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Cooper-pair binding energy

What you are seeing: two electrons added to a filled Fermi sea bind into a Cooper pair with energy Ebind=2ℏωDe−2/N(0)VE_{bind} = 2\hbar\omega_D e^{-2/N(0)V}. Exponentially small at weak coupling, the binding is nonperturbative in the attractive interaction strength.

Figure 1. Left: binding energy Eb=2ℏωDe−2/N(0)VE_b = 2\hbar\omega_D e^{-2/N(0)V} on a log scale as a function of N(0)VN(0)V, with live readout driven by sliders for coupling VV, density of states N(0)N(0), and Debye cutoff ℏωD\hbar\omega_D. Right: pair wavefunction amplitude ∣g(ξ)∣|g(\xi)| in the energy shell around the Fermi surface, peaking near ξ=0\xi = 0 with width set by the cutoff ℏωD\hbar\omega_D. Ashcroft-Mermin Ch. 34.
V (coupling)0.30
N(0) (DOS)1.00
ℏω_D (cutoff)1.00

WHAT TO TRY

  • Vary each control and watch the rail readouts respond.
  • Compare the diagnostic plot against the live scene.