At low temperatures, isotropic two-dimensional Fermi systems with quadratic band touching are unstable with respect to a symmetry-broken ground state when interactions are taken into account. Here, the authors demonstrate that in a setup with threefold symmetry only - as realized, for instance, in bilayer graphene - interactions induce a splitting of the quadratic band touching into four Dirac cones. This shifts the instability to a finite interaction strength and leads to a quantum critical point with emergent Lorentz invariance.
At low temperatures, isotropic two-dimensional Fermi systems with quadratic band touching are unstable with respect to a symmetry-broken ground state when interactions are taken into account. Here, the authors demonstrate that in a setup with threefold symmetry only - as realized, for instance, in bilayer graphene - interactions induce a splitting of the quadratic band touching into four Dirac cones. This shifts the instability to a finite interaction strength and leads to a quantum critical point with emergent Lorentz invariance.
We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal, symmetry and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analog of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the ``fixed-point collision,'' according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.
Quadratic band touching (QBT) points are widely observed in two- and three-dimensional (2D and 3D) materials, including bilayer graphene and Luttinger semimetals, and attract significant attention from theory to experiment. However, even in its simplest form, the 2D checkerboard lattice QBT model, the phase diagram characterized by temperature and interaction strength, still remains unknown beyond the weak-coupling regime. Intense debates persist regarding the existence of various interaction-driven insulating states in this system. To address these uncertainties, we employ thermal tensor network simulations, specifically exponential tensor renormalization group and tangent space tensor renormalization group, along with density matrix renormalization group calculations to provide a comprehensive finite-temperature phase diagram for this model and shed light on previous ambiguities. Notably, our findings reveal the emergence of a robust bond-nematic Dirac semimetal phase with distinct thermodynamic properties that set it part from the nematic insulating state and other symmetry-broken states. This previously overlooked feature is found to be ubiquitous in interacting QBT systems. We also discuss the implications of these results for experimental systems such as bilayer graphene and iridate compounds.
We revisit the effect of local interactions on the quadratic band touching (QBT) of Bernal stacked bilayer graphene models using renormalization group (RG) arguments and quantum Monte Carlo simulations of the Hubbard model. We present an RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased quantum Monte Carlo simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite U/t, despite the U=0 hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with a dynamical critical exponent z=1 as expected for 2+1 d Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.
We revisit the effect of local interactions on the quadratic band touching (QBT) of the Bernal honeycomb bilayer model using renormalization group (RG) arguments and quantum Monte Carlo (QMC) simulations. We present a RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead, they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased QMC simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite U/t despite the U=0 hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with (2+1)D Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.
We study the phase diagram of interacting spinless fermions on the honeycomb bilayer at charge neutrality using large-scale quantum Monte Carlo simulations. In the noninteracting limit, the low-energy spectrum features quadratically dispersing bands that touch at the corners of the hexagonal Brillouin zone. Weak to intermediate interactions induce a splitting of each of the quadratic band touching points into four Dirac points, located along high-symmetry directions of the reciprocal lattice. Strong interactions lead to the formation of a layer-polarized charge density wave, which spontaneously breaks the Z2 layer inversion symmetry and opens an insulating gap in the spectrum. We show that the semimetal-to-insulator quantum phase transition as a function of interaction is continuous and characterized by emergent relativistic symmetry. Our results for the values of the correlation-length exponent ν, the order-parameter anomalous dimension ηϕ, and the fermion anomalous dimension ηψ agree with those of the theoretically predicted 2+1D Gross-Neveu-Ising universality class with eight two-component Dirac fermions within less than 5\%\ deviation. We also determine the crossover scale as a function of interaction strength between the nonrelativistic semimetal state at high temperatures, characterized by dynamical critical exponent z=2, and the Dirac semimetal state at intermediates temperatures, characterized by z=1. Further reducing the temperature below the crossover scale at a fixed value of the interaction strength above the quantum critical point results in a classical ordering transition in the 2D Ising universality class.
We argue that a three-dimensional electronic system with the Fermi level at the quadratic band touching point such as HgTe could be unstable with respect to the spontaneous formation of the (topological) Mott insulator at arbitrary weak long-range Coulomb interaction. The mechanism of the instability can be understood as the collision of Abrikosov's non-Fermi liquid fixed point with another, quantum critical, fixed point, which approaches it in the coupling space as the system's dimensionality $d\ensuremath{\rightarrow}{d}_{\text{low}}+$, with the ``lower critical dimension'' 2<dlow<4. Arguments for the existence of the quantum critical point based on considerations in the large-N limit in d=3, as well as close to d=2, are given. In the one-loop calculation we find that dlow=3.26, and thus above, but not far from three dimensions. This translates into a temperature or energy window (Tc, T∗) over which the non-Fermi liquid scaling should still be observable, before the Mott transition finally takes place at the critical temperature ${T}_{\mathrm{c}}\ensuremath{\sim}{T}_{*}\mathrm{exp}[\ensuremath{-}zC/({d}_{\text{low}}\ensuremath{-}d{)}^{1/2}]$. We estimate $C=\ensuremath{\pi}/1.1$, dynamical critical exponent $z\ensuremath{\approx}1.8$, and the temperature scale ${k}_{B}{T}_{*}\ensuremath{\approx}(4m/{m}_{\text{el}}{ϵ}^{2})13.6\text{ }\text{ }\mathrm{eV}$, with m as the band mass and ϵ as the dielectric constant.