Cooperon diagram

The Cooperon diagram (often just called Cooperon ) occurs in connection with the analytical theory of electrical conductivity ("resistance theory") of non- superconducting metals, see for example web link 1. The name "Cooperon diagram" is nevertheless intended to refer to Leon Remember Neil Cooper , who ultimately gave rise to the explanation of the phenomenon of superconductivity through the formation of so-called Cooper pairs ( BCS theory ). Overall, it is hereby reminded that the so-called. Ladder diagrams (English: ladder diagrams )., The Cooper effect represent diagrammatically in superconductivity, very closely with the so-called crossed diagrams are related to normal in the resistance theory metals for the effects of so-called weak localization are responsible. So it is a question of cross-relationships within different branches of theoretical solid-state physics.

From ladder diagrams to crossed diagrams

The relationship leading to the formation of Cooper pairs ladder diagrams (English: ladder diagrams of ladder = the ladder) and in the conventional - but beyond the lowest order - occurring resistance theory crossed diagrams is as follows:

The ladder diagrams consist of two parallel horizontal lines, for example from (x _i , 0) to (x _f , 0) or from (x _i , y) to (x _f , y) , interrupted at certain intervals by vertical interaction. The vertical interaction leads, as with a horizontal ladder, from the lower horizontal line to the corresponding point on the parallel horizontal line. The cause of this interaction is ultimately not essential in superconductivity; mostly (comparatively!) very slow or static processes that z. B. are linked to quantized vibrations (" phonons ").

In the resistance theory of normal metals, however, the vertical lines represent the impurity scattering ( Impurities ), and the circuit diagrams represent here the usual effects that are summarized in the conventional theory of electrical resistance (so-called. Drude model ).

The non-conventional effects (so-called “coherence phenomena”, “weak localization” and others) are related to the “ crossed diagrams ”. These arise from the ladder diagrams by reversing the direction of movement of the top line: The top line no longer goes from left to right, but vice versa from right to left. This creates interactions that no longer lead, as before, from a point “ bottom left ” (or “bottom right”) to the corresponding point “ top left ” (or “top right”), but it goes “totally cross-over”, for example from "bottom left" to "top right" (or from "bottom right" to "top left").

The terms that correspond to the resulting “crossed diagrams” lead, as already mentioned, to the phenomenon of weak localization. In the case of time reversal invariance, they are equivalent to the Cooper diagrams (ladder diagrams) and are also referred to as "Cooperon diagrams".

Concrete pictures

A ladder diagram in superconductivity theory is something like the following

                  i' _______________________________________________________ f'
                             A_p      B_p..    |..      B'_p..   A'_p
                             |        |        |        |        |
                             |        |        |        |        |
                  i  ________A________B..______|________B'.._____A'_________ f

This means, for example: One electron goes on a path from i ("initial") to f ("final"), the other on a parallel path from i 'to f'. Only "vertical" scattering processes take place (e.g. from A to A_p). The vertical lines represent e.g. B. Scattering processes of the electron on quantized lattice vibrations (phonons). One electron emits a phonon; the other adsorbs it again

The corresponding Cooperon diagram from the theory of the electrical resistance of common metals is:

            (f->) f' _______________________________________________________ i'
                             A'_p     B'_p.. ..|        B_p..    A_p
                             |        |        |        |        |
                             |        |        |        |        |
                  i  ________A________B..______|________B'.._____A'_________ f   (-> f')

where the vertical lines do not represent phonons, but rather the scattering of impurities . The situation is now more complicated: while electron 1 moves from i to f , electron 2 moves in the opposite direction from f '(≈f) to i' (≈i) , and while the first electron initially interacts with impurity A - for example with a positive effect - the second electron also interacts - despite the inverse direction of movement - with the same impurity as the first, with its effect being compensated. Because of this "anti-correlation" (compensation effect!), The given diagrams contribute at most to the so-called "(weak) localization" of the electrons.

The crossed diagrams from resistance theory are created by reversing the direction of movement of the upper horizontal line in the last diagram, i.e. again i with i '(and A with A _p - now without a dash! - and B ... with B _{p ...} - also without a stroke -) connects, etc., and at the same time to transition to a consistent single particle, in which now a single electron corresponding to the Schrödinger equation a closed curve passes while at diametrically opposite points correlated (or rather anti-correlated corresponding to) the usual diagrammatic rules! so It is scattered that the contributions to conductivity are as small as possible, i.e. that the resistance is as large as possible.

So the crossed diagrams are ( cum grano salis ):

                   final _______|||_______ initial' ( $\leftarrow$ , Weg 2)
                            ..B  |  B'..
            $(\downarrow )$        |||        $\nwarrow \nearrow$        |||        $(\uparrow )$ 
                                $\nearrow \nwarrow$ 
                 initial  __..B__|__B'..__ final'   ( $\to$ , Weg 1).
                                |||

References and comments

↑ Dieter Vollhardt , Peter Wölfle : Scaling Equations from a Self-Consistent Theory of Anderson Localization . In: Physical Review Letters . tape 48 , no. 10 , March 8, 1982, p. 699-702 , doi : 10.1103 / PhysRevLett.48.699 .
↑ In order to immediately clear up a possible misunderstanding that arises from the fact that there are two different words in German ('der' or 'die') that are translated the same way in English ('the'): In the theory of superconductivity or the conductivity theory it comes (English: the conductivity) to the concept of (electric) conducting, at the so-called ladder diagrams however, not 'the' leader, but to 'the' leader. ( s : the ladder).

Web links

Dieter Vollhardt: A didactic representation of the Anderson localization (including Cooperon and "crossed" diagrams; PDF; 524 kB)

[1] Dieter Vollhardt , Peter Wölfle : Scaling Equations from a Self-Consistent Theory of Anderson Localization . In: Physical Review Letters . tape 48 , no. 10 , March 8, 1982, p. 699-702 , doi : 10.1103 / PhysRevLett.48.699 .

[2] In order to immediately clear up a possible misunderstanding that arises from the fact that there are two different words in German ('der' or 'die') that are translated the same way in English ('the'): In the theory of superconductivity or the conductivity theory it comes (English: the conductivity) to the concept of (electric) conducting, at the so-called ladder diagrams however, not 'the' leader, but to 'the' leader. ( s : the ladder).