Fermion-induced quantum criticality with two length scales in Dirac systems

The quantum phase transition to a ${\mathbb{Z}}_3$-ordered Kekulé valence bond solid in two-dimensional Dirac semimetals is governed by a fermion-induced quantum critical point, which renders the putatively discontinuous transition continuous. We study the resulting universal critical behavior in terms of a functional RG approach, which gives access to the scaling behavior on the symmetry-broken side of the phase transition, for general dimensions and number of Dirac fermions. In particular, we investigate the emergence of the fermion-induced quantum critical point for spacetime dimensions $2<D<4$. We determine the integrated RG flow from the Dirac semimetal to the symmetry-broken regime and analyze the underlying fixed-point structure. We show that the fermion-induced criticality leads to a scaling form with two divergent length scales, due to the breaking of the discrete ${\mathbb{Z}}_3$ symmetry. This provides another source of scaling corrections, besides the one stemming from being in the proximity to the first-order transition.

Figure 1

Honeycomb lattice and Kekulé dimerization. (a) Honeycomb lattice with sublattices $A$ and $B$ represented by the open and filled circles. (b) Dimerization pattern of the Kekulé VBS state, where thick red lines mark stronger bonds and thinner gray lines mark weaker bonds. (c) Brillouin zone of the honeycomb lattice with the Dirac points $K,K^{\prime }$ where the Kekulé pattern opens a mass gap.

Figure 2

Stability exponent ${\theta }_2$ for the U(1)-symmetric non-Gaussian fixed point of the ${\mathbb{Z}}_3$ GNY model. In the purely bosonic limit, this corresponds to the O(2)-symmetric Wilson-Fisher FP. We show different small numbers of fermion flavors $N_f$ as a function of the dimension. From top to bottom, we show $N_f\in \{0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1,2\}$. As soon as fermions are added, this FP becomes stable below some critical dimension. The results have been calculated in $\mathrm{LPA}8{}^{\prime }$.

Figure 3

Schematic flow of the cubic coupling as a function of a tuning parameter ${\tilde{m}}^2$. ${\tilde{m}}^2>0$ indicates the system is in the symmetric regime and ${\tilde{m}}^2<0$ that it is in the symmetry-broken regime. Vertical dotted lines are ${\tilde{m}}^2=\pm \infty$. We show four trajectories for identical initial $g$ and different values of ${\tilde{m}}^2$.

Figure 4

Flow of the quartic coupling into the different regimes: (1) For $g\equiv 0$ the flow generically starts in the symmetric regime in the ultraviolet and can be fine-tuned to approach the FIQCP. Later, it may enter the SSB regime (as indicated by the change of color from blue to red) and flows to the NG fixed point, where it remains (light red line). (2) For small $g\ne 0$ the flow trajectory is almost identical during the entire flow. Only in the deep IR it departs from the NG fixed point due to the dangerously irrelevant direction corresponding to the cubic operator. (3) There can also be flows which remain completely in the symmetric regime and no symmetry breaking occurs. In this case the flow approaches the DSM fixed point in the deep IR as indicated by the light blue line. In the insets, we schematically show the shape of the effective potential in the different regimes.

Figure 5

Flow of the dimensionful masses $m_{L,T}^2$, cf. Eqs. (24) and (25), in the symmetry-broken regime. The red and blue curves correspond to the masses of one of the trajectories that escape the NG fixed point (i.e., when $g\ne 0)$ and the black dashed curve corresponds to the flow of the purely U(1)-symmetric system with otherwise identical initial conditions. At the beginning of the flow all masses are identical and they eventually split when approaching the IR. In the IR both masses flow to nonzero values $m_L^2(k=0)\ne 0,m_T^2(k=0)\ne 0$.