We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the non-interacting level, we show that a QBCP exists and is topologically stable for a Berry flux {-+}2{pi}, if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak shortrange repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.
A (single flavor) quadratic band crossing in two dimensions is known to have a generic instability towards a quantum anomalous Hall (QAH) ground state for infinitesimal repulsive interactions. Here we introduce a generalization of a quadratic band crossing which is protected only by rotational symmetry. By focusing on the representative case of a parabolic and flat band touching, which also allows for a straightforward lattice realization, the interaction induced nematic phase becomes the dominant instability in certain parts of the phase diagram already at weak coupling, by competing successfully with the QAH insulator. The full phase diagram of the model, together with its topological properties, is mapped out using a perturbative renormalization-group, strong-coupling analysis, the mean-field theory. Interestingly, the Berry flux varies continuously in the single flavor limit with various control parameters.
We study the quantum phases driven by interaction in a semimetal with a quadratic band touching at the Fermi level. By combining the density matrix renormalization group (DMRG), analytical power expanded Gibbs potential method, and the weak coupling renormalization group, we study a spinless fermion system on a checkerboard lattice at half-filling, which has a quadratic band touching in the absence of interaction. In the presence of strong nearest-neighbor (
V1) and next-nearest-neighbor (
V2) interactions, we identify a site nematic insulator phase, a stripe insulator phase, and a phase separation region, in agreement with the phase diagram obtained analytically in the strong coupling limit (i.e., in the absence of fermion hopping). In the intermediate interaction regime we establish a quantum anomalous Hall phase in the DMRG as evidenced by the spontaneous time-reversal symmetry breaking and the appearance of a quantized Chern number
C=1. For weak interaction we utilize the power expanded Gibbs potential method that treats
V1 and
V2 on equal footing, as well as the weak coupling renormalization group. Our analytical results reveal that not only the repulsive
V1 interaction, but also the
V2 interaction (both repulsive and attractive), can drive the quantum anomalous Hall phase. We also determine the phase boundary in the ${V}_{1}\text{\ensuremath{-}}{V}_{2}$ plane that separates the semimetal from the quantum anomalous Hall state. Finally, we show that the nematic semimetal, which was proposed for $|{V}_{2}|\ensuremath{\gg}{V}_{1}$ at weak coupling in a previous study, is absent, and the quantum anomalous Hall state is the only weak coupling instability of the spinless quadratic band touching semimetal.
We analyze the effect of disorder on the weak-coupling instabilities of a quadratic band crossing point (QBCP) in two-dimensional Fermi systems, which, in the clean limit, display interaction-driven topological insulating phases. In the framework of a renormalization group procedure, which treats fermionic interactions and disorder on the same footing, we test all possible instabilities and identify the corresponding ordered phases in the presence of disorder for both single-valley and two-valley QBCP systems. We find that disorder generally suppresses the critical temperature at which the interaction-driven topologically nontrivial order sets in. Strong disorder can also drive a topological phase transition into a topologically trivial insulating state.
Abstract Interaction-driven topological phases significantly enrich the class of topological materials and thus are of great importance. Here, we study the phase diagram of interacting spinless fermions filling the two-dimensional checkerboard lattice with a quadratic band touching (QBT) point. By developing new diagnosis based on the state-of-the-art density-matrix renormalization group and exact diagonalization, we determine accurate quantum phase diagram for such a system at half-filling with three distinct phases. For weak nearest-neighboring interactions, we demonstrate the instability of the QBT towards an interaction-driven spontaneous quantum anomalous Hall (QAH) effect. For strong interactions, the system breaks the rotational symmetry realizing a nematic charge-density-wave (CDW) phase. Interestingly, for intermediate interactions we discover a symmetry-broken bond-ordered critical phase sandwiched in between the QAH and CDW phases, which splits the QBT into two Dirac points driven by interaction. Instead of the direct transition between QAH and CDW phases, our identification of an intermediate phase sheds new light on the theoretical understanding of the interaction-driven phases in QBT systems.
Motivated by recent advances in cold atomic systems, we study the equilibrium and quench properties of two-dimensional fermions with quadratic band touching at the Fermi level, in the presence of infinitely long-range interactions. Unlike when only short-range interactions are present, both nematic and quantum anomalous Hall (QAH) states appear at weak interactions, separated by a narrow coexistence region, whose boundaries mark second- and third-order quantum phase transitions. After an interaction quench, the QAH order exhibits three distinct regions: persistent or damped oscillations and exponential decay to zero. In contrast, the nematic order always reaches a nonzero stationary value through power-law damped oscillations, due to the interplay between the symmetry of the interaction and the specific topology of the quadratic band touching.