They assume that there is singlet pairing along each bond on the square lattice at some temperature which linearly decreases with increasing doping x. A key ingredient is the coupling between neighbouring horizontal and vertical bonds. This coupling C is assumed to have a negative sign to produce d-wave pairing. Furthermore, to produce a superfluid density which for small x scales x it is assumed that C is also proportional to x. [They also give a simple argument suggesting that this C scales with t' the diagonal hopping in an underlying band structure].
From this simple model they can extract some of the key phenomenology of the cuprates, including the doping dependence of the transition temperature. Coupling the order parameter fluctuations to electrons produces a self energy and spectral density consistent with ARPES experiments, including the existence of Fermi arcs.
A few things I found interesting
- Quantum fluctuations are argued to be important in the underdoped andoverdoped regime. Indeed Tc goes to zero for small non-zero dopings.
- Past the optimum Tc the superfluid density decreases with decreasing doping.
- I would be interested to see these relatively simple ideas extended to the half-filled case corresponding to organic charge transfer salts. Then, x will have to replaced with something like U-Uc where Uc is the critical value of the Hubbard U for the Mott transition.