We study the RKKY interaction of magnetic impurities in the $\ensuremath{\alpha}\ensuremath{-}{\mathcal{T}}_{3}$ model which hosts pseudospin-1 fermions with two dispersive and one flat bands. By using the effective low-energy Hamiltonian we calculate the RKKY coupling for impurities placed on the same or different sublattices. We find that there are three types of interaction, which depend on the model parameter defining the relative strength of hoppings between sublattices, two of them can be reduced to graphene case while the third one is new and is due to the presence of a flat zero-energy band. We derive general analytical expressions for the RKKY interaction in terms of Mellin-Barnes type integrals and analyze different limiting cases. The cases of finite chemical potential and temperature, as well as asymptotic at large distances, are considered. We show that the interaction between impurities located at different rim sites displays a very strong temperature dependence at small doping being a direct consequence of the flat band. The subtleties of the theorem for signs of the RKKY interaction at zero doping, as applied to the ${\mathcal{T}}_{3}$ lattice, related to the existence of a dispersionless flat band are discussed.

RKKY interaction in a doped pseudospin-1 fermion system at finite temperature

Transition metal dichalcogenides (TMDCs) are ideal candidates to explore the manifestation of spin-valley physics under external stimuli. In this study, we investigate the influence of strain on the spin and orbital angular momenta, effective g-factors, and Berry curvatures of several monolayer TMDCs (Mo and W based) using a full ab initio approach. At the K-valleys, we find a surprising decrease of the conduction band spin expectation value for compressive strain, consequently increasing the dipole strength of the dark exciton by more than one order of magnitude (for ∼1%–2% strain variation). We also predict the behavior of direct excitons g-factors under strain: tensile (compressive) strain increases (decreases) the absolute value of g-factors. Strain variations of ∼1% modify the bright (A and B) excitons g-factors by ∼0.3 (0.2) for W (Mo) based compounds and the dark exciton g-factors by ∼0.5 (0.3) for W (Mo) compounds. Our predictions could be directly visualized in magneto-optical experiments in strained samples at low temperature. Additionally, our calculations strongly suggest that strain effects are one of the possible causes of g-factor fluctuations observed experimentally. By comparing the different TMDC compounds, we reveal the role of spin–orbit coupling (SOC): the stronger the SOC, the more sensitive are the spin-valley features under applied strain. Consequently, monolayer WSe2 is a formidable candidate to explore the role of strain on the spin-valley physics. We complete our analysis by considering the side valleys, Γ and Q points, and by investigating the influence of strain in the Berry curvature. In the broader context of valley- and strain-tronics, our study provides fundamental microscopic insights into the role of strain in the spin-valley physics of TMDCs, which are relevant to interpret experimental data in monolayer TMDCs as well as TMDC-based van der Waals heterostructures.

First-principles insights into the spin-valley physics of strained transition metal dichalcogenides monolayers

Flat-band systems are a promising platform for realizing exotic collective ground states with spontaneously broken symmetry because the electron-electron interactions dominate the kinetic energy. A state of particular interest would be the chased after interlayer-phase-coherent exciton condensate but the conventional treatments of the effect of thermal and quantum fluctuations suggest that this state might be either unstable or fragile. In this work, using double twisted bilayer graphene heterostructures as an example, we show that the quantum metric of the Bloch wave functions can strongly stabilize the exciton condensate and reverse the conclusion that one would draw using a conventional approach. The quantum metric contribution arises from interband terms, and flat-bands are most commonly realized by engineering multiband systems. Our results therefore suggest that in many practical situations the quantum metric can play a critical role in determining the stability of exciton condensates in double layers formed by two-dimensional systems with flat-bands.

Quantum-metric-enabled exciton condensate in double twisted bilayer graphene

The interaction energy of the indirect-exchange or Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic spins localized on lattice sites of the $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ model is calculated using linear response theory. In this model, the AB-honeycomb lattice structure is supplemented with C atoms at the centers of the hexagonal lattice. This introduces a parameter $\ensuremath{\alpha}$ for the ratio of the hopping integral from hub to rim and that around the rim of the hexagonal lattice. A valley and $\ensuremath{\alpha}$-dependent retarded Green's function matrix is used to form the susceptibility. Analytic and numerical results are obtained for undoped $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ when the chemical potential is finite and also in the presence of an applied magnetic field. We demonstrate the anisotropy of these results when the magnetic impurities are placed on the A, B, and C sublattice sites. Additionally, a comparison of the behavior of the susceptibility of $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathcal{T}}_{3}$ with graphene shows that there is a phase transition at $\ensuremath{\alpha}=0$.

Effect of magnetic field and chemical potential on the RKKY interaction in the α − T 3 lattice

Flat-band systems are a promising platform for realizing exotic collective ground states with spontaneously broken symmetry because the electron-electron interactions dominate over the kinetic energy. A collective ground state of particular interest is the chased-after exciton condensate (EC). However, in flat-band systems other collective ground states can compete with an EC phase, and the conventional treatment of the effect of thermal and quantum fluctuations predicts the EC phase should be unstable. Here, using double-twisted bilayer graphene (TBLG) heterostructures as an example, we show that, for realistic interaction strengths, the EC phase is favored with respect to other TBLG's phases---orbital magnetism and superconductivity---when the TBLGs have opposite doping, and that the quantum metric of the Bloch wave functions stabilizes the EC, reversing the conclusion that would be drawn from the conventional approach in which quantum metric contributions are neglected. Our results suggest that the quantum metric plays a critical role in determining the stability of exciton condensates in double layers formed by systems with flat bands. 

/pdf/quantum-metric-enabled-exciton-condensate-in-double-twisted-28hjzt3p.pdf

In many situations, excitons---bound particle-hole pairs above an insulating ground state---carry an electric dipole moment, allowing them to be manipulated via coupling to an electric field. For two-dimensional systems, we demonstrate that this property of an exciton is uniquely determined by the quantum geometry of its eigenstates, and demonstrate its intimate connection with a quantity which we call the quantum geometric dipole. We demonstrate that this quantity arises naturally in the semiclassical equations of motion of an exciton in an electric field, adding a term additional to the anomalous velocity coming from Berry's curvature. In a uniform electric field, this contributes a drift velocity to the exciton akin to that expected for excitons in crossed electric and magnetic fields, even in the absence of a real magnetic field. We compute the quantities relevant to semiclassical exciton dynamics for several interesting examples of bilayer systems with weak interlayer tunneling and Fermi energy in a gap, where the exciton may be sensibly described as a two-body problem. These quantities include the exciton dispersion, its quantum geometric dipole, and its Berry's curvature. For a simple example of two gapped-graphene layers in a vanishing magnetic field, we demonstrate that there is a nonvanishing quantum geometric dipole when the layers are different, e.g., have different gaps, but vanishes when the layers are identical. We further analyze examples in the presence of magnetic fields, allowing us to examine cases involving graphene, in which a gap is opened by Landau level splitting. Heterostructures involving transition metal dichalcogenides materials are also considered. In each case, the quantum geometric dipole and Berry's curvatures play out in different ways. In some cases, the lowest energy exciton state is found to reside at finite momentum, with interesting possible consequences for Bose condensation in these systems. Additionally, we find situations in which the quantum geometric dipole increases monotonically with exciton momentum, suggesting that the quantum geometry can be exploited to produce photocurrents from initially bound excitons with electric fields, without the need to overcome an effective barrier via tunneling or thermal excitation. We speculate on further possible effects of the semiclassical dynamics in geometries where the constituent layers are subject to the same or different electric fields.

Quantum geometric exciton drift velocity

We study RKKY interactions for magnetic impurities on graphene in situations where the electronic spectrum is in the form of Landau levels. Two such situations are considered: non-uniformly strained graphene, and graphene in a real magnetic field. RKKY interactions are enhanced by the lowest Landau level, which is shown to form electron states binding with the spin impurities and add a strong non-perturbative contribution to pairwise impurity spin interactions when their separation $R$ no more than the magnetic length. Beyond this interactions are found to fall off as $1/R^3$ due to perturbative effects of the negative energy Landau levels. Based on these results, we develop simple mean-field theories for both systems, taking into account the fact that typically the density of states in the lowest Landau level is much smaller than the density of spin impurities. For the strain field case, we find that the system is formally ferrimagnetic, but with very small net moment due to the relatively low density of impurities binding electrons. The transition temperature is nevertheless enhanced by them. For real fields, the system forms a canted antiferromagnet if the field is not so strong as to pin the impurity spins along the field. The possibility that the system in this latter case supports a Kosterlitz-Thouless transition is discussed.

RKKY interactions in graphene Landau levels

RKKY Interaction in Graphene Landau Levels

Plasmons are usually described in terms of macroscopic quantities such as electric fields and currents. However, as fundamental excitations of metals, they are also quantum objects with internal structure. We demonstrate that this can induce an intrinsic dipole moment which is tied to the quantum geometry of the Hilbert space of plasmon states. This quantum geometric dipole offers a unique handle for manipulation of plasmon dynamics via density modulations and electric fields. As a concrete example, we demonstrate that scattering of plasmons with a nonvanishing quantum geometric dipole from impurities is nonreciprocal, skewing in different directions in a valley-dependent fashion. This internal structure can be used to control plasmon trajectories in two dimensional materials.

Jinlyu Cao

Papers

Quantum geometric exciton drift velocity

RKKY interactions in graphene Landau levels

RKKY Interaction in Graphene Landau Levels

Quantum Internal Structure of Plasmons.