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I am trying to calculate the spin susceptibility of BCS superconductors at finite temperatures. It's well known that for conventional superconductors, the transverse NMR relaxation $1/T_1$ exhibits a coherent peak just below the transition temperature $T_c$, while is exponentially surpressed as $e^{-\Delta(0)/T}$ at quite low temperatures.

Specifically, the relaxation time $1/T_1$ is related to the spin-spin correlation function $\chi^{-+}(q,\omega+i0^+)=\langle S^{-}S^{+}\rangle$ throuth $$ 1/T_1 = T \lim_{\omega\to0}\ 2|A_{hf}|^2\int\frac{\mathrm{d}^2q}{(2\pi)^2}\frac{\text{Im}\left[\chi^{-+}(q,\omega+i0^+)\right]}{\omega} $$ where the hyperfine coupling $A_{hf}$ is assumed momentum-independent, and the factor 2 enumerates the two axes perpendicular to the external field direction. Also, the spacial dimension is assumed to be 2.

The four-point correlation function on the lhs can be carried out by considering a mean-field BCS Hamiltonian, where the vertex correction is omitted and only the bubble diagram contributes. This eventually leads to the following formula for the relaxation time, see also e.g. Eq.(12) in this paper, $$ \begin{aligned} 1/T_1 &= T |A_{hf}|^2 \int\frac{\mathrm{d}\epsilon}{2\pi} \left[N(\epsilon)\right]^2 \left(-\frac{d n_F}{d\epsilon}\right)\\ &= |A_{hf}|^2 \int\frac{\mathrm{d}\epsilon}{4\pi} \frac{\left[N(\epsilon)\right]^2}{1+\cosh(\beta\epsilon)}. \end{aligned} $$

So the remaining task is to evaluate the density of states $N(\epsilon)=\int\frac{\mathrm{d}^2p}{(2\pi)^2}A(p,\omega)$ for BCS superconductors, which can be worked out following the steps in this post. By assuming the dispersion of free electrons as $\xi_k=\frac{k^2-k_F^2}{2m}$, the final result is $$ N(\omega) = \left\{ \begin{aligned} 0,\qquad\qquad & \qquad |\omega|<\Delta, \\ \frac{m|\omega|}{\sqrt{\omega^2-\Delta^2}},\qquad\qquad & \qquad \Delta<|\omega|<\sqrt{\Delta^2+\epsilon_F^2},\\ \frac{m}{2}\left(\frac{|\omega|}{\sqrt{\omega^2-\Delta^2}}+sgn(\omega)\right),& \qquad |\omega|>\sqrt{\Delta^2+\epsilon_F^2}, \end{aligned} \right. $$ where $\epsilon_F=\frac{k_F^2}{2m}$ and $\Delta$ denotes the BCS superconducting gap.

Here comes my questions:

  1. The density of states $N(\omega)$ becomes divergent at the gap edge when $|\omega|\to\Delta$ is approched from high frequenies, which is as expected. However, it seems disconcerting that the integral we encountered when computing $1/T_1$, which evloves $[N(\epsilon)]^2$, also diverges at $|\omega|=\Delta$. So my first question is: whether the integral diverges or not? I have tried computing the integral numerically by simply doing summations or using Monte Carlo, both of them yields unreliable results, e.g. badly scaled to the cutoff or suffer from insufficiency of convergence.

  2. I understand that as $T_c$ is approached from below, the decrease of BCS gap function $\Delta(T)$ may lead to the growth of $1/T_1$. But how can one justify the existence of coherent peak? E.g., a larger $1/T_1$, compared with the normal state value at $T_c$, can be reached just below $T_c$. Also, I am wondering how the exponential behavior of $1/T_1\sim e^{-\Delta(0)/T}$ can be derived as $T\to0$.

Any comments would be really appreciated.

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I found a useful lecture note on the nuclear relaxation of BCS superductors. It states that the integral above does diverge logarithmically at the gap edge due to the singularity in the superconducting density of states. However, this divergence can be regularized by manually adding a broadening to the electronic spectral function, e.g. a finite lifetime $\gamma$, which can be assumed of no temperature dependence in the temperature range we concerned, $$ \begin{aligned} A(k,\omega) &= -2\ \text{Im} \frac{1}{\omega-\xi_k+i\gamma-\frac{\Delta^2}{\omega+\xi_k}}\\ &=-2\ \text{Im}\frac{\omega+\xi_k}{\omega^2-\xi_k^2-\Delta^2+i\gamma(\omega+\xi_k)}. \end{aligned} $$ The density of states can still be analytically obtained, $$ N(\omega) = \frac{m}{\pi} \text{Im}\left\{\frac{1}{2\sqrt{(\omega+i\frac{\gamma}{2})^2-\Delta^2}} \left[-(\omega+\xi_+)\ln(-\epsilon_F-\xi_+)+(\omega+\xi_-)\ln(-\epsilon_F-\xi_-)\right]\right\} $$ with $\xi_\pm=i\frac{\gamma}{2}\pm\sqrt{(\omega+i\frac{\gamma}{2})^2-\Delta^2}$.

Now in the presence of $\gamma$, the divergence is eliminated and the integral over frequencies can be carried out numerically in a stable manner, see the figure below.

pic

where I have assigned $\Delta(0)/\epsilon_F=0.25$, $\frac{2\Delta(0)}{T_c}=3.528$ and $\Delta(T)=\Delta(0)\tanh\left(1.74\sqrt{\frac{T}{T_c}-1}\right)$. The last two constraints are required by the self-consistent BCS theory. It's found that a small $\gamma$ amplifies the coherent peak, while storng dampings, e.g. relatively large $\gamma$, can destroy the peak. One can also observe the exponential behavior of $1/T_1$ at low temperatures.

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  • $\begingroup$ For interested readers, chapter 14.7 in Coleman's book Introduction to many body physics also provides a comprehensive introduction to this topic :) $\endgroup$ Apr 17 at 6:35

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