We employ the cluster slave-spin method to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $delta$ and coupling strength $U$ being its parameters. In addition to a crossover reflected in the
behavior of the antiferromagnetic gap $Delta_{text{AFM}}$, this property can also be observed in the energetics of the cluster slave-spin Hamiltonian -- the antiferromagnetism at small $U$ is due to the potential energy gain while that in the strong coupling limit is driven by the kinetic energy gain, which is consistent with the results from the cluster dynamical mean-field theory calculation and the quantum Monte Carlo simulation. We find the interaction $U_{c}$ for the crossover in the AFM state, separating the weak- and strong- coupling regimes, almost remains unchanged upon doping, and it is smaller than the critical coupling strength $U_{text{Mott}}$ for the first-order metal-insulator Mott transition in the half-filled paramagnetic state. At half-filling, a relationship between the staggered magnetization $M$ and $Delta_{text{AFM}}$ is established in the small $U$ limit to nullify the Hartree-Fock theory, and a first-order Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at $U_{text{Mott}}=10t$. Finally, an overall phase diagram in the $U$-$delta$ plane is presented, which is composed of four regimes: the antiferromagnetic insulator, the antiferromagnetic metal with the compressibility $kappa>0$ or $kappa<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the antiferromagnetic metal to the paramagnetic metal, (ii) between the antiferromagnetic metal phases with positive and negative $kappa$, and (iii) separating the antiferromagnetic insulating phase from the antiferromagnetic metal phase.
The Kondo and Periodic Anderson models describe many of the qualitative features of local moments coupled to a conduction band, and thereby the physics of materials such as the heavy fermions. In particular, when the exchange coupling $J$ or hybridiz
ation $V$ between the moments and the electrons of the metallic band is large, singlets form, quenching the magnetism. In the opposite, small $J$ or $V$, limit, the moments survive, and the conduction electrons mediate an effective interaction which can trigger long range, often antiferromagnetic, order. In the case of the Kondo model, where the moments are described by local spins, Nozi`eres considered the possibility that the available conduction electrons within the Kondo temperature of the Fermi surface would be insufficient in number to accomplish the screening. Much effort in the literature has been devoted to the study of the temperature scales in the resulting `exhaustion problem, and how the `coherence temperature where a heavy Fermi liquid forms is related to the Kondo temperature. In this paper, we study a version of the Periodic Anderson model in which some of the conduction electrons are removed in a way which avoids the fermion sign problem and hence allows low temperature Quantum Monte Carlo simulations which can access both singlet formation and magnetic ordering temperature scales. We are then able to focus on a somewhat different aspect of exhaustion physics than previously considered: the effect of dilution on the critical $V$ for the singlet-antiferromagnetic transition.
New Dirac points appear when periodic potentials are applied to graphene, and there are many interesting effects near these new Dirac points. Here we investigate the $textit{Zitterbewegung}$ effect of fermions described by a Gaussian wave packet in g
raphene superlattice near new Dirac points. The $textit{Zitterbewegung}$ near different Dirac points has similar characteristics, while Fermions near new Dirac points have different group velocities in both $x$- and $y$-direction, which causes the different properties of the $textit{Zitterbewegung}$ near new Dirac points. We also investigate the $textit{Zitterbewegung}$ effect influenced by all Dirac points, and get the evolution with changing potential. Our intensive results suggest that graphene superlattice may provide an appropriate system to study $textit{Zitterbewegung}$ effect near new Dirac points experimentally.
We perform a systematic study of the Zitterbewegung effect of fermions, which are described by a Gaussian wave with broken spatial-inversion symmetry in a three-dimensional low-energy Weyl semimetal. Our results show that the motion of fermions near
the Weyl points is characterized by rectilinear motion and Zitterbewegung oscillation. The ZB oscillation is affected by the width of the Gaussian wave packet, the position of the Weyl node, and the chirality and anisotropy of the fermions. By introducing a one-dimensional cosine potential, the new generated massless fermions have lower Fermi Velocities, which results in a robust relativistic oscillation. Modulating the height and periodicity of periodic potential demonstrates that the ZB effect of fermions in the different Brillouin zones exhibits quasi-periodic behavior. These results may provide an appropriate system for probing the Zitterbewegung effect experimentally.
We propose a tunable electronic band gap and zero-energy modes in periodic heterosubstrate-induced graphene superlattices. Interestingly, there is an approximate linear relation between the band gap and the proportion of inhomogeneous substrate (i.e.
, percentages of different components) in the proposed superlattice, and the effect of structural disorder on the relation is discussed. In inhomogeneous substrate with equal widths, zero-energy states emerge in the form of Dirac points by using asymmetric potentials, and the positions of Dirac points are addressed analytically. Further, the Dirac point exists at $mathbf{k}=mathbf{0}$ only for specific potentials; every time it appears, the group velocity vanishes in $k_y$ direction and the resonance occurs. For general cases that inhomogeneous substrate with unequal widths, a part of zero-energy states are described analytically, and differently, they are not always Dirac points. Our prediction may be realized on the heterosubstrate such as SiO$_2$/BN type.
We investigate the electronic Bloch oscillation in bilayer graphene gradient superlattices using transfer matrix method. By introducing two kinds of gradient potentials of square barriers along electrons propagation direction, we find that Bloch osci
llations up to terahertz can occur. Wannier-Stark ladders, as the counterpart of Bloch oscillation, are obtained as a series of equidistant transmission peaks, and the localization of the electronic wave function is also signature of Bloch oscillation. Forthermore, the period of Bloch oscillation decreases linearly with increasing gradient of barrier potentials.
In a pristine monolayer graphene subjected to a constant electric field along the layer, the Bloch oscillation of an electron is studied in a simple and efficient way. By using the electronic dispersion relation, the formula of a semi-classical veloc
ity is derived analytically, and then many aspects of Bloch oscillation, such as its frequency, amplitude, as well as the direction of the oscillation, are investigated. It is interesting to find that the electric field affects the component of motion, which is non-collinear with electric field, and leads the particle to be accelerated or oscillated in another component.
The NMR relaxation rate and the static spin susceptibility in graphene are studied within a tight-binding description. At half filling, the NMR relaxation rate follows a power law as $T^2$ on the particle-hole symmetric side, while with a finite chem
ical potential $mu$ and next-nearest neighbor $t$, the $(mu+3t)^2$ terms dominate at low excess charge $delta$. The static spin susceptibility is linearly dependent on temperature $T$ at half filling when $t=0$, while with a finite $mu$ and $t$, it should be dominated by $(mu+3t)$ terms in low energy regime. These unusual phenomena are direct results of the low energy excitations of graphene, which behave as massless Dirac fermions. Furthermore, when $delta$ is high enough, there is a pronounced crossover which divides the temperature dependence of the NMR relaxation rate and the static spin susceptibility into two temperature regimes: the NMR relaxation rate and the static spin susceptibility increase dramatically as temperature increases in the low temperature regime, and after the crossover, both decrease as temperature increases at high temperatures. This crossover is due to the well-known logarithmic Van Hove singularity in the density of states, and its position dependence of temperature is sensitive to $delta$.
