Quadratic Band Crossing Point Research Articles

Favorable phase transitions induced by spinful electron–electron interactions in two-dimensional semimetal with a quadratic band crossing point

We study the effects of marginally spinful electron–electron interactions on the low-energy instabilities and favorable phase transitions in a two-dimensional (2D) spin-1/2 semimetal that owns a quadratic band crossing point (QBCP) parabolically touched by the upper and lower bands. In the framework of a renormalization group procedure, all sorts of interactions are treated on the equal footing to derive the coupled energy-dependent evolutions of all interaction couplings that govern the low-energy properties. Deciphering the essential physical information of such flows, we at first find that the tendencies of interaction parameters fall into three categories including Limit case, Special case, and General case based on the initial conditions. In addition, the 2D QBCP system is attracted to several distinct kinds of fixed points (FPs) in the interaction-parameter space, namely FP1+/FP2−, FP1±/ FP2±/FP3±, and FP1±/FP3±/FP41,42,43± with the subscripts characterizing the features of FPs for the Limit, Special, and General cases, respectively. Furthermore, as approaching these FPs, we demonstrate that the spinful fermion–fermion interactions can induce a number of favorable instabilities accompanied by certain phase transitions. Specifically, the quantum anomalous Hall (QAH), quantum spin Hall (QSH), and nematic (Nem.) site(bond) states are dominant for FP1±, FP2±, and FP3±, respectively. Rather, QSH becomes anisotropic nearby FP41,42,43± with one component leading and the others subleading. Besides, Nem.site(bond), chiral superconductivity, and nematic-spin-nematic (NSN.) site(bond) are subleading candidates around these FPs. Our findings provide valuable insights for further research into the 2D QBCP and similar systems.

Interaction-driven topological phase transition in monolayer CrCl2 (pyrazine) 2

The quadratic band crossing points (QBCPs) at the Fermi level in two dimensions have been proposed to be unstable under electron-electron interactions. The possible interaction-driven states include the quantum anomalous Hall (QAH) state and various nematic ordered states. In this paper, motivated by the discovery of ferromagnetic van der Waals layered metal-organic framework ${\mathrm{CrCl}}_{2}{(\mathrm{pyrazine})}_{2}$, we theoretically propose that a single layer of ${\mathrm{CrCl}}_{2}{(\mathrm{pyrazine})}_{2}$ might realize one or some of these interaction-driven states based on the QBCP protected by ${C}_{4}$ symmetry. By introducing short-range density-density type repulsion interactions into this system, we have found the phase diagram depending on different interaction ranges and strengths. The exotic phases include the staggered chiral flux state manifesting the QAH effect, the site-nematic insulator, and the site-nematic Dirac semimetal state. The QAH state is robust against perturbations breaking the QBCP but it is weakened by increasing temperature. The metal-organic framework is tunable by changing the transition-metal elements, which might improve the gap size and stability of this interaction-induced QAH state.

Correlation Renormalized and Induced Spin-Orbit Coupling

Interplay of spin-orbit coupling (SOC) and electron correlation generates a bunch of emergent quantum phases and transitions, especially topological insulators and topological transitions. We find that electron correlation will induce extra large SOC in multi-orbital systems under atomic SOC and change ground state topological properties. Using the Hartree–Fock mean field theory, phase diagrams of px /py orbital ionic Hubbard model on honeycomb lattice are well studied. In general, correction of strength of SOC δ λ ∝ (Uʹ–J). Due to breaking down of rotation symmetry, form of SOC on multi-orbital materials is also changed under correlation. If a non-interacting system is close to fermionic instability, spontaneous generalized SOC can also be found. Using renormalization group, SOC is leading instability close to quadratic band-crossing point. Mean fields at quadratic band-crossing point are also studied.

Correlation renormalized and induced spin-orbit coupling

Abstract Interplay of spin-orbit coupling (SOC) and electron correlation generates a bunch of emergent quantum phases and transitions, especially topological insulators (TI) and topological transitions. In this work, we find that electron correlation will induce extra large SOC in multi-orbital systems under atomic SOC and change ground state topological properties. Using Hartree-Fock mean field theory, phase diagrams of $p_{x}/p_{y}$ orbital ionic Hubbard model on honeycomb lattice are well studied. In general, correction of strength of SOC $\delta \lambda \propto (U'-J)$. Due to breaking down of rotation symmetry, form of SOC on multi-orbital materials are also changed under correlation. If non-interacting system is close to fermionic instability, spontaneous generalized SOC can also be found. Using renormalization group, SOC is leading instability close to quadratic band-crossing point. Mean fields at quadratic band-crossing point are also studied.

Unprotected quadratic band crossing points and quantum anomalous Hall effect in FeB2 monolayer

Quadratic band crossing points (QBCPs) and quantum anomalous Hall effect (QAHE) have attracted the attention of both theoretical and experimental researchers in recent years. Based on first-principle calculations, we find that the FeB$_2$ monolayer is a nonmagnetic semimetal with QBCPs at $K$. Through symmetry analysis and $\mathbf{k}\cdot\mathbf{p}$ invariant theory, we find that the QBCP is not protected by rotation symmetry and consists of two Dirac points with same chirality (Berry phase of $2\pi$). Once introducing Coulomb interactions, we find that there is a spontaneous-time-reversal-breaking instability of the spinful QBCPs, which gives rise to a $C=2$ QAH insulator with orbital moment ordering.

Topological Insulating Phase in Single-Layer Pentagonal Covalent Organic Frameworks: A Reticular Design Using Metal Phthalocyanine

Organic topological insulators have received tremendous attention lately due to their designability and highly chemical diversity. In this work, we propose an isoreticular series of single-layer π-conjugated covalent organic frameworks with the Cairo pentagonal tiling, termed mcm-ZnPc-1, mcm-ZnPc-2, mcm-ZnPc-3, and mcm-ZnPc-3N, where mcm is the crystal net’s symbol and ZnPc stands for zinc phthalocyanine. First-principles calculations using density functional theory show that mcm-ZnPc-1 and mcm-ZnPc-2 are trivial insulators with band gaps of 0.96 and 1.18 eV, respectively. Interestingly, expanding the π-conjugated system of mcm-ZnPc-2, followed by a chemical substitution, results in mcm-ZnPc-3 and mcm-ZnPc-3N with a distorted Dirac point and a quadratic band-crossing point in their band structures, respectively. The topological analyses indicate that mcm-ZnPc-3N is a Z2 topological insulator with a nontrivial gapless edge state. Our tight-binding model for the pentagonal lattice suggests that the topologically trivial-to-nontrivial transition can be attributed to the sufficient electronic coupling between two adjacent linkers and to the fact that the band-crossing point locates at the Fermi level. Remarkably, we show that replacing Zn in mcm-ZnPc-3N with Cd and Hg can substantially enlarge the nontrivial band gap from 0.3 meV up to 10 meV due to strong spin–orbit coupling strengths although a biaxial strain is necessary to tune the Fermi level of mcm-HgPc-3N in order to enter its topological insulating phase. Our proposed materials are predicted to be dynamically, thermally, and mechanically stable, thus paving the way for their experimental realization. This work not only introduces the pentagonal lattice as a building motif for framework structures but also demonstrates a strategy to engineer the exotic band structure of organic materials.

Twisted bilayer graphene in a parallel magnetic field

We study the effect of an in-plane magnetic field on the noninteracting dispersion of twisted bilayer graphene. Our analysis is rooted in the chirally symmetric continuum model, whose zero-field band structure hosts exactly flat bands and large energy gaps at the magic angles. At the first magic angle, the central bands respond to a parallel field by forming a quadratic band crossing point (QBCP) at the moir\'e Brillouin zone center. Over a large range of fields, the dispersion is invariant with an overall scale set by the magnetic field strength. For deviations from the magic angle and for realistic interlayer couplings, the motion and merging of the Dirac points lying near charge neutrality are discussed in the context of the symmetries, and we show that small magnetic fields are able to induce a qualitative change in the energy spectrum. We conclude with a discussion on the possible ramifications of our study to the interacting ground states of twisted bilayer graphene systems.

Quantifying the fragility of unprotected quadratic band crossing points

We examine a basic lattice model of interacting fermions that exhibits quadratic band crossing points (QBCPs) in the non-interacting limit. In particular, we consider spinless fermions on the honeycomb lattice with nearest neighbor hopping $t$ and third-nearest neighbor hopping $t''$, which exhibits fine-tuned QBCPs at the corners of the Brillouin zone for ${t'' = t/2}$. In this situation, the density of states remains finite at the Fermi level of the half-filled band and repulsive nearest-neighbor interactions $V$ lead to a charge-density-wave (CDW) instability at infinitesimally small $V$ in the random-phase approximation or mean-field theory. We examine the fragility of the QBCPs against dispersion renormalizations in the ${t\mbox{-}t''\mbox{-}V}$ model using perturbation theory, and find that the $t''$-value needed for the QBCPs increases with $V$ due to the hopping renormalization. However, the instability toward CDW formation always requires a nonzero threshold interaction strength, i.e., one cannot fine-tune $t''$ to recover the QBCPs in the interacting system. These perturbative arguments are supported by quantum Monte Carlo simulations for which we carefully compare the corresponding threshold scales at and beyond the QBCP fine-tuning point. From this analysis, we thus gain a quantitative microscopic understanding of the fragility of the QBCPs in this basic interacting fermion system.

From quantum anomalous Hall phases to topological metals in interacting decorated honeycomb lattices

An analysis of the stability of topological states induced by Coulomb repulsion on decorated honeycomb lattices is presented. Based on a mean-field treatment of a spinless extended Hubbard model on the decorated honeycomb lattice we show how the quantum anomalous Hall (QAH) phase is a robust topological phase which emerges at various electron fillings and involves either quadratic band crossing points (QBCP) or Dirac points. The topological QAH phase is also found to be most stable against thermal fluctuations up to moderate temperatures when the Coulomb repulsion is maximally frustrated and at half filling. We show how a topological metal can be induced from the QAH for certain electron doping ranges. Electrons on the Fermi surface of such metallic states are characterized by having nonzero Berry phases which can give rise to nonquantized intrinsic Hall conductivities.

Search for plasmons in isotropic Luttinger semimetals

Luttinger semimetals include materials like gray tin (α-Sn) and mercury telluride, which are three-dimensional gapless semiconductors having a quadratic band crossing point (QBCP). Due to a growing interest in QBCPs and new experimental efforts, it is essential to study the finite-temperature properties of such systems. In this paper, we investigate the emergence of plasmons in the presence of Coulomb interactions in isotropic Luttinger semimetals, for zero as well as generic nonzero temperatures. When the Fermi level lies right at the QBCP, which is the point where twofold degenerate conduction and valence bands touch each other quadratically, we find that plasmons cannot appear at zero temperature. However, for nonzero temperatures, thermal plasmons are generated. Whether they are long-lived or not depends on the values of temperature, effective electron mass and effective fine-structure constant, and the number of fermion flavors. We also numerically estimate the behavior of the inelastic scattering rate at nonzero temperatures, as a function of energy, where the signatures of the QBCP thermal plasmons show up as a sharp peak. Our results will thus serve as a guide to experimental probes on these systems.

Fate of interaction-driven topological insulators under disorder

We analyze the effect of disorder on the weak-coupling instabilities of 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 non-trivial order sets in. Strong disorder can also cause a topological phase transition into a topologically trivial insulating state.

Engineering interaction-induced topological insulators in a3×3substrate-induced honeycomb superlattice

We consider a system of spinless fermions on the honeycomb lattice with substrate-induced modulated electrostatic potentials tripling the unit cell. The resulting non-Abelian SU(2) gauge fields act cooperatively to realize a quadratic band crossing point (QBCP). Using a combination of mean-field theory and renormalization group techniques, we show that in the QBCP regime, arbitrarily weak repulsive electronic interactions drive the system into the quantum anomalous Hall state. This proves that substrate-induced local voltages are an effective knob to induce the spontaneous formation of a topological quantum phase.

Interaction-driven topological and nematic phases on the Lieb lattice

We show that topological states are often developed in two-dimensional semimetals with quadratic band crossing points (BCPs) by electron–electron interactions. To illustrate this, we construct a concrete model with the BCP on an extended Lieb lattice and investigate the interaction-driven topological instabilities. We find that the BCP is marginally unstable against infinitesimal repulsions. Depending on the interaction strengths, topological quantum anomalous/spin Hall, charge nematic, and nematic-spin-nematic phases develop separately. Possible physical realizations of quadratic BCPs are provided.

Electronic structure of (LaNiO3)2/(LaAlO3)Nheterostructures grown along [111

The electronic structure of a LaNiO${}_{3}$ bilayer grown along the [111] direction and confined between insulating layers of LaAlO${}_{3}$ is theoretically investigated using a combination of first-principles calculations and effective multiorbital lattice models. The local density approximation (LDA) band structure is well reproduced by a tight-binding model for the Ni ${e}_{g}$ orbitals defined on the buckled honeycomb lattice. We highlight peculiar properties of this model, which include almost flat bands as well as linear and quadratic band-crossing points. The effect of local correlations is discussed within the LDA$+U$ scheme and within the Hartree-Fock approximation for interacting multiorbital lattice models. Over a wide range of interaction parameters, we find that a ferromagnetic phase is energetically favored. We discuss the possibility of additional orbital order, which could stabilize a spontaneous Chern insulator with chiral edge modes or a staggered orbital phase with a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ reconstruction of the unit cell. By studying an interacting nickel-oxygen lattice model, we find that the stability of these orbitally ordered phases also depends on the value of the charge-transfer energy. Controlling the charge-transfer energy might therefore be an important step towards engineering exotic electronic phases in certain classes of oxide heterostructures.

Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2D Fermi Systems with a Quadratic Band Crossing

We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the noninteracting level, we show that a QBCP exists and is topologically stable for a Berry flux +/-2pi 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 short-range 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.