters—distinguishes it from the generic behavior in Abelian cases, suggesting underlying structural differences in the relaxation pathways depending on the operator spectrum and fusion rules.

An immediate consequence is that engineered initialization of quantum many-body systems in high-rank or low-level non-Abelian symmetry-breaking states (such as cold-atom ladders or synthetic gauge models) could allow experimental access to these relaxation dynamics, provided the appropriate entanglement-resolved observables are measured.

7. Outlook and Open Questions

• Universality: The presence or absence of the quantum Mpemba effect for different symmetry-breaking representations suggests non-universality that warrants further investigation. It remains open whether analogous phenomena can occur beyond the specific class of primaries analyzed here, or for other non-Abelian groups.
• Extension to Mixed/Hybrid Systems: Extensions to mixed initial states, different boundary conditions, or systems with both non-Abelian and Abelian symmetry sectors could uncover further structure in entanglement asymmetry dynamics.
• Applications to Quantum Information and Experiment: The analytic solvability of non-Abelian entanglement asymmetry dynamics in the WZW model, combined with its sensitivity to both symmetry group structure and initial conditions, presents a promising tool for benchmarking quantum simulators and exploring resource theories for non-Abelian quantum information.

In summary, the entanglement asymmetry in the $\widehat{su}(N)_k$ WZW model not only quantifies the subsystem-resolved restoration of non-Abelian symmetry after explicit breaking, but also establishes a controlled setting in which exceptional quantum relaxation phenomena, such as the quantum Mpemba effect, can be rigorously analyzed and potentially observed (Fujimura et al., 6 Sep 2025).