Upper bounds on the superfluid stiffness and superconducting $T_c$: Applications to twisted-bilayer graphene and ultra-cold Fermi gases

TL;DR

This paper derives universal upper bounds on the superfluid stiffness and critical temperature $T_c$ for superconductors, with applications to twisted bilayer graphene and ultra-cold Fermi gases, emphasizing the role of phase fluctuations.

Contribution

It introduces rigorous upper bounds on superfluid stiffness applicable to multi-band systems in any dimension, providing new insights into $T_c$ limits beyond mean field theory.

Findings

For 2D systems, $k_B T_c \,\leq\, E_F/8$, an exact bound.

Applied bounds constrain $T_c$ in twisted bilayer graphene close to experimental values.

Bound analysis offers insights into the 3D $T_c$ upper limits.

Abstract

Understanding the material parameters that control the superconducting transition temperature $T_c$ is a problem of fundamental importance. In many novel superconductors, phase fluctuations determine $T_c$, rather than the collapse of the pairing amplitude. We derive rigorous upper bounds on the superfluid phase stiffness for multi-band systems, valid in any dimension. This in turn leads to an upper bound on $T_c$ in two dimensions (2D), which holds irrespective of pairing mechanism, interaction strength, or order-parameter symmetry. Our bound is particularly useful for the strongly correlated regime of low-density and narrow-band systems, where mean field theory fails. For a simple parabolic band in 2D with Fermi energy $E_F$, we find that $k_BT_c \leq E_F/8$, an exact result that has direct implications for the 2D BCS-BEC crossover in ultra-cold Fermi gases. Applying our multi-band bound to magic-angle twisted bilayer graphene (MA-TBG), we find that band structure results constrain the maximum $T_c$ to be close to the experimentally observed value. Finally, we discuss the question of deriving rigorous upper bounds on $T_c$ in 3D.