Blume-Emery-Griffiths model on the honeycomb lattice

Exact results are obtained for a spin-1 system on the honeycomb lattice with the Blume-Emery-Griffiths Hamiltonian –/kT =J _i,j S _i S _j +Ki,jS _i ² _j ² – _i S _i ² +HS _i subject to the constraintK=–ln coshJ. ForJ>0, the system behaves like a spin-1/2 Ising ferromagnet with the free energy analytic everywhere except at the first-order phase boundaryH=0, tanhJ<(2+e )/ . Derivatives of the free energy across this boundary are discontinuous and we obtain the exact expression for the spontaneous magnetization. ForJ<0, the system can be transcribed into an antiferromagnetic spin-1/2 Ising model in a real magnetic field, and from this equivalence portions of the exact phase boundary are determined.     

Half-filled Kondo lattice on the honeycomb lattice

The unique linear density of state around the Dirac points for the honeycomb lattice brings much novel features in strongly correlated models. Here we study the ground-state phase diagram of the Kondo lattice model on the honeycomb lattice at half-filling by using an extended mean-field theory. By treating magnetic interaction and Kondo screening on an equal footing, it is found that besides a trivial discontinuous first-order quantum phase transition between well-defined Kondo insulator and antiferromagnetic insulating state, there can exist a wide coexistence region with both Kondo screening and antiferromagnetic orders in the intermediate coupling regime. In addition, the stability of Kondo insulator requires a minimum strength of the Kondo coupling. These features are attributed to the linear density of state, which are absent in the square lattice. Furthermore, fluctuation effect beyond the mean-field decoupling is analyzed and the corresponding antiferromagnetic spin-density-wave transition falls into the O(3) universal class. Comparatively, we also discuss the Kondo necklace and the Kane-Mele-Kondo (KMK) lattice models on the same lattice. Interestingly, it is found that the topological insulating state is unstable to the usual antiferromagnetic ordered states at half-filling for the KMK model. The present work may be helpful for further study on the interplay between conduction electrons and the densely localized spins on the honeycomb lattice.     

Topological states in the Hofstadter model on a honeycomb lattice

We provide a detailed analysis of a topological structure of a fermion spectrum in the Hofstadter model with different hopping integrals along the $x,y,z$-links ($t_x=t,t_y=t_z=1$), defined on a honeycomb lattice. We have shown that the chiral gapless edge modes are described in the framework of the generalized Kitaev chain formalism, which makes it possible to calculate the Hall conductance of subbands for different filling and an arbitrary magnetic flux ϕ. At half-filling the gap in the center of the fermion spectrum opens for $t>t_c=2^{\varphi }$, a quantum phase transition in the 2D-topological insulator state is realized at $t_c$. The phase state is characterized by zero energy Majorana states localized at the boundaries. Taking into account the on-site Coulomb repulsion U (where $U<<1$), the criterion for the stability of a topological insulator state is calculated at $t<<1$, $t\sim U$. Thus, in the case of $U>4\mathrm{\Delta }$, the topological insulator state, which is determined by chiral gapless edge modes in the gap Δ, is destroyed.     

BCS-BEC crossover on the two-dimensional honeycomb lattice

The attractive Hubbard model on the honeycomb lattice exhibits, at half filling, a quantum critical point between a semimetal with massless Dirac fermions and an s-wave superconductor (SC). We study the BCS-BEC crossover in this model away from half filling at zero temperature and show that the appropriately defined crossover line (in the interaction-density plane) passes through the quantum critical point at half filling. For a range of densities around half filling, the "underlying Fermi surface" of the SC, defined as the momentum space locus of minimum energy quasiparticle excitations, encloses an area which changes nonmonotonically with interaction. We also study fluctuations in the SC and the semimetal, and show the emergence of an undamped Leggett mode deep in the SC. Finally, we consider possible implications for ultracold atoms in optical lattices and the high temperature SCs.     

Critical behaviour of the hard-core lattice gas on the honeycomb lattice

The hard triangle lattice-gas model (lattice-gas on the honeycomb lattice with first neighbour exclusion) is studied by the phenomenological renormalization method. The critical activity is found to be z = 7.85 and the critical exponents suggest that this model belongs to the 2-D Ising universality class.     

Dynamic scaling behaviors of the restricted-solid-on-solid model on honeycomb and square-octagon lattice substrates

The dynamic scaling behaviors of the restricted-solid-on-solid (RSOS) model on two new types of substrate, which are honeycomb and square-octagon lattice substrates, are studied by means of Kinetic Monte Carlo simulations. The growth exponent β and the roughness exponent α defined, respectively, by the surface width via W ~ t ^β and the saturated width via W _sat ~ L ^α , L being the system size, were obtained by a power-counting analysis. Our simulation results show that the Family-Vicsek scaling is still satisfied. However, the structures of the substrates indeed affect the dynamic behavior of the growth model. The values of the roughness exponents fall between regular and fractal lattices. Deeper analysis show that the coordination number of the substrates play an crucial role.     

A honeycomb lattice model simulating the surface states of topological insulators

A honeycomb lattice model exhibiting the quantum spin-Hall effect is proposed, where the low-energy properties of the electrons are mainly determined by the energy spectrum in the vicinity of the Γ point, for suitable parameters. The nontrivial topology of the energy bands is revealed by calculating the Chern numbers, Berry curvature distribution, and edge state spectrum. We further show that in the continuum limit, the model Hamiltonian is equivalent to the effective model for the surface states in thin films of three-dimensional topological insulators. As a consequence, this lattice model provides a useful tool for numerical simulation of the physical properties of the surface states.     

Monte Carlo study of the antiferromagnetical Ising model on a centred honeycomb lattice

We use the Monte Carlo method to study an antiferromagnetical Ising spin system on a centred honeycomb lattice, which is composed of two kinds of 1/2 spin particles A and B. There exist two different bond energies J_A-A and J_A-B in this lattice. Our study is focused on how the ratio of J_A-B to J_A-A influences the critical behaviour of this system by analysing the physical quantities, such as the energy, the order parameter, the specific heat, susceptibility, {etc} each as a function of temperature for a given ratio of J_A-B to J_A-A. Using these results together with the finite-size scaling method, we obtain a phase diagram for the ratio J_A-B / J_A-A. This work is helpful for studying the phase transition problem of crystals composed of compounds.     

Spin excitations and thermodynamics of the antiferromagnetic Heisenberg model on the layered honeycomb lattice

We present a spin-rotation-invariant Green-function theory for the dynamic spin susceptibility in the spin-1/2 antiferromagnetic Heisenberg model on a stacked honeycomb lattice. Employing a generalized mean-field approximation for arbitrary temperatures, the thermodynamic quantities (two-spin correlation functions, internal energy, magnetic susceptibility, staggered magnetization, Néel temperature, correlation length) and the spin-excitation spectrum are calculated by solving a coupled system of self-consistency equations for the correlation functions. The temperature dependence of the magnetic (uniform static) susceptibility is ascribed to antiferromagnetic short-range order. The Néel temperature is calculated for arbitrary interlayer couplings. Our results are in a good agreement with numerical computations for finite clusters and with available experimental data on the β-Cu₂V₂O₂ compound.     

Mott-Hubbard transition and antiferromagnetism on the honeycomb lattice

The Hubbard model is investigated for a halffilled honeycomb lattice, using a variational method. Two trial wave functions are introduced, the Gutzwiller wave function, well suited for describing the “metallic” phase at small U and a complementary wave function for the insulating regime at large values of U. The comparison of the two variational ground states at the mean-field level yields a Mott transition at U _c /t ≈ 5:3. In addition, a variational Monte Carlo calculation is performed in order to locate the instability of the “metallic” wave function with respect to antiferromagnetism. The critical value U _m/t ≈ 3:7 obtained in this way is considered to be a lower bound for the true critical point for antiferromagnetism, whereas there are good arguments that the mean-field value U _c/t ≈ 5:3 represents an upper bound for the Mott transition. Therefore the “metal”- insulator transition for the honeycomb lattice may indeed be simultaneously driven by the antiferromagnetic instability and the Mott phenomenon.     

Power-law spin correlations in a perturbed spin model on a honeycomb lattice

We consider the spin-1/2 model on the honeycomb lattice in the presence of a weak magnetic field hα < 1. Such a perturbation destroys the exact integrability of the model in terms of gapless fermions and static Z2 fluxes. We show that it results in the appearance of a long-range tail in the irreducible dynamic spin correlation function: ∝ h(z)(2)f(t,r), where f(t,r) ∝ [max(t,r)]-4 is proportional to the density polarization function of fermions.     

Zitterbewegung in the honeycomb photonic lattice

The propagation of a wave packet in a honeycomb photonic lattice has been studied using the time-dependent wave packet dynamics. It is found that the wave packet, superposed from the positive and negative energy modes at the vicinity of the two inequivalent Dirac points, can transform into a double-ring structure, which is caused by the interference between the two positive and negative energy modes around the Dirac points and is closely related to the Zitterbewegung (ZB). Also, a possible way to detect the ZB effect is proposed in the honeycomb photonic lattice.     

The frustrated Heisenberg antiferromagnet on the honeycomb lattice: J1-J2 model

We study the ground-state phase diagram of the frustrated spin-[Formula: see text] antiferromagnet with J(2) = xJ(1) > 0 (J(1) > 0) on the honeycomb lattice, using the coupled-cluster method. We present results for the ground-state energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. We find a paramagnetic PVBC phase for x(c(1)) < x < x(c(2)), where x(c(1)) ≈ 0.207 ± 0.003 and x(c(2)) ≈ 0.385 ± 0.010. The transition at x(c(1)) to the Néel phase seems to be a continuous deconfined transition (although we cannot exclude a very narrow intermediate phase in the range 0.21 ? x ? 0.24), while that at x(c(2)) is of first-order type to another quasiclassical antiferromagnetic phase that occurs in the classical version of the model only at the isolated and highly degenerate critical point [Formula: see text]. The spiral phases that are present classically for all values x > 1/6 are absent for all x ? 1.     

Spectrum of the non-abelian phase in Kitaev’s honeycomb lattice model

The spectral properties of Kitaev’s honeycomb lattice model are investigated both analytically and numerically with the focus on the non-abelian phase of the model. After summarizing the fermionization technique which maps spins into free Majorana fermions, we evaluate the spectrum of sparse vortex configurations and derive the interaction between two vortices as a function of their separation. We consider the effect vortices can have on the fermionic spectrum as well as on the phase transition between the abelian and non-abelian phases. We explicitly demonstrate the 2ⁿ-fold ground state degeneracy in the presence of 2n well separated vortices and the lifting of the degeneracy due to their short-range interactions. The calculations are performed on an infinite lattice. In addition to the analytic treatment, a numerical study of finite size systems is performed which is in exact agreement with the theoretical considerations. The general spectral properties of the non-abelian phase are considered for various finite toroidal systems.