s± pairing near a Lifshitz transition

Observations of robust superconductivity in some of the iron based superconductors in the vicinity of a Lifshitz point where a spin density wave instability is suppressed as the hole band drops below the Fermi energy raise questions for spin-fluctuation theories. Here we discuss spin-fluctuation pairing for a bilayer Hubbard model, which goes through such a Lifshitz transition. We find s± pairing with a transition temperature that peaks beyond the Lifshitz point and a gap function that has essentially the same magnitude but opposite sign on the incipient hole band as it does on the electron band that has a Fermi surface.

with t ⊥ /t = 3.5 and μ set so that the site filling 〈 n〉 = 1.05 is shown in Fig. 1(a). If the filling is kept constant as t ⊥ /t is increased, the system has a Lifshitz transition such that for t ⊥ > 3.67 the hole Fermi surface at the Γ point disappears as illustrated in Fig. 1(b). We are interested in studying the pairing for parameters such that the spin density wave (SDW) instability is suppressed by this Lifshitz transition. In a random phase approximation (RPA) the spin susceptibility is given by   Here T is the temperature, G 0 (k, ω n ) = (iω n − ξ k ) −1 and ω n = (2n + 1)πT and Ω m = 2mπT are the usual fermionic and bosonic Matsubara frequencies. For a fixed filling, as t ⊥ /t is increased and the Lifshitz transition is approached, χ 0 which peaks near wavevector (π, π, π), decreases. For 〈 n〉 = 1.05, we take U = 2.4t so that the SDW instability determined from Eq. (3) is suppressed by the Lifshitz transition as shown in Fig. 2. With this suppression of the SDW order, one can imagine that superconductivity may appear following the usual paradigm. However, the Lifshitz transition that has suppressed the SDW instability can also lead to a suppression of the s ± pairing associated with the scattering of pairs between the electron Fermi surface and the incipient hole band. For a fixed pairing strength, T c decreases as the hole band moves below the Fermi energy 18 .
k n n n n n n , n and determine T c from the temperature at which the leading eigenvalue of Eq. (5) goes to 1. Here we use a spin-fluctuation mediated interaction 19 , Here the first term in the effective interaction is the bare interaction U which is momentum independent. The effect of this term is small due to the sign change of the gap between the two bands. In Eq.
n k n n n , 0 n Note that we keep the Fermi surface unchanged in the dressed Green's function. For 〈 n〉 = 1.05 and U = 2.4t, the resulting value of T c , interpolated from the temperature at which λ crosses 1, is plotted in Fig. 2 as a function of t ⊥ /t. As shown in this figure, after the SDW instability is suppressed by the Lifshitz transition, a pairing transition occurs at a T c which peaks as t ⊥ /t increases and then falls off as the hole band is pushed further below the Fermi energy.
The momentum dependence of the superconducting gap function Δ (k, ω = πT) ≡ Φ (k, πT)/Z(k, πT) is shown in Fig. 3. This is an A 1g (s ± ) state in which the sign of Δ changes between k z = 0 (bonding) and k z = π (antibonding) bands. One can see that the magnitudes of the two gaps Δ (k x , k y , k z = 0) and Δ (k x , k y , k z = π) are comparable even though the hole band is below the Fermi energy.
In order to understand the peak in T c that occurs as the hole band drops below the Fermi energy, it is useful to separately examine the dependence of T c on the changes in n n that occur as the T c at which the eigenvalue of Eq. (5) goes to 1 as a functional of χ and the pair propagator GG. We can calculate the variation in T c due to the change in χ with GG unchanged when t ⊥ increases by Δ t ⊥ , c GG c c and the variation due to the change in pair propagator GG when χ is unchanged and t ⊥ increases by Δ t ⊥ , c c c We set Δ t ⊥ = 0.01t. The results of the calculation are shown in Fig. 4. Here one sees that the initial increase in T c arises from both the changes in χ and GG. The latter effect is associated with an increase in the quasi-particle spectral weight Z −1 (k, ω) on the electron Fermi surface that occurs as the hole band drops below the Fermi energy. This increase in the quasi-particle spectral weight initially ameliorates the decrease in T c resulting from the submergence of the hole band. The initial positive contribution associated with the variation in χ reflects the change in the frequency structure of the spin-fluctuations. As the hole band drops below the Fermi energy, a gap opens in the low energy q z = π spin fluctuation spectrum and spectral weight is transfered to higher energies as shown in Fig. 5, which leads to stronger pairing 20 . The ultimate decrease in T c is due to the decrease of the pair propagator n n as t ⊥ increases and the hole band drops further below the Fermi energy, as well as the decreasing strength of the spin-fluctuations.

Discussion
To conclude, we have studied a two-layer Hubbard model with parameters chosen so that a SDW instability is suppressed by a Lifshitz transition in which the hole band at the Γ point drops below the Fermi energy. Here, we have kept the site filling fixed and varied the interlayer hopping to tune the system through the Lifshitz point. For a physical system this might by obtained via strain 11 . Following the suppression of the SDW order, we find the onset of an s ± superconducting state whose transition temperature T c initially increases as the system is pushed beyond the Lifshitz point by further increasing t ⊥ /t. We find that this increase in T c is associated with both an increase in the quasi-particle spectral weight and an increase in the strength of the pairing interaction, which are related to the incipient hole band and the resulting change in the spectral distribution of the spin-fluctuations. We find that the gap function on the incipient hole band is similar in magnitude but of the opposite sign to that on the electron band which crosses the Fermi surface. Here one sees that initially as t ⊥ /t increases beyond 3.7 and the SDW order is suppressed, both the change in χ and the change in GG lead to an increase in T c . Then, as t ⊥ /t increases further and the hole band drops deeper below the Fermi energy, the changes in both χ and GG lead to a reduction in T c .   χ(π, π, π, Ω) at T = 0.1t versus Ω for U = 2.4t for different values of t ⊥ , which continuously increases in units of 0.01t from t ⊥ = 3.7t (red curve)to t ⊥ = 4t (blue curve). The thick line represents the value of t ⊥ = 3.77t corresponding to the maximum value of T c . As t ⊥ increases and the hole band drops below the Fermi energy, the spin fluctuation spectral weight is shifted to higher frequencies, of order 5 to 10T c , where it is more effective for pairing. However, as t ⊥ increases further and the spectral weight moves to still higher frequencies, the strength of the pairing decreases and this combined with the suppression of the pair propagator GG leads to a rapid decrease of T c .