Twofold Spontaneous Symmetry Breaking in the Heavy-Fermion Superconductor ${\mathrm{UPt}}_3$

The field-orientation dependent thermal conductivity of the heavy-fermion superconductor ${\mathrm{UPt}}_3$ was measured down to very low temperatures and under magnetic fields throughout the distinct superconducting phases: $B$ and $C$ phases. In the $C$ phase, a striking twofold oscillation of the thermal conductivity within the basal plane is resolved reflecting the superconducting gap structure with a line of node along the $a$ axis. Moreover, we find an abrupt vanishing of the oscillation across a transition to the $B$ phase, as a clear indication of a change of gap symmetries. We also identify extra two line nodes below and above the equator in both $B$ and $C$ phases. From these results together with the symmetry consideration, the gap function of ${\mathrm{UPt}}_3$ is determined as a $E_{1u}$ representation characterized by a combination of two line nodes at the tropics and point nodes at the poles.

Figure 1

Magnetic field dependence of the thermal conductivity $\kappa (H)/T$ along the $c$ and $b$ axes at various temperatures. The open and closed arrows represent the $B\to C$ transitions $H_{\mathrm{BC}}$ and the upper critical fields $H_{c2}$, respectively. Inset: temperature dependence of $\kappa (T)/T$ under zero field and at 3 T for $H\parallel b$. The dashed line shows $\kappa /T=L_0/\rho$ ($L_0$: the Lorentz number) obtained from the normal-state resistivity $\rho$ at 3 T using the Wiedemann-Franz law.

Figure 2

Angular variation of the thermal conductivity $\kappa (\varphi )$ normalized by ${\kappa }_n$ at 50 mK as a function of the azimuthal angle $\varphi$ for $|{\mu }_{0}H|=(\mathrm{a})$ 3.0, (b) 1.0, and (c) 0.5 T, respectively. $\kappa (\varphi )$ is measured with rotating $H$ within the $ab$ plane (the polar angle $\theta =90°$) as schematically shown in (d), where $\varphi$ and $\theta$ are measured from the $a$ and $c$ axes, respectively, and $q$ is injected along the $c$ axis. The solid lines show the twofold component in $\kappa (\varphi )/{\kappa }_n$. The open circles represent $\kappa (\varphi )/{\kappa }_n$ at 1 T obtained under the field cooling condition at every angle. (e) Field variation of the twofold amplitude $|C_{2\varphi }/{\kappa }_n|$ at 50 mK at $\theta =90°$ (solid circles) and 63° (open circle), respectively.

Figure 3

Angular variation of the thermal conductivity $\kappa (\theta )$ normalized by ${\kappa }_n$ at 50 mK as a function of the polar angle $\theta$ for $|{\mu }_{0}H|=(\mathrm{a})$ 1.5, (b) 1.0, and (c) 0.5 T, respectively. The $\kappa (\theta )/{\kappa }_n$ curves measured by rotating $H$ [inset of (a)] within the $ac$ plane (open circles) and the $bc$ plane (closed circles) are simultaneously plotted. Inset of (c): $\Delta \kappa (\theta )/{\kappa }_n\equiv \mathbf{(}\kappa (\theta )-{\kappa }_0-{\kappa }_{2\theta }\mathbf{)}/{\kappa }_n$ vs $\theta$ plot at 50 mK at 0.5 T, where ${\kappa }_0$ is a $\theta$-independent term and ${\kappa }_{2\theta }=C_{2\theta }\cos 2\theta$ is a twofold component. (d) The phase diagram of ${\mathrm{UPt}}_3$ with the three superconducting phases, labeled $A$, $B$, and $C$, for $H\parallel b$. The open and closed circles represent $H_{\mathrm{BC}}$ and $H_{c2}$, respectively, deduced from the present measurements. The schematic shapes of the gap symmetries for each phase are shown.

Figure 4

$\theta$ dependence of the thermal conductivity difference obtained by subtracting $\kappa (\theta ,\varphi =0°)$ from $\kappa (\theta ,\varphi =90°)$ denoted by the open and closed circles, respectively, in Fig. 3a. Right axis: the density of states difference normalized at $\theta =90°$ (arbitrary scale) along the vertical nodal and antinodal scannings for three possible gap functions in the $C$ phase: 1. The present $E_{1u}\mathbf{(}{k}_b(5k_c^2-1)\mathbf{)}$, 2. $E_{1g}(k_b{k}_c)$, 3. $E_{2u}(k_a{k}_b{k}_c)$. Those gap structures are sketched in the inset.