Merging Of Dirac Points Research Articles

Merging of Dirac points through uniaxial modulation on an optical lattice

We analyze the scenario of modulating the potential strength of bound atoms in an optical honeycomb lattice patterned by an electric field to emulate uniaxial strain. This modulation can be achieved by a combination of the strength of the patterned electric field and gauge vector effects using the Floquet approach. We show that such a modulation allows one to follow through a topological transition between a semi-metal and a band insulator, when two non-equivalent K points merge as a function of the electric field strength. We explicitly compute the wavefunctions for the moving K points and the Chern numbers up to the transition. Anisotropic effective masses and the insulating gap are described close to the semimetal-insulator transition.

Magnetic Ordering, Anomalous Lifshitz Transition, and Topological Grain Boundaries in Two-Dimensional Biphenylene Network.

We study the electronic properties of a new planar carbon crystal formed through networking biphenylene molecules. Novel electronic features among carbon materials such as zone-center saddle point and peculiar type-II Dirac fermionic states are shown to exist in the low-energy electronic spectrum. The type-II state here has a nearly flat branch and is close to a transition to type I. Possible magnetic instabilities related to low-energy bands are discussed. Furthermore, with a moderate uniaxial strain, a pair of Dirac points merge with the zone-center saddle point, realizing concurrent Lifshitz transitions of van Hove singularity as well as pair annihilation of the Dirac fermions. A new effective Hamiltonian encompassing all distinctive low-energy states is constructed, revealing a finite winding number of the pseudospin texture around the Dirac point, quantized Zak phases, and topological grain boundary states.

Electric field induced band tuning, optical and thermoelectric responses in tetragonal germanene: a theoretical approach.

In this article, we have systematically explored the electronic, optical and thermoelectric properties of tetragonal germanene (T-Ge) using first principles calculations. The ground state geometry of pristine T-Ge is buckled and exhibits nodal line semi-metallic behaviour. In addition, we have proposed a tight binding (TB) model Hamiltonian that efficiently explains the emergence of double Dirac points at the Fermi level of T-Ge. Furthermore, a hopping relation has been explored at which both Dirac points merge and then annihilate resulting in a direct band gap at the Γ point. To exploit the buckling of the system, we have employed a transverse electric field, which invariably breaks the sublattice symmetry and removes the degeneracies at the Fermi surface. Furthermore, the band gap at the Dirac points varies linearly with the external electric field strength. Our TB Hamiltonian adequately satisfies the first principles results even in the presence of an external electric field. Moreover, we have found that T-Ge offers efficient tuning of band gaps at the Dirac points compared to other buckled systems viz. hexagonal silicene and germanene. In addition, the optical behaviour of T-Ge has been explained in accordance with the electronic states of the system. The strong optical responses in a low energy region make the material efficient for optical nanodevice applications. Moreover, T-Ge shows relatively better thermoelectric behaviour than graphene. Therefore, the external electric field induced tunable band gap and intriguing low energy optical signals pave the way to choose T-Ge as a smart choice for optoelectronic device applications. Finally we have suggested probable routes for experimental realization of the T-Ge structure.

Signatures of merging Dirac points in optics and transport

We consider the optical and transport properties in a model two-dimensional Hamiltonian which describes the merging of two Dirac points. At low energy, in the presence of an energy gap parameter $\Delta$, there are two distinct Dirac points with linear dispersion, these are connected by a saddle point at higher energy. As $\Delta$ goes to zero, the two Dirac points merge and the resulting dispersion exhibits semi-Dirac behaviour which is quadratic in the $x$-direction ("nonrelativistic") and linear the $y$-direction ("relativistic").In the clean limit for each direction ($x,y$) the contribution of the intraband and interband optical transitions are both given by universal functions of photon energy $\Omega$ and chemical potential $\mu$ normalized to the energy gap. We provide analytic formulas for both small and large $\Omega/2\Delta$ and $\mu/\Delta$ limits. These define, respectively, Dirac and semi-Dirac-like regions. For $\Omega/2\Delta$ and $\mu/\Delta$ of order one, there are deviations from these asymptotic behaviors. Considering optics and also transport, such as dc conductivity, thermal conductivity and the Lorenz number, such deviations provide signatures of the evolution from the Dirac to the semi-Dirac regime as the gap $\Delta$ is varied.

Non-Hermitian perspective of the band structure in heavy-fermion systems

We analyze a two-dimensional Kondo lattice model with special emphasis on non-Hermitian properties of the single-particle spectrum, following a recent proposal by Kozii and Fu. Our analysis based on the dynamical mean-field theory elucidates that the single-particle spectral weight shows the exceptional points (EPs). Correspondingly, the spectral weight exhibits the band touching, resulting in a structure similar to the "Fermi arc". Furthermore, we find an intriguing phenomenon arising from the periodicity of the lattice. The EPs generated by two distinct Dirac points merge and change into a hybrid point which vanishes as the exchange interaction is increased. Accordingly, the paramagnetic phase in the low temperature region shows a significant difference from non-interacting fermions: the imaginary part of the self-energy yields the "Fermi loop" without any defective points.

Dynamical merging of Dirac points in the periodically driven Kitaev honeycomb model

We study the effect of a half wave rectified sinusoidal electromagnetic (EM) wave on the Kitaev honeycomb model with an additional magneto-electric coupling term {arising due to induced polarization of the bonds. Within the framework of Floquet analysis, we show that merging of a pair of Dirac points in the gapless region of the Kitaev model leading to a semi-Dirac spectrum is indeed possible} by externally varying the amplitude and the phase of the EM field.

Merging of the Dirac points in electronic artificial graphene

Theory predicts that graphene under uniaxial compressive strain in an armchair direction should undergo a topological phase transition from a semimetal into an insulator. Due to the change of the hopping integrals under compression, both Dirac points shift away from the corners of the Brillouin zone towards each other. For sufficiently large strain, the Dirac points merge and an energy gap appears. However, such a topological phase transition has not yet been observed in normal graphene (due to its large stiffness) neither in any other electronic system. We show numerically and analytically that such a merging of the Dirac points can be observed in electronic artificial graphene created from a two-dimensional electron gas by application of a triangular lattice of repulsive antidots. Here, the effect of strain is modeled by tuning the distance between the repulsive potentials along the armchair direction. Our results show that the merging of the Dirac points should be observable in a recent experiment with molecular graphene.

Band structure of ABC-trilayer graphene superlattice

We investigate the effect of one-dimensional periodic potentials on the low energy band structure of ABC trilayer graphene first by assuming that all the three layers have the same potential. Extra Dirac points having the same electron hole crossing energy as that of the original Dirac point are generated by superlattice potentials with equal well and barrier widths. When the potential height is increased, the numbers of extra Dirac points are increased. The dispersions around the Dirac points are not isotropic. It is noted that the dispersion along the ky direction for kx = 0 oscillates between a non-linear dispersion and a linear dispersion when the potential height is increased. When the well and barrier widths are not identical, the symmetry of the conduction and valence bands is broken. The extra Dirac points are shifted either upward or downward depending on the barrier and well widths from the zero energy, while the position of the central Dirac point oscillates with the superlattice potential height. By considering different potentials for different layers, extra Dirac points are generated not from the original Dirac points but from the valleys formed in the energy spectrum. Two extra Dirac points appear from each pair of touched valleys, so four Dirac points appeared in the spectrum at particular barrier height. By increasing the barrier height of superlattice potential two Dirac points merge into the original Dirac point. This emerging and merging of extra Dirac points is different from the equal potential case.

Moving and merging of Dirac points on a square lattice and hidden symmetry protection

First, we study a square fermionic lattice that supports the existence of massless Dirac fermions, where the Dirac points are protected by a hidden symmetry. We also consider two modified models with a staggered potential and the diagonal hopping terms, respectively. In the modified model with a staggered potential, the Dirac points exist in some range of magnitude of the staggered potential, and move with the variation of the staggered potential. When the magnitude of the staggered potential reaches a critical value, the two Dirac points merge. In the modified model with the diagonal hopping terms, the Dirac points always exist and just move with the variation of amplitude of the diagonal hopping. We develop a mapping method to find hidden symmetries evolving with the parameters. In the two modified models, the Dirac points are protected by this kind of hidden symmetry, and their moving and merging process can be explained by the evolution of the hidden symmetry along with the variation of the corresponding parameter.

Merging of Dirac points and Floquet topological transitions in ac-driven graphene

We investigate the effect of an in-plane AC electric field coupled to electrons in the honeycomb lattice and show that it can be used to manipulate the Dirac points of the electronic structure. We find that the position of the Dirac points can be controlled by the amplitude and the polarization of the field for high frequency drivings, providing a new platform to achieve their merging, a topological transition which has not been observed yet in electronic systems. Importantly, for lower frequencies we find that the multi-photon absorptions and emissions processes yield the creation of additional pairs of Dirac points. This provides an additional method to achieve the merging transition by just tuning the frequency of the driving. Our approach, based on Floquet formalism, is neither restricted to specific choice of amplitude or polarization of the field, nor to a low energy approximation for the Hamiltonian.

Interlayer magnetoresistance in multilayer Dirac electron systems: motion and merging of Dirac cones

We theoretically study the effect of the motion and the merging of Dirac cones on the interlayer magnetoresistance in multilayer graphene-like systems. This merging, which can be induced by a uniaxial strain, gives rise in a monolayer Dirac electron system to a topological transition from a semi-metallic phase to an insulating phase whereby Dirac points disappear. Based on a universal Hamiltonian, proposed to describe the motion and the merging of Dirac points in two-dimensional Dirac electron crystals, we calculate the interlayer conductivity of a stack of deformed graphene-like layers using the Kubo formula in the quantum limit where only the contribution of the n = 0 Landau level is relevant. A crossover from a negative to a positive interlayer magnetoresistance is found to take place as the merging is approached. This sign change of the magnetoresistance can also result from a coupling between the Dirac valleys, which is enhanced as the magnetic field amplitude increases. Our results describe the behavior of the magnetotransport in the organic conductor α-(BEDT)2I3 and in a stack of deformed graphene-like systems. The latter can be simulated by optical lattices or microwave experiments in which the merging of Dirac cones can be observed.

Friedel oscillations at the Dirac cone merging point in anisotropic graphene and graphenelike materials

We study the Friedel oscillations induced by a localized impurity in an anisotropic graphenelike structure. We focus on the limit when the two inequivalent Dirac points merge. We find that, in this limit, the Friedel oscillations manifest very peculiar features, such as a strong asymmetry and an atypical inverse square-root decay. Our calculations are performed using both a $T$-matrix approximation and a tight-binding exact diagonalization technique. They allow us to numerically obtain the local density of states as a function of energy and position as well as an analytical form of the Friedel oscillations in the continuum limit. The two techniques yield results that are in excellent agreement, confirming the accuracy of such methods to approach this problem.

Interaction-induced merging of Dirac points in non-Abelian optical lattices

We study the properties of an ultracold Fermi gas loaded in a square optical lattice and subjected to an external and classical non-Abelian gauge field. We calculate the energy spectrum of the system and show that the Dirac points in the energy spectrum will remain quite stable under onsite interaction of certain strength. Once the on-site interaction grows stronger than a critical value, the Dirac points will no longer be stable and merge into a single hybrid point. This mergence implies a quantum phase transition from a semimetallic phase to a band insulator. The on-site interaction between ultracold fermions could be conveniently controlled by Feshbach resonances in current experiments. We proposed that this remarkable interaction induced mergence of Dirac points may be observed in the ultracold fermi gas experiments.

Merging Dirac points and topological phase transitions in the tight-binding model on the generalized honeycomb lattice

Moving, merging and annihilating Dirac points are studied theoretically in the tight-binding model on honeycomb lattice with up-to third-nearest-neighbor hoppings. We obtain a rich phase diagram of the topological phase transitions in the parameter space of direction-dependent hoppings. We obtain the conditions for the three Dirac points to merge and for the tricritical points. We find that only very small third-nearest-neighbor hoppings are enough for the existence of the merging of three-Dirac-points and the tricritical points, if the system is sufficiently anisotropic. The density of states is obtained to be $D(\epsilon) \propto |\epsilon|^{1/3}$ when three Dirac points merge, and $D(\epsilon) \propto |\epsilon|^{1/4}$ at the tricritical points. It is possible to realize these topological phase transitions in the ultracold atoms on the optical lattice, strained monolayer graphene or strained bilayer graphene.

Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice

Dirac points are central to many phenomena in condensed-matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the electrons behave as relativistic Dirac fermions. In solids, the rigid structure of the material determines the mass and velocity of the electrons, as well as their interactions. A different, highly flexible means of studying condensed-matter phenomena is to create model systems using ultracold atoms trapped in the periodic potential of interfering laser beams. Here we report the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice. Using momentum-resolved interband transitions, we observe a minimum bandgap inside the Brillouin zone at the positions of the two Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking inversion symmetry. Moreover, changing the lattice anisotropy allows us to change the positions of the Dirac points inside the Brillouin zone. When the anisotropy exceeds a critical limit, the two Dirac points merge and annihilate each other-a situation that has recently attracted considerable theoretical interest but that is extremely challenging to observe in solids. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials in which the topology of the band structure is crucial, but also provide an avenue to exploring many-body phases resulting from the interplay of complex lattice geometries with interactions.

Semi-Dirac point in the Hofstadter spectrum

The spectrum of tight binding electrons on a square lattice with half a magnetic flux quantum per unit cell exhibits two Dirac points at the band center. We show that, in the presence of an additional uniaxial staggered potential, this pair of Dirac points may merge into a single one, with a topological transition towards a gapped phase. At the transition, the spectrum is linear in one direction and quadratic in the other one (a spectrum recently named "hybrid" or "semi-Dirac"). This transition is studied in the framework of a general Hamiltonian describing the merging of Dirac points. The possibility of creating gauge fields for cold atoms in optical lattices may offer the first opportunity to observe this merging of Dirac points and the hybrid dispersion relation.

A universal Hamiltonian for motion and merging of Dirac points in a two-dimensional crystal

We propose a simple Hamiltonian to describe the motion and the merging of Dirac points in the electronic spectrum of two-dimensional electrons. This merging is a topological transition which separates a semi-metallic phase with two Dirac cones from an insulating phase with a gap. We calculate the density of states and the specific heat. The spectrum in a magnetic field B is related to the resolution of a Schrodinger equation in a double well potential. They obey the general scaling law e_n \propto B^{2/3} f_n(Delta /B^{2/3}. They evolve continuously from a sqrt{n B} to a linear (n+1/2)B dependence, with a [(n+1/2)B]^{2/3} dependence at the transition. The spectrum in the vicinity of the topological transition is very well described by a semiclassical quantization rule. This model describes continuously the coupling between valleys associated with the two Dirac points, when approaching the transition. It is applied to the tight-binding model of graphene and its generalization when one hopping parameter is varied. It remarkably reproduces the low field part of the Rammal-Hofstadter spectrum for the honeycomb lattice.

Merging of Dirac points in a two-dimensional crystal

We study under which general conditions a pair of Dirac points in the electronic spectrum of a two-dimensional crystal may merge into a single one. The merging signals a topological transition between a semimetallic phase and a band insulator. We derive a universal Hamiltonian that describes the physical properties of the transition, which is controlled by a single parameter, and analyze the Landau-level spectrum in its vicinity. This merging may be observed in the organic salt $\ensuremath{\alpha}\ensuremath{-}{(\text{BEDT-TTF})}_{2}{\text{I}}_{3}$ or in an optical lattice of cold atoms simulating deformed graphene.