I mentioned in passing a worrying quantum Monte Carlo study published in PRB in 1999

Correlated wave functions and the absence of long-range order in numerical studies of the Hubbard model

M. Guerrero, G. Ortiz, and J. E. Gubernatis

The graph below shows the distance dependence of the pairing correlation function in the d-wave channel. If superconductivity occurs it should lend to a non-zero value equal to the square of the superconducting order parameter.

It certainly looks like it tends to zero at large distances.

However, careful examination shows that it seems to have a non-zero value of order 0.001.

Perhaps, that is just a finite size effect.

But, we should ask, "

**How big do we expect the long-range correlations, i.e. the magnitude of the square of the order parameter d, to be?**"

A cluster DMFT calculation on the doped Hubbard model (in the PRB below) gives a value of order 0.03 for the order parameter d. This means d^2 ~ 0.001 consistent with the QMC study which claims no superconductivity!

Anomalous superconductivity and its competition with antiferromagnetism in doped Mott insulators

S. S. Kancharla, B. Kyung, D. Sénéchal, M. Civelli, M. Capone, G. Kotliar, and A.-M. S. Tremblay

Similar issues arise when assessing the results of

Absence of Superconductivity in the Half-Filled Band Hubbard Model on the Anisotropic Triangular Lattice

R. T. Clay, H. Li, and S. Mazumdar

If I take the order parameter estimated by a RVB calculation reported in this PRL (by Ben Powell and myself) and square its value it predicts a long-range pairing correlation (~0.001) comparable to the extremely small values found in the numerical study claiming absence of superconductivity.

Clay, Li, and Mazumdar also mentioned the problematic observation that the pairing correlation they calculated did not increase with the Hubbard U. However, my previous post discussed how Scalapino and collaborators argued this is because one needs to factor in the quasi-particle renormalisation Z that also occurs with increasing U. For the half-filled Hubbard model this probably leads to an order of magnitude enhancement of the pairing as U increases towards the Mott insulating phase, since Z decreases from 1 to 0.3 and the renormalised P_d scales with 1/Z^2.

So, I remain to be convinced that superconductivity does not occur in the Hubbard model, both upon doping the Mott insulator or at half-filling near the band-width controlled Mott transition.

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