Pauli-limited superconductivity with classical magnetic fluctuations

We examine the effect of classical magnetic fluctuations on the phase diagram of paramagnetically-limited two-dimensional superconductors under a Zeeman magnetic field. We derive the free energy expansion in powers of the superconducting order parameter and analyze the character of the normal-superconducting transition. While the transition is of the second order for all temperatures in the absence of magnetic fluctuations, we find that proximity to magnetism drives both the transition into the uniform state and that into the modulated Fulde-Ferrell-Larkin-Ovchinnikov state to first order at intermediate temperatures. We compute the thermodynamic signatures of the normal-superconducting transition along the upper critical field.

Figure 1

(Color online) Optimal wave vector $q_0\left[b_c\left(T\right)\right]$ (solid) and quartic coefficient $\gamma \left[B_c\left(T\right)\right]$ (dot dashed) obtained by unrestricted maximization of $b_c\left(t\right)$ for LO state. Quartic coefficient for small-$q$ expansion (dashed) is also shown. Upper/lower panels are for $s$- and $d$-wave ($q$ along node) gaps.

Figure 2

(Color online) The normal-to-superconducting transition in $s$-wave (upper) and $d$-wave (lower) superconductors under a Zeeman field. Modulation is along gap node for $d$-wave. Magnetic fluctuations, $\tilde{\eta }$, modify the $B_{c2}$ transition, showing four distinct regions (for increasing $T$): second order into modulated, first order into modulated, first order into uniform, and second order into uniform states.

Figure 3

(Color online) Transition lines for $s$-wave and $d$-wave with nodally oriented $\mathbf{Q}$. Upper curves are in the absence of fluctuations and lower curves are for $\tilde{\eta }\tilde{\chi }=0.5$.

Figure 4

(Color online) Quartic Landau coefficient evaluated along $B_c\left(T\right)$ for $s$-wave and $d$-wave symmetries with LO modulation for $t<t_P$ and uniform state for $t>t_P$.

Figure 5

(Color online) $S$-wave $b_c\left(t\right)$ for both constant $\chi \left(t\right)$ and ${\chi }_1\equiv \chi \left(t_c\right)$ for $t_c=0.76$. Upper and lower curves are for ${\chi }_1$ and $\chi \left(t\right)$, respectively. Since $\tilde{\eta }{\chi }_1<\tilde{\eta }\chi \left(t\right)$ at $t<t_c$, the critical field is suppressed less for ${\chi }_1$. As $\tilde{\eta }\chi <1$ for both cases, the region of first-order transitions is larger for ${\chi }_1$.

Figure 6

(Color online) Optimal wave vector at N-SC transition for $s$-wave with $\tilde{\eta }\tilde{\chi }=0.4$. Small-$q$ approximation predicts $t_{\text{LO},grad}$ and $t_{\text{LO},grad}^{\star }$. Upswing in $q$ below $t_{\text{LO},grad}^{\star }$ indicates breakdown of small-$q$ expansion where ${\alpha }_4\propto \nu \to 0$ (Ref. 44).

Figure 7

(Color online) Specific heat jump at second-order N-SC transition for (a) $s$-wave and (b) $d$-wave symmetries with $\tilde{\eta }\tilde{\chi }=0.0$ and 0.3. Inset: specific heat jump $\Delta C$ at second-order N-SC transition.

Figure 8

(Color online) Thermodynamics at first-order N-SC transition for $s$-wave with $\tilde{\eta }\tilde{\chi }=0.28$, 0.32, and 0.36. Shown are (a) decrease in entropy and (b) decrease in susceptibility. Temperatures $t_P^{\star }$, $t_{\text{LO}}$, and $t_{\text{LO}}^{\star }$ are labeled for $\tilde{\eta }\tilde{\chi }=0.28$.

Figure 9

(Color online) Temperature derivatives of quadratic and quartic Landau coefficients for $d$-wave at fixed $b_{c,d}$ and $q_{0,d}$ in zero-temperature limit. Main figure shows ${\alpha }_{\text{LO}}^{\prime }$ and $-{\gamma }_{\text{LO}}^{\prime }$, both of which limit to zero at $t=0$. Inset: ${\alpha }_{\text{LO}}^{″}\approx 4.54$ at $t=0$.