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.
Highlights
Our work is motivated by the fundamental question, what limits the superconducting (SC) transition temperature Tc? Within BCS mean-field theory, and its extensions like Eliashberg theory, the amplitude of the SC order parameter is destroyed by the breaking of pairs, and Tc scales with the pairing gap Δ
The material parameters that control the mean-field Tc are the electronic density of states (DOS) at the chemical potential Nð0Þ and the effective interaction, determined by the spectrum of fluctuations that mediate pairing
Irrespective of that, our results suggest that magic-angle twisted bilayer graphene (MA TBG) is a strongly correlated SC in a phase fluctuation dominated regime
Summary
Our work is motivated by the fundamental question, what limits the superconducting (SC) transition temperature Tc? Within BCS mean-field theory, and its extensions like Eliashberg theory, the amplitude of the SC order parameter is destroyed by the breaking of pairs, and Tc scales with the pairing gap Δ. BCS theory-based intuition suggests that narrow bands have a large DOS Nð0Þ and lead to high-temperature superconductivity Is this true, or do phase fluctuations limit the Tc? We note that ultracold Fermi gases in the strongly interacting regime of the BCS-BEC crossover [9,10] exhibit experimental values [11] of kBTc=EF larger than those observed in the solid state. All of these observations raise the question of ultimate limits on the Tc of a superconductor or paired superfluid. We show that the presence of nonuniversal prefactors in the relation between Tc and Ds, as well their scaling behavior near a SC quantum critical point, poses challenges in deriving a rigorous bound in 3D
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