# Spin Fluctuations in the Cuprates

In the history of the cuprate superconductors, many predictions have been put forth, but there was one non-trivial prediction that has stood out among the rest. This is the prediction of the $d_{x^2-y^2}$ order parameter symmetry from spin-fluctuation models that were put forth before its experimental verification.

The idea is quite simple, and you can read more about it in this set of lecture notes by A.J. Leggett, where he lays out the concepts very well. I summarize the main points below.

In the cuprates, the Fermi surface is usually assumed to look like so, which has been determined by ARPES experiments:

Schematic of Fermi Surface as Determined by ARPES

One also knows that the antiferromagnetic phase in the parent compound looks like so:

(a) Real space representation of the antiferromagnetic parent phase. (b) Reciprocal space representation with Q representing the antiferromagnetic Bragg wavevector.

i)     Now, one can see that the points on the Fermi surface close to ($\pi$, 0) and (0, $\pi$) are the ones connected by the antiferromagnetic Bragg wavevector, Q. One would then predict a singlet pair wavefunction, as those points on the Fermi surface would be expected to exhibit the largest gap.

ii)     The other input is that scattering should not change the sign of the pair wave function, $F_\textbf{k} = orbital\times spin$, which comprises the orbital and spin components. Since the spin part is a singlet, (i.e. $1/\sqrt{2} (\uparrow_1\downarrow_2-\downarrow_1\uparrow_2)$), it will change sign when the pair interacts through a spin-fluctuation. Therefore, to keep $F_\textbf{k}$ invariant, the orbital part must also change sign under the scattering/interaction of wavevector Q.

The two criteria leave $d_{x^2-y^2}$ symmetry as the only option, and hence spin-fluctuation theories explicitly predict this symmetry.

Obviously, this does not mean that spin-fluctuation theories are correct, but it is worth noting that they have made a non-trivial prediction.

While this historical note is well-known to those have been studying high-temperature superconductivity since its discovery, those of us who were born around the same time as the discovery of the cuprates sometimes lose this kind of historical context.

Images are taken from the lecture notes by A.J. Leggett linked above.