Bipartite entanglement and surface criticality: The extra contribution of non-ordinary edge in entanglement

Abstract

Recent works on the scaling behaviors of entanglement entropy at the SO(5) deconfined quantum critical point (DQCP) sparked a huge controversy. Different bipartitions gave out totally different conclusions for whether the DQCP is consistent with a unitary conformal field theory. In this work, we connect two previously disconnected fields -- the many-body entanglement and the surface criticality -- to reveal the behaviors of entanglement entropy in various bipartite scenarios, and point out that only the ordinary bipartition purely reflects the criticality of the bulk; otherwise, the extra gapless edge mode will also contribute to the entanglement. We have demonstrated that the correspondence between the entanglement spectrum and the edge energy spectrum still approximately persists even at a bulk-gapless point, thereby influencing the behavior of entanglement entropy. Our results establish that boundary conditions induced by the cut are decisive for entanglement-based probes and provide practical protocols to separate bulk from boundary contributions.

Figures (3)

• Figure 1: Lattices of $J$-$Q_3$ model with standard/tilted cut considered in this work. The system sizes are set the same vertically and horizontally. The black dots refers to the spins where the edge excitation/entanglement spectra are derived. (a)(b) $J$-$Q_3$ model with standard cut/bipartition. (c)(d) $45^\circ$ tilted cut/bipartition. The boundaries in (a)(c) are vertically open and horizontally periodic. (b)(d) are periodic in both directions, with the red dashed lines referring to the entanglement subsystem edges.
• Figure 2: Finite-size scaling of observables for $J$-$Q_3$ model. Top row: standard (axis–aligned) cut (in blue); bottom row: $45^{\circ}$ tilted cut (in red). Columns (left to right) show (a)(g): surface correlations $C_{\parallel}$ at $Q = 2.5$ in VBS phase with open boundary condition ("real cut"), (b)(c)(d)(h)(i)(j): surface correlation $C_{\parallel}$, surface–bulk correlations $C_{\perp}$, and Binder cumulant $U_{2}$ at $Q = Q_c$ with real cut, (e)(f)(k)(l): $C_{\parallel}$ and $U_2$ of entanglement–Hamiltonian ("fake-cut") at $Q = Q_c$, respectively. In the VBS phase, $C_{\parallel}$ decays to $0$ (a): exponentially and (g): algebraically. At the putative DQCP, in the standard geometry $C_{\parallel}$ and $U_{2}$ vanish algebraically, while the tilted edge saturates $C_{\parallel}\neq0$ and $U_{2}\to1$ when $L \to \infty$, signaling an extraordinary order that is mirrored in the entanglement Hamiltonian edge as well. The system sizes of the figures are (a): $L = 8 \sim 32$, (g): $L = 32 \sim 80$, (b)(c)(d)(h)(i)(j): $L = 16 \sim 80$, (e)(f)(k)(l): $L = 16 \sim 48$, respectively. Solid lines are least–squares fits described in the text.
• Figure 3: (a)(b) Edge excitation spectra with open boundary condition (real cut) and (c)(d) entanglement spectra (fake cut) of $J$-$Q_3$ model near the entanglement boundary at the putative DQCP. (a)(c) are using the standard cutting (bipartition), while (b)(d) are $45^\circ$ tilted. The system size $L = 48$ for all spectra. The system inverse temperature is $\beta = 96$ for excitation spectra and $\beta = 80$ for entanglement spectra. Note that (a)(b) are plotted using $\omega$ with the dimension of energy, while (c)(d) are plotted using a dimensionless variable $\omega_E$, resulting in the difference in vertical scales.