Superconductivity with repulsion: a variational approach

TL;DR

This work addresses the stability of superconductivity in multiband systems with repulsive channels by identifying that the conventional mean-field free energy $F_{ m MF}$ is not variational. By applying Bogoliubov's variational principle, the authors derive a bounded free energy $F_{ m Var}=F_{ m MF}+igra H-H_{ m MF}igra_{ m MF}$ that preserves the BCS gap equations as a saddle-point condition and ensures a true minimum even when repulsion is present. Through a single-band and a two-band $s_ ext{±}$ toy model, they show that the instability predicted by $F_{ m MF}$ is resolved by $F_{ m Var}$, with the attractive/repulsive channels defined by the full kernel $oldsymbol{V}oldsymbol{ it$Pi}$ rather than $oldsymbol{V}$ alone. The framework is basis-agnostic and provides a consistent platform for analyzing fluctuations, potentially extendable to Eliashberg theory, thus clarifying conceptual ambiguities raised by prior works and enabling reliable analysis of multiband superconductors with competing interactions.

Abstract

We revisit the stability of the superconducting state within mean-field theory in the presence of repulsive pairing interactions, focusing on multiband systems where such channels naturally arise. We show that, when repulsion is present, the self-consistent BCS solution appears as a saddle point of the conventional mean-field free energy, casting doubt on its physical stability. We show that this pathology is an artifact of using a non-variational functional. Recasting the problem with Bogoliubov's variational principle restores a free energy that is bounded from below and places the BCS solution at a genuine minimum. Using a two-band toy model relevant to iron-based superconductors, we demonstrate the stability of the s$_\pm$ state and clarify how projection schemes that rely only on the interaction matrix can misidentify the attractive eigenmode that drives pairing. Our results clarify the instability issue highlighted by Aase et al. and provide a consistent foundation for analyzing fluctuations in the presence of repulsive interactions.

Figures (3)

• Figure 1: Energy landscape of a single-band superconductor versus the dimensionless gap $\Delta/V$. Panels (a) and (b) display the attractive ($V>0$) and repulsive ($V<0$) case, respectively. $F_{\text{MF}}$, dashed line and $F_{\text{Var}}$, solid line, both renormalized to the hopping $t$. The self-consistent BCS gap $\bar{\Delta}$ satisfies the saddle point condition for both functionals. In (a) $\bar{\Delta}$ sits at a true minimum of both $F_{\text{MF}}$ and $F_{\text{Var}}$ with the variational correction being only quantitative. In (b) $F_{\text{MF}}$ is unbounded from below, while $F_{\text{Var}}$ remains bounded and attains its global minimum at $\bar{\Delta}=0$. Parameters $t=1$ eV, $V=\pm 2 t$, $T=0$.
• Figure 2: Free energy landscape for a two-band model with purely interband repulsion, $V_{12}=V_{21}=V<0$. (a) Color map of $F_{\rm MF}/t$ in terms of $(\Delta_{1}/V ,\Delta_{2}/V)$ from Eq. \ref{['F_MF']}. (b) Variational free energy $F_{\rm Var}/t$, obtained from Eqs. \ref{['F_Var']} and \ref{['spm_Hav']}. Black diagonal lines show the attractive and repulsive fluctuating modes around the BCS solution as defined by the diagonalization of the full kernel. $F_{\rm MF}$ is unbounded from below along the repulsive direction, placing the BCS $s_{\pm}$ solution at a saddle point, $F_{\rm Var}$ instead remains bounded and exhibits a true minimum at the BCS solution. (c,d) Value of $F_{\rm MF}$ (dashed) and $F_{\rm Var}$ (solid) along the attractive and repulsive directions, respectively. These cuts mirror the single-band behavior of Fig. \ref{['1band_FMF']}. For the attractive mode the variational correction is only quantitative, whereas along the repulsive mode it cures the unboundedness of $F_{\rm MF}$ and restores a stable minimum. Parameters $t_{1}=t_{2}=1$ eV, $V_{12}=V_{21}=V= -2 t$, $T=0$.
• Figure 3: Identification of the attractive mode in the two-band system with inequivalent bands and purely interband repulsion. (a) Color map of $F_{\text{MF}}(\Delta_1, \Delta_2)$ normalized to $t_1$, showing the $s_\pm$ solution. The solid line, $\Delta_1 = -\alpha, \Delta_2$, with $\alpha = \sqrt{\Pi_2/\Pi_1}$, corresponds to the attractive eigenmode of the full kernel $\hat{V}\hat{\Pi}$, the critical mode defining the SC instability. The dashed line, $\Delta_1 = -\Delta_2$, corresponds to the "attractive" eigenmode of the interaction matrix $\hat{V}$. This subspace does not include the $s_\pm$ solution and does not align to the attractive mode direction. Even requiring the diagonal projection to pass through the saddle-point solution would still fail to identify the instability line. As discussed in the text, the two attractive subspaces coincide only when the bands are equivalent. (b) Cuts of $F_{\text{MF}}/t_1$ along the two directions. The diagonalization of the full BCS kernel identifies the true attractive mode, while the projection based solely on $\hat{V}$ does not contain the BCS solution. Parameters $t_1=3t_2=1$ eV, $V = - 2 t_1$, $T = 0$.