Novel $H_{\rm c2}$ suppression mechanism in a spin triplet superconductor -- Application to UTe$_2$--

TL;DR

This work proposes a novel $H_{c2}$ suppression mechanism for spin-triplet superconductors with equal-spin pairing, showing that coupling between Cooper-pair polarization and field-induced magnetization can reduce $H_{c2}$ from the orbital limit to very small values. Applying the theory to UTe$_2$, it reproduces axis- and pressure-dependent $H_{c2}$ data and identifies a nonunitary equal-spin triplet state with pairing $(oldsymbol{ m eta})=(oldsymbol{ m eta}_b+ioldsymbol{ m eta}_c)k_a$ under finite SOC. The analysis connects a d-vector rotation, SOC strength, and the tetra-critical point to the observed multiple $H$–$T$ phase diagrams, unifying suppression and enhancement phenomena under a single framework. These results imply a general mechanism for diagnosing spin structure in triplet superconductors and highlight the role of DOS asymmetry and induced moments in heavy-fermion systems.

Abstract

A novel $H_{\rm c2}$ suppression mechanism is theoretically proposed in a spin triplet superconductor (SC) with equal spin pairs. We show that the upper critical field $H_{\rm c2}$ can be reduced from the orbital depairing limit $H^{\rm orb}_{\rm c2}$ to arbitrarily small value, keeping the second order phase transition nature. This mechanism is sharply different from the known Pauli-Clogston limit for a spin singlet SC where the reduction is limited to $\sim$0.3$H^{\rm orb}_{\rm c2}$ with the first order transition when the Maki parameter goes infinity. This novel $H_{\rm c2}$ suppression mechanism is applied to UTe$_2$, which is a prime candidate for a spin triplet SC, to successfully analyze the $H_{\rm c2}$ data for various crystalline orientations both under ambient and applied pressure, and to identify the pairing symmetry. It is concluded that the non-unitary spin triplet state with equal spin pairs is realized in UTe$_2$, namely $(\hat b+i\hat c)k_a$ in $^3$B$_{\rm 3u}$ which is classified under finite spin orbit coupling scheme.

Figures (14)

• Figure 1: Schematic $H$-$T$ phase digram for $H\parallel b$-axis machida5machida6. In the A$_1$(A$_2$) phase the spin polarization $\bold S$ points to the antiparallel (parallel) direction along the $a$-axis at low fields and turns to the parallel (antiparallel) to the $b$-axis in higher fields above the d-vector rotation field $H_{\rm rot}$ denoted by TCP. $H_{\rm m}$ is the first order metamagnetic transition. The dotted line inside the A$_1$ phase is the hypothetical transition line for the A$_2$ phase.
• Figure 2: (a) Schematic figure to explain the $H_{\rm c2}$ suppression at $T=0$. $H_{\rm c2}$ is reduced from $H^{\rm orb}_{\rm c2}$ by the amount of $H_{\rm eff}=H+\Delta H$ with $\Delta H=\alpha_0\kappa M(H)$. $M(H)\propto H$ is shown below by the red line. (b) $H_{\rm c2}(T)$ is reduced from $H^{\rm orb}_{\rm c2}(T)$ by the amount of $\Delta H$ at $T=0$ and by $\Delta T=\kappa M(H)$ along the $T$-axis. $\Delta H$, $\Delta T$, and $H^{\rm orb}_{\rm c2}(T)$ form a triangle in the $H$-$T$ plane. $\alpha_0$=$|(dH^{\rm orb}_{\rm c2}/dT)_{T_{\rm c}}|$.
• Figure 3: Comparison of $H_{\rm c2}$ with $H^{\rm orb}_{\rm c2}$ for all field orientations from $a$-axis$\rightarrow$$b$$\rightarrow$$c$$\rightarrow$$a$$\rightarrow$$b$-axis (left to right). $H^{\rm orb}_{\rm c2}$ is estimated from the initial slopes by $H^{\rm orb}_{\rm c2}=1.8|(dH_{\rm c2}/dT)_{T_{\rm c}}|$. The color regions indicate the differences between them. The gray (brown) areas show the regions for $H_{\rm c2}<H^{\rm orb}_{\rm c2}$ ($H_{\rm c2}>H^{\rm orb}_{\rm c2}$). The data (dot points) are taken from Aoki et al hc2'. $H_{\rm c2}$ for $H\parallel b$ comes from Refs. [rosuel] and [sakai].
• Figure 4: (a) The difference between $H_{\rm c2}$ and $H^{\rm orb}_{\rm c2}$ shown in Fig. \ref{['hc2']} by the gray and brown regions is compared with the theoretical calculations of $\Delta H=\alpha_0\kappa M(H)$ for various angles indicated by arrows. The up arrows (down arrows) show the suppressed (enhanced) $H_{\rm c2}$. $\alpha_0=|(dH_{\rm c2}/dT)_{T_{\rm c}}|$, $\kappa=2.7$K/$\mu_{\rm B}$, and $M(H)$ from (b). (b) The magnetization curves $M(H)$ for three principal axes taken from Miyake, et al miyakemiyake2. We ignore the renormalization of $\alpha_0$ for simplicity and clarity.
• Figure 5: (a) $H^a_{\rm c2}(T)$ for the $a$-axis: $H^{\rm orb}_{\rm c2}(T\rightarrow0)$ tends to $\sim$30T. The red triangle shows the $H_{\rm c2}$ reduction. The data are taken from Tokiwa et al tokiwa. Note that a slight enhancement of $H^a_{\rm c2}(T)$ in the high field region is due to the metamagnetic transition along the $a$-axis above 8T tokiwashimizu4. (b) $H^b_{\rm c2}(T)$ for the $b$-axis: The red triangle shows the $H_{\rm c2}$ reduction. TCP at 14T denotes the teta-critical point where the four second order transitions meet, corresponding to the d-vector rotation point. The spin polarization $\bf S$ antiparallel to the $a$-axis at low $H$. The A$_1$ phase changes into the state with $\bf S$ being antiparallel to the $b$-axis above TCP. The positive sloped $H^b_{\rm c2}(T)$ in the A$_2$ phase above 14T with $\bf S$ parallel to the $b$-axis is enhanced with the rate denoted by the triangle with brown color there. $H_{\rm m}$ shows the meta-magnetic transition where A$_2$ terminates. The data points come from the experiments sakai. (c) $H^c_{\rm c2}(T)$ for the $c$-axis: The red triangle shows the $H_{\rm c2}$ enhancement. Above 4T denoted by kink, $\bf S$ changes from $a$-antiparallel to $c$-parallel. $H^{\rm orb}_{\rm c2}(T\rightarrow0)\sim$12T is enhanced. The data are taken from Tokiwa et al tokiwa.