Superconducting Classes in the Extended Hubbard Model

Abstract

This paper describes group theoretical classification of superconducting states (SC) in the extended Hubbard model with on-site repulsion (U), nearest neighbor attraction (V) and nearest neighbour exchange interaction (J) on the two-dimensional square lattice using the mean field approach. By decomposing the pairing interaction into irreducible parts; A1g, B1g and Eu of D4h point symmetry, we have derived two singlet SCs (s-wave and d-wave) from A1g and B1g, eight triplet SCs from Eu. The first three types of triplet SC have pairing by electrons with antiparallel spin, the second two types have pairing by electrons with equal spin and the last three types are non-unitary and have pairing by only up-spin electrons. We showed that three non-unitary states have to be accompanied with a ferromagnetic order from the structure of the maximal little groups. We performed numerical studies for these SCs. For parameters and electron density favorable for the ferromagnetic order, a non-unitary SC coexistent with ferromagnetism is most stable.

Glide reflection symmetry, Brillouin zone folding, and superconducting pairing for theP4/nmmspace group

“…Since both α, β terms acquire the same orbital parity factor and each state gets shifted by one unit of the 1-Fe unit cell we have ∆ A e e e,αβ = ∆ B e e e,αβ . Plugging this into (22) and setting (−1) α = (−1) β , we see that all EO and OE mixed terms cancel and we get only same glide parity terms.…”

“…More precisely, the tensor irreducible representations of the space group used to classify the pairing are the irreducible representations of the group of the wave-vector labeled by total momentum k k k +(−k k k) = (0, 0) (For a general argument, see Ref. 22). Given this, we know that the irreducible representations of the D 4h point group are either even or odd under inversion.…”

Group Theoretical Analysis of the Vortex Lattice States in the Attractive Hubbard Model

“…In a previous paper, 5), 6) we gave a group theoretical classification of superconducting states in the extended Hubbard model with an attractive interaction on a two-dimensional square lattice using the Hartree-Fock-Bogoliubov equation in the absence of a magnetic field. In the previous paper, 5), 6) we derived two singlet 8Cs (s-wave and d-wave) and eight triplet 8Cs including three non-unitary states.…”

“…Thus we cannot straightforwardly apply the previous group theoretical analysis 5) to this case. Noting that the translation of crystal lattice acts as a projective representation on the space of the mean-field Hamiltonian H m , we have obtained a group theoretical classifcation of 8C states (vortex lattice states) in the presence of a magnetic field for the extended Hubbard model.…”

“…In this case we obtained three types of symmorphic tetragonal vortex lattice states with invariance group G i = PiL(i = 1, 3,5), in which the center of point rotations is at a lattice point of the crystal lattice. We determined the locations and winding numbers of vortex lines for these states using symmetries of Gi.…”