On edge states in superconductors with time inversion symmetry breaking

Abstract

Superconducting states with different internal topology are discussed for the layered high-T c materials. If the time inversion symmetry is broken, the superconductivity is determined not only by the symmetry of the superconducting state, but also by the topology of the ground state. The latter is determined the integer-valued momentumspace topological invariant N . The current carrying boundary between domains with different N (N 2 = N 1 ) is considered. The current is produced by fermion zero modes, which number per one layer is 2(N 2 − N 1 ).Recently a new phase transition in high temperature superconductor was reported on in Ni-doped Bi 2 Sr 2 CaCu 2 0 8 [1]. It occurs at low temperature, T ∼ 200 mK and seems to correspond to the opening of the gap in the superconductor which is gapless (contains lines of nodes) above transition. Since the gap usually appears in superconductor with lines of nodes if the time inversion (T ) symmetry is violated, this suggests that the spontaneous breaking of the T -symmetry occurs in the new transition. Such transition can be caused by interaction with magnetic impurities.Some evidence of the additional symmetry breaking with opening of the gap in a pure Bi 2 Sr 2 CaCu 2 0 8 in the presence of magnetic field was reported in [2]. The d x 2 −y 2 + id xy complex order parameter was suggested in [3] to describe this experiment. The possibility of the broken T -symmetry was discussed also within the grain boundaries and twin boundaries [4,5].Since the broken T -symmetry becomes popular now for the high temperature superconductors, we discuss some consequences of the broken Tsymmetry in D=2 spatial dimension relevant for layered superconductors. These consequences are similar to that in thin films of superfluid 3 He-A and other 2D systems where T -symmetry is broken. They are described in terms of the integer-valued topological invariant of the ground state [6,7,8,9,10], which gives rise to chiral edge states on the boundary of superconductor and in some cases to quantization of Hall conductivity. We dicuss here this topological invariant for the layered superconductors.

Chirald-wave superconductivity in doped graphene

“…With the quantized value of the spin Hall conductance being a universal property, which cannot change within the same topological phase, the result must valid for any finite size of the d xy component [118]. The thermal Hall conductance similarly measures the transverse heat current as a function of a temperature gradient.…”

“…For systems classified with a Chern number the difference between the number of right and left moving edge modes is equivalent to the change in Chern number across the interface [123]. For a chiral d-wave superconductors this means two co-propagating, or chiral, edge modes crossing the bulk gap [118].…”

“…the first Chern number developed for quantum Hall systems. However, the non-trivial topology is more easily visualized by instead using a Skyrmion number formula, which gives the Chern number as [116,117,118,119]:…”

“…Many time-reversal symmetry breaking superconducting states are, however, not chiral. For example, in the other often discussed fully gapped time-reversal breaking state in the cuprates, d x 2 −y 2 + is, them-vector just oscillates between two extrema when moving around the k-point origin and does not cover the full circle, thus giving N = 0 [118]. In a multiband superconductor, the Skyrmion winding numbers have to be calculated for each individual band crossing the Fermi level and can then add up to zero [120].…”

“…The two co-propagating, or chiral, edge states of a d x 2 −y 2 + id xy -wave superconductor carry a spontaneous edge current [47,118,109]. For chiral d-wave superconducting graphene the edge current has been calculated using the charge continuity equation in combination with the Heisenberg equation for the particle number [28].…”