The nontrivial ground state topology in the coexistence phase of chiral d-wave superconductivity and 120-degree magnetic order on a triangular lattice

Abstract

The Z 2 topological invariant is defined in the chiral d-wave superconductor having a triangular lattice in the presence of the 120 degrees magnetic ordering. By analyzing the Z 2 invariant, we determine the conditions of implementing nontrivial phases in the model with regard to superconducting pairings between nearest and next nearest neighbors. It is often supposed in such system that the pairing parameter between nearest neighbors should be equal to zero due to influence of the intersite Coulomb interaction. We show that taking into account even weak pairings in the first coordination sphere leads to the disappearance of the gapless excitations of the bulk spectrum in the wide region of the parameter space. Thus topological invariants can be defined in such region. Solving the problem of open edges it is shown that the zero energy modes are realized basically in the topologically nontrivial phases. Such zero modes are topologically protected Majorana modes. A connection between the Z 2 invariant calculated at the symmetric points of the Brillouin zone with respect to the electron-hole symmetry and the integer topological invariant of the ground state of the 2D lattice expressed in terms of the Green functions is established in the presence of noncollinear magnetic ordering.

Aspects of topological superconductivity in 2D systems: noncollinear magnetism, skyrmions, and higher-order topology

“…In conclusion, the 2D system can be divided into a set of independent 1D systems having the appropriate quantum number k 2 . For the fixed k 2 , it is possible to determine the 1D topological invariant called the Majorana number [7] and to obtain conditions for the formation of the Majorana edge states in the transverse direction [34,93].…”

“…For the quadratic Hamiltonian, the parity of the ground state is determined only by filling of the states in specific points of the Brillouin zone. Consider the 2D triangular lattice with the 120-degree spin ordering and chiral d-wave superconductivity [92,93] being described by Hamiltonian (3). For this case, the indices f and m in Hamiltonian (3) label the sites of the triangular lattice and the wave-vector of the 120 • spin structure is Q = (2π/3, 2π/3).…”

Stability of the coexistence phase of chiral superconductivity and noncollinear spin ordering with a nontrivial topology and strong electron correlations

“…To determine the structure of the Majorana mode, we use a method similar to that used for models disregarding interactions [17]. We consider a system with the triangular lattice containing a finite number (N 1 ) of sites along the direction of the translation vector a 1 , whereas periodic boundary conditions are imposed along the a 2 direction (cylindrical geometry).…”

“…According to this definition and symmetry reasons [17], the Majorana mode in the cylindrical geometry occurs at…”

“…However, the further analysis showed that chiral superconductivity does not coexist with stripe spin ordering, but coexists with a magnetic order corresponding to a 120 • structure [16]. The conditions for implementation of the Majorana modes in this coexistence phase are defined [17]. The Z 2 and Z invariants are calculated for the quadratic Hamiltonian and it is shown that topologically nontrivial phases with an odd value of the Z invariant correspond to the parametric regions with the Majorana modes.…”