The edge states properties of the topological insulator with fermion path nonanalyticity points located at the edge.

The edge states properties of a finite-size 2D topological insulator (TI) with the open boundary conditions in both directions are studied. It is shown that fermion path nonanalyticity points appearing on the edge of finite-size TI cause a nonuniform edge state distribution along the TI edge. The character of this distribution depends substantially on the edge state's energy position in the bulk band gap. The edge state with the energy in the middle of the gap tends to be located in the corners between two nonparallel edges of the system, while the edge states with the energy close to the bulk band edge avoid the corners. The monotonic transition from one space distribution of the edge state wave function to another along the edge state's band is observed. The mentioned edge state's structure leads to the nonlinear current-voltage characteristic of the square-shaped TI with contacts connected with the corners of the device. It opens the possibility to control the current with the gate voltage and is useful for construction the nano-size devices. For the edge states with energy in the middle of a bulk gap, the vortex structure of the probability flux located near corners is found.