Spin-Lattice Relaxation in Two-Dimensional Superconducting BKT Transition

TL;DR

This work examines how spin-lattice relaxation $1/T_1T$ can signal the Berezinskii–Kosterlitz–Thouless (BKT) transition in two-dimensional superconductors, extending beyond conventional transport probes. By coupling a 2D BdG description to phase fluctuations governed by a classical XY model and performing Monte Carlo sampling on a $72 \times 72$ lattice, the authors locate $T_{\rm BKT}$ from the universal jump of the superfluid density and compute $1/T_1T$ from Green's functions across $s$- and $d$-wave pairings. They find a Hebel–Slichter–like peak in $1/T_1T$ near $T_{\rm BKT}$ for $s$-wave, driven by coherence peaks in the DOS, while no such peak appears near $T_{\rm BCS}$. In contrast, $d$-wave shows no HS peak; the zero-frequency DOS vanishes below $T_{\rm BKT}$ and rises above it, with the derivative $\partial A_\omega(\omega=0)/\partial T$ peaking near $T_{\rm BKT}$. The Knight shift exhibits a kink at $T_{\rm BCS}$, providing an additional spectroscopic signature. Together, these results offer a practical route to detect the BKT transition and distinguish pairing symmetry via NMR-like probes in 2D superconductors and related systems with strong phase fluctuations.

Abstract

Two-dimensional superconductors undergo a Berezinskii-Kosterlitz-Thouless transition driven by vortex-antivortex unbinding, yet experimental signatures beyond transport remain limited. Here, we show that the spin-lattice relaxation rate provides a direct probe of this transition. In a 2-dimensional $s$-wave superconductor, $1/T_1T$ develops a Hebel-Slichter-like peak around $T_{\rm{BKT}}$, originating from the emergence of coherence peaks in the density of states, while no peak appears at the pair formation scale $T_{\rm{BCS}}$. We further extend our analysis to the $d$-wave superconductor. Our results highlight spin-lattice relaxation rate as a sensitive tool to detect the superconducting BKT transition and open routes to exploring its manifestation in unconventional pairing states.

Figures (7)

• Figure 1: Schematic diagram of the characteristic temperatures $T_{\mathrm{BKT}}$ and $T_{\rm{BCS}}$ for the superconducting transition in a phase-fluctuation-dominated unconventional superconductor.
• Figure 2: (a) Temperature dependence of the superfluid density for s-wave pairing. The black dot-dashed curve denotes the universal scaling law $\rho_s(T)/\rho_s(0) \sim 2T/\pi J_{\theta}$, with its intersection with the superfluid curve indicating the BKT transition temperature. Blue squares represent the superfluid density of bosons in the XY model, while red circles show the fermionic superfluid density computed from the BdG Hamiltonian. Although the fermionic and bosonic superfluid densities deviate away from $T_{\rm{BKT}}$, all three curves intersect near $T_{\rm{BKT}}=0.1$. The black dashed line indicates the location of the BKT transition. (b) Temperature dependence of the spin–lattice relaxation rate, showing a Hebel–Slichter coherence peak near $T_{\rm{BKT}}$. (c) Density of states (DOS) at different temperatures. The U-shaped gap vanishes above the transition temperature. (d) Temperature dependence of the zero-frequency DOS $A_{\omega}(\omega=0)$ and its derivative $\frac{\partial A_{\omega}(\omega=0)}{\partial T}$. Below $T_{\rm{BKT}}$ the U-shaped gap persists and the zero-frequency DOS remains zero. Above the transition temperature, it increases rapidly.
• Figure 3: The temperature-dependent gap magnitude $|\Delta|(T)$ (a) and the corresponding spin-lattice relaxation rate $1/T_1T$ (b) for a system with a $|\Delta|(T)$ determined by self-consistent gap equation. $|\Delta|(T)$ is denoted red squares, and $1/T_1T$ by green circles. The BKT transition temperature $T_{\rm{BKT}}$ is indicated by the black dashed line, while the mean-field temperature $T_{\rm{BCS}}$, corresponding to gap closure, is marked by the red dashed line.
• Figure 4: (a) Temperature dependence of the superfluid density for d-wave pairing. All three curves intersect near $T_{\rm{BKT}}=0.1$. The black dashed line indicates the location of the BKT transition. (b) Temperature dependence of the spin–lattice relaxation rate $1/T_1T$ and its derivative with temperature $\frac{\partial (1/T_1T)}{\partial T}$, showing no Hebel–Slichter coherence peak near $T_{\rm{BKT}}$. (c) DOS at different temperatures. (d) Temperature dependence of the zero-frequency DOS and its derivative. Below $T_{\rm{BKT}}$ the V-shaped gap persists and the zero-frequency DOS remains nearly zero. Above the transition temperature, it increases rapidly.
• Figure S1: Waterfall plot of the DOS for s-wave pairing at various temperatures. Each curve represents the local DOS along a horizontal cut across the lattice. As temperature increases, the superconducting gap gradually decreases. Bound states emerge near $T_c$, as highlighted by the red dashed box. Above $T_c$, the gap is smeared out by thermal effects: the U-shaped gap closes and the coherence peak characteristic of s-wave pairing disappears.