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.
We give group theoretical classification of superconducting (SC) states with a single SC order parameter in the extended Hubbard model with uniform lattice distortion on the two-dimensional square lattice, by using the mean field approach. We obtain two singlet SCs (s-wave and d-wave) and eight triplet SCs by decomposing the pairing interaction into irreducible components (A1g, B1g and Eu) of point symmetry D4h. We show that in the singlet SC state with a single SC order parameter (s-wave or d-wave) no anisotropic lattice distortion occurs. In the case of triplet SC states, two SC states among eight must be accompanied with anisotropic lattice distortion. The symmetry of SC order parameter uniquely determines the lattice symmetry. We have performed numerical calculations for these SC states, and shown the possibility that one of the SC states with anisotropic lattice distortion becomes most stable of all SC states in the extended Hubbard model. We also discuss singlet SC states with multiple SC order parameters (s-wave and d-wave). It is shown that the anisotropic distortion must occur in (s+d)-SC states and not in (s+id)-SC state.
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 φ through a unit cell of the crystal lattice is φ0/p2 (φ0=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 D4. 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 doubly degenerate.
Possible ordered states in the 2D extended Hubbard model with on-site (U>0) and nearest-neighbor (V) interaction are examined near half filling, with emphasis on the effect of finite V. First, the phase diagram at absolute zero is determined in the mean field approximation. For
V<0, a state where d_{x^{2}-y^{2}}-wave superconductivity (dSC), commensurate spin-density-wave (SDW) and
π-triplet pair coexist is seen to be stabilized. Here, the importance of
π-triplet pair on the coexistence of dSC and SDW is indicated. This coexistent state is hampered by the phase separation (PS), which is generally expected to occur in the presence of finite-range attractive interaction, but survives. For V>0, a state where commensurate charge-density-wave (CDW), SDW and ferromagnetism (FM) coexist is seen to be stabilized. Here, the importance of FM on the coexistence of CDW and SDW is indicated. Next, in order to examine the effects of fluctuation on each mean field ordered state, the renormalization group method for the special case that the Fermi level lies just on the saddle points, (
π,0) and (0,
π), is applied. The crucial difference from the mean field result is that superconductivity can arise even for U>0 and
V≥0, where the superconducting gap symmetry is d_{x^{2}-y^{2}}-wave for U>4V and s-wave for U<4V. Finally, the possibilities that the mean field coexistent states survive in the presence of fluctuation are discussed.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (
J) and correlated hopping (
K) interaction. Detailed numerical analysis of
s-,
p-, and
d-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (
t) and next-nearest- (
t2) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the
d-wave and the extended
s-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the
tt2 ratio, giving in particular for
t2=0 s-wave (
d-wave) pairing as most favorable in the low (high) density limit, but increasing
t2 can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects
s-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
We study the effect of the nearest neighbor anisotropic exchange interaction on superconductivity in extended Hubbard model in a square lattice. This Hamiltonian is one of a base model to describe various effects on triplet superconductors, e.g. Sr 2 RuO 4 , because the ferromagnetic exchange is advantagous for triplet pairing. Within a one-band meanfield approximation, we calculate the critical temperatures of spin singlet and triplet superconductivity. We obtain a phase diagram in the longitudinal vs transverse exchange couplings and we show how the stable pairing state is controlled by the anisotropy of the exchange interaction.