Persistent current in 2D topological superconductors

Abstract

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 symmetry 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 2π-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 4π-phase periodic persistent current along the boundaries. The regions in the space of parameters, which correspond to the emergence of 2π- and of 4π-harmonics, are numerically determined. The peculiarities in the persistent current behavior are studied.

The quantum Hall effect at a weak magnetic field

“…A rational flux φ = p q Let us consider the q-subbans, splitting from one Bloch band for a rational flux φ = p q , here p and q are coprime integers. At q > 2 a magnetic field breaks a time reversal symmetry, the CI state is realized [9][10][11][12]. In this case a topological trivial Bloch band splits into q-magnetic subbands with non-trivial topological index C α (α is the index of a subband in a fine structure of the spectrum).…”

“…A non-trivial topology of the ground state provides the quantization of the Hall conductance, which can be interpreted as the Chern number. In the case of a rational magnetic flux, a topological state of the Harper-Hofstadter Hamiltonian is characterized by the Chern numbers and chiral edge gapless modes, realized the CI state [9][10][11][12]. Filled r-bands with the Chern numbers C α yield a Hall conductance σ xy = (e 2 /h)C r , C r = r α=1 C α .…”

Edge modes in the Hofstadter model of interacting electrons

“…Topological states are described in the framework of the band theory, while the topological invariant characterizes the class of a topological insulator or a topological superconductor [1][2][3]. A ground state of the Chern systems is characterized by the Chern number, it is an integer topological number which is well defined in the insulator (superconductor) state for the band isolated from all other bands [4][5][6]. A nontrivial topology of the ground state provides the quantization of the Hall conductance, which can be interpreted as the Chern number.…”