In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hoppin...In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hopping integrals which are the physical origin of cohesion of atoms. For thesimple case of s-valent system, the inversion of the many-body potentials are discussed in detail by using the lattice inversion method.展开更多
An analytic expression for π and π* electronic structure of graphene is derived within the tight-binding approximation. Including up to fifth-nearest neighbors, the tight-binding description of electronic dispersio...An analytic expression for π and π* electronic structure of graphene is derived within the tight-binding approximation. Including up to fifth-nearest neighbors, the tight-binding description of electronic dispersion quite accurately reproduces the first-principle calculation result over the entire Brillouin zone. The maximal deviation of the fifth-nearest tight-binding result from the first-principle result is only 6 meV for π band, and 25 meV for π* band. This 25 meV deviation is only one-tenth of the maximal deviation of the third-nearest tight-binding result. It is more important that the fitted parameters exponentially approach to zero as the distance between interacting atoms increases.展开更多
Dirac states composed of Px,y orbitals have been reported in many two-dimensional (2D) systems with honeycomb lattices recently. Their potential importance has aroused strong interest in a comprehensive understandin...Dirac states composed of Px,y orbitals have been reported in many two-dimensional (2D) systems with honeycomb lattices recently. Their potential importance has aroused strong interest in a comprehensive understanding of such states. Here, we construct a four-band tight-binding model for the Px,y-orbital Dirac states considering both the nearest neighbor hopping interactions and the lattice-buckling effect. We find that Px,y-orbital Dirac states are accompanied with two addi- tional narrow bands that are flat in the limit of vanishing n bonding, which is in agreement with previous studies. Most importantly, we analytically obtain the linear dispersion relationship between energy and momentum vector near the Dirac cone. We find that the Fermi velocity is determined not only by the hopping through n bonding but also by the hopping through ~ bonding of Px,y orbitals, which is in contrast to the case of pz-orbital Dirac states. Consequently, Px,y-orbital Dirac states offer more flexible engineering, with the Fermi velocity being more sensitive to the changes of lattice constants and buckling angles, if strain is exerted. We further validate our tight-binding scheme by direct first-principles calcula- tions of model-materials including hydrogenated monolayer Bi and Sb honeycomb lattices. Our work provides a more in-depth understanding of Px,y-orbital Dirac states in honeycomb lattices, which is useful for the applications of this family of materials in nanoelectronics.展开更多
In this paper we investigate the influence of the next-nearest-neighbor coupling on the spectrum of plasmon excitations in graphene. The nearest-neighbor tight-binding model was previously considered to calculate the ...In this paper we investigate the influence of the next-nearest-neighbor coupling on the spectrum of plasmon excitations in graphene. The nearest-neighbor tight-binding model was previously considered to calculate the plasmon spectrum in graphene [1]. We extend these results to the next-nearest-neighbor tight-binding model. As in the calculation of the nearest-neighbor model, our approach is based on the numerical calculation of the dielectric function and the loss function. We compare the plasmon spectrum of the two models and discuss the differences in the dispersion.展开更多
The time-dependent density functional-based tight-bind (TD-DFTB) method is implemented on the multi-core and the graphical processing unit (GPU) system for excited state calcu-lations of large system with hundreds...The time-dependent density functional-based tight-bind (TD-DFTB) method is implemented on the multi-core and the graphical processing unit (GPU) system for excited state calcu-lations of large system with hundreds or thousands of atoms. Sparse matrix and OpenMP multithreaded are used for building the Hamiltonian matrix. The diagonal of the eigenvalue problem in the ground state is implemented on the GPUs with double precision. The GPU- based acceleration fully preserves all the properties, and a considerable total speedup of 8.73 can be achieved. A Krylov-space-based algorithm with the OpenMP parallel and CPU acceleration is used for finding the lowest eigenvalue and eigenvector of the large TDDFT matrix, which greatly reduces the iterations taken and the time spent on the excited states eigenvalue problem. The Krylov solver with the GPU acceleration of matrix-vector product can converge quickly to obtain the final result and a notable speed-up of 206 times can be observed for system size of 812 atoms. The calculations on serials of small and large systems show that the fast TD-DFTB code can obtain reasonable result with a much cheaper computational requirement compared with the first-principle results of CIS and full TDDFT calculation.展开更多
Density functional based tight binding (DFTB) model is employed to study the sp3-to-sp2 transformation of diamond-like carbon at elevated temperatures. The understanding could lead to the direct-growth of graphene on ...Density functional based tight binding (DFTB) model is employed to study the sp3-to-sp2 transformation of diamond-like carbon at elevated temperatures. The understanding could lead to the direct-growth of graphene on a wide variety of substrates.展开更多
The discovery of graphene has led to the devotion of intensive efforts,theoretical and experimental,to produce two-dimensional(2D)materials that can be used for developing functional materials and devices.This work pr...The discovery of graphene has led to the devotion of intensive efforts,theoretical and experimental,to produce two-dimensional(2D)materials that can be used for developing functional materials and devices.This work provides a brief review of the recent developments in the lattice models of 2D Dirac materials and their relevant real material counterparts that are crucial for understanding the origins of 2D Dirac cones in electronic band structures as well as their material design and device applications.We focus on the roles of lattice symmetry,atomic orbital hybridization,and spin-orbit coupling in the presence of a Dirac cone.A number of lattice models,such as honeycomb,kagome,ruby,star,Cairo,and line-centered honeycomb,with different symmetries are reviewed based on the tight-binding approach.Inorganic and organic 2D materials,theoretically proposed or experimentally synthesized to satisfy these 2D Dirac lattice models,are summarized.展开更多
We propose a new generalized Su–Schrieffer–Heeger model with hierarchical long-range hopping based on a onedimensional tetratomic chain. The properties of the topological states and phase transition, which depend on...We propose a new generalized Su–Schrieffer–Heeger model with hierarchical long-range hopping based on a onedimensional tetratomic chain. The properties of the topological states and phase transition, which depend on the cointeraction of the intracell and intercell hoppings, are investigated using the phase diagram of the winding number. It is shown that topological states with large positive/negative winding numbers can readily be generated in this system. The properties of the topological states can be verified by the ring-type structures in the trajectory diagram of the complex plane. The topological phase transition is strongly related to the opening(closure) of an energy bandgap at the center(boundaries) of the Brillouin zone. Finally, the non-zero-energy edge states at the ends of the finite system are revealed and matched with the bulk–boundary correspondence.展开更多
Molecular simulation finds application in a wide range of research fields based on life and materials sciences.It helps comprehend and predict the chemical and physical properties of substances;thus,it is useful in di...Molecular simulation finds application in a wide range of research fields based on life and materials sciences.It helps comprehend and predict the chemical and physical properties of substances;thus,it is useful in directing R&D and industrial production.In this special issue,we focus on molecular simulations in material sciences.Molecular simulation employs computational models from microscopic to mesoscopic levels,which is reflected in this special issue.For example,Liu et al.1 reported modulation of catalytic activity for CO2 hydrogenation using quantum density functional theory(DFT).Yin et al.2 parameterized a semiempirical density functional tight-binding(DFTB)model to study deposition of carbon on copper surface.At the atomic level,Ren et al.展开更多
Analytical formulas for evaluating the relation of carrier density and Fermi energy for semiconductors with a tight-binding band have been proposed. The series expansions for a carrier density with fast convergency ha...Analytical formulas for evaluating the relation of carrier density and Fermi energy for semiconductors with a tight-binding band have been proposed. The series expansions for a carrier density with fast convergency have been obtained by means of a Bessel function. A simple and analytical formula for Fermi energy has been derived with the help of the Gauss integration method. The results of the proposed formulas are in good agreement with accurate numerical solutions. The formulas have been successfully used in the calculation of carrier density and Fermi energy in a miniband superlattice system. Their accuracy is in the order of 10-5.展开更多
The wave function temporal evolution on the one-dimensional (ID) lattice is considered in the tight-binding approxi- mation. The lattice consists of N equal sites and one impurity site (donor). The donor differs f...The wave function temporal evolution on the one-dimensional (ID) lattice is considered in the tight-binding approxi- mation. The lattice consists of N equal sites and one impurity site (donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time -t1/3 and its amplitude decreases ,- t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.展开更多
Taking into account both the intrinsic curvature and Zeeman effects, persistent currents in a multi-walled carbon nanotorus are explored by using a supercell method, within the tight-binding formalism. It is shown tha...Taking into account both the intrinsic curvature and Zeeman effects, persistent currents in a multi-walled carbon nanotorus are explored by using a supercell method, within the tight-binding formalism. It is shown that in the absence of the Zeeman effect, the intrinsic curvature induces some dramatic changes in energy spectra and thus changes in the shape of the flux-dependent current. A paramagnetism diamagnetism transition is observed. With consideration of the Zeeman splitting energy, the period of persistent current is destroyed, and a diamagnetism-paramagnetism transition is obtained at high magnetic field. In addition, we further explore the effect of external electric field energy (Eef) on persistent current, indicating that it changes unmonotonously with Eef.展开更多
Chemical modification and vertical stacking of two-dimensional materials are promising techniques for new nanoelectronic devices. We present Density Functional Tight Binding(DFTB) calculations of a field-effect device...Chemical modification and vertical stacking of two-dimensional materials are promising techniques for new nanoelectronic devices. We present Density Functional Tight Binding(DFTB) calculations of a field-effect device,based on lateral and vertical heterostructures of 2D materials. The device consists of a phosphorene channel protected by graphene sheets, which work as contacts and are divided into the source and drain by local hydrogenation of graphene, which gives insulating graphane. In this device composed of only 3 layers, single sheets of graphene-graphane can work as both leads and oxide gate, while also acting as protective layers for a phosphorene channel. We show how for perfect vd W heterostructures of graphane/phosphorene/graphane and graphene/phosphorene/graphene the Schottky barrier is deeply influenced by normal electric fields, and we characterize electronic transport of such a device. Finally, we characterize phosphorene channel doping and defects, which, at very high densities in the transport direction, enables transport inside the phosphorene bandgap.展开更多
In this work,we develop a new many-body potential for alpha-hafnium(α-Hf)based on the second moment approximation of tight-binding(TB-SMA)theory by introducing an additional Heaviside step function into the potential...In this work,we develop a new many-body potential for alpha-hafnium(α-Hf)based on the second moment approximation of tight-binding(TB-SMA)theory by introducing an additional Heaviside step function into the potential model and a new analytical scheme of density function.All the parameters of the new potential have been systematically evaluated by fitting to ground-state properties including cohesive energy,lattice constants,elastic constants,vacancy formation energy,structure stability and equation of state.By using the present model,the melting point,melt heat,thermal expansion coefficient,point defects,and low-index surface energies ofα-Hf were calculated through molecular dynamics simulations.Comparing with experiment observations from others,it is shown that these properties can be reproduced reasonably by the present model,some results being more consistent to the experimental data than those by previous suggested models.This indicates that this work is sutiable in TB-SMA potential for hexagonal close packed metals.展开更多
Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewe...Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients—one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum—are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.展开更多
文摘In this paper, we deduce the analytical form of many-body interatomic potentials based on the Green's function in tight-binding representation. The many-body potentials are expressed as the functions of the hopping integrals which are the physical origin of cohesion of atoms. For thesimple case of s-valent system, the inversion of the many-body potentials are discussed in detail by using the lattice inversion method.
基金Supported from the Scientific Research Foundation of Henan University of Science and Technology under Grant Nos.2008ZY036Student Research Training Program 2009178, and 2009183
文摘An analytic expression for π and π* electronic structure of graphene is derived within the tight-binding approximation. Including up to fifth-nearest neighbors, the tight-binding description of electronic dispersion quite accurately reproduces the first-principle calculation result over the entire Brillouin zone. The maximal deviation of the fifth-nearest tight-binding result from the first-principle result is only 6 meV for π band, and 25 meV for π* band. This 25 meV deviation is only one-tenth of the maximal deviation of the third-nearest tight-binding result. It is more important that the fitted parameters exponentially approach to zero as the distance between interacting atoms increases.
基金Project supported by the National Key Research and Development Projects of China(Grant No.2016YFA0202300)the National Natural Science Foundation of China(Grant No.61390501)+1 种基金the Science Fund from the Chinese Academy of Sciences(Grant No.XDPB0601)the US Army Research Office
文摘Dirac states composed of Px,y orbitals have been reported in many two-dimensional (2D) systems with honeycomb lattices recently. Their potential importance has aroused strong interest in a comprehensive understanding of such states. Here, we construct a four-band tight-binding model for the Px,y-orbital Dirac states considering both the nearest neighbor hopping interactions and the lattice-buckling effect. We find that Px,y-orbital Dirac states are accompanied with two addi- tional narrow bands that are flat in the limit of vanishing n bonding, which is in agreement with previous studies. Most importantly, we analytically obtain the linear dispersion relationship between energy and momentum vector near the Dirac cone. We find that the Fermi velocity is determined not only by the hopping through n bonding but also by the hopping through ~ bonding of Px,y orbitals, which is in contrast to the case of pz-orbital Dirac states. Consequently, Px,y-orbital Dirac states offer more flexible engineering, with the Fermi velocity being more sensitive to the changes of lattice constants and buckling angles, if strain is exerted. We further validate our tight-binding scheme by direct first-principles calcula- tions of model-materials including hydrogenated monolayer Bi and Sb honeycomb lattices. Our work provides a more in-depth understanding of Px,y-orbital Dirac states in honeycomb lattices, which is useful for the applications of this family of materials in nanoelectronics.
文摘In this paper we investigate the influence of the next-nearest-neighbor coupling on the spectrum of plasmon excitations in graphene. The nearest-neighbor tight-binding model was previously considered to calculate the plasmon spectrum in graphene [1]. We extend these results to the next-nearest-neighbor tight-binding model. As in the calculation of the nearest-neighbor model, our approach is based on the numerical calculation of the dielectric function and the loss function. We compare the plasmon spectrum of the two models and discuss the differences in the dispersion.
基金The project was supported by National Natural Science Foundation of China(21573201)the Ministry of Science and Technology of China(2016YFA0200604)and the Special Program for Applied Research on Super Computation of the National Nature Science Foundation of China-Guangdong Joint Fund(U1501501)~~
文摘The time-dependent density functional-based tight-bind (TD-DFTB) method is implemented on the multi-core and the graphical processing unit (GPU) system for excited state calcu-lations of large system with hundreds or thousands of atoms. Sparse matrix and OpenMP multithreaded are used for building the Hamiltonian matrix. The diagonal of the eigenvalue problem in the ground state is implemented on the GPUs with double precision. The GPU- based acceleration fully preserves all the properties, and a considerable total speedup of 8.73 can be achieved. A Krylov-space-based algorithm with the OpenMP parallel and CPU acceleration is used for finding the lowest eigenvalue and eigenvector of the large TDDFT matrix, which greatly reduces the iterations taken and the time spent on the excited states eigenvalue problem. The Krylov solver with the GPU acceleration of matrix-vector product can converge quickly to obtain the final result and a notable speed-up of 206 times can be observed for system size of 812 atoms. The calculations on serials of small and large systems show that the fast TD-DFTB code can obtain reasonable result with a much cheaper computational requirement compared with the first-principle results of CIS and full TDDFT calculation.
文摘Density functional based tight binding (DFTB) model is employed to study the sp3-to-sp2 transformation of diamond-like carbon at elevated temperatures. The understanding could lead to the direct-growth of graphene on a wide variety of substrates.
基金the National Natural Science Founda-tion of China(No.12074215)and Taishan Scholar Program of Shandong Province.
文摘The discovery of graphene has led to the devotion of intensive efforts,theoretical and experimental,to produce two-dimensional(2D)materials that can be used for developing functional materials and devices.This work provides a brief review of the recent developments in the lattice models of 2D Dirac materials and their relevant real material counterparts that are crucial for understanding the origins of 2D Dirac cones in electronic band structures as well as their material design and device applications.We focus on the roles of lattice symmetry,atomic orbital hybridization,and spin-orbit coupling in the presence of a Dirac cone.A number of lattice models,such as honeycomb,kagome,ruby,star,Cairo,and line-centered honeycomb,with different symmetries are reviewed based on the tight-binding approach.Inorganic and organic 2D materials,theoretically proposed or experimentally synthesized to satisfy these 2D Dirac lattice models,are summarized.
基金Project supported by the National Natural Science Foundation of China(Grant No.11405100)the Natural Science Basic Research Program in Shaanxi Province of China(Grant Nos.2022JZ-02,2020JM-507,and 2019JM-332)+1 种基金the Doctoral Research Fund of Shaanxi University of Science and Technology in China(Grant Nos.2018BJ-02 and 2019BJ-58)the Youth Innovation Team of Shaanxi Universities.
文摘We propose a new generalized Su–Schrieffer–Heeger model with hierarchical long-range hopping based on a onedimensional tetratomic chain. The properties of the topological states and phase transition, which depend on the cointeraction of the intracell and intercell hoppings, are investigated using the phase diagram of the winding number. It is shown that topological states with large positive/negative winding numbers can readily be generated in this system. The properties of the topological states can be verified by the ring-type structures in the trajectory diagram of the complex plane. The topological phase transition is strongly related to the opening(closure) of an energy bandgap at the center(boundaries) of the Brillouin zone. Finally, the non-zero-energy edge states at the ends of the finite system are revealed and matched with the bulk–boundary correspondence.
文摘Molecular simulation finds application in a wide range of research fields based on life and materials sciences.It helps comprehend and predict the chemical and physical properties of substances;thus,it is useful in directing R&D and industrial production.In this special issue,we focus on molecular simulations in material sciences.Molecular simulation employs computational models from microscopic to mesoscopic levels,which is reflected in this special issue.For example,Liu et al.1 reported modulation of catalytic activity for CO2 hydrogenation using quantum density functional theory(DFT).Yin et al.2 parameterized a semiempirical density functional tight-binding(DFTB)model to study deposition of carbon on copper surface.At the atomic level,Ren et al.
文摘Analytical formulas for evaluating the relation of carrier density and Fermi energy for semiconductors with a tight-binding band have been proposed. The series expansions for a carrier density with fast convergency have been obtained by means of a Bessel function. A simple and analytical formula for Fermi energy has been derived with the help of the Gauss integration method. The results of the proposed formulas are in good agreement with accurate numerical solutions. The formulas have been successfully used in the calculation of carrier density and Fermi energy in a miniband superlattice system. Their accuracy is in the order of 10-5.
文摘The wave function temporal evolution on the one-dimensional (ID) lattice is considered in the tight-binding approxi- mation. The lattice consists of N equal sites and one impurity site (donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time -t1/3 and its amplitude decreases ,- t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10674113)Program for New Century Excellent Talents in University of China (Grant No. NCET-06-0707)+2 种基金Foundation for the Author of National Excellent Doctoral Dissertation of China (Grant No. 200726)Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 06A071)partially by Hunan Provincial Innovation Foundation For Postgraduate (Grant No. S2008yjscx06)
文摘Taking into account both the intrinsic curvature and Zeeman effects, persistent currents in a multi-walled carbon nanotorus are explored by using a supercell method, within the tight-binding formalism. It is shown that in the absence of the Zeeman effect, the intrinsic curvature induces some dramatic changes in energy spectra and thus changes in the shape of the flux-dependent current. A paramagnetism diamagnetism transition is observed. With consideration of the Zeeman splitting energy, the period of persistent current is destroyed, and a diamagnetism-paramagnetism transition is obtained at high magnetic field. In addition, we further explore the effect of external electric field energy (Eef) on persistent current, indicating that it changes unmonotonously with Eef.
基金supported through the German Research Foundation within the project “Straintronics of imperfect quasi-two-dimensional materials: coplanar vs lamellar heterostructures” (CU 44/43).
文摘Chemical modification and vertical stacking of two-dimensional materials are promising techniques for new nanoelectronic devices. We present Density Functional Tight Binding(DFTB) calculations of a field-effect device,based on lateral and vertical heterostructures of 2D materials. The device consists of a phosphorene channel protected by graphene sheets, which work as contacts and are divided into the source and drain by local hydrogenation of graphene, which gives insulating graphane. In this device composed of only 3 layers, single sheets of graphene-graphane can work as both leads and oxide gate, while also acting as protective layers for a phosphorene channel. We show how for perfect vd W heterostructures of graphane/phosphorene/graphane and graphene/phosphorene/graphene the Schottky barrier is deeply influenced by normal electric fields, and we characterize electronic transport of such a device. Finally, we characterize phosphorene channel doping and defects, which, at very high densities in the transport direction, enables transport inside the phosphorene bandgap.
基金supported by the National Natural Science Foundation of China(Grant Nos.51071018 and 51271018)
文摘In this work,we develop a new many-body potential for alpha-hafnium(α-Hf)based on the second moment approximation of tight-binding(TB-SMA)theory by introducing an additional Heaviside step function into the potential model and a new analytical scheme of density function.All the parameters of the new potential have been systematically evaluated by fitting to ground-state properties including cohesive energy,lattice constants,elastic constants,vacancy formation energy,structure stability and equation of state.By using the present model,the melting point,melt heat,thermal expansion coefficient,point defects,and low-index surface energies ofα-Hf were calculated through molecular dynamics simulations.Comparing with experiment observations from others,it is shown that these properties can be reproduced reasonably by the present model,some results being more consistent to the experimental data than those by previous suggested models.This indicates that this work is sutiable in TB-SMA potential for hexagonal close packed metals.
基金supported by the Universidad del Pais Vasco/Euskal Herriko Unibertsitatea (Grant No. UFI 11/55)the Ministerio de Economia y Competitividad (Grant No. FIS2012-36673-C03-03)+2 种基金the Basque Government (Grant No. IT472-10)the Helmholtz Gemeinschaft Deutscher-Young Investigators Group (Grant No. VH-NG-717, Functional Nanoscale Structure and Probe Simulation Laboratory)the Impuls und Vernetzungsfonds der HelmholtzGemeinschaft Postdoc Programme
文摘Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients—one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum—are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between s and p orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.