A unified theory of quantum critical points beyond the conventional Landau-Ginzburg-Wilson paradigm remains unknown. According to Landau cubic criterion, phase transitions should be first-order when cubic terms of order parameters are allowed by symmetry in the Landau-Ginzburg free energy. Here, from renormalization group analysis, we show that second-order quantum phase transitions can occur at such putatively first-order transitions in interacting two-dimensional Dirac semimetals. As such type of Landau-forbidden quantum critical points are induced by gapless fermions, we call them fermion-induced quantum critical points. We further introduce a microscopic model of SU(N) fermions on the honeycomb lattice featuring a transition between Dirac semimetals and Kekule valence bond solids. Remarkably, our large-scale sign-problem-free Majorana quantum Monte Carlo simulations show convincing evidences of a fermion-induced quantum critical points for N = 2, 3, 4, 5 and 6, consistent with the renormalization group analysis. We finally discuss possible experimental realizations of the fermion-induced quantum critical points in graphene and graphene-like materials.Quantum phase transitions are governed by Landau-Ginzburg theory and the exceptions are rare. Here, Li et al. propose a type of Landau-forbidden quantum critical points induced by gapless fermions in two-dimensional Dirac semimetals.
According to the Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization-group (RG) analysis that continuous transition can happen at putatively first-order
Z3 transitions in two-dimensional (2D) Dirac semimetals and such non-Landau phase transitions were dubbed ``fermion-induced quantum critical points'' (FIQCPs) [Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao, Nat. Commun. 8, 314 (2017)]. The RG analysis, controlled by the
1/N expansion with
N the number of flavors of four-component Dirac fermions, shows that the FIQCP occurs for $N\ensuremath{\ge}{N}_{c}$. Previous quantum Monte Carlo simulations of a microscopic model of
SU(N) fermions on the honeycomb lattice showed that the FIQCP occurs at the transition between Dirac semimetals and Kekule valence bond solids for $N\ensuremath{\ge}2$. However, the precise value of the lower bound
Nc has not been established. Especially, the case of
N=1 has not been explored by studying microscopic models so far. Here, by introducing a generalized
SU(N) fermion model with
N=1 (namely, spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of the FIQCP for
N=1. Consequently, our results suggest that the FIQCP can occur in 2D Dirac semimetals for all positive integers $N\ensuremath{\ge}1$.
In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and
Z3-ordered phases (e.g., Kekule valence-bond solid), where cubic terms of the order parameter are allowed in the quantum Landau-Ginzberg theory and the transitions are putatively first order. From large-
N renormalization-group (RG) analysis, we find that fermion-induced quantum critical points (FIQCPs) [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)] occur when
N (the number of flavors of four-component Dirac fermions) is larger than a critical value
Nc. Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for
Nc, i.e.,
Nc>1/2. (Here the ``1/2'' flavor of four-component Dirac fermions is equivalent to one flavor of four-component Majorana fermions). Moreover, we show that the emergence of two length scales is a typical phenomenon of FIQCPs and obtain two different critical exponents, i.e., $\ensuremath{\nu}\ensuremath{\ne}{\ensuremath{\nu}}^{\ensuremath{'}}$, by large-
N RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.
Fluctuations can change the phase transition properties drastically. An example is the fermion-induced quantum critical point (FIQCP), in which fluctuations of the massless Dirac fermions turn a putative Landau-de Gennes first-order phase transition (FOPT) with a cubic boson interaction into a continuous one. However, for the Landau-Devonshire theory, which characterizes another very large class of FOPTs, its fate under the coupling with extra fluctuations has not been explored. Here, we discover a new type of FIQCP, in which the Dirac fermion fluctuations round the boson Landau-Devonshire FOPT into a continuous phase transition. By using the functional renormalization group analyses, we determine the condition for the appearance of this FIQCP. Moreover, we point out that the present FIQCP can be a supersymmetric critical point. We finally show that the low-temperature phase diagram can provide distinct experimental evidences to detect this FIQCP.
This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.
Fluctuations can change the phase transition properties drastically. An example is the fermion-induced quantum critical point (FIQCP), in which fluctuations of the massless Dirac fermions turn a putative Landau--de Gennes first-order phase transition (FOPT) with a cubic boson interaction into a continuous one. However, for the Landau-Devonshire theory, which characterizes another very large class of FOPTs, its fate under the coupling with extra fluctuations has not been explored. Here, we discover another type of FIQCP, in which the Dirac fermion fluctuations round the boson Landau-Devonshire FOPT into a continuous phase transition. By using the functional renormalization group analyses, we determine the condition for the appearance of this FIQCP. Moreover, we point out that the present FIQCP can be a supersymmetric critical point. We finally show that the low-temperature phase diagram can provide distinct experimental evidences to detect this FIQCP.
Topology plays a cardinal role in explaining phases and quantum phase transitions beyond the Landau-Ginzburg-Wilson paradigm. In this study, we formulate a set of models of Dirac fermions in 2+1 dimensions with SU(
N)
×SU(2)
×U(1) symmetry that have the potential to host critical points described by field theories with topological terms. For
N=2 it shows a rich phase diagram containing semimetallic, quantum spin Hall insulating, Kekulé valence bond solid and s-wave superconducting phases and features multiple Landau-Ginzburg-Wilson phase transitions driven by interaction strength. At
N=1 a deconfined quantum critical point is observed. At
N=2 one expects the critical theory to correspond to a level 2 Wess-Zumino-Witten theory in 2+1 dimensions. Here the numerical results however show a strong first order transition. Another transition can be governed by a topological
θ-term which is rendered irrelevant for even values of
N thus leading to Landau-Ginzburg-Wilson behaviour.