Entanglement in the $θ$-vacuum

Abstract

We compute the entanglement entropy and the entanglement spectrum of the vacuum state in the massive Schwinger model at a finite $θ$ angle. The $θ$ term is implemented through a chirally rotated lattice Hamiltonian that preserves the periodicity in $θ$ already at the operator level and maintains the correct massless limit without $θ$-dependent lattice artifacts. We clarify the physical origin of entanglement entropy enhancement at $θ=π$ by relating it to the competition between distinct electric-flux vacuum branches. We show that the peak near $θ=π$ persists across the range of masses studied and corresponds to the point of maximal competition between distinct vacuum branches with opposite electric-field orientation, where quantum fluctuations due to fermion pair creation are maximized. While this entropy enhancement is generic, a pronounced narrowing of the entanglement gap occurs only near the critical mass ratio $m/g\simeq0.33$. Using the Bisognano--Wichmann (BW) theorem, we construct a lattice BW entanglement Hamiltonian and compare it with the exact modular Hamiltonian obtained from the reduced density matrix. We observe agreement between these Hamiltonians in the infrared sector, indicating that the entanglement Hamiltonian is well approximated by a spatially weighted microscopic Hamiltonian. These results establish entanglement observables as sensitive probes of the $θ$-dependent vacuum structure and highlight the chirally rotated formulation as a natural framework for open boundary conditions. Additionally, we discuss possible applications to entanglement in topological insulators and quantum wires.

Figures (15)

• Figure 1: Vacuum energy branches $E_n(\theta) \!\propto\! (\theta + 2\pi n)^2$ of the Schwinger model, shown as dashed parabolas. The physical vacuum energy $E_{\mathrm{vac}}(\theta)$ (solid red) is obtained by taking the lower envelope of these branches. As $\theta$ varies, the system switches between neighboring branches (dashed gray), leading to degeneracy at $\theta = \pi$.
• Figure 2: Potential $V(\phi)$ of the bosonized massive Schwinger model for different values of the dimensionless parameter $\kappa=e^\gamma m/g$, for fixed $\theta=\pi$, where CP is preserved and degenerate vacua appear in the weak-coupling limit.
• Figure 3: Ground-state observables of the massive Schwinger model as functions of the topological angle $\theta$ in the small-mass regime $m/g = 0.05$ ($N=1500$). Panel (a) shows the ground-state energy difference $\Delta E_0(\theta)=E_0(\theta)-E_0(0)$, (b) the half-chain entanglement entropy $S_{\rm EE}$, (c) the spatially averaged electric field $\langle E\rangle$, and (d) the chiral condensate difference $\Delta\Sigma(\theta)=\Sigma(\theta)-\Sigma(0)$. All quantities vary smoothly over the full $\theta$ interval. The entropy develops a broad maximum at $\theta=\pi$, while the remaining observables exhibit regular $2\pi$-periodic behavior.
• Figure 4: Ground-state observables as functions of $\theta$ for $m/g = 0.33$ ($N=1500$), close to the critical region. Panel (a) shows $\Delta E_0(\theta)$, (b) $S_{\rm EE}$, (c) $\langle \bar{E}\rangle$, and (d) $\Delta\Sigma(\theta)$, defined as in Fig. \ref{['GSenergy005']}. Compared to the small-mass case, the energy develops a nonanalytic slope at $\theta=\pi$. This is accompanied by a rapid variation of the electric field and a sharply localized peak in the entanglement entropy.
• Figure 5: Ground-state observables as functions of $\theta$ for $m/g = 0.42$ ($N=1500$). Panel (a) shows $\Delta E_0(\theta)$, (b) $S_{\rm EE}$, (c) $\langle \bar{E}\rangle$, and (d) $\Delta\Sigma(\theta)$, defined as in Fig. 3. The nonanalytic structure at $\theta=\pi$ persists beyond the critical mass. The entanglement entropy continues to exhibit a narrow maximum, though the peak is less pronounced than for $m/g=0.33$.