Inhomogeneous time-reversal symmetry breaking in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$

We show that the observed time-reversal symmetry breaking (TRSB) of the superconducting state in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ can be understood as originating from inhomogeneous strain fields near edge dislocations of the crystal. Specifically, we argue that, without strain inhomogeneities, ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ is a single-component, time-reversal symmetric superconductor, likely with $d_{x^2-y^2}$ symmetry. However, due to the strong strain inhomogeneities generated by dislocations, a slowly decaying subleading pairing state contributes to the condensate in significant portions of the sample. As it phase winds around the dislocation, time-reversal symmetry is locally broken. Global phase locking and TRSB occur at a sharp Ising transition that is not accompanied by a change of the single-particle gap and yields a very small heat capacity anomaly. Our model thus explains the puzzling absence of a measurable heat capacity anomaly at the TRSB transition in strained samples and the dilute nature of the time-reversal symmetry broken state probed by muon spin rotation experiments. We propose that plastic deformations of the material may be used to manipulate the onset of broken time-reversal symmetry.

Figure 1

(a) Edge dislocation in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ inducing local breaking of time-reversal symmetry due to inhomogeneous strain. Near the dislocation, large strain fields mix a primary $d_{x^2-y^2}$ pairing state with other subleading symmetry channels, e.g. a $g_{xy(x^2-y^2)}$ pairing state [16]. As a result, $d\pm g$ pairing emerges on opposite sides of the dislocation, forcing the relative phase $\phi \left(\mathit{r}\right)$ to wind in between, thus breaking time-reversal symmetry locally. (b) Time-reversal symmetry breaking appears in the bulk system as a stand-alone phase transition where the local phase windings are locked through the long-range decay of strain fields that couple dislocations.

Figure 2

Monte-Carlo simulation to minimize the Ginzburg-Landau free energy, Eq. (1), for the two order parameters in the presence of a dislocation. (a) Strain-induced onset of the imaginary part of the order parameter ${\psi }_g$ at four locations (see inset) around a dislocation. Because ${\psi }_d$ is already present at $T<T_c$, this onset locally breaks time-reversal symmetry at $T_{\mathrm{TRSB}}^{★}$. Note that both these temperatures are much larger than the temperature $T_c^{<}$ at which the homogeneous $g$-wave order parameter would onset. (b),(c) Order parameter distribution near the dislocation for fixed temperature [red arrow in (a)] and for ${\lambda }_x=8$ and ${\lambda }_0=5$ respectively, see Eq. (7).

Figure 3

Amplitude (circle size) and phase (color) of the strain-induced $g$-wave component in proximity to a dislocation. Here we consider ${\lambda }_x$ to be the dominant strain-superconductivity coupling, as shown in Fig. 2. Inhomogeneous TRSB occurs when the phase winding (right, middle, and bottom) is energetically favored over the vanishing of the $g$-wave component. Away from the dislocation the magnitude of the $g$ component is small and a direct sign change is always preferred (right, top, cut at $z=25b$). Both degenerate solutions—distinguished by the pseudospin $\sigma$, see Eq. (4), and reflecting the Ising symmetry of the TRSB order—are shown.

Figure 4

(a) Evolution of $T_{\mathrm{TRSB}}^{★}$ upon varying the interaction parameters ${\lambda }_0$, ${\lambda }_z$, and ${\lambda }_x$. Recall that $T_c$ ($T_c^{<}$) is the temperature at which a homogeneous ${\psi }_d$ (${\psi }_g$) component sets in, and $T<T_c^{<}$ results in a uniform time-reversal symmetry broken state. The analytic results (gray curve) indicate that $T_{\mathrm{TRSB}}^{★}\left({\lambda }_{0,z}\right)$ deviates quadratically away from $T_c^{<}$. Meanwhile, $T_{\mathrm{TRSB}}^{★}\left({\lambda }_x\right)$ deviates from $T_c^{<}$ only above a threshold value. (b) Phase diagram as function of homogeneous $B_{1g}$ strain and temperature. It is evaluated for ${\lambda }_x=8.75$ and for $T_c\left(0\right)=T_{\mathrm{TRSB}}^{★}\left(0\right)$. The analytic relation $T_c\left(\epsilon \right)-T_c\left(0\right)={(\delta T^2+{\epsilon }^2)}^{1/2}$ is shown in black.

Figure 5

Mapping of the physical system to an effective coarse-grained model. Panel (a) shows the Ising domain walls associated with the Ising TRSB order parameter of Eq. (4), which proliferate along a dislocation line according to the 1D Ising model. Panel (b) illustrates the coupling between multiple dislocations, which can form open lines connecting the surfaces of the sample of closed loops. As a result, as shown in panel (c), the critical behavior of the model is the same as that of an anisotropic 2D Ising model with exchange parameters $J$ and $J/\eta$, with $\eta \gg 1$.

Figure 6

Entropy change $C\left(T\right)/T$ per Ising pseudospin (in units of $k_B/J$) of the anisotropic Ising model with $J^{\prime }/J=0$ (red) and 0.01 (blue). This illustrates our approach to estimate the total weight of the heat-capacity anomaly at the transition for finite $J^{\prime }$ from the transferred calorimetric weight in $C_{\mathrm{Ising},d=1}/T$.