Emergent SO(5) Symmetry at the Néel to Valence-Bond-Solid Transition

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 $\mathrm{SO}(4)=[\mathrm{SU}(2)\times \mathrm{SU}(2)]/{\mathbb{Z}}_2$ 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.

Figure 1

The joint probability distribution $P({\tilde{N}}_x,{\tilde{\phi }}_x)$, after rescaling $N_x$ and ${\phi }_x$ to have unit variance, in a critical system of size $L=100$. The upper plane shows the contour plot.

Figure 2

Main panel: variance ratio ${\sigma }_{\phi }/{\sigma }_N$ plotted against $J$ for various $L$. Curves cross at $J_c$ as expected from SO(5) symmetry. Inset: same quantity as a function of $L$ for several $J$ around $J_c\simeq 0.0885$ (key in Fig. 3).

Figure 3

Main panel: $F_2^4$ [Eq. (4)] versus $L$ for a few $J$ values close to $J_c\simeq 0.0885$. Dashed lines: values in the Néel phase, at the SO(5)-symmetric critical point, and in the U(1) and ${\mathbb{Z}}_4$-symmetric regimes of the VBS phase. Inset: $J$ dependence of $F_2^4$ for various $L$ (key in Fig. 2).

Figure 4

Correlators $G_{\mathcal{O}}(r)$ for operators ${\phi }_x{N}_z$, ${\phi }_x{\phi }_y$, and ${\mathcal{O}}_J$ in a system of size $L=100$. (The $G_{\mathcal{O}}$ have been normalized to agree at $r=10$.) Inset: $G_{{\mathcal{O}}_J}$ for various $L$.

Figure 5

Conjectured phase diagram for the fully SO(5)-symmetric $\mathrm{NL}\sigma \mathrm{M}$ with the WZW term. The fixed point on the left also governs the deconfined critical point, where SO(5) symmetry is emergent. Moving away from $J_c$ introduces the relevant symmetry-breaking perturbation ${\mathcal{O}}_J$ (not shown), leading to Néel or VBS order.