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I have been reading various articles on Renormalization group where they compute the flow of some parameter which becomes increasingly attractive and then say that parameter is responsible for Cooper pairing and relate it to some $D_{x^2-y^2}$ symmetry.

This reminds me of one of the spherical harmonics but when they were calculating flow they were indexing parameters by linear momentum so I'm not sure what the relationship would be if $D_{x^2-y^2}$ was a reference to spherical harmonics.

I believe my closest guess is that it has to do with the symmetry of the pairing function $V(\theta_1,\theta_2)$ but I'm not certain.

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  • $\begingroup$ Which articles? $\endgroup$
    – Qmechanic ♦
    Jul 28, 2015 at 23:35
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    $\begingroup$ Never see $d_{x^2+y^2}$. $x^2+y^2$ has $s$ wave symmetry. You probably mean $d_{x^2-y^2}$. $\endgroup$
    – Meng Cheng
    Jul 28, 2015 at 23:49
  • $\begingroup$ Phys. Rev. B 61 20 (2000) Zanchi and Schulz is the only one I have printed off but I found it in a bunch when looking quickly at some references from this article and the references from Shankar's 1994 article: R. Shankar, Rev.Mod. Phys. 66 129 (1994) $\endgroup$
    – Shane P Kelly
    Jul 28, 2015 at 23:53
  • $\begingroup$ You are right Meng Cheng. That's what I meant. $\endgroup$
    – Shane P Kelly
    Jul 28, 2015 at 23:55
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    $\begingroup$ It's basically the expectation value of the order parameter $c_{\mathbf{k}\uparrow}c_{-\mathbf{k},\downarrow}$ on the superconducting ground state. $\endgroup$
    – Meng Cheng
    Jul 29, 2015 at 0:59

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Every phase transition has an order parameter: something that vanishes above the transition temperature and is finite below.

In superconductors, the order parameter is a complex quantity related to the superconducting gap: $\Delta = |\Delta| e^{i \phi}$.

In BCS theory, there is a self-consistent equation for the gap:

$\Delta_k = -\sum_q V_{kq} \frac{\Delta_q}{2E_q} \tanh \frac{E_q}{2 KT}$.

At this point BCS made the assumption that $V_{kq} = -V$ (the interaction potential is momentum independent and attractive) leading to $\Delta_k = \Delta_q = \Delta$ (i.e., also momentum independent and constant). The gap has the same value at all positions of the Fermi surface. It is the so-called $s$-wave (isotropic) gap.

Now, if one does not make the approximations in $V_{kq}$ then funny stuff happens. For instance, if $V$ is positive (hence repulsive) you can still have superconductivity! But this means that the gap must change sign (note the minus sign in the self-consistent equation). And this change will happen in momentum space directions where the interaction potential is large.

A particular case of this gap sign change is an order parameter with $d_{x^2 - y^2}$ symmetry. This is a gap that will change sign at $(k_x,k_y) = (\pm\pi,\pm\pi)$ directions in momentum space. It is the currently mostly accepted symmetry for cuprate superconductors.

This symmetry is not necessarily that of the paring potential. What is required is that the pairing potential must have a large value close to $(\pm\pi,\pm\pi)$ directions.

The still unanswered question is what is the pairing potential, though.

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