Entanglement Asymmetry and Quantum Mpemba Effect for Non-Abelian Global Symmetry

Abstract

Entanglement asymmetry is a measure that quantifies the degree of symmetry breaking at the level of a subsystem. In this work, we investigate the entanglement asymmetry in $\widehat{su}(N)_k$ Wess-Zumino-Witten model and discuss the quantum Mpemba effect for SU$(N)$ symmetry, the phenomenon that the more symmetry is initially broken, the faster it is restored. Due to the Coleman-Mermin-Wagner theorem, spontaneous breaking of continuous global symmetries is forbidden in $1+1$ dimensions. To circumvent this no-go theorem, we consider excited initial states which explicitly break non-Abelian global symmetry. We particularly focus on the initial states built from primary operators in the fundamental and adjoint representations. In both cases, we study the real-time dynamics of the Rényi entanglement asymmetry and provide clear evidence of quantum Mpemba effect for SU$(N)$ symmetry. Furthermore, we find a new type of quantum Mpemba effect for the primary operator in the fundamental representation: increasing the rank $N$ leads to stronger initial symmetry breaking but faster symmetry restoration. Also, increasing the level $k$ leads to weaker initial symmetry breaking but slower symmetry restoration. On the other hand, no such behavior is observed for adjoint case, which may suggest that this new type of quantum Mpemba effect is not universal.

Figures (13)

• Figure 1: The sketch of subregions $A$ and $B$ in two-dimensional Euclidean spacetime. Here subregion $B$ is defined as the total system excluding the subregion $A$.
• Figure 2: The symmetry operators on an $n$-sheeted manifold can be topologically deformed. For example, the symmetry operator $U_A(g_1)$ on the first sheet can be moved to the second sheet and fuse with the symmetry operator $U_A(g_2)$ there. As a result, the operators on the second sheet transform as ${\cal O}_2 \to U_{\Sigma}^{\dagger}(g_1) {\cal O}_2 U_{\Sigma}(g_1)$ The same applies to the remaining sheets.
• Figure 3: Real-time dependence of $\text{EA}_2$ for $\tilde{x}_0 < 0$ in the large interval limit $\tilde{\tau}_0 \to 0$ (left panel) and its quasi-particle interpretation (right panel). In the right panel, the symbol "$\star$" denotes the operator insertion point $(x_0,0)$, and red line indicates the subregion $A$. The trajectories of the quasi-particles are depicted by the blue lines, and the horizontal and vertical axes represent space and real time directions, respectively.
• Figure 4: The real-time dependence of $\text{EA}_2$ for $\tilde{x}_0 > 0$ in the large interval limit $\tilde{\tau}_0 \to 0$ (left panel) and its quasi-particle interpretation (right panel). Here, we assume $\tilde{x}_0 < 1 /2$ for simplicity.
• Figure 5: Numerical plot of $\text{EA}_2$ for $(N,k) = (2,1)$ obtained from the analytical expression for $\tilde{\tau}_0 = 0.01, \ 0.1,\ 0.2$, plotted as a function of the dimensionless time $\tilde{t}$. The left panel corresponds to $\tilde{x}_0 = -0.5$ and the right panel to $\tilde{x}_0 = 0.1$. A smaller value of $\tilde{\tau}_0$ preserves the sharp feature predicted by the quasi-particle picture, while larger values smooth out the profiles.