This project will investigate a simple question: why does superconductivity disappear as we raise the temperature? To understand this question one needs to understand that two ingredients are required for superconductivity (i) the electrons must pair up into “Cooper pairs” to overcome the Pauli exclusion principle and (ii) long-range phase coherence must exist between the Copper pairs, i.e. the wavefunction must have the same phase in macroscopically separated parts of the sample. In the BCS theory of superconductivity the Cooper pairs of electrons fall apart at the critical temperature T_c. This is almost certainly what happens at T_c in conventional superconductors such as tin and lead. But another possibility exists – fluctuations in the phase can eliminate the phase coherence and allow a state in which there are Cooper pairs, but there is no superconductivity.

In this project you will learn the methods of quantum field theory and apply the imaginary time path integral formulation of quantum mechanics to the problem of superconductivity in the Hubbard model. This will allow you to go beyond the mean field approximation to the slave boson approach to the Hubbard model. You will calculate the relationship between the penetration depth, λ, (an experimental measure of how strong the phase coherence is) and T_c. This has recently been measured in organic superconductors and it was found that T_c ~ λ^-3 [1], this result currently has no theoretical explanation [1] – and is incompatible with BCS theory (see figure 6 of Ref. 1). Indeed the simplest version of the phase fluctuation theory overestimates this trend (giving T_c ~ λ^-2). This experiment therefore clearly requires a subtle explanation.

[1] For a review see B. J. Powell and R. H. McKenzie J. Phys.: Condens. Matter 18 (2006) R827.