With a generic lattice model for electrons occupying a semi-infinite crystal with a hard surface, we study the eigenstates of the system with a bulk band gap (or the gap with nodal points). The exact solution to the wave functions of scattering state
s is obtained. From the scattering states, we derive the criterion for the existence of surface states. The wave functions and the energy of the surface states are then determined. We obtain a connection between the wave functions of the bulk states and the surface states. For electrons in a system with time-reversal symmetry, with this connection, we rigorously prove the correspondence between the change of Kramers degeneracy of the surface states and the bulk time-reversal $Z_2$ invariant. The theory is applicable to systems of (topological) insulators, superconductors, and semi-metals. Examples for solving the edge states of electrons with/without the spin-orbit interactions in graphene with a hard zigzag edge and that in a two-dimensional $d$-wave superconductor with a (1,1) edge are given in appendices.
We present a formalism to calculate the orbital magnetization of interacting Dirac fermions under a magnetic field. In this approach, the divergence difficulty is overcome with a special limit of the derivative of the thermodynamic potential with res
pect to the magnetic field. The formalism satisfies the particle-hole symmetry of the Dirac fermions system. We apply the formalism to the interacting Dirac fermions in graphene. The charge and spin orderings and the exchange interactions between all the Landau levels are taken into account by the mean-field theory. The results for the orbital magnetization of interacting Dirac fermions are compared with that of non-interacting cases.
By taking into account the charge and spin orderings and the exchange interactions between all the Landau levels, we investigate the integer quantum Hall effect of electrons in graphene using the mean-field theory. At the fillings $ u = 4n+2$ with $n
= 0, 1, cdots$, the system is in the high-symmetry state with the Landau levels four-fold degenerated. We show that with doping the degenerated lowest empty levels can be sequentially filled one level by one level, the filled level is lower than the empty ones because of the symmetry breaking. This result explains the step $Delta u$ = 1 in the integer quantized Hall conductivity of the experimental observations. We also present in the supplemental material a high efficient method for dealing with huge number of the Coulomb couplings between all the levels.
Using a four-band Hamiltonian, we study the phase boundary of spin-polarized-current state (SPCS) of interacting electrons in bilayer graphene. The model of spin-polarized-current state has previously been shown to resolve a number of experimental pu
zzles in bilayer graphene. The phase boundaries of the SPCS with and without the external voltage between the two layers are obtained in this work. An unusual phase boundary where there are two transition temperatures for a given carrier concentration is found at finite external voltage. The physics of this phenomenon is explained.
By taking into account the possibility of all the intralayer as well as the interlayer current orderings, we derive an eight-band model for interacting electrons in bilayer graphene. With the numerical solution to the model, we show that only the cur
rent orderings between the same sublattice sites can exist within the range of the physical interacting strength. This result confirms our previous model of spin-polarized-current phase for the ground-state of interacting electrons in bilayer graphene that resolves a number of experimental puzzles.
We propose a model of spin-polarized-current state for electrons in bilayer graphene. The model resolves the puzzles as revealed by experiments that (a) the energy gap $E_{rm gap}$ of the insulating ground state at the charge neutrality point (CNP) c
an be closed by a perpendicular electric field of either polarity, (b) $E_{rm gap}$ increases significantly with increasing the magnetic field $B$, (c) the particle-hole spectrum is asymmetric in the presence of $B$, (d) there is a peak structure in the electric conductivity at small $B$ at the CNP, and (e) there are quantum Hall states stemming from lifting of degeneracy in the lowest Landau level. The model predicts that the ground state of the system close to the CNP is a ferrimagnet at finite $B$ and the Hall current is spin polarized.
We present a super-high-efficiency approximate computing scheme for series sum and discrete Fourier transform. The summation of a series sum or a discrete Fourier transform is approximated by summing over part of the terms multiplied by corresponding
weights. The calculation is valid for the function under the transform being piecewise smooth in the continuum variable. The scheme reduces significantly the requirement for computer memory storage and enhances the numerical computation efficiency without losing accuracy. By comparing with the known results of examples, we show the accuracy and the efficiency of the scheme. The efficiency can be higher than $10^6$ for the examples.
Based on the four-band continuum model, we study the ordered-current state (OCS) for electrons in bilayer graphene at the charge neutrality point. The present work resolves the puzzles that (a) the energy gap increases significantly with increasing t
he magnetic field $B$, (b) the energy gap can be closed by the external electric field of either polarization, and (c) the particle-hole spectrum is asymmetric in the presence of $B$, all these as observed by the experiment. We also present the prediction of the hysteresis energy gap behavior with varying $B$, which explains the existing experimental observation on the electric conductance at weak $B$. The large energy gap of the OCS is shown to originate from the disappearance of Landau levels of $n$ = 0 and 1 states in conduction/valence band. By comparing with the existing models and the experiments, we conclude that the OCS is a possible ground state of electrons in bilayer graphene.
With the two-band continuum model, we study the broken inversion and time-reversal symmetry state of electrons with finite-range repulsive interactions in bilayer graphene. With the analytical solution to the mean-field Hamiltonian, we obtain the ele
ctronic spectra. The ground state is gapped. In the presence of the magnetic field $B$, the energy gap grows with increasing $B$, in excellently agreement with the experimental observation. Such an energy gap behavior originates from the disappearance of a Landau level of $n$ = 0 and 1 states. The present result resolves explicitly the puzzle of the gap dependence of $B$.
On a lattice model, we study the possibility of existence of gapped broken inversion symmetry phase (GBISP) of electrons with long-range Coulomb interaction in bilayer graphene using both self-consistent Hartree-Fock approximation (SCHFA) and the ren
ormalized-ring-diagram approximation (RRDA). RRDA takes into account the charge-density fluctuations beyond the mean field. While GBISP at low temperature and low carrier concentration is predicted by SCHFA, we show here the state can be destroyed by the charge-density fluctuations. We also present a numerical algorithm for calculating the self-energy of electrons with the singular long-range Coulomb interaction on the lattice model.
