Spin-fluctuation–mediated pairing in multiband superconductors
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Transcript of 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)