Spin-fluctuation–mediated pairing in multiband superconductors

Spin-fluctuation–mediated pairing in multiband superconductors

Maxim M. Korshunov

[ФМЯ-Октябрь-2012]

Divergence of the electron-electron scattering vertex

  =

Determines Copper instability

𝑉 (𝜔 ,𝑘 )≈ 𝜆=−𝑔𝑒− h𝑝2

For , :

Hubbard model

RPA Spin susceptibility

N.F. Berk and J.R. Schrieffer, PRL 17, 433 (1966)

Utk

𝐻=∑𝑓 ,𝜎

(𝜀−𝜇)𝑛𝑓 𝜎+ ∑𝑓 ,𝑔 ,𝜎

𝑡 𝑓𝑔𝑎 𝑓 𝜎+ 𝑎𝑔𝜎+𝑈∑

𝑓

𝑛𝑓 ↑𝑛𝑓 ↓

Effective interaction from spin fluctuations (Berk and Schrieffer 1966, Scalapino et al. 1989-1996)

Vs

c0

c0

c0

Magnetic instability:

Spin-fluctuation mediated interaction

Numerical results (FLEX)

D. Manske, Theory of Unconventional Superconductors, Springer-Verlag (2004)

S. Graser, T.A. Maier, P.J. Hirschfeld, and D.J. Scalapino, NJP 11, 025016 (2009)

Effective dimensionless coupling

D.J. Scalapino, E. Loh Jr., and J.E. Hirsch, PRB 34, 8190 (1986)

From the stationary condition one can find the eigenvalue problem:

𝐻=𝐻0+𝐻 𝑖𝑛𝑡 5-orbital tight-binding model

S. Graser, T.A. Maier, P.J. Hirschfeld, and D.J. Scalapino, NJP 11, 025016 (2009)

Matrix elements:

Interactions in band space become effectively momentum-dependent!

Matrix elements: orbitals → bands

𝐻 𝑖𝑛𝑡=𝑈∑𝑖 ,𝑚

𝑛𝑖𝑚↑𝑛𝑖𝑚↓+ 𝐽 ∑𝑖 ,𝑚<𝑛

∑𝜎 ,𝜎 ′

𝑑𝑖𝑛 𝜎† 𝑑𝑖𝑚𝜎 ′

† 𝑑𝑖𝑛 𝜎 ′𝑑𝑖𝑚𝜎+𝑈 ′ ∑𝑖 ,𝑚<𝑛

𝑛𝑖𝑛𝑛𝑖𝑚+ 𝐽 ′ ∑𝑖 ,𝑚≠𝑛

𝑑𝑖𝑛↑† 𝑑𝑖𝑛↓

† 𝑑𝑖𝑚↓𝑑𝑖𝑚↑

𝑐𝐤 𝜆𝜎=∑𝑛

𝑎𝐤𝑛 ,𝜆𝑑𝐤𝑛𝜎

Multiorbital dynamical spin susceptibility

Diagrams for

RPA for

is a matrix with indices

where “–” is due to the bubble

RPA for

We assume that

RPA for

where “–” is due to the bubble

RPA for

Similarly,

RPA for

If

Check: single-band case

That is exactly the single-band RPA solution for

Interactions in the RPA

Few easy steps to get pairing

1. Calculate pairing interaction in the orbital basis

2. Transform to the band basis

3. Solve eigenvalue problem and find the gap function and the corresponding eigenvalue

S. Graser, T.A. Maier, P.J. Hirschfeld, and D.J. Scalapino, NJP 11, 025016 (2009)A.F. Kemper et al., NJP 12, 073030 (2010)