Exactly solvable model of topological insulator realized on spin-1/2 lattice

In this paper we propose an exactly solvable model of a topological insulator defined on a spin-1/2 square decorated lattice. Itinerant fermions defined in the framework of the Haldane model interact via the Kitaev interaction with spin-1/2 Kitaev sublattice. The presented model, whose ground state is a non-trivial topological phase, is solved exactly. We have found out that various phase transitions without gap closing at the topological phase transition point outline the separate states with different topological numbers. We provide a detailed analysis of the model's ground-state phase diagram and demonstrate how quantum phase transitions between topological states arise. We have found that the states with both the same and different topological numbers are all separated by the quantum phase transition without gap closing. The transition between topological phases is accompanied by a rearrangement of the spin subsystem's spectrum from band to flat-band states.Comment: 8 pages, 9 figure

Giant magnetoresistance of edge current between fermion and spin topological systems

A spin-$\frac{1}{2}$ subsystem conjoined along a cut with a subsystem of spinless fermions in the state of topological insulator is studied on a honeycomb lattice. The model describes a junction between a 2D topological insulator and a 2D spin lattice with direction-dependent exchange interactions in topologically trivial and nontrivial phase states. The model Hamiltonian of the complex system is solved exactly by reduction to free Majorana fermions in a static $\mathbb{Z}_2$ gauge field. In contrast to junctions between topologically trivial phases, the junction is defined by chiral edge states and direct interaction between them for topologically nontrivial phases. As a result of the boundary interaction between chiral edge modes, the edge junction is defined by the Chern numbers of the subsystems: such the gapless edge modes with the same (different) chirality switch on (off) the edge current between topological subsystems. The sign of the Chern number of spin subsystem is changed in an external magnetic field, thus the electric current strongly depends both on a direction and a value of an applied weak magnetic field. We have provided a detailed analysis of the edge current and demonstrate how to switch on (off) the electric current in the magnetic field.Comment: 4 pages, 9 figure

Edge modes in the Hofstadter model of interacting electrons

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 spectrum 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.Comment: 7 pages, 5 figure