Theoretical Studies on Off-Axis Phase Diagrams and Knight Shifts in UTe\(_2\): Tetra-Critical Point, d-Vector Rotation, and Multiple Phases

Inspired by recent remarkable sets of experiments on UTe \(_2\) $_2^{}$ : discoveries of the fourth horizontal internal transition line running toward a tetra-critical point (TCP) at H = 15 T, the off-axis high-field phases, and abnormally large Knight shift (KS) drop below \(T_\textrm{c}\) $T_{\text{c}}$ for \(H \parallel a\) $H\Vert a$ -magnetic easy axis, we advance further our theoretical work on the field (H)-temperature (T) phase diagram for \(H \parallel b\) $H\Vert b$ -magnetic hard axis which contains a positive sloped \(H_\textrm{c2}\) $H_{\text{c2}}$ departing from TCP. A nonunitary spin-triplet pairing with three components explains these experimental facts simultaneously and consistently by assuming that the underlying normal electron system with a narrow bandwidth characteristic to the Kondo temperature \(\sim\) $\sim$ 30 K unsurprisingly breaks the particle-hole symmetry. This causes a special invariant term in Ginzburg–Landau (GL) free energy functional which couples directly with the 5f magnetic system, giving rise to the \(T_\textrm{c}\) $T_{\text{c}}$ splitting and ultimately to the positive sloped \(H_\textrm{c2}\) $H_{\text{c2}}$ and the horizontal internal transition line connected to TCP. The large KS drop can be understood in terms of this GL invariance whose coefficient is negative and leads to a diamagnetic response where the Cooper pair spin is antiparallel to the applied field direction. The present scenario also accounts for the observed d-vector rotation phenomena and off-axis phase diagrams with extremely high \(H_\textrm{c2}\) $H_{\text{c2}}$ \(\gtrsim\) $\gtrsim$ 70 T found at angles in between the b- and c-axes and between the bc-plane and a-axis, making UTe \(_2\) $_2^{}$ a fertile playground for a possible topological superconductor.