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

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.

💡 Research Summary

The paper addresses a long‑standing controversy surrounding the scaling of entanglement entropy (EE) at the SO(5) deconfined quantum critical point (DQCP) of the J‑Q₃ model. Different bipartitions—an axis‑aligned “standard” cut and a 45° tilted cut—have previously yielded contradictory conclusions about whether the DQCP is described by a unitary conformal field theory (CFT). By bridging many‑body entanglement theory with the field of surface criticality, the authors demonstrate that the discrepancy originates from the boundary conditions imposed by the cut, not from a failure of bulk CFT.

The model studied is the spin‑½ J‑Q₃ Hamiltonian on a square lattice, with antiferromagnetic Heisenberg coupling J and a six‑spin Q term. At Q≈1.4915 the system exhibits an emergent SO(5) symmetry that unifies antiferromagnetic (AFM) and valence‑bond‑solid (VBS) order parameters, providing a candidate DQCP. Large‑scale stochastic series expansion quantum Monte‑Carlo (SSE‑QMC) simulations are performed with inverse temperature β=2L to access ground‑state properties. Two geometries are considered: (i) a standard cut that follows the lattice axes, and (ii) a 45° tilted cut that cuts across the VBS columnar patterns. For each geometry the authors study both a real physical edge (open boundary) and a “fake” edge defined by the entanglement Hamiltonian (EH) obtained via the multi‑replica trick.

Surface observables include the parallel surface spin‑spin correlation C∥(L), the perpendicular surface‑bulk correlation C⊥(L), and the surface Binder cumulant U₂(L). In the VBS phase (Q>Qc) the standard cut shows exponential decay of C∥, indicating a gapped surface, while the tilted cut displays algebraic decay (C∥∝L^p with p≈‑0.085), revealing a gapless surface despite a gapped bulk. At the putative DQCP the standard cut exhibits ordinary surface criticality: C∥∝L^{-(1+η∥)} with η∥≈2.36, C⊥∝L^{-(1+η⊥)} with η⊥≈1.305, and U₂→0, consistent with a disordered surface. Remarkably, these exponents satisfy a new scaling relation 2η⊥=η∥+η, where η≈0.26 is the bulk anomalous dimension, defining a previously unknown surface universality class for the SO(5) DQCP.

Conversely, the tilted cut yields extraordinary surface criticality: C∥ approaches a finite constant (≈0.043) and U₂→1, signalling long‑range order confined to the edge. The extracted exponents η∥≈0.43 and η⊥≈‑0.461 are outside any known O(N) surface classifications, confirming that the tilted geometry imposes a boundary condition that simultaneously breaks the emergent SO(5) symmetry at the edge while preserving bulk criticality.

The authors then turn to the entanglement Hamiltonian. By measuring surface correlations of the EH (the “fake” cut) they find that the same ordinary/extraordinary dichotomy appears: the standard EH shows decaying C∥ and vanishing U₂, whereas the tilted EH displays a finite C∥ and U₂≈1. This demonstrates that the extra gapless edge mode present in the tilted geometry contributes directly to the entanglement spectrum (ES) and EE, just as a physical edge would.

To test the Li–Haldane–Poilblanc conjecture, the low‑lying ES obtained from the EH is compared with the dynamical structure factor of a real physical edge for both geometries. The spectra match closely in both cases, indicating that even at a bulk‑gapless critical point the ES faithfully reproduces the edge excitation spectrum. Hence, the EE measured across a tilted cut is not a clean probe of bulk SO(5) criticality; it is polluted by boundary contributions arising from the extraordinary surface state.

The paper concludes with several practical implications. First, only an ordinary (axis‑aligned) bipartition yields EE that reflects pure bulk CFT data, making it the appropriate choice for diagnosing conformal invariance at a DQCP. Second, the tilted cut, while geometrically “fair” with respect to the four columnar VBS patterns, actually introduces a non‑trivial boundary fixed point that adds a universal logarithmic term to EE, potentially misleading interpretations. Third, the persistence of the Li–Haldane–Poilblanc correspondence at criticality provides a powerful tool to separate bulk and boundary contributions by comparing ES with edge spectra. Finally, the authors propose a protocol: use standard cuts for bulk CFT verification, and employ tilted cuts when one wishes to explore surface critical phenomena such as extraordinary transitions.

Overall, the work resolves the EE scaling controversy at the SO(5) DQCP by revealing the decisive role of boundary conditions, introduces a new surface universality class, and establishes a concrete methodology for disentangling bulk and edge effects in entanglement‑based studies of quantum criticality.