Fluctuation-induced continuous transition and quantum criticality in Dirac semimetals

We establish a scenario where fluctuations of new degrees of freedom at a quantum phase transition change the nature of a transition beyond the standard Landau-Ginzburg paradigm. To this end, we study the quantum phase transition of gapless Dirac fermions coupled to a ${\mathbb{Z}}_3$ symmetric order parameter within a Gross-Neveu-Yukawa model in 2+1 dimensions, appropriate for the Kekulé transition in honeycomb lattice materials. For this model, the standard Landau-Ginzburg approach suggests a first-order transition due to the symmetry-allowed cubic terms in the action. At zero temperature, however, quantum fluctuations of the massless Dirac fermions have to be included. We show that they reduce the putative first-order character of the transition and can even render it continuous, depending on the number of Dirac fermions $N_f$. A nonperturbative functional renormalization group approach is employed to investigate the phase transition for a wide range of fermion numbers and we obtain the critical $N_f$, where the nature of the transition changes. Furthermore, it is shown that for large $N_f$ the change from the first to second order of the transition as a function of dimension occurs exactly in the physical 2+1 dimensions. We compute the critical exponents and predict sizable corrections to scaling for $N_f=2$.

Figure 1

(a) Sketch of the LG free energy $F_{\text{LG}}\left[\varphi \right]=F_0+{r\left|\varphi \right|}^2+g({\varphi }^3+{\varphi }^{*3})+\lambda {\left|\varphi \right|}^4$ across the phase transition, where the dotted (dashed) line corresponds to the unordered (ordered) phase. (b) Sketch of the Kekulé valence-bond solid state on the honeycomb lattice. Darker and lighter red bonds mark different hopping amplitudes. The pattern leads to a tripled unit cell and reduced rotational symmetry in the Kekulé phase. (c) Schematic phase diagram for the quantum semi-metal to Kekulé VBS transition in two-dimensional Dirac semimetals. With the functional RG approach, we determine the number of critical fermion flavors for fermion-induced quantum criticality to occur at $N_{f,c}\approx 1.9$. We conclude that the Kekulé quantum transition in graphene can be expected to be continuous and give rise to quantum critical behavior.

Figure 2

(a) Schematic FRG diagrams contributing to the flow of the mass parameter $m^2$. (b) Diagrams contributing to the flow of the cubic coupling $g$. (c) Diagrams contributing to the flow of the quartic coupling $\lambda$. The crossed circle represents the scale derivative of the corresponding propagator. Importantly, the propagator lines are dressed propagators.

Figure 3

The two largest critical exponents as function of the space-time dimension for fixed $N_f=1$ and $N_f=10$. Our FRG result is compared to an expansion around the upper critical dimension $D=4-\epsilon$ to first order in $\epsilon$. In contrast to this perturbative calculation, we find that the decisive second largest exponent remains positive in three space-time dimensions for $N_f=1$.

Figure 4

Largest two critical exponents in $D=2+1$ for different $N_f$. The first critical exponent determines the correlation length exponent $\theta =1/\nu$. The second decides over the order of the phase transition. If ${\theta }_2<0$, the transition is continuous. The different lines mark different orders of the expansion: light blue, blue, red, light red corresponds to $\mathrm{LPA}4{}^{\prime },\mathrm{LPA}6{}^{\prime },\mathrm{LPA}8{}^{\prime }$, and $\mathrm{LPA}12{}^{\prime }$, respectively. $\mathrm{LPA}{N}^{\prime }$ denotes that the order parameter potential is expanded up to the ${\varphi }^N$ coupling. For large $N_f$, the critical exponents approach the values ${\theta }_i=D-(i+1)$, see text above Eq. (39).

Figure 5

Regime where the phase transition becomes of second order, for varying space-time dimension $D$ and fermion flavor number $N_f$. The red regime denotes the FRG result from this work. It is compared to the result obtained in the vicinity of the upper critical dimension $D=4-\epsilon$ [11].