In the framework of mean field approach, we study topological Mott transition in a two band model of spinless fermions on a square lattice at half filling. We consider the combined effects of the on-site Coulomb repulsion and the spin-orbit Rashba co
upling. The spin-orbit Rashba coupling leads to a distinct phase of matter, the topological semimetal. We are talking about a new type of phase transition between the non-topological insulator state and topological semimetal state. New phase state is characterized by the zero energy Majorana states, which there are in defined region of the wave vectors and are localized at the boundaries of the sample. The zero energy Majorana states are dispersionless (they can be considered as flat bands), the Chern number and Hall conductance are equal to zero (note in two dimensional model).(note in two dimensional model).
Using mean field approach, we provide analytical and numerical solution of the symmetric Anderson lattice for arbitrary dimension at half filling. The symmetric Anderson lattice is equivalent to the Kondo lattice, which makes it possible to study the
behavior of an electron liquid in the Kondo lattice. We have shown that, due to hybridization (through an effective field due to localized electrons) of electrons with different spins and momenta $textbf{k}$ and $textbf{k}+overrightarrow{pi}$, the gap in the electron spectrum opens at half filling. Such hybridization breaks the conservation of the total magnetic momentum of electrons, the spontaneous symmetry is broken. The state of electron liquid is characterized by a large Fermi surface. A gap in the spectrum is calculated depending on the magnitude of the on-site Coulomb repulsion and value of s-d hybridization for the chain, as well as for square and cubic lattices. Anomalous behavior of the heat capacity at low temperatures in the gapped state, which is realized in the symmetric Anderson lattice, was also found.
Using a weak limit for the hopping integral in one direction in the Hofstadter model, we show that the fermion states in the gaps of the spectrum are determined within the Kitaev chain. The proposed approach allows us to study the behavior of Chern i
nsulators (CI) in different classes of symmetry. We consider the Hofstadter model on the square and honeycomb lattices in the case of rational and irrational magnetic fluxes $phi$, and discuss the behavior of the Hall conductance at a weak magnetic field in a sample of finite size. We show that in the semiclassical limit at the center of the fermion spectrum, the Bloch states of fermions turn into chiral Majorana fermion liquid when the magnetic scale $ frac{1}{ phi} $ is equal to the sample size N. We are talking about the dielectric-metal phase transition, which is determined by the behavior of the Landau levels in 2D fermion systems in a transverse magnetic field. When a magnetic scale, which determines the wave function of fermions, exceeds the size of the sample, a jump in the longitudinal conductance occurs. The wave function describes non-localized states of fermions, the sample becomes a conductor, the system changes from the dielectric state to the metallic one. It is shown, that at $1/phi>$N the quantum Hall effect and the Landau levels are not realized, which makes possibility to study the behavior of CI in irrational magnetic fluxes.
We provide a detailed analysis of a realization of chiral gapless edge modes in the framework of the Hofstadter model of interacting electrons. In a transverse homogeneous magnetic field and a rational magnetic flux through an unit cell the fermion s
pectrum splits into topological subbands with well-defined Chern numbers, contains gapless edge modes in the gaps. It is shown that the behavior of gapless edge modes is described within the framework of the Kitaev chain where the tunneling of Majorana fermions is determined by effective hopping of Majorana fermions between chains. The proposed approach makes it possible to study the fermion spectrum in the case of an irrational flux, to calculate the Hall conductance of subbands that form a fine structure of the spectrum. In the case of a rational flux and a strong on-site Hubbard interaction $U$, $ U >4 Delta $ ($ Delta $ is a gap), the topological state of the system, which is determined by the corresponding Chern number and chiral gapless edge modes, collapses. When the magnitude of the on-site Hubbard interaction changes, at the point $ U = 4 Delta $ a topological phase transition is realized, i.e., there are changes in the Chern numbers of two subbands due to their degeneration.
A junction between two boundaries of a topological superconductor (TSC), mediated by localized edge modes of Majorana fermions, is investigated. The tunneling of fermions across the junction depends on the magnetic flux. It breaks the time-reversal s
ymmetry at the boundary of the sample. The persistent current is determined by the emergence of Majorana edge modes. The structure of the edge modes depends on the magnitude of the tunneling amplitude across the junction. It is shown that there are two different regimes, which correspond to strong and weak tunneling of Majorana fermions, distinctive in the persistent current behavior. In a strong tunneling regime, the fermion parity of edge modes is not conserved and the persistent current is a $2pi$-periodic function of the magnetic flux. When the tunneling is weak the chiral Majorana states, which are propagating along the edges have the same fermion parity. They form a $4pi$-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of $2pi$- and of $4pi$-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied.
We study the behavior of spinless fermions in superconducting state, in which the phases of the superconducting order parameter depend on the direction of the link. We find that the energy of the superconductor depends on the phase differences of the
superconducting order parameter. The solutions for the phases corresponding to the energy minimuma, lead to a topological superconducting state with the nontrivial Chern numbers. We focus our quantitative analysis on the properties of topological states of superconductors with different crystalline symmetry and show that the phase transition in the topological superconducting state is result of spontaneous breaking of time-reversal symmetry in the superconducting state. The peculiarities in the chiral gapless edge modes behavior are studied, the Chern numbers are calculated.
In the framework of Hofstadter`s approach we provide a detailed analysis of a realization of exotic topological states such as the Chern insulator with large Chern numbers. In a transverse homogeneous magnetic field a one-particle spectrum of fermion
s transforms to an intricate spectrum with a fine topological structure of the subbands. In a weak magnetic field $H$ for a rational magnetic flux, a topological phase with a large Chern number is realized near the half filling. There is an abnormal behavior of the Hall conductance ${sigma_{xy}simeq frac{e}{2 H}}$. At half-filling, the number of chiral Majorana fermion edge states increases sharply forming a new state, called the chiral Majorana fermion liquid.
