Superconductivity due to massless boson exchange in the strong-coupling limit

We solve the problem of fermionic pairing mediated by a massless boson in the limit of large coupling constant. At weak coupling, the transition temperature is exponentially small and superconductivity is robust against phase fluctuation. In the strong coupling limit, the pair formation occurs at a temperature of the order of the Fermi energy, however, the actual transition temperature is much smaller due to phase and amplitude fluctuations of the pairing gap. Our model calculations describe superconductivity due to color magnetic interactions in quark matter and in systems close to a ferromagnetic quantum critical point with Ising symmetry. Our strong-coupling results are, however, more general and can be applied to other systems as well, including the antiferromagnetic exchange in 2D used for description of the cuprates.

Figure 1

(Color online) Schematic phase diagram for the superconducting transition temperature $T_c$ and the pairing instability temperature $T_{\text{pair}}$ as function of the dimensionless coupling constant $g$. While $T_c\simeq T_{\text{pair}}$ for weak coupling, an intermediate regime $T_c<T<T_{\text{pair}}$ with phase (and amplitude) fluctuations occurs in the strong-coupling limit.

Figure 2

Characteristic energy scales for the weak- $(g⪡1)$, intermediate- $(g\sim 1)$, and strong- $(g⪢1)$ coupling limit. While at weak coupling, ${\omega }_0\simeq E_F∕g$ is large compared to $E_F$ and thus irrelevant, it emerges as a new low-energy scale in the strong coupling limit.

Figure 3

(Color online) Particle-particle response function ${\chi }_{pp}(\omega ,T)$ for $g=0.1$ as function of $T$ for various energies. Inset: ${\chi }_{pp}(\omega ,T)$ as as function of energy for $T=1.001T_c$.

Figure 4

Leading correction of the fermion boson vertex ${\Gamma }_{k,q}(\omega ,\Omega )$. The solid lines stand for the fermionic propagator and the wiggly lines for the bosonic propagator, respectively.