Fluctuation and strain effects in a chiral $p$-wave superconductor

For a tetragonal material, order parameters of $p_x$ and $p_y$ symmetry are related by rotation and hence have the same $T_{\mathrm{c}}$ at a mean-field level. This degeneracy can be lifted by a symmetry-breaking field, such as (uniaxial) in-plane strain, such that at $T_{\mathrm{c}}$, the order parameter is only of $p_x$ or $p_y$ symmetry. Only at a lower temperature also the respective other order parameter condenses to form a chiral $p$-wave state. At the mean-field level, the derivative of $T_{\mathrm{c}}$ with strain is discontinuous at zero strain. We analyze the consequences of (thermal) fluctuations on the strain-temperature phase diagram within a Ginzburg-Landau approach. We find that the order-parameter fluctuations can drive the transition to be weakly first order, rounding off this discontinuity. We discuss the possibility of a second-order transition into a nonsuperconducting time-reversal-symmetry-breaking phase and consequences for the spin-triplet superconductor ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$.

Figure 1

(a) Mean-field temperature-strain phase diagram for the weak-coupling parameters $b_2=-b_3=2/3b_1$ and (b) phase diagram for $s=0$ with respect to the fourth-order terms of the free energy. For parameters inside the dashed triangle, fluctuations cannot drive the superconducting transition.

Figure 2

Zero-strain order parameter for $\alpha =0.05$ and various TRS-breaking couplings $\beta /\alpha$. For $\beta /\alpha =0.75$, the points where the metastable solutions disappear are indicated as well (dashed lines). Note that for any finite $\beta /\alpha >0$, the transition becomes (weakly) first order.

Figure 3

The minimal ratio $\alpha /\beta$ needed in order to have a second-order transition into a nonsuperconducting TRS-breaking phase. The shaded region denotes $\alpha /\beta <1$, where the action Eq. (1) becomes unstable.

Figure 4

Temperature-strain phase diagram for $b_2=5\times {10}^{-4}{b}_1$ and different values of $b_3$. The upper lines denote $T_{\mathrm{c}}^{\left(1\right)}$, where the system enters a $p_x$ ($p_y$)-wave state, and the lower lines $T_{\mathrm{c}}^{\left(2\right)}$, where it enters the chiral state. Note that the temperature is scaled with respect to the second-order transition temperature $T_{\mathrm{SC}}^{\left(0\right)}$.

Figure 5

Temperature-strain phase diagram for finite $b_2$ and $b_3$ showing no cusp at zero strain due to the first-order transition directly into a chiral state (thick lines) driven by the TRS-breaking fluctuations. Note that the temperature scale is with respect to the second-order transition temperature $T_{\mathrm{SC}}^{\left(0\right)}$.