We study the scaling behavior of the Rényi entanglement entropy with smooth boundaries at the putative deconfined critical point separating the Néel antiferromagnetic and valence-bond-solid states of the two-dimensional J-Q_{3} model. We observe a subleading logarithmic term with a coefficient indicating the presence of four Goldstone modes, signifying the presence of an SO(5) symmetry at the transition point, which spontaneously breaks into an O(4) symmetry in the thermodynamic limit. This result supports the conjecture that an SO(5) symmetry emerges at the transition point, but reveals the transition to be weakly first-order. We demonstrate, through this Letter, a novel approach to detect emergent continuous symmetry and, more importantly, identify weakly first-order phase transitions efficiently, which have been notoriously challenging for conventional methods.
We study the Rényi entanglement entropy (EE) of the two-dimensional J-Q model, the emblematic quantum spin model of deconfined criticality at the phase transition between antiferromagnetic and valence-bond-solid ground states. State-of-the-art quantum Monte Carlo calculations of the EE reveal critical corner contributions that scale logarithmically with the system size, with a coefficient in remarkable agreement with the form expected from a large-N conformal field theory with SO(N=5) symmetry. However, details of the bipartition of the lattice are crucial in order to observe this behavior. If the subsystem for the reduced density matrix does not properly accommodate valence-bond fluctuations, logarithmic contributions appear even for cornerless bipartitions. We here use a 45° tilted cut on the square lattice. Beyond supporting an SO(5) deconfined quantum critical point, our results for both the regular and tilted cuts demonstrate important microscopic aspects of the EE that are not captured by conformal field theory.
The deconfined quantum critical point (DQCP) in square lattice S=1/2 quantum antiferromagnets has been extensively studied with a large body of evidence pointing to a weakly first-order transition scenario. Recent studies, which focused on entanglement at this nearly continuous DQCP in square lattice J-Q models, have observed conflicting bipartite entanglement entropy (EE) scaling behavior. One bipartition choice gave scaling coefficients in remarkable agreement with predictions from the unitary CFT corresponding to the putative DQCP. While another equally natural choice gave scaling coefficients in complete violation of unitary CFT that may be attributed to lack of scale invariance at the known weakly first-order behavior of the model. This motivates the exploration of DQCP behavior via entanglement measures in lattice models with distinct crystalline symmetries. Here we study a S=1/2 honeycomb model that hosts a nearly continuous transition between Néel and valence-bond-solid ground states relevant to probing DQCP. Using large-scale quantum Monte Carlo simulations, we compute the Rényi EE for a variety of bipartitions and test the CFT based description of the DQCP on the honeycomb lattice. For smooth bipartitions, we find no evidence of logarithmic corrections, in accordance with CFT, thereby essentially ruling out contributions from Goldstone modes. For subsystems with corners, CFT predicts universal logarithmic contributions, which we extract for corners with 60 and 120 degree angles and find close agreement with an emergent SO(5) CFT. While we observe scaling consistent with a critical system in the majority of cases, we also demonstrate an intriguing counterexample of the hexagon subsystem that exhibits a subtle period three oscillation. This results in three separate finite-size series, where the sign of the logarithmic term apparently changes depending on the series.
Recent numerical and theoretical studies on the two-dimensional
J-
Q3 model suggests that the deconfined quantum critical point is actually a
SO(5)-symmetry-enhanced first-order phase transition that is spontaneously broken to
O(4). However, this conclusion has mainly relied on finite-size scaling of the entanglement entropy, lacking direct evidence from physical observables.} Here, we investigate the dynamical spectra of spin and bond operators at the deconfined critical point of the
J-
Q3 model using large-scale quantum Monte Carlo simulations, and contrasting them with the well-established
O(3) Wilson-Fisher criticality in the
J1-
J2 Heisenberg model. Although both models exhibit two gapless magnon modes in the Néel phase, their critical behaviors diverge strikingly. At the
J1-
J2 critical point, the Higgs mode becomes gapless, yielding three gapless modes that reflect the full restoration of the
O(3) symmetry. {In the
J-
Q3 model, we instead observe four gapless transverse modes at the either side of the transition. This spectral feature, together with the entanglement entropy results, provides direct evidence for the weakly first-order scenario that the deconfined quantum critical point exhibits an emergent
SO(5) symmetry that spontaneously breaks to
O(4).
We show numerically that the "deconfined" quantum critical point between the Néel antiferromagnet and the columnar valence-bond solid, for a square lattice of spin 1/2, has an emergent SO(5) symmetry. This symmetry allows the Néel vector and the valence-bond solid order parameter to be rotated into each other. It is a remarkable (2+1)-dimensional analogue of the SO(4)=[SU(2)×SU(2)]/Z(2) symmetry that appears in the scaling limit for the spin-1/2 Heisenberg chain. The emergent SO(5) symmetry is strong evidence that the phase transition in the (2+1)-dimensional system is truly continuous, despite the violations of finite-size scaling observed previously in this problem. It also implies surprising relations between correlation functions at the transition. The symmetry enhancement is expected to apply generally to the critical two-component Abelian Higgs model (noncompact CP(1) model). The result indicates that in three dimensions there is an SO(5)-symmetric conformal field theory that has no relevant singlet operators, so is radically different from conventional Wilson-Fisher-type conformal field theories.
We systematically investigate the scaling behaviors of the disorder operator and the entanglement entropy (EE) of the easy-plane JQ (EPJQ) model at its transitions between the antiferromagnetic XY ordered phase (AFXY) and the valence bond solid (VBS) phase. We find there exists a tiny yet finite value of the order parameters at the AFXY-VBS phase transition points of the EPJQ model, and the finite order parameter is strengthened as anisotropy $\mathrm{\ensuremath{\Delta}}$ varies from the Heisenberg limit ($\mathrm{\ensuremath{\Delta}}=1$) to the easy-plane limit ($\mathrm{\ensuremath{\Delta}}=0$). This observation provides evidence that the N\'eel-VBS transition in the JQ model setting evolves from weak to prominent first-order transition as the system becomes anisotropic. Furthermore, both EE and disorder operator with smooth boundary cut exhibit anomalous scaling behavior at the transition points, resembling the scaling inside the Goldstone mode (AFXY) phase, and the anomalous scaling becomes strengthened as the transition becomes more first order. In particular, for $\mathrm{\ensuremath{\Delta}}\ensuremath{\le}0.3$, the obtained log-coefficients of EE converge to 0.5 which is the same as the contribution from one Goldstone mode in the N\'eel phase. For $\mathrm{\ensuremath{\Delta}}>0.3$, the log-coefficients are smaller and our findings might suffer from strong finite-size effects due to the fact that the remaining N\'eel order here is quite tiny.
We systematically investigate the finite size scaling behavior of the Rényi entanglement entropy (EE) of several representative 2d quantum many-body systems between a subregion and its complement, with smooth boundaries as well as boundaries with corners. In order to reveal the subleading correction, we investigate the quantity “subtracted EE” S^s(l) = S(2l) - 2S(l) S s ( l ) = S ( 2 l ) − 2 S ( l ) for each model, which is designed to cancel out the leading perimeter law. We find that (1) for a spin-1/2 model on a 2d square lattice whose ground state is the Neel order, the coefficient of the logarithmic correction to the perimeter law is consistent with the prediction based on the Goldstone modes; (2) for the (2+1)d ( 2 + 1 ) d O(3) Wilson-Fisher quantum critical point (QCP), realized with the bilayer antiferromagnetic Heisenberg model, a logarithmic subleading correction exists when there is sharp corner of the subregion, but for subregion with a smooth boundary our data suggests the absence of the logarithmic correction to the best of our efforts; (3) for the (2+1)d ( 2 + 1 ) d SU(2) J-Q _2 2 and J-Q _3 3 model for the deconfined quantum critical point (DQCP), we find a logarithmic correction for the EE even with smooth boundary.