[Anderson]:
I do not, however, accept that Scalapino’s calculations, refined as they are, come near to settling the question he has raised. The numerical analysis of Scalapino’s reference 7, and the analysis of experiments in his references 8 and 9, all have, logically, two pieces. To take ref 7 for definiteness, Scalapino’s group does two computations. The first, while difficult, is logically unimpeachable: It is a carefully designed simulation of the properties of the Hubbard model which we both agree is by far the main Hamiltonian candidate. Sure enough, they find d-wave superconductivity and other physical properties that agree with experiments—and also with much simpler mean field theories (1).
In step 2, they attempt to derive from the measured quantities a number of theoretical constructs, such as the "pairing interaction vertex," assuming that the underlying theory is the conventional Feynman-Dyson diagram theory as adapted for condensed matter problems in the 1960s. Thus, the procedure, far from being a purely direct computational result, is a theoretical construct with a very relevant input of unproven assumptions. The interaction vertex which is derived does not look at all like the J term in the Hamiltonian; it is much smaller than it should be, at high energies, growing to its full strength only at the lowest energies.
This is an unlikely result. It says that somehow all the low-frequency spin fluctuations have killed this giant interaction at the high-frequency end, but left it intact at low frequency to do its work on the pairing gap. It amounts to replacing my "elephant" J with almost nothing but its indirect consequences. It also contradicts the simplest mean field theory of the t-J model (ref 1, called because of its simplicity the "Plain Vanilla" theory).
If I found a result which so blatantly did not make physical sense, I might have questioned the method rather than attacking those of us who seem to have found the right answers by doing things otherwise. It is very attractive to abandon these particular methodological assumptions, because the same set of assumptions, applied in the normal state of the same materials, have had zero success in describing its unique and very anomalous properties.
[Scalapino]:
While there is a growing consensus that superconductivity in the high Tc cuprates arises from strong short-range Coulomb interactions between electrons rather than the traditional electron-phonon interaction, the precise nature of the pairing interaction remains controversial (1). This is the case even among those who agree that the essential physics of the cuprates is contained in the Hubbard model (Perspectives, "Is there glue in cuprate superconductors?", by P. W. Anderson, 22 June 2007, p. 1705). For example, both Anderson’s resonating-valence-based (RVB) theory (2) and the spin-fluctuation exchange theory (3, 4) lead to a short-range interaction which forms d x2−y2 pairs. However, the dynamics of the two interactions differ.
In the RVB picture, the superconducting phase is envisioned as arising out of a Mott-liquid of fluctuating singlet pairs. These pairs are bound by a superexchange interaction J which is proportional to t2/U. Here t is the effective hopping matrix element between adjacent sites and U is an onsite Coulomb interaction. J is determined by the virtual hopping of an electron of a given spin to an adjacent site containing an electron with an opposite spin (5). Thus the dynamics of J involves virtual excitations above the Mott gap which is set by U, and the pairing interaction is essentially instantaneous. In this case, as Anderson recently discussed (6), one would not speak of a pairing glue.
In the spin-fluctuation exchange picture, the pairing is viewed as arising from the exchange of particle-hole spin 1 fluctuations whose dynamics reflect the frequency spectrum seen in inelastic magnetic neutron scattering. This spectrum covers an energy range which is small compared with U or the bare bandwidth 8t. In this case, the pairing interaction is retarded and in analogy to the traditional phonon mediated pairing, one says that the spin-fluctuations provide the pairing glue.
Thus, the question of whether there is pairing glue in the cuprates is a question about the dynamics of the pairing interaction. It offers a way of distinguishing different theories. For the Hubbard model, recent numerical calculations (7) have shown that the strongest pairing occurs for U of order the bandwidth 8t. This is also thought to be the parameter regime appropriate to the cuprates. In this regime, these calculations find that the dynamic dependence of the pairing interaction is the same as that of the dynamic spin susceptibility. Thus, there is pairing glue in the Hubbard model.
Of course, the ultimate question is: What does the experiment tell us about the dynamics of the pairing interaction? Just as the spatial structure of the pairing interaction can be determined from the k-dependence of the superconducting gap, the dynamics of the interaction is reflected in the frequency dependence of the gap. In addition, if the interaction is retarded, that is delayed for a time of order ћ/(2J), the gap will have both a real and an imaginary component. This frequency structure of the gap is reflected in a variety of experiments and analysis of structure in the angular resolved photoemission spectrum (8), and the infrared conductivity (9) have suggested that the dynamics is indeed determined by spin-fluctuations. However, as opposed to the Hubbard model, real materials have phonons and alternative explanations have also been proposed (10). Thus, while there is pairing glue in the Hubbard model, more experimental work is needed to settle the question of whether there is glue in the cuprate superconductors.
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