Emergent magnetic pseudogap from phase fluctuations and hierarchy of scales in two-dimensional superconductors

Abstract

Preformed pairs and phase fluctuations are believed to play a vital role in predicting the charge pseudogap in the normal state of two-dimensional superconductors. In this work, we extend this idea and further identify the emergent magnetic pseudogap from pure phase fluctuations without invoking any competing order. We examine the NMR relaxation rate $1/T_1T$ by evaluating the bubble contribution and leading-order vertex correction within perturbation theory. It is found that the magnetic pseudogap, manifesting as a smooth suppression of $1/T_1T$ in the normal state, is characterized by a temperature scale $T_\text{mPG}$ distinct from the superconducting gap $\Delta_\text{SC}$ and transition temperature $T_c$. The onset scales of both charge and magnetic pseudogap are dominated by the competition of BKT correlation length $\xi(T)$ and BCS coherence length $\xi_\text{BCS}$. Moreover, the vertex correction is shown to be irrelevant for $d$-wave pairing, while it becomes prominent in $s$-wave systems and drives a coherent enhancement of $1/T_1T$ at lower temperatures just above $T_c$. We attribute this normal-state enhancement of $1/T_1T$ to the diverging coherence peak at the $s$-wave superconducting gap edge, which shares the same spirit as the celebrated Hebel-Slichter peak in the BCS theory. Analogous to the coherent Hebel-Slichter peak, regularization by Fermi-liquid-like scatterings is important and is characterized by a scattering length $\ell$. The normal-state coherent enhancement of $1/T_1T$ is hence described by the competition of $\xi(T)$ and $\ell$, through which the coherence scale $T_\text{coh}$ is determined. As a result, the complete evolution of $1/T_1T$ is understood quantitatively in a unified picture as the interplay among hierarchy of scales $\xi(T)$, $\xi_\text{BCS}$ and $\ell$.