Dirac Fermions with Competing Orders: Non-Landau Transition with Emergent Symmetry

We consider a model of Dirac fermions in $2+1$ dimensions with dynamically generated, anticommuting SO(3) Néel and $Z_2$ Kekulé mass terms that permits sign-free quantum Monte Carlo simulations. The phase diagram is obtained from finite-size scaling and includes a direct and continuous transition between the Néel and Kekulé phases. The fermions remain gapped across the transition, and our data support an emergent SO(4) symmetry unifying the two order parameters. While the bare symmetries of our model do not allow for spinon-carrying $Z_3$ vortices in the Kekulé mass, the emergent SO(4) invariance permits an interpretation of the transition in terms of deconfined quantum criticality. The phase diagram also features a tricritical point at which the Néel, Kekulé, and semimetallic phases meet. The present sign-free approach can be generalized to a variety of other mass terms and thereby provides a new framework to study exotic critical phenomena.

Figure 1

The model of fermions on the honeycomb lattice with hopping $t$ and Hubbard repulsion $U$, coupled to Ising spins on the lattice bonds (the solid circles) with exchange $J$ and transverse field $h$. The couplings ${\xi }_{ij}=0$, $\pm \xi$ have a Kekulé modulation. The unit cell (shaded blue) contains six fermionic sites and nine Ising spins. The lattice vectors are ${\overrightarrow{A}}_1$ and ${\overrightarrow{A}}_2$.

Figure 2

Phase diagram with semimetallic, antiferromagnetic (AFM), and Kekulé-ordered (KVBS) phases from QMC simulations at $T=0.05$. Circles (diamonds) indicate the onset of long-range Néel (Kekulé) order; open (solid) symbols are critical values based on $L=3$ and 6 ($L=6$ and 9); see the text.

Figure 3

Correlation ratios for (a) antiferromagnetic order and (b) Kekulé order across the AFM-KVBS transition at $U=6$. The crossings yield a critical value of $1/h_c\approx 0.29$.

Figure 4

(a) Free-energy derivative $\partial F/\partial h$ and (b) single-particle gap ${\mathrm{\Delta }}_{\mathrm{sp}}$ at the Dirac point. Here, $U=6$.

Figure 5

Ratio of the standard deviations of the AFM and KVBS order parameters. (Inset) Joint (normalized) probability distribution of the two order parameters near the critical point ($1/h_c=0.29$) for $L=6$. Here, $U=6$.