Higher-Order Fermion Interactions in BCS Theory

TL;DR

The work investigates higher-order fermionic deformations in BCS-like phase transitions, focusing on an octic interaction and its effect in a multiflavor setting. By introducing Hubbard-Stratonovich fields and integrating out $N$ fermions, the authors derive a one-loop effective action with a gap equation and a free-energy containing a key parameter $c=\lambda/g^3$; they identify a critical coupling $c_0=g\nu\beta_0$ separating second-order (mean-field) and first-order transitions, with a special case at $c=c_0$ yielding a $1/4$-order parameter exponent. Numerical analysis confirms the analytic picture: for $c<c_0$ the transition is second-order with conventional or deformed gap-temperature behavior, while for $c>c_0$ the gap becomes multivalued and the system undergoes a first-order transition at $T_c^{\star}$. The results imply that higher-order fermionic interactions can significantly modify gap dynamics and transition order, with potential implications for multiband and type-1.5 superconductors, and motivate further microscopic derivations of the effective couplings.

Abstract

We investigate the impact of higher-order fermionic deformations in multiflavor Bardeen-Cooper-Schrieffer (BCS) theory. Focusing specifically on the 6- and 8-fermion interactions, we show that these terms can have significant consequences on the dynamics of the system. In certain regions of parameter space, the theory continues to exhibit second-order phase transitions with mean-field critical exponents and the same critical temperature; however, the temperature dependence of the superconducting gap can deviate markedly from conventional BCS behavior. In other regions, the theory exhibits first-order phase transitions or second-order phase transitions with non-mean field exponents. We conclude by discussing potential phenomenological applications of these theories.

Figures (3)

• Figure 1: 4- and 8-point functions.
• Figure 2: $\Delta$vs.$T$ for $c=0$ (standard BCS, in red), then $c=0.8\, c_0$ (black), $c=c_0$ (dashed, blue), $c=1.2\, c_0$ (green). Here $g\nu=0.3$, $\omega_{\star}=100$.
• Figure 3: a) Free energy vs.$T$ (same conventions as in figure 1). b) Enlarged view of the free energy in the case $c=1.2\, c_0$. The lower branch in Fig. 1 corresponds to the branch with positive free energy. The upper branch in figure 1 has negative free energy until $T^{\star}_{c}\approx 4.3$, where a first-order phase transition takes place. For $T>T^{\star}_{c}$, $\Delta=0$.