Andersen Group El.-Phon. QMC C60 Resistivity saturation

Pseudgap in high temperature superconductors

We have studied the pseudogap seen in cuprates and organic systems [κ-(BEDT-TTF)2 X], using the dynamical cluster approximation (DCA). In this approach a cluster with Nc atoms is embedded in a self-consistent bath.
The smallest cluster showing a pseudogap is Nc=4. To analyze the results, we have simplified this to a four level model, including the cluster levels with k vectors (π,0) and (0,π), but neglecting the essentially filled or empty (0,0) and (π,π) levels. Each cluster level is coupled to a bath level. For small U, each cluster level forms a Kondo state with its bath, leading to a Kondo peak at the Fermi energy. As U is increased, it becomes favorable to form a bound state between the cluster levels. Then interference effects suppress the Fermi energy peak and transfer weight to excited neutral cluster states, leading to a pronounced pseudogap structure. This happens when the lowest cluster state is nondegenerate. Changing the parameters so that the lowest cluster state is a triplet, the Kondo character remains even for large U. This, however, does not apply to the compounds studied here.
We have then performed DCA calculation for Nc=8, where both =(π,0) and (π/2,π/2) occur. The postscript file shows correlation functions, Cij=〈 n_i n_j〉-〈 n_i〉 〈 n_j〉, where i and j stand for orbital and spin indices. The results are obtained for the first DCA iteration, when the bath is fully metallic. For small U the (π,0)up(π,0)down curve bends down as expected when (π,0) forms a Kondo state with its bath. For larger U the curve turns upwards. This is due to the formation of a bound, nondegenerate, state in the (π,0)-(0,π) space, as in the four level model. This is supported by other correlation functions in this space. At this point a pseudogap starts to form for (π,0). For larger U a similar behavior is seen in the (π/2,π/2) space, and a pseudogap forms also in this space. The decisive difference is that the coupling to the bath is much stronger in the (π/2,π/2) space and therefore a Kondo state remains favorable up to larger U. The stronger coupling is due to the stronger dispersion in the band structure at (π/2,π/2).
The lowest neutral cluster state is a singlet with a substantial local d-wave pairing. The excited neutral states forming the pseudogap are triplets or singlets with a smaller d-wave pairing. The pseudogap is then related to breaking preformed d-wave pairs.
Although the pseudogap is due to electronic correlation, we find that there is nevertheless a substantial isotope effect.

G. Sangiovanni and O. Gunnarsson:
Isotope effect in the pseudogap state of high-temperature copper-oxide superconductors,
Phys. Rev. B, Rapid Commun. 84, 100505 (2011).

J. Merino and O. Gunnarsson:
Pseudogap in cuprate and organic superconductors,

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For further information contact Olle Gunnarsson (, Jaime merino ( or Giorgio Sangiovanni (

Max-Planck Institut für Festkörperforschung
Postfach 800 665 D-70506 Stuttgart

Last Update: September 2012
Andersen Group