Nonordinary criticality at the edges of planar spin-1 Heisenberg antiferromagnets

Dangling edge spins of dimerized two-dimensional spin-1 Heisenberg antiferromagnets are shown to exhibit nonordinary quantum critical correlations, akin to the scaling behavior observed in recently explored spin-1/2 systems. Based on large-scale quantum Monte Carlo simulations, we observe remarkable similarities between these two cases and examine the crossover to the fundamentally distinct behavior in the one-dimensional limit of strongly coupled edge spins. We complement our numerical analysis by a cluster mean-field theory that encompasses the qualitatively similar behavior for the spin-1 and the spin-1/2 cases and its dependence on the spatial edge spin configuration in a generic way.

Figure 1

Columnar-dimer lattice with nondangling (N) edge spins (top edge) and dangling (D) edge spins (bottom edge). Solid (open) circles show bulk (edge) spins, and thick red (thin black) lines denote intra- (inter)dimer couplings $J_D\left(J\right)$.

Figure 2

Correlations parallel, $C_{\parallel }(L/2)$ (top panel), perpendicular, $C_{\perp }(L/2)$ (middle panel), and the staggered susceptibility ${\chi }_s$ (bottom panel) as functions of $L$ for $S=1$ and $S=1/2$ for the two different edge configurations. The dashed lines indicate the fits as described in the text.

Figure 3

Two different perturbations that we applied to the dangling edge spins. In panel (a), the intraedge couplings get rescaled by a factor of $(1+\kappa )$, whereas in panel (b), additional spins interact with the edge spins via a Heisenberg coupling of strength $\kappa J_D$. In both cases, $\kappa =0$ corresponds to the original model. In both panels, the varied couplings are shown by dashed lines. In panel (b), the half-open circles denote the additional spins that couple each to an original dangling edge spin.

Figure 4

Intraedge correlations $C_{\parallel }(L/2)$ as functions of $L$ for the perturbation shown in Fig. 3 for the cases of $S=1/2$ (top panel) and $S=1$ (bottom panel).

Figure 5

String order correlations $\mathcal{S}(L/2)$ as functions of $1/L$ for the perturbation shown in Fig. 3 for the $S=1$ system. Also shown are the results for an isolated one-dimensional spin-1 Heisenberg chain.

Figure 6

Intraedge correlations $C_{\parallel }(L/2)$ as functions of $L$ for the perturbation shown in Fig. 3 for the cases of $S=1/2$ (top panel) and $S=1$ (bottom panel). In both panels, the dashed lines indicate a power-law decay corresponding to the ordinary case of the nondangling edge.

Figure 7

Structure of the MF decoupling of the columnar-dimer lattice model for the case of dangling edge spins. The lattice is considered semi-infinite in the vertical ($y$) direction and has a horizontal extent that is specified by the number $L$ of edge spins ($L=4$ in the figure) where periodic boundary conditions are considered. The opaque dimers are part of the MF Hamiltonian. The remaining degrees of freedom are given by translation symmetry. The bottom row shows the special construction to treat the dangling edge-spin condition.

Figure 8

Comparison between the numerical solution of the lattice model MF theory in Eq. (11) (solid lines) to the analytical MF theory of the $\mathrm{O}\left(n\right)$ model (dashed lines) for $\tau =-6\times {10}^{-4}$ and $L=400$ for different values of spin $S$. The left (right) panel shows the nondangling (dangling) edge-spin case.

Figure 9

Binder ratio $Q$ of the spin-1 Heisenberg model on the columnar-dimer lattice for different system sizes $L$ as a function of $J/J_D$. Close to the crossing point, the data has been approximated by polynomials of degree $d$ (shown here for $d=3$).

Figure 10

Uniform susceptibility ${\chi }_u$ multiplied by system size $L$ of the spin-1 Heisenberg model on the columnar-dimer lattice for different $L$'s as a function of $J/J_D$. Close to the crossing point, they have been approximated by polynomials of degree $d$ (shown here for $d=3$).

Figure 11

Crossing points in the Binder ratio $Q$ and the uniform susceptibility ${\chi }_{u}L$ between the fitting polynomials for systems with linear sizes $L$ and $2L$ vs the inverse linear system size $1/L$ for the spin-1 Heisenberg model on the columnar-dimer lattice. The fitting polynomial has degree $d$, and the range of system sizes is $L=30,40,50,60$. The connecting lines are a guide for the eye.

Figure 12

Spin-correlation function $C_{\parallel }(L/2)$ and string order correlation function $\mathcal{S}(L/2)$ along the dangling edge spins on the spin-1 Heisenberg model on the columnar-dimer lattice (denoted as D) at a values of $J/J_D=0.1$ inside the bulk-disordered regime and for an isolated one-dimensional spin-1 Heisenberg chain for comparison (denoted as chain) vs the inverse linear system size $1/L$. For these QMC simulations, $T=J/\left(2L\right)$ was used, and lines are guides to the eye.