The ground state of the diatomic molecules in nature is inevitably bonding and its first excited state is antibonding. We demonstrate theoretically that for a pair of distant adatoms placed buried in 3D-Dirac semimetals, this natural order of the states can be reversed and antibonding ground state occurs at the lowest energy of the so-called bound states in the continuum. We propose experimental protocol with use of STM-tip to visualize the topographic map of the local density of states on the surface of the system to reveal the emerging Physics.Introduction.-Three-dimensional Dirac semimetals (3D-DSMs) such as Cd 3 As 2 and Na 3 Bi[1-5] represent novel class of functional materials constituting 3D analogous of gapless graphene [6][7][8]. The band structure of 3D semimetals contains the set of the so-called Dirac points in which conduction and valence bands touch and effective mass becomes zero. Around these points the dispersion of quasiparticles corresponds to those of massless relativistic Dirac particles which result in series of unusual properties of these materials such as linear magnetoresistance, unprecedented Shubnikov-de Haas oscillations and ultrahigh carrier mobility [9][10][11].In this work, we predict one more interesting feature of such materials. Namely, if we consider buried pair of distant adatoms in the bulk of a 3D-DSM as depicted at Fig.1, the ground state of this molecular system formed from bound states in the continuum (BICs) of the adatoms [12][13][14] will be characterized by antibondingtype orbital. This differs from the natural order in diatomic molecules where the ground state is of bondingtype in vast majority of cases and formation of antibonding ground state till now was reported only in systems of artificially fabricated InAs and Ge/Si p-type quantum dots for certain values of the inter-dot separations [15,16]. The behavior we report is a unique effect arising from long-range correlations between distant adatoms mediated by bulk fermions in 3D-DSMs. To detect the predicted effect, we propose to use the experimental approach developed in Ref. [17] for imaging isodensity contours of molecular states by scanning tunneling microscope (STM)-tip, as outlined at Fig.1.The Model.-For theoretical analysis of two adatoms buried inside 3D-DSM, as depicted at Fig.1, we employ an Anderson-like Hamiltonian [18][19][20]

We show that fermion charge fractionalization can take place in a recently proposed chiral gauge model for graphene even in the absence of Kekulé distortion in the graphene honeycomb lattice. In this model, electrons couple in a chiral way to a pseudomagnetic field with a vortex profile in such a way that it can be used to describe the influences of topological defects, such as disclinations, on the electronic states. We also extend the model by adding the coupling of fermions to an external magnetic field and show that the fermion charge can be fractionalized by means of only gauge potentials. It is shown that the chiral fermion charge can also have fractional value. We also relate the fractionalization of the fermion charge to the parity anomaly in an extended quantum electrodynamics, which involves vector and axial-vector gauge fields.

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Abstract. We solve the generalized relativistic harmonic oscillator in 1 + 1 dimensions in the presence of a minimal length. Using the momentum space representation, we explore all the possible signs of the potentials and discuss their bound-state solutions for fermions and antifermions. Furthermore, we also find an isolated solution from the Sturm-Liouville scheme. All cases already analyzed in the literature, are obtained as particular cases.