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

Abstract

This paper describes a group theoretical classification of superconducting states (vortex lattice states) in a unifom magnetic field of the extended Hubbard model with on-site attraction (U < 0), or nearest-neighbor attraction (V < 0) on a two-dimensional square lattice.Using symmetries of the magnetic translation and the tetragonal rotation of the system, we obtain invariance groups of vortex lattice states. When the magnetic flux <p through a unit cell of the crystal lattice is <PO/p2 (<Po = ch/2e is the flux quantum, p is an integer) we obtained three types of symmorphic tetragonal vortex lattice states, in which the center of point rotations is at a lattice site of the crystal lattice, corresponding to the irreducible representation of the tetragonal symmetry D 4 . For these three states, locations and winding numbers of vortex lines are determined by the symmetries of their invariance group. From these three symmorphic states we derived three (two) types of non-symmorphic vortex lattice states in which the center of point rotations is at the center of a plaquette (link) of the crystal lattice. We show that d-wave, s-wave and p-wave local order parameters inevitably coexist due to the invariance magnetic translation group L of the vortex lattice state. From the irreducible projective representations (two-dimensional) of L, we show that all energy bands of the quasiparticles are at least dOD bly degenerate.

Orbital-resolved vortex-core states in FeSe superconductors: A calculation based on a three-orbital model

“…It is not the symmetry group of Abrikosov lattice in which only single vortex is stabilized within one magnetic unit cell. Breakthrough of this difficulty was presented by M. Ozaki et al 32 . In their work the magnetic translation group describing single vortex was discovered to be a subgroup of direct product of conventional magnetic translation group and gauge transformation group U(1).…”

“…Instead of doing calculations of two vortices in one magnetic unit cell, we follow the method given by M. Ozaki et al 32,33 , in which only single vortex structures are calculated in one magnetic unit cell, so that the calculated results can be classified by irreducible representations of magnetic translation group. The numerical calculations in previous works, as mentioned above [25][26][27] , are mostly carried out for two vortices in one magnetic unit cell.…”

“…The modification of pairing order parameters accounts for eliminating the mixing of different pairing states under the action of magnetic translation group. The mathematical interpretation of doing this is essentially searching for gauge transformed order parameters, which span a representation of magnetic translation group 32,33 . The gauge transformation, carried out by phase φ(i, j), has different definition with respect to anisotropic s and d x 2 −y 2 wave pairing states, whereas in the case of isotropic s wave pairing it is trivial.…”

“…and the resultant transformation of order parameter is 31 {T ( R λ )}: N v = 2, the group condition of magnetic translation group, which is the symmetry group of Abrikosov lattice, is that only single magnetic flux ϕ 0 = hc 2e is contained in one magnetic unit cell 32,33 , i.e., N v = 1. It has been pointed out that d x 2 −y 2 ∼ cos(k x ) − cos(k y ) wave order parameters will mix with extended s * ∼ cos(k x ) + cos(k y ), p x ∼ i sin(k x ), and p y ∼ i sin(k y ) wave order parameters under operation of magnetic translation group 32,33 .…”

“…It has been pointed out that d x 2 −y 2 ∼ cos(k x ) − cos(k y ) wave order parameters will mix with extended s * ∼ cos(k x ) + cos(k y ), p x ∼ i sin(k x ), and p y ∼ i sin(k y ) wave order parameters under operation of magnetic translation group 32,33 . Such a mixing originates from the fact the symmetry group of normal state Hamiltonian contains a local gauge transformation generated by the vector potential of a magnetic field.…”