Spin-Fluctuation Mediated Pairing

Antiferromagnetic Spin Susceptibility → d-wave Gap Symmetry

Gap Δ(k) on Brillouin Zone

Spin Susceptibility χ(q,0)

AFM Correlation ξ (in a) ξ = 2.5 a

Coupling Constant g g = 0.80

Next-NN Hopping t'/t t'/t = -0.25

Chemical Potential μ/t μ/t = -1.0

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Spin susceptibility:

χ(q) = χ₀/(1 + ξ²(q−Q)²)

Effective pairing:

V(k,k') = −g²χ(k−k')

Gap equation:

Δ(k) = −Σ_{k'} V(k,k') Δ(k')/2E(k')

d-wave solution:

Δ(k) = Δ₀(cos kx − cos ky)

Nodes at:

kx = ±ky

In cuprate superconductors, the electron-phonon mechanism is too weak. Instead, virtual spin fluctuations peaked near Q=(π,π) provide a net attractive interaction in the d_{x²−y²} channel. Pairs on the same sublattice repel (at q=0) but attract via AFM fluctuations. The d-wave gap has four nodes on the Fermi surface.