Néel-State to Valence-Bond-Solid Transition on the Honeycomb Lattice: Evidence for Deconfined Criticality

We study a spin-$1/2$ SU(2) model on the honeycomb lattice with nearest-neighbor antiferromagnetic exchange $J$ that favors Néel order and competing six-spin interactions $Q$ that favor a valence-bond-solid (VBS) state in which the bond energies order at the “columnar” wave vector $\mathbf{K}=(2\pi /3,-2\pi /3)$. We present quantum Monte Carlo evidence for a direct continuous quantum phase transition between Néel and VBS states, with exponents and logarithmic violations of scaling consistent with those at analogous deconfined critical points on the square lattice. Although this strongly suggests a description in terms of deconfined criticality, the measured threefold anisotropy of the phase of the VBS order parameter shows unusual near-marginal behavior at the critical point.

Figure 1

The honeycomb lattice has a two-site basis (labeled $A$ and $B$) and elementary Bravais lattice translations ${\widehat{e}}_1$ and ${\widehat{e}}_2$, with distances from origin specified in units of ${\widehat{e}}_1$ and ${\widehat{e}}_2$. Three types of bonds (labeled 0, 1, 2), oriented along the three principal directions, “belong” to each Bravais lattice site. Columnar and plaquette VBS order at wave vector $\mathbf{K}$ correspond to different choices of the order-parameter phase, with solid filled dimers on a link $⟨ij⟩$ denoting high (low) values of $⟨P_{⟨ij⟩}⟩$ in the columnar (plaquette) state. Here, three domain walls meet at the core of the $Z_3$ vortex, which carries a net $S=1/2$ spin. Also shown is a depiction of the six-spin interaction terms in $H$.

Figure 2

Binder cumulants of ${\overrightarrow{M}}_s$ and $E_{\Psi }$ as a function of $q$ for different sizes $L$ (symbols), fit to a polynomial in $(q-q_{cD/N})L^{1/{\nu }_{D/N}}$ (lines) with best-fit values ${\nu }_N=0.5080$, ${\nu }_D=0.5237$, $q_{cN}=1.1912$, and $q_{CD}=1.1892$. Best-fit values are for the $L\ge 48$ part of the displayed data.

Figure 3

Scaling collapse of Binder cumulants of ${\overrightarrow{M}}_s$ and $E_{\Psi }$, using values of $q_{cD}$, $q_{cN}$, ${\nu }_D$, and ${\nu }_N$ quoted in legend of Fig. 2. Similar collapses for $⟨{\overrightarrow{M}}_s^2⟩L^{1+{\eta }_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}}$, $⟨|\Psi {|}^2⟩L^{1+{\eta }_{\mathrm{VBS}}}$ are also displayed, obtained using the following best-fit values: $q_{cN}=1.1956$, ${\nu }_N=0.5003$, ${\eta }_{\mathrm{N}\acute{\mathrm{ee}}\mathrm{l}}=0.3539$ ($⟨{\overrightarrow{M}}_s^2⟩$), $q_{cD}=1.1864$, ${\nu }_D=0.558$, and ${\eta }_{\mathrm{VBS}}=0.25$ ($⟨|\Psi {|}^2⟩$). Best-fit values are for the $L\ge 48$ part of the displayed data.

Figure 4

${\rho }_{s}L$ does not obey standard quantum-critical scaling ${\rho }_{s}L=h\mathbf{(}(q-q_{cN})L^{1/{\nu }_N}\mathbf{)}$ with dynamical exponent $z=1$ (for instance, see drift in $\beta =2L$ data shown in top inset). In contrast ${\rho }_{s}L/\log (L/L_0)$ with $L_0=0.37$ shows excellent scaling. Symbols are QMC data, and lines are best fit to this modified scaling form, with $q_{cN}\approx 1.190(2)$ and ${\nu }_N\approx 0.54(2)$ in agreement with our $T=0$ results. Bottom inset: Temperature dependence of ${\chi }_u/T$ close to criticality. In the Néel phase ($q=1.18$), QMC data (symbols) are well fit by ${\chi }_u/T=a+b/T$, whereas on the VBS side ($q=1.2$), a sharp drop is observed as expected. Close to criticality ($q=1.19$), QMC data are better fit by ${\chi }_u/T=c+d\log (J/T)$. Lines are fits to the above forms with $a=0.024$, $b=0.0005$, $c=0.022$, and $d=0.0024$.

Figure 5

The dimensionless $Z_3$-anisotropy parameter $W_3$ scales to zero with increasing $L$ value in the Néel phase but grows with size in the columnar VBS phase. Top inset zooms in on the behavior of near-critical systems, which display nearly scale-independent behavior. Bottom inset: histogram of $E_{\Psi }$ for $L=36$ at $q=1.184$ close to $q_{cD}$. The brightness of each color patch reflects the weight.