Nonunitary spin-triplet superconductors in Zeeman magnetic field

Abstract

We study spin-triplet superconductivity with both unitary and nonunitary pairing in the presence of an external Zeeman magnetic field. Within a mean-field framework, we exactly diagonalize the Bogoliubov-de Gennes Hamiltonian and derive general expressions for the quasiparticle spectrum, superconducting gap, critical temperature, and spin magnetization, valid for arbitrary magnetic-field strengths and temperatures. We analyze in detail the nonlinear spin susceptibility and the field evolution of the superconducting gap and transition temperature, highlighting qualitative differences between unitary and nonunitary pairing states. Our results are broadly applicable to a wide range of materials, including systems with both weak and strong spin-orbit coupling. We show that systematic measurements of the critical temperature and spin susceptibility as functions of the magnitude and orientation of the magnetic field provide a powerful means to identify the structure of the spin-triplet order parameter, and we discuss implications of our findings for candidate materials such as 4Hb-TaS$_2$ and PrOs$_4$Sb$_{12}$.

Figures (4)

• Figure 1: The dependence of the superconducting critical temperature $T_c$ on the magnitude of the Zeeman magnetic field $H$ for the case of a unitary spin-triplet superconductor, ${\bf q}_{\bf k}=0$. All the parameters are measured in the units of the critical temperature at zero magnetic field $T_{c0}$. (a) The curves are obtained by numerically evaluating Eq. \ref{['SelfEq']} for the gap function in $E_u$ representation of the group $O_h$, see Sec. \ref{['Applications']}. The results are in perfect agreement with Eq. \ref{['Analytics']}. (b) Numerical evaluation of Eq. \ref{['Analytics']} at $\zeta\ge0.82$. Multiple solutions for $T_c$ at a given $H$ indicate the metastable phase and the first-order transition.
• Figure 2: $E_u$ representation of the cubic point group $O_h$. (a) The dependence of the nonlinear spin susceptibility $\chi_{xx}$, measured in units of the normal-state value $\chi_n$, on the magnitude of the Zeeman magnetic field $H_x$, for different choices of the order parameter $(\eta_1,\eta_2)$. The figure is numerically evaluated by differentiating Eq. \ref{['MExplicit']}. (b) The dependence of the superconducting gap $\Delta$ on the Zeeman field $H_x$. The figure is numerically evaluated from the gap equation \ref{['SelfEq']}. The peak in $\chi_{xx}$ appears exactly at fields where $\Delta$ decreases abruptly. $\Delta$ and $\mu_B H_x$ are measured in units of the zero-field transition temperature $T_{c0}$. The temperature is taken to be $T=T_{\mathrm{c0}}/6$.
• Figure 3: The dependence of parameter $\zeta$, Eq. \ref{['Analytics']}, on the magnetic field direction, ${\mathbf H}=H(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$. The vector ${\mathbf d}_{\bf k}$ here is determined by the $E_u$ representation of $O_h$ with $(\eta_1,\eta_2) = (-1,\sqrt{3})/2$.
• Figure 4: Same as Fig. \ref{['Fig:Eu']}, but for the $E_{2u}$ irreducible representation of the $D_{6h}$ point group instead of the $E_u$ representation of $O_h$.